ISSN 004-8917
AUSTRALIAN
JOURNAL Volume 24, Number 2, May 1992
SPECIAL ISSUE ON ALGORITHMS
41 Guest Editorial
42 Algorithmic Issues in the Geometry of Planar Linkage Movement S WHITESIDES
51 Algorithms for Groups J CANNON and GEORGE HA VAS
61 Parallel Search Algorithms for Trees and Graphs PCHAUDHUR1
70 An Overview of Adaptive Sorting A MOFFA T and O PETERSSON
SPECIAL FEATURES
50 Book Review Information
78 Book Reviews
Published for Australian Computer Society Incorporated Registered by Australia Post, Publication No. NBG 1124 AUSTRALIAN
Editor Computer R COOK, The University of Queensland ———■«™——«■—— iiiiiiiiii iiiiiiiiiiii Editorial Co-ordinator LOIS FORDHAM, The University of 77ie Australian Computer Journal is an official publication Queensland of the Australian Computer Society Incorporated. Associate Editors CONTRIBUTIONS to ensure uniformity of style. Referees may MJ LAWRENCE, University of NSW suggest major revisions to be performed by the All material for publication should be sent to: author. C ANDREWS, Mr Rob Cook, Editor Proofs and Reprints: Page proofs of Papers DN WILSON, University of Technology, Australian Computer Journal and Short Communications are sent to the Sydney Centre for Information Technology Research authors for correction prior to publication. Fifty K RAYMOND, The University of Queensland The University of Queensland, Qld 4072 copies of reprints will be supplied to authors J ZELEZNIKOW, Latrobe University without charge. 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The theme of this special issue of the journal is algorithms, paper, by Chaudhuri, outlines how some of the fundamen the “central concept of computer science”1. Research in tal methods for graph search can be paralellised. Algorithms is having a resurgence in Australia, and the If there are any classic topics in computer science, then four papers in this volume exhibit current popular themes certainly sorting is among them. The search for a good of the research. sorting algorithm began as soon as the first machines were The first paper falls within the subfield of computa powered up; today, a section on sorting is an integral part tional geometry, an area which has been very active over of every computer science curriculum. Many proposals for the last few years. Its popularity is driven by the increasing adaptive sorting algorithms have been made in the last 10 number of computational problems with a geometric fla years; these are algorithms which sort a list faster if the list vour: in Robotics, Geographical Information Systems, is already “partially sorted”. Moffat and Petersson discuss Graphics and Visualisation. Whitesides presents the com the competing formal notions of “partially sorted” and putational geometry approach to motion planning for lin give an introduction to algorithms which exploit this kages in a restricted environment. “presortedness”. There has always been a strong collaboration between mathematicians and computer scientists working in algo REFERENCE rithms. In the past, the main flow of ideas has been for 1 Les Goldschlager and Andrew Lister, Computer Science, A Modem mathematics to computer science. The second paper, by Introduction, Prentice Hall International Series in Computer Science, 2nd edition 1988, page xi. Cannon and Havas shows, that the flow has become bidi rectional. The paper surveys algorithmic methods which have been successful in group theory, one of the fundamen Peter Eades Alistair Moffat tal branches of algebra. Professor and Head Senior Lecturer Algorithms research has often focused on improving of Computer Science Department of Computer the resource efficiency of computation. It is clear that for The University of Science many problems, future improvements in time efficiency Newcastle The University of can only be achieved with massive parallelism. The third Melbourne
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THE AUSTRALIAN COMPUTER JOURNAL, VOL. 24, No. 2, MAY 1992 41 1 INTRODUCTION Algorithmic Issues This paper considers the planning of planar placements and motions for linkages, especially chains of links. A linkage consists of a collection of rigid rods of given in the Geometry of lengths. Rods may be fastened together at their endpoints, which act like planar universal joints. When the linkage is placed in the plane, its rods are allowed to cross over one Planar Linkage another. As the linkage moves, the rods may rotate freely about their endpoints, subject only to the constraint that Movement distances between certain joints are determined by the existence of rods between them. Sue Whitesides A configuration or placement of a linkage is a specifica School of Computer Science tion of the exact location in the plane of all the joints of the McGill University, Montreal, Canada H3A 2A7 linkage. (This specification could be made in several ways, such as by giving the coordinates of some joints and the angular openings of others.) A motion or movement of a linkage refers to the existence of continuous functions of time that specify constraint-respecting locations for all the A linkage is a collection of rigid rods that are fastened joints. together at their endpoints, about which they may rotate freely. A chain of links is a simple example of such a lin 1.1 The Peaucellier Inversor kage. This paper is an introduction to algorithmic and Figure 1 shows the Peaucellier Inversor, invented in 1864. complexity issues in the placement and movement of This linkage, when properly designed and set up, contains planar chains and linkages. (The treatment is entirely geo a joint C that draws a straight line segment and that cannot metric; dynamics of linkages are not considered) deviate from this line segment. Two of its joints O and A Keywords: algorithms, motion planning, robotics, lin are thumbtacked to the plane so that their locations do not kage, computational geometry, complexity theory. change, although angles with vertices at these joints may CR Categories: F.2. change. Every configuration to which the linkage can be moved from the first placement shown in Figure 1 posi tions C on the straight line segment joining points P and Q. This linkage carries out physically the mathematical transformation I called inversion with respect to a circle. By definition, I maps an arbitrary point (p,@) to the point (1 Ip, ©). The origin “goes to infinity”. It is well known that I maps circles through the origin tckstraight lines. In the case of the Peaucellier Inversor, the' circle through the origin is the circle centred at A on which joint*# is con strained to lie. One can prove that O, B and C remain collinear as the linkage moves, and that there is a constant (which can be chosen to be 1 by a suitable change of scale) such that the distance from O to B remains equal to that constant divided by the distance from O to C. See Courant and Robbins (1969) and Eves (1965) for more details. Hence, C must lie at the image of B under the inversion I
|f Copyright 1992, Australian Computer Society Inc. General permission to republish, but not for profit, all or part of this material is granted, provided that the A CJ’s copyright notice is given and that reference is made to the publication, to its date of issue, and to the fact that reprint ing privileges were granted by permission of the Australian Computer Society Inc.
Manuscript received November 1991. Figure 1. The Peaucellier Inversor.
42 THE AUSTRALIAN COMPUTER JOURNAL, VOL. 24, No. 2, MAY 1992 MOVEMENT ALGORITHMS
with respect to the circle (not shown) of radius 1 centred at Although we are focusing on planar linkage problems, O. As B moves on the circle centred at A, C must move we note that RECONFIGURATION and PLACEMENT along the image under / of this circle, which is a straight problems can be posed for objects other than linkages, in line. This straight line is the line through P and Q. Con dimension 3 as well as in the plane. (The RECONFIGU stants imposed by the linkage prevent C from travelling RATION problem for arbitrary objects is sometimes beyond either Pot Q. called the piano movers’ problem, or FINDPATH. See for example Alt and Yap (1990a), Alt and Yap (1990b), 1.2 Some Basic Problems LaTombe (1991), Schwartz and Yap (1987). Whitesides A chain is a sequence of n links Lj, L2,Ln, consecutively (1985) for surveys and Schwartz and Yap (1987), connected together at their endpoints. We denote the end Schwartz, Sharir and Hopcroft (1987) for edited works.) points of Lj by Aj_; and A,-, and the length of L, by See Figure 2. A basic question to ask about the placement and 1.3 What Follows movement of linkages and chains is the following: The rest of the paper is organised as follows. Section 2 gives examples of some very conceptually simple prob lems involving almost trivial linkages, namely chains of n links, that are nevertheless hard from the point of view of computational complexity. Section 3 makes the point that algebraic techniques provide a way to solve, at least in principle, most linkage movement problems. Section 4 Figure 2. A chain. makes the point that if an algorithm designer knows that no linkage with more than a prespecified number N of RECONFIGURATION: links will ever have to be considered, then the designer can GIVEN: A linkage and two configurations for it. use algebraic techniques to provide a polynomial time QUESTION: Is there a motion of the linkage between the algorithm for that particular bound N. This state of affairs, two configurations? If so, what is an example of such a that simple n-link chain problems can be difficult when no motion? bound is put on n, whereas movement problems for much Of course RECONFIGURATION may be asked sub more complicated linkages admit polynomial time solu ject to additional constraints such as: tion provided that the number of links is at most some N, — the linkage must avoid collision with given obstacles; motivates Section 5. Section 5 gives examples of some — certain joints must remain fixed at certain locations in linkage movement problems for linkages with an arbi the plane. trary, unbounded number n of links that nevertheless Let us say that two configurations of a linkage are admit polynomial time solution. Section 6 summarises and equivalent if there is a constraint-satisfying motion raises some directions for further inquiry. between the two. For example, the two configurations of the Peaucellier Inversor shown in Figure 1 are equivalent. 2 RULER FOLDING Clearly equivalence defines an equivalence relation on Some deceptively simple problems for n-link chains are configurations. Hence RECONFIGURATION can be difficult in the sense of complexity theory. Here is an rephrased, “Given two configurations of a linkage, and example — see Hopcroft, Joseph and Whitesides (1985). possibly some constraints, are the configurations equival RULER FOLDING: ent?” The notion of equivalence classes of configurations GIVEN: Positive integer lengths l\, l2, ..., /n of an n-link suggests a related problem: chain (carpenter’s ruler), and a positive integer k. PLACEMENT: QUESTION: Can the ruler be folded into length at most k! GIVEN: A linkage together with some constraints (such as In other words, is there a configuration of the chain that certain joints having certain locations). positions all the joints in a closed interval of length kl QUESTION: Are there any configurations that satisfy these constraints? If so, how many equivalence classes of 2.1 RULER FOLDING is NP-complete configurations are there, and what is a representative con Because the basic idea is useful for other placement ques figuration for each of these classes? tions, we repeat from Hopcroft, Joseph and Whitesides For example, the Peaucellier Inversor has configura (1985) the easy proof that RULER FOLDING is NP- tions for which C does not lie on line segment PQ, although complete. Clearly the problem belongs to NP. The proof these configurations are not equivalent to the ones shown that it is NP-complete uses a reduction from the classic in Figure 1. One could assemble the device so that E and D NP-complete SET PARTITION problem — see Garey coincide. This would make several of the links coincide. In and Johnson (1979). particular, links DC and EC would coincide and would be SET PARTITION: able to turn freely about their coincident joints D and E GIVEN: A collection of n not necessarily distinct positive
THE AUSTRALIAN COMPUTER JOURNAL, VOL. 24, No. 2, MAY 1992 43 MOVEMENT ALGORITHMS integer weights wh w2,..., wn. 2,2 An Application to a Graph Embedding Problem QUESTION: Can the collection of weights be partitioned The NP-completeness of RULER FOLDING can be used into two subcollections A and B so that to show the NP-hardness of other geometric problems, such as the following graph embedding problem, des tV,- = ^WjeB tVj ? cribed in Johnson (1982) and proved NP-hard by a reduc Here is the basic idea of the reduction. From the given tion from 3-SAT in Saxe (1980). (Note that in this prob weights for SET PARTITION, we want to design a ruler lem, the graph embedding is allowed to have edges that together with a folding length k. The ruler should fold into cross.) length at most k if and only if the weights can be equally WEIGHTED GRAPH EMBEDDABILITY: partitioned. First, make a ruler whose n links correspond to GIVEN: A graph G = (V,E), and a positive integer weight the n given weights: set length 4 = weight wt. To this ruler, w(u,v) for each edge (u,v) in E attach two more links La and Lb at the beginning and two QUESTION: Is there a function/ not necessarily one-to- more links Ly and Lz at the end, forming an n+4-link chain one, mapping the vertices in V to the plane so that for each La,Lb,Lh...,Ln,Ly,Lz. The lengths of the four added links are edge (u,v) in E, the Euclidean distance between flu) and assigned the values fly) equals the weight w(u,v)? 4 = ly = 2? J Wj and 4 = 4= 2lb = 2ly. Note that it is not clear that this problem belongs to NP. Finally, choose k = la = 4- In other words, the ruler is to be The answer to an instance could be yes, yet the points fly) folded into an interval of length equal to its two longest could have non-rational coordinates. Hence quickly links. See Figure 3. checkable verifications of yes instances might have to work by some other means than exhibiting a set of points that respect the specified edge weights. L = 2lIt = 5>i ly = Zw; It = 2Zwj •------•------*«---•------• Here is a way to reduce SET PARTITION to WEIGH Wj w„ TED GRAPH EMBEDD ABILITY using the same idea as Figure 3. The ruler designed from the weights. the reduction from SET PARTITION to RULER FOLD ING. First, make a ruler whose link lengths correspond to In any successful folding of the ruler, La and Lz must be the weights of SET PARTITION (don’t add any extra links placed on top of each other, and consequently, Lb and Ly this time). Suppose that the SET PARTITION instance has will have coincident endpoints located at the midpoints of n weights, so that the ruler has n links and hence n+1 joints. La and Lz. Note that these coincident endpoints are just the Observe that the weights can be separated into two equal- first and last joints of the ruler whose links correspond to weight piles if and only if the ruler can be folded into a the weights from SET PARTITION. If the n+4-link ruler straight line segment so that its first and last joints are can be folded, an equal-weight partition of the weights can placed at the same position. be read off from the folding: weights whose corresponding Instead of requiring that the ruler be folded into a links point to the right (the higher indexed endpoint being staight line segment, require that it be placed inside a right of the lower indexed endpoint) form one part of this planar region that is so thin that any placement of the ruler partition, and the remaining weights form the other part. with A0 at An must be just a slight perturbation of a folding Similarly, a folding of the ruler can be read off from an of the ruler into a line segment that places A0 at An. To do equal-weight partition of the weights. See Figure 4. this, imagine that P and Q are points in the plane that are a large distance d apart. Imagine that Pis connected to A; by a two-link chain P, Xb Ab where the links are both of length R for some large R This means that A,- must be located in a circle of radius r = 2R centred at P. ~ Now imagine that Q is connected to A; by a two-link chain of the same description. Then A; must be located in a circle of radius r = 2R centred at Q. If 2r is a little greater than d, the circles will intersect, and the ruler will be confined to this narrow intersection. By an appropriate Figure 4. Correspondence between foldings and equal-weight choice of formulas for d and r, one can ensure that these partitions. values can be computed rapidly and that the ruler can be folded onto a line segment with Ao at A„ if and only if it can SHORT-LINK RULER FOLDING is the restriction of be placed in the intersection of the circles with Ao at A„. RULER FOLDING to rulers whose n link lengths 4 satisfy See Figure 5. One suitable choice for r and d is some bound 4 < M A dynamic programming algorithm solves the restricted problem in linear time O(n) (see Hop r=(l + 64n2(24)2)/(8n)andd=(64n2(24)2- l)/(4n). croft, Joseph and Whitesides, 1985).
44 THE AUSTRALIAN COMPUTER JOURNAL, VOL. 24, No. 2, MAY 1992 MOVEMENT ALGORITHMS
\~*------d------►]
Figure 5. Approximating a line segment. Figure 6. A disc interchange problem.
It is not difficult to design a weighted graph G = (V,E) that how to pose this problem algebraically, in terms of multi- encodes the geometric situation just described. variable polynomial equations and inequalities. To specify that the two discs should not collide with 3 THE ALGEBRAIC APPROACH each other, we write an inequality that forces their centres Complex algorithms have been designed that are so to keep distance at least 1 apart: general that they can solve a wide variety of placement and motion planning problems. In addition to linkages, these (*i — xi)2 + (yi — yi)1 > 1- algorithms can handle objects with telescoping pieces, Multiplying this out, we obtain a polynomial inequality in objects consisting of several parts, and many others. In the four variables xi,yi,x2,y2: fact, the algorithms can handle any geometry problem that X\ — 2x1*2 + xi2 + y\ — +y-i — i > o. can be expressed in terms of logical expressions involving multivariable polynomial equations and inequalities with To specify that Dx should not collide with either vertical rational coefficients. Explanation of how these algorithms wall, we want actually work is not given here. For expositions of this, see Xy > 1/2 and*i < 7/2. LaTombe (1991), Schwartz and Yap (1987), Whitesides (1985). The inequalities yi > 1/2 andyx < 7/2 3.1 Several Examples will keep Dx from colliding with the horizontal walls. To suggest just what a large class of problems this is, we Similarly, the constraints on D2 are expressed by give several examples of how to formulate simple motion planning problems in terms of algebra. None of the exam x2 > 1 /2, x2 < 7/2, y2 > 1 /2 and y2 < 7/2. ples that follow involves a placement problem or a 3- dimensional problem, but the reader should have no diffi Any solution (x^y^jj) to this system of multivariable culty seeing that such problems can indeed be formulated polynomial inequalities gives a collision-free placement of algebraically after reading the examples below. the discs. Furthermore, from any collision-free placement EXAMPLE 1: interchanging two disks of the discs, one can read off the coordinates of their Our first example is the formulation of a motion coor centres to obtain a solution to the system of inequalities. dination problem in which the object to be moved consists The set of all solutions (x1,y1,x2,y2) is a set S of points in a of two circular discs Dx and D2 each of radius 1/2. Let 4-dimensional configuration space. However, this set of Oq,yi) and (x2,y2) denote the coordinates of the centres of points need not be a connected set. It may consist of more D\ and D2, respectively. Suppose that the discs are con than one component. Hence the RECONFIGURATION fined inside a square box of side 4, with vertices at (0,0), question, rephrased algebraically, becomes, “Do the initial (4,0), (0,4) and (4,4). The initial configuration of the two- and final configurations belong to the same connected part object is given by component of S ?” In other words, do (3 /4,13 /4,13 /4,3/4) and (13 /4,3/4,3/4,l 3 /4) belong to the same component of C*i,Jh) = (3/4,13/4) and (x^) = (13/4,3/4). SI We ask if there is a collision-free motion interchanging the * two discs. See Figure 6. EXAMPLE 2: a telescoping link Of course just by looking at the picture, we can see that Without setting up a complete motion planning prob the discs can be interchanged. Our purpose here is to see lem, let us note that it is easy to describe a telescoping link
THE AUSTRALIAN COMPUTER JOURNAL, VOL. 24, No. 2, MAY 1992 45 MOVEMENT ALGORITHMS in terms of polynomial inequalities. Suppose the link is a bounded by a constant times n, where n is the length of the line segment whose length varies between 1 and 3. Let input. For a simple instance of a geometric problem formu (xi,y1) and denote the endpoints of this line segment. lated algebraically, this n could be very large indeed, so Then the distance between the endpoints must be a that the algorithm would take a long time in practice, even number in the range 1 to 3. In other words, though its running time increases only linearly with prob lem size. (xi — x2)2 + (yi — y2)2 > 1 and The most meaningful formulas for the running times of (*i — x2)2 + (yi —y2)2 ^ 9, the general algorithms based on an algebraic approach are which can be multiplied out to obtain polynomial given as functions of several parameters (each of which is inequalities. bounded above by the length of the input): the number of variables, the number of polynomials involved in the logi EXAMPLE 3: a chain of links cal expressions, the maximum degree of the polynomials, At this point, it should be clear that a chain of n links and the number of bits involved in writing down the coeffi that are allowed to cross over one another can be modelled cients of the polynomials. Such formulas show how the by a system of polynomial equations in 2(n+l) variables running time depends on each of these parameters. x0yo >■■■> xt»yn- One has only to specify, for l
46 THE AUSTRALIAN COMPUTER JOURNAL, VOL. 24, No. 2, MAY 1992 MOVEMENT ALGORITHMS
algorithms involves dealing with round-off errors. It is typically necessary to extend the precision of the computa tions in order to ensure correct branching on tests like, “Does point p lie above, on, or below line L?” Any analysis that estimates the running time of a correct implementa tion of an algorithm on a real sequential machine must take into account the extra amount of time required to handle numerical details. At present, the subject of robust computational geometry is still in its infancy. Hence the reader is warned that algorithms regarded as fast with Figure 7. Placement for a 5-link chain. respect to a real RAM model of computation may or may not give rise to useful implementations. QUESTION: Can the chain be placed inside the polygon if crossing links are allowed? EXAMPLE 1: moving a single link It is easy to see how to formulate this problem algebrai One of the most basic of all motion planning problems cally, using at most 202 variables. The algebraic question is the following reconfiguration problem. will be whether the system of polynomial equations and LADDER MOVING: inequalities has any solutions. This can be answered in GIVEN: The length of a single link (closed line segment); polynomial time in the length of the algebraic description. two configurations of the link; and for each polygon in a set The algebraic description will contain equations specify of simple polygonal obstacles for the link, an ordered list of ing the link lengths, and for each joint Aj and for each edge the coordinates of the vertices of the polygon. of the polygon, an inequality specifying that the joint must QUESTION: Can the link be moved between the two lie on the correct side of the line determined by the edge. given configurations? As the previous section pointed out, the constraint that Schwartz and Sharir (1983) showed that this problem links must not cross can be handled algebraically. Can the can be solved in 0(n5) running time, where n is the total reader use that idea to formulate algebraically the FEW number of vertices of all the polygonal obstacles and a real LINKS problem above for simple polygons that need not RAM model of computation is assumed. Several algo be convex? rithms with better running times were proposed. Ke and O’Rourke (1987) proved a lower bound of fl(n2) for 4.2 Specialised Algorithms for Problems Involving ladders moving in the plane and a lower bound of D(n4) for Few Links ladders moving past polyhedral obstacles in 3-dimensions. The general algebraic approach enables us to see easily In Ke and O’Rourke (1987), best known upper bounds of that many problems involving few links can be solved in 0(n2bgri) for the planar problem and 0(n5logri) for the polynomial time. However, the algebraic algorithms do' 3-dimensional problem are cited. not provide reasonable running times in practice. Hence the search is on for finding fast specialised algorithms for EXAMPLE 2: a telescoping link all kinds of motion planning problems involving a fixed or Halperin and Sharir (1990) considered several few- bounded number of links. The hope is that customised degrees-of-freedom motion planning problems. For algorithms will be more practical than ones arising from a example, they considered RECONFIGURATION for a general approach. two-link arm moving in the plane amidst point obstacles. It is important to introduce a few grains of salt at this In general, the term “arm” refers to a chain of links with point about what is meant by a fast algorithm. In the field one endpoint fixed down. Here, the arm consists of two of computational geometry, a real RAM model of compu links op andpq. Joint o is fixed at the origin, op can extend tation is often adopted. In this model, it is assumed that or shrink arbitrarily, pg has a fixed, given length, and both data can be any real numbers (not just rational ones), that links can rotate arbitrarily. Using ideas from Halperin, any real number requires only one unit of storage, and that Overmars and Sharir (1988), they gave an 0(n2log2ri) arithmetic operations on real numbers (and often other algorithm for determining the answer to the yes-no ques operations such as extracting square roots) take only one tion of whether there is a motion taking the one given unit of time. configuration to the other. When the running time of an algorithm is said to be in 0(ri) with respect to real RAM, the n does not refer to the EXAMPLE 3: a design problem number of bits of data given, since all real numbers are Suzuki and Yamashita (1991) have proposed problems viewed as having the same length. Rather, the h measures that do not have the general format of either RECON the number of data items. Of course this model of compu FIGURATION or PLACEMENT problems. As they point tation is highly unrealistic. Implementation of geometric out, these problems concern the analysis of given movable
THE AUSTRALIAN COMPUTER JOURNAL, VOL, 24, No. 2, MAY 1992 47 MOVEMENT ALGORITHMS objects in given environments. Their objective is to study of describing motions that carry out reconfigurations once design problems when the environment is given, but not the it has been determined that these exist. We omit the des movable object. One wants to design a movable object that cription of how this is done in algebraic approaches. A can attain some goal relative to the given environment. purely geometric algorithm that produces a motion as Here is an example of a problem studied by Suzuki and output can do this by giving a sequence of “simple Yamashita. Recall that an “arm” is a chain of links with motions”. Here, what is considered a simple motion is one endpoint fixed down. Suppose an arbitrary n-sided entirely up to the algorithm designer. The only require convex polygon is given. Design a k-link arm (which must ment is that there be no ambiguity about the motion being remain inside the polygon), a position for its fixed end described. point, and an initial configuration of the arm so that the Figures 8 and 9 illustrate two possible concepts of following goal is achieved: the tip of the arm can reach simple motion for a chain of links. every point inside the polygon. An arm that attains this In Figure 8, the joints between S and V are the only goal is said to “manage” the given polygon. They show, joints that change position during the simple motion. The among other results, that it can be decided whether an angles at joints S, T, U and V are the only ones that change. arbitrary n-sided convex polygon is manageable by a 2- Joint U must not be straight or folded in order to apply link arm. When a polygon is manageable in this way, they simple motion at this combination of joints. Also, the have an algorithm that can design an appropriate arm in chains of links between 5 and T, T and U, and U and V must 0(n2) time. have all intermediate joints straight initially; furthermore, In posing problems of this type, there is no need to these joints are to remain straight. The simple motion restrict the number of links of a linkage, or for that matter, consists of rotating T clockwise or counterclockwise about to consider only linkages. Suppose we have in mind a type S while adjusting the joint angles at U and V so that the of movable object (discs, chains of arbitrary numbers of two-segment “elbow” [T,U],[U,V\ tracks the motion of T. links, telescoping 2-link arms, etc.) and a design goal for Stopping conditions can be specified that will make pre that object (reaching all points in its environment with cise the final configuration of the chain at the termination some specified point on the object, being able to move of the simple motion. between any two given configurations, etc.). Then we can pose the following question relative to that class of objects and that type of design goal. DESIGN: GIVEN: An environment for the movable object. QUESTION: Can a movable object and a placement for that object be computed so that the design goal is attained? Here the instances of the problem consist of descrip Figure 8. The first type of simple motion. tions of only the obstacles; the algorithm designer knows Alternatively, one might hold fixed the angles at all but the design goal and the class of movable objects to five joints S, T, U, V and W. Initially,lie angled atT and V consider. must not be folded or straight. The locations of all joints We close this section by mentioning that few-link prob except those between S and W are to remain fixed. The lems are of interest in computer ariimation and human simple motion consists of moving U along some specified figure simulation. See, for example, Korein (1985). straight line until some ^topping criterion is satisfied. See 5 FAST ALGORITHMS FOR n-lirik PROBLEMS Figure 9. When a linkage moveihent problem does not require that the number of links be at most some given constant, the number of variables in an algebraic formulation is not bounded, and so the algebraic approach does not yield a polynomial time algorithm. We will refer to linkage prob lems that impose no bound on the number of links as ra-Iink problems. Some n-link problems are hard from the com plexity point of view (RULER FOLDING is one of many ' SSs examplel). On the other hand, some rc-link problems admit Figure 9. A second type of ample motion. polynomial time solution without appealing to real RAM, although not via the algebraic approach (SHORT-LINK Of course, there is a difference between the number of RULER FOLDING is an example). simple motions output by an algorithm and the time it takes to compute that sequence. The time to compute the 5.1 Simple Motions sequence is at least as great as the number of simple Before proceeding further, we consider briefly the problem motions in the sequence.
48 THE AUSTRALIAN COMPUTER JOURNAL, VOL. 24, No. 2, MAY 1992 MOVEMENT ALGORITHMS
5.2 Examples of n-link Problems Admitting Polynomial Time Solution One example of an n-link problem that nevertheless admits polynomial time solution with respect to a real RAM model of computation appears in Hopcroft, Joseph and Whitesides (1985). Here, the yes-no RECONFIGU RATION problem for an arm confined to a circle is solved by an algorithm running in 0(ri) time. In case reconfigura Figure 10. Two equivalent configurations. tion is possible, an 0(ti3) algorithm outputs a sequence of 0(n3) simple motion steps that do the reconfiguration. This This means there is a motion between any pair of given result was improved by Kantabutra and Kosaraju (1986), configurations. See Figure 10. who gave an 0(n) algorithm that computes a sequence of From any initial configuration, one can quickly com 0(n) simple motion steps. pute a sequence of simple motions taking the chain into a We consider one last example (see Lenhart and White- convex shape. Similarly, one can quickly compute a sides, 1991) of an n-link reconfiguration problem that can sequence of motions taking the desired final configuration be solved by a polynomial time algorithm with respect to a into a convex shape. If these two convex shapes are in the real RAM model of computation. Suppose we are given a same equivalence class, which can be determined by the chain of n links, a fixed position in the plane for the first simple test above, one can quickly compute a sequence of joint, a fixed position in the plane for the last joint, and two simple motions taking the one convex shape to the other. If configurations of the chain having the first and last joints the two convex shapes are in different equivalence classes, at the specified locations. We want to know whether the there is no motion between the two original given configu chain can be reconfigured, and if so, how to move between rations of the chain. the two configurations. The links are allowed to cross. It turns out that chains of links having both endpoints 6 CONCLUSION fixed can have at most two equivalence classes of configu We have considered three broad types of algorithmic prob rations (see Lenhart and Whitesides, 1991). If there are lems for linkages, namely RECONFIGURATION, two classes of configurations, then each configuration in PLACEMENT, and briefly, DESIGN. We have consi the one class is the mirror image, with respect to the line dered the possibility of bounding the length of the links, through the fixed points A0 and An, of a configuration in and have seen an example where this makes a hard prob the other class. lem easy from the point of view of computational The intuition is very roughly this. Imagine that the chain complexity. has been pulled out into a convex polygonal shape (indeed it We have also considered problems with an unbounded is always possible to do this), with a link of the appropriate number n of links as well as problems where the number of length inserted between the fixed endpoints. Now it can be’ links is at most some specified bound. When the number of thought of as a polygon. Of course 3-sided polygons, or links is bounded, we indicated that general algebraic triangles, are rigid; they have two equivalence classes of methods often provide a way to see that the problem can be embeddings, which are mirror images. Similarly, an n- solved in polynomial time. sided polygon that has three long sides joined together by We have remarked that there are two vastly different very short chains of several very tiny links looks so much models of computation commonly used to study these like a triangle that it, too, has just two equivalence classes problems, and that fast algorithms on real RAM machines of embeddings. Every polygon can be thought of as con may or may not have practical implementations. sisting of its three longest links, connected together by Finally, we saw examples M fast (on real RAM) geo chains, perhaps not so short, of smaller links. Depending metric algorithms both for few-link problems and for n- on the lengths of these connecting chains, the polygon link problems. either behaves like a triangle and has two equivalence Some topics touched on in this paper that suggest future classes of embeddings, or has enough flexibility to have research directions are: 1) finding fast, geometric solutions just one class of embeddings. for few-link problems; 2) finding implementation methods The criterion for determining whether there are one or for geometric algorithms that take numerical issues into two equivalence classes of configurations is simple. If the consideration; 3) increasing knowledge of 3-dimensional lengths of the second and third longest links sum to more problems; 4) considering design as well as reconfiguration than half of the sum of all the link lengths, then there are and placement problems; 5) understanding what factors two equivalence classes. Otherwise, there is only one equi can make n-link motion planning easy. > valence class. For example, if the links are of lengths 5,3, 7, 2 and 4, the chain has only one equivalence class of ACKNOWLEDGEMENT configurations because (5 + 4) < (1/2) (5 + 3 + 7 + 2 + 4). The author thanks Peter Eades and Godfried Toussaint for
THE AUSTRALIAN COMPUTER JOURNAL, VOL 24, No. 2, MAY 1992 49 MOVEMENT ALGORITHMS
their encouragement to survey this topic, and the anony LaTOMBE, J. (1991): Robot Motion Planning, Kluwer Academic Pub mous referee who pointed out the relevance of reference lishers, 1991. LENHART, W. and WHITESIDES, S. (1991): “Turning a polygon Saxe (1980) in Johnson (1982). inside-out”, Proc. of the third Canadian Conference on Computational Geometry, Vancouver, 6-10 August, 1991, pp. 66-69. SAXE, J.B. (1980): Two Papers on Graph Embedding Problems. Report No. CMU-CS- 80-102, Department of Computer Science, Camegie- REFERENCES Mellon University, Pittsburgh PA. ALT, H. and YAP, C.K. (1990a): “Algorithmic aspects of motion plan SCHWARTZ, J.T. and SHARIR, M. (1983): “On the piano movers’ ning: a tutorial, part 1”, Algorithms Review, Vol. 1, No. 1, pp. 43-60. problem: II. general techniques for computing topological properties ALT, H. and YAP, C.K. (1990b): “Algorithmic aspects of motion plan of real algebraic manifolds”, Advances in AppL Math, Vol. 4, pp. ning: a tutorial, part 2”, Algorithms Review, Vol 1, No. 2, pp. 61-78. 298-351. CANNY, J.F. (1988): The Complexity of Robot Motion Planning. Series: SCHWARTZ, J.T. and YAP, C.K. eds. (1987): Algorithmic and Geometric ACM Doctoral Dissertation Award: 1987, MIT Press, Cambridge. Robotics, Lawrence Earlbaum Assoc., Hillsdale, New Jersey. COURANT, R. and ROBBINS, H. (1941 renewed 1969): What is Mathe SCHWARTZ, J„ SHARIR,M. and HOPCROFT, J. eds. (1987): Planning, matics? Oxford University Press, pp. 155-157. Geometry and Complexity of Robot Motion. Ablex Publishing Corp., EVES, H. (1963 and 1965): A Survey of Geometry, Vols. 1 and 2, Allyn Norwood, New Jersey. and Bacon Inc., Boston. SCHWARTZ, J. and SHARIR, M. (1983): “On the piano movers’ prob GAREY, M. and JOHNSON, D. (1979): Computers and Intractability a lem: V. the case of a rigid polygonal body moving amidst polygonal Guide to the Theory of NP-Completeness. W.H. Freeman, San barriers,” Comm. Pure and AppL Math., Vol. 5, No. XXXVI, pp. Francisco. 345-398. HALPERIN, D. and SHARIR, M. (1990): Improved Combinatorial SUZUKI, I. and YAMASHITA, M. (1991): Designing Multi-Link Robot Bounds and Efficient Techniques for Certain Motion Planning Prob Arms in a Convex Polygon. Technical Report TR-90-10-01, lems with Three Degrees of Freedom. Technical Report 176/90, the Department of Electrical Engineering and Computer Science, The Moise and Frida Eskenasy Institute of Computer Science, Tel Aviv University of Wisconsin-Milwaukee, 1991. To appear in Proc. 29th University. Allerton Conference on Communication, Control and Computing, Uni HALPERIN, D„ OVERMARS, M.H. and SHARIR, M. (1988): Efficient versity of Illinois, Urbana, 2-4 October, 1991. Motion Planning for an L-shaped Object. Technical Report 418, WHITESIDES, S.H. (1985): “Computational Geometry and Motion Courant Institute of Mathematical Sciences, New York University. Planning” in Computational Geometry, ed. G.T. Toussaint, Elsevier HOPCROFT, J., JOSEPH, D. and WHITESIDES, S. (1985): “On the Science, Amsterdam, pp. 377-427. movement of robot arms in 2-dimensional bounded regions,” SIAMJ. Comput., Vol. 14, No. 2, pp. 315-333. JOHNSON, D.S. (1982): “The NP-completeness column: an ongoing BIOGRAPHICAL NOTE guide”, Journal of Algorithms, Vol. 3, pp. 89-99. Sue Whitesides holds a Masters Degree in Electrical Engi KANTABUTRA, V. and KOSARAJU, S.R. (1986): “New algorithms for neering from Stanford University and a PhD in Mathematics multilink robot arms,” J. Computer and System Sciences 32, pp. 136-153. from the University ofWisconsin. She was on the faculty of KE, Y. and O’ROURKE, J. (1987): Lower Bounds on Moving a Ladder in Tufts University in Massachusetts and Dartmouth College in Two and Three Dimensions. Report JHU-87/14, Department of New Hampshire before coming to McGill University in Computer Science, Johns Hopkins University, Baltimore. KOREIN, J.U. (1985): A Geometric Investigation of Reach. Series: ACM 1983. Her interests lie in geometry, algorithms and discrete distinguished dissertations: 1984, MIT Press, Cambridge. mathematics.
BOOK REVIEW INFORMATION % ' i There have been a number of mattere of concern recently regarding book reviews for i.e. do not have any immediate merit as ‘respectable’, seyious works disseminating the Journal, and I thought I would take a small amount of space toexplain the system a the .science or art of computing. For example, the most often rejected fooks are little, and give voice to some of the concerns. se which are simply handbooks for third party applications software. :h book is then ‘matched’ with a reviewer whose nominated areas of interest CURRENT PROBLEMS tbasically align with those of the book. A letter of offer is sent to the potential Our major current problem is the backlog of reviews we have awaiting publication. The reviewer. If no reply is received, then either a reminder offer is sent if the book is so size of the Journal and its cost has necessitated a page limit being imposed on each issue, well matched to the reviewer as to be hard to neglect, or a negative response is with the consequence that review space is generally tmall per issue. This has meant that assumed. If a book is not accepted after three offers then it is rejected from the there has been a growing lag time between receipt of a review and its publication For process. this I apologise, but ask reviewers to accept that no-one is being singled out for delay 3 Once a book is accepted, it is sent to the reviewer with a guide for preparing the —it is a general problem. Couple this with the increasing number of good books being review. The time in which the review is to be expected is nominated by the reviewer. received, and the problem will not get smaller. At present I haw about 50 books If the review is not received within that time, then reminders are issued in three awaiting distribution, all of them of interest to readers. ~ grades — ‘gentle’ nudge(!), ‘firm’ nudge(!) and ‘please return the book’. If the book In the interests of expediting the publication of as many reviews as possible, I will is still not returned, nor the review received, the publisher is notified that the therefore have to be quite ruthless in insisting on a 300-350 word limit on reviews. If this reviewer has ‘let the side down’ and a review will not appear. Reviewer history is can resonably be achieved by a little editing, then I will do that locally, however,if also taken into account in allocating books. substantial changes would be needed, then the review will be retuned to the author for 4 After the review is received, it is edited if necessary, then typeset, and then shortening. published at a time governed entirely by space available in the Journal. This latter One further problem currently is the matching of books with reviewers — especially constraint has been recently the most severe. in the mojre esoteric areas of the discipline. I am very dependent on the accuracy of areas 5 After publication, copies of the review are sent to the reviewer and the book of interest nominated by reviewers on their interest pro-forma. I also hope to publish a supplier. ^ list of ney books received in subsequent journals so that if particular readers have an It is the adopted policy that only reviews received in accordance with the above interest in a book then they may offer to review it. procedure are published. Unsolicited reviews are not published.
HOW THE SYSTEM WORKS CONCLUSION A short/explanation of how the book review section works may assist readers and It is hoped that the above explanation will help readers and reviewers to understand the reviewers in understanding some of our difficulties — and also to appreciate in more way book reviews work, and that those that are currently frustrated by the slowness of detail what is required when a book is offered for review. The process is as follows: publication will sense an equal frustration on the part of the editors. 1 Books are received directly from publishers and suppliers. They are not solicited. Dr Chris Andrews They are examined as to their suitability for this Journal, and rejected if unsuitable, Associate Editor, Book Reviews
SO THE AUSTRALIAN COMPUTER JOURNAL, VOL. 24, No. 2, MAY 1992 1 INTRODUCTION Algorithms for Perhaps one of the more unexpected discoveries made by mathematicians over the past two decades has been the existence of powerful algorithmic methods in many Groups branches of algebra. Traditionally, the goal of research in algebra has been the discovery of classification theorems John Cannon and George Havas* which attempt to characterize all algebraic structures School of Mathematics and Statistics satisfying a particular set of axioms. However, with the University of Sydney, NSW 2006 growth of interest in algebraic computation (driven, in * Key Centre for Software Technology large part, by a desire to construct symbolic solutions for Department of Computer Science various types of differential equations), mathematicians in University of Queensland, Queensland 4072 the early 1970s were led to discover new approaches to such fundamental problems as computing the greatest common divisor of two integral polynomials and finding the irreducible factors of an integral polynomial. Since Group theory is a particularly fertile field for the design of then several hundred new algorithms have been developed practical algorithms. Algorithms have been developed in various areas of algebra. It is becoming increasingly across the various branches of the subject and they find clear that most, if not all, branches of algebra have a rich wide application. Because of its relative maturity, compu algorithmic content. tational group theory may be used to gain insight into the Compared with most branches of algebra, the algo general structure of algebraic algorithms. This paper exam rithmic content of group theory, computational group the ines the basic ideas behind some of the more important ory, has reached an advanced state of development, both in algorithms for finitely presented groups and permutation terms of the range and sophistication of the algorithms and groups, and surveys recent developments in these fields. in terms of their effectiveness in solving worthwhile prob Keywords and Phrases: Algorithms, group theory, pres lems. Thus, practical algorithms have been designed for entations, permutations, data structures, complexity. computing detailed information concerning the structure, CR Categories: 1.1, G.2, G.4, F.2. representations and extensions of various types of finite group. Techniques have also been developed for studying finitely-presented (fp) infinite groups. Programs imple menting group theoretic algorithms find application not only in the study of groups directly but also in many of the other branches of mathematics which use group theoretic methods. These include coding theory, design theory, dif ferential equations, discrete Fourier transform theory, finite geometry, graph theory, harmonic analysis, mathematical crystallography, number theory and topology. A group may be specified in a number of different ways: in terms of a finite presentation, as a group of permutations or matrices, or as the group of automorphisms of a combi natorial structure such as a block design, geometry or graph. Experience has shown that, as a general rule, the most powerful algorithms are those designed with a par ticular form of group specification in mind, eg. permuta tion groups. The major areas of activity in computational group theory are finitely presented groups, permutation groups, matrix groups, finite p-groups, finite soluble groups and representation theory. It is our aim to introduce the reader Copyright® 1992, Australian Computer Society Inc. General permission to some of the basic ideas that underpin the design of to republish, but not for profit, all or part of this material is granted, provided that the ACJ’s copyright notice is given and that reference is algorithms for group theory. A good introduction to group made to the publication, to its date of issue, and to the fact that reprint theory is provided by Rotman (1973). For reasons of brev ing privileges were granted by permission of the Australian Computer ity we restrict ourselves to fp-groups and permutation Society Inc. groups, two areas where the problems and approaches are Manuscript received 15 November 1991. quite different. Even within these areas, we make no
THE AUSTRALIAN COMPUTER JOURNAL, VOL. 24, No. 2, MAY 1992 51 ALGORITHMS FOR GROUPS
attempt at completeness. A forthcoming book of Sims (to 2.2 Coset Enumeration and Related Algorithms appear) provides a thorough introduction to the theory of Given an fp-group G,G = CtI,...rcr| R[,...,R) (where Ri,...,Rs algorithms for fp-groups, while a recent book of Butler are words in the generators Xi ,...rcr), and given a subgroup H (1991) gives an elementary introduction to computational of G, H = (h\,...,ht> (where hu...,htare also words in the methods for permutation groups. Both of these books generators), classical coset enumeration procedures include some historical information. An early account of attempt to construct a permutation representation for G, algorithms for p-groups and soluble groups is given by corresponding to the action of Gby (right) multiplication Laue, Neubiiser and Schoenwaelder (1984), while recent on the (right) cosets of H, by means of a trial-and-error accounts of work in computational representation theory process. The cosets are traditionally identified with the may be found in Lux and Pahlings (1991), Michler (1990) integers 1where coset 1 always corresponds to the and Schneider (1990). Important collections of papers on given subgroup H. A new coset k is defined as the image of computational group theory appear in Leech (1970), some existing coset i under (right) multiplication by some Atkinson (1984), Cannon (1990) and Cannon (1991b). generator xj of G or by an inverse xy\ The cosets must Computational group theory has been applied to an satisfy the following conditions: enormous variety of problems. Noteworthy achievements (a) coset 1 must be mapped to itself by each of include the classification of the 4-dimensional crystallo (b) each coset j must be mapped to itself by each of the graphic groups (Brown, Billow, Neubiiser, Wondratschek defining relators /?i,...,Rs and by each product xyxi'- and Zassenhaus, 1978) and the construction of sporadic The action of the G-generators on the cosets is stored in simple groups (Higman and Sims, 1968;Sims, 1973;Leon a two-dimensional array known as a coset table. Enforce and Sims, 1977; Soicher, 1990). Many published applica ment of rules (a) and (b) yields values for some hitherto tions prior to 1984 may be found in a bibliography by unknown coset table entries (deductions) and, also, the Felsch, an early version of which has been published identification of cosets which have been multiply defined (Felsch, 1978). A sample of more recent applications may (coincidences). The procedure terminates when be found in the bibliography by Cannon (1991a), which (i) for each coset i, the action of each generator xj and contains a partial listing of papers citing the computer inverse xy' on i is known; and algebra system Cayley. (ii) rules (a) and (b) are satisfied. This procedure was used extensively in hand computa 2 FINITELY PRESENTED GROUPS tion prior to the development of computers. Beginning in 2,1 Introduction 1952, different versions of the procedure have been A well-known theorem asserts that, in general, the word adapted for machine computation and it is perhaps the problem for fp-groups is undecidable (see Rotman, 1973, most widely applied technique in computational group chapter 12). Consequently, algorithms for fp-groups are theory. However, despite its antiquity, our understanding fundamentally different in nature to algorithms for groups of the relationship between a given presentation for G and given in some concrete form (eg. permutation groups or the performance of a particular version of coset enumera groups of matrices over finite fields). tion when applied to that presen tationis poor. Introductory Let G and A" be two fp-groups. Typical of the elemen descriptions of the procedure are giveh by Cannon, Dim- tary questions mathematicians wish to answer about fp- ino, Havas and Watson (1973), Johnson (1980), Leech groups are the following: (1970b, 1984) and Neubiiser (1982). Sims (to appear) — Is G the trivial group? # gives a formal account pf coset enumeration in terms of — Is G finite? •t' |J automata and rational languages. — If G is infinite, is it free? " Performance of the procedure is very sensitive to — If G is finite, what is its order and structure? changes in the'rules used to introduce new cosets. Because — What are the abelian (nilpotent, soluble, perfect) quo of the many possible variations in the manner in which tients of G? new cosets are introduced, coset enumeration must be — Is G abelian (nilpotent, soluble, perfect)? regarded as a family of procedures rather than as a single — Can we construct a small degree permutation represen procedure. For a given coset enumeration procedure, there tation for G? is no computable bound, in terms of length of input and a — Can w|e construct a small degree matrix representation hypothetical index, to the number of cosets \yhich need to for G f ver some given field? be defined in the coset enumeration process to complete — Are the groups G and K isomorphic? the enumeration. (The existence of such a bound would The algorithms used to compute with fp-groups may be violate the unsolvability of the word problem for finitely conveniently described under three headings: Todd- presented groups.) Further, Sims (to appear) has proved Coxeter or coset enumeration based methods; Knuth- that there does not exist a polynomial bound, in terms of Bendix or term-rewriting methods; and quotient group the maximum number of cosets defined, for the number of methods. coset tables which may be constructed using simple coset
52 THE AUSTRALIAN COMPUTER JOURNAL, VOL 24, No, 2, MAY 1992 ALGORITHMS FOR GROUPS table operations similar to those employed in a coset enu “large” subgroup of G and L is a small subgroup in which meration procedure. This result indicates that the running detailed structural computation is possible. Since the time of a coset enumeration procedure, as a function of number of double cosets HxL is often a small fraction of available space, may be unpleasant. the number of single cosets Hx, this techinque, when appli Cannon et al (1973) identify a number of factors that cable, offers potentially great space savings. affect the efficiency of an enumeration. The next signifi As noted at the outset, the classical coset enumeration cant study of coset enumeration procedures was carried procedure constructs a permutation representation for G out by Sims (to appear). Recently, after extensive experi on the cosets of H. Linton (1991b) describes a version of mentation, Havas introduced new coset enumeration coset enumeration which constructs a matrix representa strategies, which exhibit dramatically better performance tion for G over a designated field k (usually a finite field). than previous versions when applied to many “difficult” In the simplest interpretation, Linton’s algorithm con enumerations. The performance of a coset enumeration structs the permutation module corresponding to the program, in doing a particular enumeration, is measured in action of G on the cosets of a subgroup H. However, since large part by the maximum number of cosets that are the algorithm works by constructing representations of the simultaneously defined and by the total number of cosets group algebra kG, given a suitable choice of ideal genera defined during the course of the enumeration. The maxi tors in kG, it is possible, for example, to directly construct mum gives a direct measure of the storage requirements. constituents of a permutation module for G. For example, consider the presentation
THE AUSTRALIAN COMPUTER JOURNAL, VOL. 24, No. 2, MAY 1992 53 ALGORITHMS FOR GROUPS produced by the low index algorithm, is examined for the (1979) and Le Chenadec (1986), while Sims (to appear) presence of infinite factors. investigates the procedure in detail. Having the coset table for a subgroup Hof the fp-group A major success of the KB-procedure in group theory G enables us to construct a presentation for H. A lemma of was its application by Sims (1987) to verify nilpotency of Schreier describes a generating set for H in terms of the an fp-group. The nilpotent quotient algorithm (see below) generators of G and a system of coset representatives for H is used initially to construct a polycyclic presentation for in G. (Such coset representatives may be read off the coset the nilpotent quotient Q of G. Using the relations of this table for H in G.) The Reidemeister rewriting process presentation as an initial set of rewrite rules, and a special permits us to rewrite the relators of G and their conjugates term-ordering, Sims was able to develop an effective algo as words in these Schreier generators. The ensuing relators rithm for verifying the triviality of the kernel of the quo constitute a presentation for H. Again, the coset table tient Q. (If the kernel is nontrivial, the algorithm fails to contains all the information needed to perform Reidemeis terminate.) ter rewriting. Details are given by Johnson (1980) and A major advantage of the KB-procedure over coset Neiibuser (1982). A variation of the Reidemeister rewrit enumeration is that it may sometimes be used to construct ing process, which rewrites //-elements given as words in a confluent presentation for an infinite group. In the case of the generators of G as words in an arbitrary generating set a finite fp-group, coset enumeration is usually the most for H, was described by Benson and Mendelson (1966) efficient method for constructing a confluent presentation. (see also Neubiiser, 1982). Using this theoretical basis, Epstein, Holt and Rees (1991) describe a practical algo programs for constructing presentations of subgroups of rithm based on the KB-procedure for constructing a solu finite index in fp-groups have been implemented by a tion to the word problem for groups known as automatic number of workers, including Havas (1974) and Arrell and groups. This class of groups has solvable word problem Robertson (1984). and includes many important families of groups which Subgroup presentations produced by a Reidemeister- arise naturally in geometry and topology (e.g. hyperbolic Schreier process generally involve large numbers of and Euclidean groups). generators and relators and are poorly suited for human or Recently, Holt and Rees (to appear a) developed a computer use. A theorem of Teitze asserts that, given program which endeavours to determine whether or not presentations for two isomorphic groups, repeated appli two fp-groups G and K are isomorphic. The program cation of three simple rules (Tietze transformations) will alternates between attempting to construct an isomor suffice to transform one presentation into the other. How phism between G and K, and attempting to prove non ever, there is no general algorithm for identifying the order isomorphism by discovering a structural difference. The in which the Tietze transformation rules should be applied. Knuth-Bendix procedure is used to construct a reduction Presentation simplification programs which take “bad” machine for each of G and K These two reduction presentations and produce “good” presentations for machines are used to systematically construct homomor- groups have been developed by Havas, Kenne, Richardson phisms 6: G—ATand then to test each such homomorphism and Robertson (1984) and Robertson (1988). 6 for the properties of being surjective and injective. The As noted at the outset, coset enumeration over a sub nonisomorphism testing relies on findipg some structural group H constructs a permutation representation for G. difference by comparing various kinds of quotients of G We can no w study the quotient of G defined by this repres and K When applied to special classes of groups, such as entation using permutation group methods. Conversely, automatic groups, it can be quite successful. For example, given a finite group G in some concrete repfesentation, we Holt and Rees report thaf it was able to quickly settle the may use the group multiplication to construct cosci tables. isomorphism question for all but two pairs in a collection For such a group G with moderate order, Cannon (1973) of about 30 pairs of link groups. It resolved the question for shows how to construct a compact presentation for G from the last two pairs with more difficulty, taking some hours a coset table so that we can then apply fp-group methods to oh a Sun 3/60. G. Sims (1991) employed the KB-procedure to deduce non-obvious relations in two groups. In each case, the 2.3 Term-rewriting Methods relations could not have been discovered using the current A specialization of the Knuth-Bendix term-rewriting generation of coset enumeration procedures. This is one of procedure (Knuth and Bendix, 1970) has been applied to the first reported instances where the KB-procedure out fp-groupsfStarting with a finite presentation for a monoid performs coset enumeration when both are potentially M, the Knuth-Bendix procedure for strings (KB-procedure) applicable. attempts to construct a confluent presentation for M A confluentpresentation for M consists of a system of rewrit 2.4 Quotient Group Methods ing rules which convert any element of M into a unique An important technique for studying an fp-group G normal form. The KB-procedure has been studied exten involves constructing homomorphic images of G, which sively. Some applications to groups are given by Gilman may be either members of some class of fp-groups having
54 THE AUSTRALIAN COMPUTER JOURNAL, VOL. 24, No. 2, MAY 1992 ALGORITHMS FOR GROUPS solvable word problem, permutation groups or matrix fully applied them to determine all groups with order groups. As noted above, a successful coset enumeration dividing 256: there are 56092 groups of order 256 of the cosets of some subgroup yields a homomorphism (O’Brien, 1990,1991). onto a permutation group, while the low index algorithm A number of variations on the original p-quotient algo systematically searches for homomorphisms into the rithm have been made. Thus, a general nilpotent quotient symmetric group Sym(ri), for small n. In this section we program (with no dependency on a prime p and allowing examine techniques for directly constructing abelian, nil- infinite quotients) was prototyped by Sims in Mathematica potent and soluble quotients of an fp-group. In each case, a and a new implementation has just been developed by confluent presentation for the quotient group is con Nickel at ANU. Havas, Newman and Vaughan-Lee structed. Note that, if the particular quotient is equal to G, (1990) have produced an analogue of the group nilpotent this effectively solves the word problem for G. quotient algorithm for graded Lie algebras. This has appli The structure of a finitely generated abelian group A cations to p-groups where the quotients are too large to be may be obtained by computing the Smith normal form of handled by the group program and has been used to inves its relation matrix (an integer matrix). Efficient algorithms tigate questions related to the Burnside problem. Finally, for computing this form for large matrices have been during a recent visit to Australia, Vaughan-Lee developed described by Havas and Sterling (1979). If the isomor a variation for finitely presented associative algebras. phism between A and its canonical form is required, the A more difficult area is the computation of soluble reduced basis algorithm of Lenstra, Lenstra and Lovasz quotients. Wamsley (1977), Leedham-Green (1984) and (1982) may be utilized, as described by Sims (to appear, Plesken (1987) have proposed generalizations of the nil- Chapter 8). Thus, given an arbitrary fp-group G, the struc potent quotient algorithm to a soluble quotient algorithm. ture of its maximal abelian quotient, GIG', is readily The Plesken algorithm has been implemented by Wegner obtained. at St Andrews as part of his PhD thesis, and has had some Hand computations in the sixties led to the develop success. Baumslag, Cannonito and Miller (1981) outlined ment by Macdonald (1974) and Wamsley (1974) of algo a method for constructing polycyclic quotients. While rithms for computing finite nilpotent p-quotients of G, their interest was purely theoretical, Sims (1990b) has wherep is a prime dividing the order of GIG'. Starting with further developed their ideas and implemented them in the an exponent-p-quotient of GIG', the algorithms succes special case of metabelian quotients. An implementation sively extend a current p-quotient H by an elementary of the general algorithm (see Sims, to appear, chapters 9 abelian group that is centralized by H. Since the extension and 10) involves a sophisticated combination of many theory is particularly simple in this situation, it is possible algorithms, including Grobner basis techniques (Buch- to design extremely effective algorithms. Nice descriptions berger, 1985). Success has been reported in specific cases of a basic algorithm are given by Newman (1976) and by Neubiiser and Sidki (1988), Newman and O’Brien (to Havas and Newman (1980). The algorithm outputs the appear) and Havas and Robertson (to appear), all using p-quotient in terms of a power-commutator presentation, a combinations of previously described programs. special confluent presentation. The p-quotient algorithm At the DIMACS Workshop on Groups and Computa relies critically on a particular string rewriting procedure tion in 1991, Holt and Rees (to appear b) demonstrated a known as commutator collection, where the rewrite rules program for finding certain quotients of finitely presented are the relations of a power-commutator presentation (see groups. A backtrack search attempts to construct a Felsch, 1976; Havas and Nicholson, 1976; Leedham- homomorphism between the given fp-group G and nomi Green and Soicher, 1990; and Vaughan-Lee, 1990b). nated permutation representations of selected finite Since the development of the original p-quotient algo groups. In particular, this program may be used to identify rithms, Havas and Newman (1980) and Vaughan-Lee small perfect groups that occur as quotients of G. This (1984) have introduced many improvements. A new work builds on the classification by Holt and Plesken implementation has been recently developed in Canberra (1989) of all perfect groups of order up to a million. by Newman and O’Brien with additional enhancements. Having found a representation of G, the program converts The algorithm has been extensively applied to the it to a regular representation, and then attempts to con investigation of Burnside groups (see Vaughan-Lee, struct larger quotients of G by performing elementary 1990a). As an illustration of the power of current imple abelian extensions using either a Reidemeister-Schreier or mentations, the p-quotient algorithm has computed a p-quotient algorithm. The portion of the subgroup lattice power-commuter presentation for the three-generator res obtained is represented graphically and may be manipu tricted Burnside group of exponent 5, a group with class 17 lated interactively. and order 52282. Building on the p-quotient algorithm, Leedham-Green and Newman designed and implemented 3 PERMUTATION GROUPS ' an algorithm for generating descriptions of p-groups. In 3.1 Introduction his PhD thesis at the Australian National University, Computational methods for permutation groups differ O’Brien significantly refined these methods and success fundamentally from those designed for fp-groups since the
THE AUSTRALIAN COMPUTER JOURNAL, VOL. 24, No. 2, MAY 1992 SS ALGORITHMS FOR GROUPS undecidability of the word problem means that many fp- over the set of elements of G. The design of effective group procedures cannot be guaranteed even to terminate. algorithms depends critically on the method chosen to Thus, if we are lucky, the fp-procedures may provide us represent this set. The representation should have the fol with some information about the overall properties of the lowing features: group under consideration. However, in the case of a per (a) it should display key aspects of the group structure; mutation group, it is possible to design fast algorithms (b) it should have the property of inheritance, thus a sub which can provide us with highly detailed information group of G should directly inherit its representation about the structure of the group. from that of G; Over the past two decades, a large number of useful (c) it should be effectively computable. permutation group algorithms have been discovered. Sims (1970,1971a, 1971b) introduced the notion of a Because the field is so extensive, this survey is restricted to base and strong generating set (BSGS) as the appropriate a few fundamental classes of algorithms and no attempt is set representation for a permutation group. A base for G is made at completeness, even in the case of the areas we do a sequence B = (pu...,/3k> of distinct elements of Cl such that consider. A more complete account of basic permutation the identity is the only element of G that fixes B pointwise. group algorithms is given by Cannon (1984) and Bosma Let G« .„>at denote the pointwise stabilizer of {au...,ak}c Cl. and Cannon (to appear). Butler (1991) provides an intro Then B defines a sequence of subgroups, G = G° > G2) > ductory account of many basic algorithms. We do not ... > Gk*l) = <1>, where G® = Gpt.. p._r A strong generating set attempt to survey the extensive literature on the complex S for G, relative to the base B, is a generating set for G ity of permutation group algorithms but rather refer the which contains generators for each subgroup in the chain. reader to Babai (1991). Given B and S, it is a straightforward matter to compute Let G be a permutation group acting on the finite set Cl the orbit A, =ft0® anda transversal (A0for G'+1) in G®, for and suppose G is given in terms of a small set X of generat i = 1 ,...,k ing permutations. The following are representative of the Knowledge of A; and G® for i = 1 ,...,k immediately gives type of information sought by permutation group theorists: us the order of G, a membership test for G, and the possibil — What are the G-invariant subsets (partitions) of the set ity of listing the elements of G without repetition. The n? central philosophy in permutation group computation is to — What is the order of the group G? assume that the group Gis represented in terms of a BSGS — Is a given element of Sym(Cl) a member of the group G? (whose construction is discussed below). Sims showed that — Find generators for the stabilizer of a sequence (set) of the availability of a BSGS enables us to design effective elements of Cl. algorithms for constructing important classes of sub — Determine the various series of characteristic sub groups and quotient groups of G. Wherever possible, algo groups of G: derived series; lower central series; upper rithms for constructing such subgroups and quotient central series. groups of G are organized in such a way that the new group — Find generators for the Sylowp-subgroup of G, where p inherits a BSGS from the known BSGS for G. For example, is a prime dividing the order of G. the backtrack algorithms for constructing subgroups, des — Compute the centralizer (normalizer, normal closure) cribed below, automatically produce a BSGS for the of a subgroup H of G. subgroup. * — Determine representatives for the conjugacy classes of The design of fast methods for constructing a BSGS for elements of G. G is one of the first problems which must be solved. In — Compute a composition series for G and determine the 1967, Sims developed a BSGS algorithm based on the isomorphism type of each composition factor. following lemma of Schreier: Let G be a group with gener The construction df a G-invariant partition of Cl pro ating set X, arid let H be a subgroup of G. If U is a right ceeds by computing the finest G-invariant partition that transversal for H in G such that U contains the identity contains a given pair of points a and fi. Consideration of all element then H is generated by the set {ux
56 THE AUSTRALIAN COMPUTER JOURNAL, VOL. 24, No. 2, MAY 1992 ALGORITHMS FOR GROUPS generators produced by the lemma. In fact, the number of done in order to establish whether or not the Cooperman- elements needed to generate the stabilizer is usually a tiny Finkelstein algorithm is a practical competitor to the fraction of the number produced by the lemma. (It is easy Brownie-Cannon-Sims algorithm. to see that at most n elements are needed to generate the Given a particular base B for G and strong generators stabilizer of a point in G.) With careful organization, the relative to B, there exist fast algorithms for computing above idea yields a BSGS algorithm that produces small strong generators for G relative to a different base B' for G. generating sets for the terms of the stabilizer chain. In Sims (1971a, 1971b) shows how to modify the strong practice, the algorithm works well for degrees less than generators when two adjacent base points are inter 100 but becomes impractical as the degree increases changed. By concatenating the new base B'onto the end of beyond a few hundred. For the case of a soluble permuta B, the desired base change may be achieved by performing tion group, Sims (1990a) describes a BSGS algorithm a succession of adjacent interchanges. Some modifications which takes account of the particular structure of these to speed up this procedure are described in Butler (1991). groups. Cooperman, Finkelstein and Luks (1989) describe a base For larger degrees a different approach is adopted. First change algorithm with running time 0(kn2). A different a “probable” BSGS for G is constructed, and then an approach to changing base is to use a probabilistic proce algorithm is applied which either verifies that the BSGS is dure similar to the random Schreier algorithm mentioned correct, or establishes that it is incomplete. A “probable” above (see Leon, 1980). Such a procedure is employed in BSGS may be constructed very quickly by using a fixed the Brownie-Cannon-Sims verification algorithm. The number of randomly chosen elements of G in place of the complexity of such algorithms has been analyzed by products ux in Schreier’s lemma (probabilistic Schreier Cooperman, Finkelstein and Sarawagi (1990). algorithm). The main inductive step in BSGS verification is the following: Suppose H is a subgroup of G-2) with a 3.3 Subgroup Constructions certified BSGS. If we can show that H = G®, then we will The base change algorithm enables us to rapidly construct have verified the correctness of the BSGS for G. In prac a BSGS for the pointwise stabilizer of a sequence of points tice, //will be the approximation to G® constructed by the from Cl. The availability of a BSGS for G enables us to test probabilistic Schreier algorithm. The first verification membership of G in polynomial time, where the degree of algorithm was developed by Sims and published by Leon the polynomial (dependent on the data structure and space (1980) and involves using the Todd-Coxeter algorithm to used) may be as low as 2. The combination of a BSGS construct a presentation for G in terms of strong genera algorithm and membership testing immediately allows us tors. A presentation is assumed to be known for H on its to construct the normal closure of a subgroup of G. The strong generating set by induction. The verification availability of a normal closure algorithm in turn enables involves comparing the index of //in G with the length of us to construct the derived series and lower central series the orbit fif. This method made it practical to construct for G, and also allows us to test subgroups for the proper BSGSs for groups having degree in the low thousands. In ties of being perfect, soluble, nilpotent, subnormal, etc. (see 1986, Brownie and Cannon implemented a new verifica Butler and Cannon, 1982). tion algorithm suggested by Sims. Instead of testing each The availability of the BSGS representation of a permu of the Schreier generators given in the above lemma for tation group provides the appropriate foundation for the membership of H, we test a much smaller subset defined in design of efficient backtrack searches for subgroups of G terms of representatives for the orbits of certain one-point whose elements satisfy some elementary property. By and two-point stabilizers. This algorithm, as implemented carefully choosing a base adapted to a particular back in Cayley, has constructed BSGSs for groups of degree up track search, we can often greatly reduce the size of the to 500000 and, given sufficient memory, is capable of search tree. The BSGS backtrack search of a permutation computing BSGSs for groups having degrees up to a group was introduced by Sims in 1970, when he described million. backtrack algorithms for computing centralizers and , The theoretical complexity of BSGS algorithms intersections of subgroups. Over the next decade, Butler, depends heavily upon the choice of data structures, partic Cannon and Sims developed backtrack searches for con ularly on the data structure used to store the transversals structing set stabilizers, normalizers, Sylow p-subgroups U©. Sims stored each transversal G® in the form of a and for testing conjugacy of elements and subgroups (see linearized tree structure known as a Schreier vector and the Sims, 1971a, 1971b; Butler, 1982, 1983, 1991). Holt running time of his original algorithm was bounded by (1991) presents a backtrack algorithm for computing sub 0(n6), where n is the degree of G. Using the so-called group normalizers which employs many additional tests to labelled branching data structure for the G®, Jerrum prune the backtrack search tree. The performance of his (1986) described a variant of the original Sims algorithm algorithm is superior in many cases to the Butler algorithm with running time 0(n5). Cooperman and Finkelstein as implemented in the Cayley library of intrinsic functions. (1991) describe an algorithm which verifies strong genera Very recent work of Leon (1991) represents a major tion in 0(n4) time. More experimental work needs to be step in the evolution of backtrack algorithms for permuta
THE AUSTRALIAN COMPUTER JOURNAL, VOL. 24, No. 2, MAY 1992 57 ALGORITHMS FOR GROUPS
tion groups. The efficiency of a backtrack search is heavily indeed contain Alt(n), deduce from these elements the dependent upon the information available to prune the primitivity of G and the presence of Jordan witnesses. This search tree. Using the idea of successive refinement of algorithm is applicable to groups having degree up to a ordered partitions, first introduced by McKay (1978, million and, if G does contain Alt(n), it usually only needs 1981) as part of his highly successful graph isomorphism to sample two or three elements in order to recognize the algorithm, Leon is able to devise new and powerful tests. fact. An early implementation of a set stabilizer algorithm As a next step, Cameron and Cannon (1991) developed based on these ideas demonstrates performance that is an algorithm for identifying any doubly transitive group. dramatically superior to the “first generation” set stabil By carefully analyzing the lengths of the orbits of a two- izer algorithm. As a result of this work we can expect a point stabilizer (three-point stabilizer in a triply transitive new generation of backtrack algorithms, exhibiting super group), the algorithm avoids having to compute the ior performance, to emerge in the near future. derived subgroup of G, except in the case of some rela Let p be a prime dividing | G|, and let P denote the Sylow tively small groups (eg. one-dimensional affine groups). p-subgroup of G. Traditional approaches to computing P At the 1985 Groups — St Andrews meeting, Neumann have involved performing a series of cyclic extensions (1986) described a practical algorithm, based on the commencing with the subgroup generated by an element O’Nan-Scott Theorem, for determining a BSGS for each of p-power order. Butler and Cannon (1989) describe an composition factor of a general permutation group G. The algorithm which employs a backtrack search for possible general strategy involves reducing to a primitive group T, extending elements. This algorithm is restricted to groups locating the socle of T, and then splitting the socle into its having degrees in the low hundreds and primes whose simple direct factors. Luks (1987) published a polynomial exponent in | G| does not exceed 16. Holt (1991) suggests time algorithm for this problem which appears to be of using his fast normalizer algorithm to locate possible theoretical interest only. Neumann’s algorithm made some extending elements. Butler and Cannon (1991) present a use of the fact that practical computation with such an recursive method based on reduction of the degree. The algorithm will be restricted to groups having degree at reduction is based on the observation that if zis ap-central most a few million. Kantor (1991) pushes this approach element having order p, then the kernel of the action of the somewhat further and shows that, except in the case of a centralizer of z on the cycles of z is a p-group. This algo few small groups, it is possible to name the composition rithm, which may involve a probabilistic search for a factors of any permutation group having degree not p-central element of G, generally runs a great deal faster exceeding 106 at the cost of computing a BSGS for G and than the backtrack method. Yet another method based on the derived subgroup of G. Thus, Kantor improves on the the use of homomorphisms has been suggested by Atkin Neumann method by avoiding the step of constructing and son and Neumann (1990). Kantor (1985) and Kantor and splitting the socle of T. It appears that the Kantor algo Taylor (1988) give polynomial time algorithms for com rithm may be generalized to groups of degree up to 108 so puting Sylow p-subgroups which appear to be of theoreti that his approach covers all permutation groups that are cal interest only. presently amenable to practical computation. Going in the other direction, Holt has developed an 3.4 Abstract Structure algorithm for constructing the first and second cohomol- Given the availability of constructions for centralizer, ogjf groups of a permutation group. This very'complex normal closure and Sylow subgroup, one can contemplate aif orithm builds on several of the permutation group algo computing a description of the abstract jtructure of G. rithms described above^ employs the nilpotent quotient Such a goal became particularly attractive with|the com algorithm and involves intricate module-theoretic calcula pletion of the classific|tion of the finite simple groups. tions (see Holt, 1984,1985a, 1985b). A very important requirement is to be able to determine •r quickly whether or not a permutation group G of degree n 4 ACKNOWLEDGEMENTS contains the alternating group Alt(n) in its natural repres Both authors were' partially supported by Australian entation. Indeed, it is desirable to recognize this situation Research Council grants. The second author was partially before incurring the expense of constructing a BSGS for G. supported by DIMACS/Rutgers-NSFSTC88-09648. The starting point is a theorem of Jordan which states that if a primitive group contains ap-cycle, wherep < n — 3 is REFERENCES a prime, then G contains Alt(n). This theorem has been ARRELL, D.G. and ROBERTSON, E.F. (1984): A modified Todd- subsequently extended by other workers to identify many Coxeter algorithm. In M.D. Atkinson (ed), pp. 27-32. ATKINSON, M.D. (1975): An algorithm for finding the blocks of a other permutation cycle types that are Jordan witnesses ior permutation group, Math. Comp. 29, pp. 911-913. Alt(n). Cameron and Cannon, in work extending over a ATKINSON, M.D. (ed) (1984): Computational Group Theory, Academic decade, have constructed a subtle probabilistic recognition Press, London. ATKINSON, M.D. and NEUMANN, P.M. 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(1984): A computer programme for the calculation of the matrix groups III; Sylow subgroups, /. Symbolic Comput. 8, pp. Schur multiplier of a finite permutation group. In M.D. Atkinson (ed), 241-252. pp. 307-319. BUTLER, G. and CANNON, J. (1991): Computing Sylow subgroups HOLT, D.F. (1985a): A computer program for the calculation of a using homomorphic images of centralizers, /. Symbolic Comput. 12, covering group of a finite group, /. Pure AppL Algebra 35, pp. pp. 443-457. 287-295. CAMERON, PJ. and CANNON, J. (1991): Fast recognition of doubly HOLT, D.F. (1985b): The mechanical computation of first and second transitive groups, /. Symbolic Comput. 12, pp. 459-474. cohomology groups,/. Symbolic Comput. 1, pp. 351-361. CANNON, J. J. (1973): Construction of defining relators for finite groups, HOLT, D.F. (1991): The computation of normalizers in permutation Discrete Math. 5, pp. 105-129. groups,/. Symbolic Comput. 12, pp. 499-516. CANNON, J.J. (1984): A computational toolkit for finite permutation HOLT, D.F. and PLESKEN, W. (1989): Perfect groups, Oxford Univer groups, Proc. Rutgers Group Theory Year 1983-1984, Cambridge sity Press, Oxford. University Press, New York,pp. 1-18. HOLT, D.F. and REES, S. (to appear a): Testing for isomorphism CANNON, J. (ed) (1990): Computational group theory I, /. Symbolic between finitely presented groups, Proc. Conf. Groups and Combina Comput. 9, numbers 5-6. torics, Durham, 1989, Cambridge University Press, Cambridge. CANNON, J.J. (1991a): A bibliography of Cayley citations, SIGSAM HOLT, D.F. and REES, S. (to appear b): A graphics system for displaying Bull 25, pp. 75-81. finite quotients of finitely presented groups, Proc. DIMA CS Workshop CANNON, J. (ed) (1991b): Computational group theory II,/. Symbolic Groups and Computation, 1991, AMS-ACM. Comput. 12, numbers 4-5. JERRUM, M.R. (1986): A compact representation for permutation CANNON, J. and BOSMA, W. (1991): A Handbook of Cayley functions, groups, /. Algorithms 7, pp. 60-78. Computer Algebra Group, University of Sydney. JOHNSON, D.L. (1980): Topics in the theory of group presentations, CANNON, J.J., DIMINO, LA., HAVAS, G. and WATSON, J.M. (1973): Cambridge University Press, Cambridge. Implementation and analysis of the Todd-Coxeter algorithm, Math. KANTOR, W.M. (1985): Sylow’s theorem in polynomial time, /. Comput. Comp. 27, pp. 463-490. System ScL 30, pp. 359-394. CONWAY, J.H. (1984): An algorithm for double coset enumeration? In KANTOR, W.M. (1991): Finding composition factors of permutation M.D. Atkinson (ed), pp. 33-37. groups of degree n < 106, /. Symbolic Comput. 12, pp. 517-526. COOPERMAN, G. and FINKELSTEIN, L. (1991): A strong generating KANTOR, W.M. and TAYLOR, D.E. (1988): Polynomial-time versions test and short presentations for permutation groups,/. Symbolic Corn- of Sylow’s theorem, /. Algorithms 9, pp. 1-17. put 12, pp. 475-497. KNUTH, D.E. and BENDIX, P.B. (1970): Simple word problems in COOPERMAN, G., FINKELSTEIN, L. and LUKS, E. (1989): Reduction universal algebra. In J. Leech (ed), pp. 263-297. of group constructions to point stabilizers, ISSAC 89, ACM Press, LAUE, R. NEUBUSER, J. and SCHOENWAELDER, U. (1984): Algo ' New York, pp. 351-356. rithms for finite soluble groups and the SOGOS system. In M.D. COOPERMAN, G„ FINKELSTEIN, L. and SARAWAGI, N. (1990): A Atkinson (ed), pp. 105-135. random base change algorithm for permutation groups, ISSAC 90, LE CHENADEC, P. (1986): Canonical forms for finitely presented alge ACM Press, New York, pp. 161-168. bras, Pitman, London. DIETZE, A. and SCHAPS, M. (1974): Determining subgroups of a given LEECH, J. (ed) (1970a): Computational problems in abstract algebra, finite index in a finitely presented group, Canadian J. Math 26, pp. Pergamon Press, Oxford. 769-782. LEECH, J. (1970b): Coset enumeration. In J. Leech (ed), pp. 21-35. EPSTEIN, D„ HOLT, D.F. and REES, S. (1991): The use of Knuth- LEECH, J. (1984): Coset enumeration. In M.D. Atkinson (ed), pp. 3-18. Bendix methods to solve the word problem in automatic groups, /. LEEDHAM-GREEN, C.R. (1984): A soluble group algorithm. In M.D. Symbolic Comput. 12, pp. 397-414. Atkinson (ed), pp. 85-101. FELSCH, V. (1976): A machine independent implementation of a collec LEEDHAM-GREEN, C.R. and SOICHER, L. (1990): Collection from tion algorithm for the multiplication of group elements, SYMSA C’76, the left and other strategies, /. Symbolic Comput. 9, pp. 665-676. ACM Press, New York, pp. 159-166. LENSTRA, A.K., LENSTRA, H.W. Jr and LOVASZ, L. (1982): Factor FELSCH, V. (1978): A KWIC indexed bibliography on the use of compu ing polynomials with rational coefficients, Math. Ann. 261, pp. ters in group theory and related topics, SIGSAM Bull 12, pp. 23-86. 515-534.
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LEON, J.S. (1980): On an algorithm for finding a base and strong SCHNEIDER, G.J. (1990): Computing with endomorphism rings of generating set for a group given by generating permutations, Math. modular representations, /. Symbolic Comput. 9, pp. 607-636. Comp. 35, pp. 941-974. SIMS, C.C. (1970): Computational methods in the study of permutation LEON, J.S. (1991): Permutation group algorithms based on partitions I: groups. In J. Leech (ed), pp. 169-183. Theory and algorithms, J. Symbolic Comput. 12, pp. 533-583. SIMS, C.C. (1971 a): Determining the conjugacy classes of a permutation LEON, J.S. and SIMS, C.C. (1977): The existence and uniqueness of a group, Computers in algebraand number theory [SIAM-AMS Proc. 4], simple group generated by (3,4)-transpositions, BulL Amer. Math pp. 191-195. Soc. 83, pp. 1039-1040. SIMS, C.C. (1971b): Computation with permutation groups, Proc. LEPIQUE, E. (1972): Ein Programm zur Berechnung von Untergruppen Second Symp. Symbolic and Algebraic Manipulation, ACM, New von gegebenen Index in endlich prasentierten Gruppen, Diplomarbeit, York, pp. 23-28. RWTH Aachen. SIMS, C.C. (1973): The existence and uniqueness of Lyons’ group, Finite LINTON, S.A. (1991a): Double coset enumeration,/. Symbolic Comput. Groups 72, North-Holland, Amsterdam, pp. 138-141. 12, pp. 415-426. SIMS, C.C. (1987): Verifying nilpotence,7. Symbolic Comput. 3, pp. LINTON, S.A. (1991b): Constructing matrix representations of finitely 231-247. presented groups, J. Symbolic Comput. 12, pp. 427-438. SIMS, C.C. (1990a): Computing the order of a solvable permutation LUKS, E.M. (1987): Computing the composition factors of a permuta group, /. Symbolic Comput 9, pp. 699-705. tion group in polynomial time, Combinatorica 7, pp. 87-99. SIMS, C.C. (1990b): Implementing the Baumslag-Cannonito-Miller poly LUX, K. and PAHLINGS, H. (1991): Computational aspects of represen cyclic quotient algorithm, J. Symbolic Comput 9, pp. 707-723. tation theory of finite groups, Representation theory of finite groups SIMS, C.C. (1991): The Knuth-Bendix procedure for strings as a substi and finite-dimensional algebras [Progress in Mathematics, Volume tute for coset enumeration, J Symbolic Comput. 12, pp. 439-442. 95], Birkhauser, Basel, pp. 37-64. SIMS, C.C. (to appear): Computation with finitely presented groups, Cam MACDONALD, I.D. (1974): A computer application to finite p-groups, bridge University Press, Cambridge. J. Austral Math Soc. 17, pp. 102-112. SOICHER, L.H. (1990): A new existence and uniqueness proof for the McKAY, B.D. (1978): Computing automorphisms and canonical label O’Nan group, Bull LondonMath Soc. 22, pp. 148-152. lings of graphs, Combinatorial mathematics [Lecture Notes in VAUGHAN-LEE, M.R. (1984): An aspect of the nilpotent quotient Mathematics 686], Springer-Verlag, Berlin, Heidelberg, New York, algorithm. In M.D. Atkinson (ed), pp. 75-83. pp. 223-232. VAUGHAN-LEE, M.R. (1990a): The restricted Burnside problem, Cla McKAY, B.D. (1981): Practical graph isomorphism, Congressus Nume- rendon, Oxford. rantium 30, pp. 45-87. VAUGHAN-LEE, M.R. (1990b): Collection from the left, J. Symbolic MENNICKE, J. (1959): Einige endliche Gruppe mit drei Erzeugenden Comput. 9, pp. 725-733. und drei Relationen, Arch Math X, pp. 409-418. WAMSLEY, J.W. (1974): Computation in nilpotent groups (theory), MICHLER, G. (1990): Some problems in computational representation Proc. Second Intemat. Conf. Theory of Groups [Lecture Notes in theory, J. Symbolic Comput. 9, pp. 571-590. Mathematics 372], Springer-Verlag, Berlin, Heidelberg, New York, NEUBUSER, J. (1982): An elementary introduction to coset-table pp. 667-690. methods in computational group theory, Groups — St Andrews 1981 WAMSLEY, J.W. (1977): Computing soluble groups, Group Theory [London Math. Soc. Lecture Note Ser. 71], Cambridge University [Lecture Notes in Mathematics 573], Springer-Verlag, Berlin, Hei Press, Cambridge, pp. 1-45. delberg, New York, pp. 126-129. NEUBUSER, J. and SIDKI, S. (1988): Some computational approaches to groups given by finite presentations. Mathematica Universitaria 7, pp. 77-120. Rio de Janeiro. NEUMANN, P.M. (1986): Some algorithms for computing with finite permutation groups, Proc. Groups — St Andrews 1985 [London Math BIOGRAPHICAL NOTE Soc. Lecture Note Ser. 121], Cambridge University Press, Cam John Cannon is a Reader in Pure Mathematics at the Univer bridge, pp. 59-92. sity of Sydney. He is leader of the Computer Algebra Group NEWMAN, M.F. (1976): Calculating presentations for certain kinds of quotient groups, SYMSAC76, ACM Press, New York, pp. 2-8. there and the principal architect of the computer algebra NEWMAN, M.F. and O’BRIEN, E.A. (to appear): A computer aided system Cayley. analysis of some finitely presented groups, J. Austral Math Soc. Qeorge Havas is a Senior Lecturer in Computer.Science at O’BRIEN, E.A. (1990): The p-group generation algorithm, J. Symbolic Comput. 9, pp. 677-698. rife University of Queensland. His previous appointments O’BRIEN, E.A. (1991): The groups of order 256, J. Algebra 143, pp. include: Lecturer in Computing at Canberra College of 219-235. I,U Advanced Education fnow University of Canberra); PLESKEN, W. (1987): Towards a soluble quotient algorithm,/. Symbolic Comput. 4, pp. 123-127, 'if Research Fellow in Mathematics at the Institute of Advanced ROBERTSON, E.F. (1988): Tietze transformations with weighted sub Studies, Australian National University; Principal Research string search. /. Symbolic Comput. 6, pp. 59-64. Scientist, Division of Computing Research, CSIRO; and ROTMAN, J.J. (1973): The theory of groups; an introduction, Allyn and Bacon, Boston. director of Research and Development, Csironet.
60 THE AUSTRALIAN COMPUTER JOURNAL, VOL. 24, No. 2, MAY 1992 1 INTRODUCTION Graph theory has applications in solving a wide range of Parallel Search problems in diverse disciplines. In the past, the study of graph theory algorithms has mostly been in an environ Algorithms for ment based on sequential computation. Recent changes in hardware technology have made it feasible to use a large number of processors that cooperate and coordinate their Trees and Graphs efforts to solve a single problem at a time, so that the elapsed execution time is reduced significantly. This has Pranay Chaudhuri turned the attention of many researchers in computer School of Computer Science and Engineering science to find efficient parallel and distributed algorithms The University of New South Wales for various graph theory problems. PO Box 1, Kensington, NSW 2033 The execution of a graph algorithm is controlled by a combination of logical, topological, quantitative, and chronological information derived from the problem, the graph, and the processing environment; this combination was referred to as the control information by Cheung Tree and graph searching are of fundamental importance, (1983). Classification of parallel and distributed graph since they form the basis for solving a wide range of other algorithms is based on the computational environment for graph theory problems Parallel algorithms developed for exercising the control information. In parallel graph algo different types of tree and graph searching problems are rithms the control information is available at one site, i.e., mainly based on shared memory model of the single- the execution of the algorithm is centrally controlled and instruction stream, multiple-data stream computer. This the processors work synchronously. On the other hand, in computational model may be viewed as a parallel variant distributed graph algorithms, which are suitable for net of the standard Random Access Machine (RAM) model of work environments, the control information is available at sequential computation and is commonly known as different sites and no single site has complete information. parallel-RAM or PRAM in the literature. This paper aims In this paper, I shall be dealing only with tree and graph in reviewing the algorithms for tree and graph searching searching problems on computational models which problems on various PRAM models. assume full synchronisation. That is, distributed graph Keywords and Phrases: parallel algorithms, graph the search algorithms will not be described here; however, an ory, binary tree, tree traversal, depth-first search, breadth- interested reader is referred to some works of this type first search, breadth-depth search, PRAM model time which include (Awerbuch, 1985a-c; Chang, 1982; complexity. Chaudhuri, 1987c, 1992b; Cheung, 1983; Cidon, 1987; CR Categories: GT: Algorithm, SD: G1.0, G2.2, F2.2. Sharma et al., 1989). This paper reviews the significant amount of research that has been reported in the literature over the past ten years on the design and analysis of parallel algorithms for searching trees and graphs. Section 2 briefly covers the computational models which are normally used for design ing parallel graph algorithms. Terminologies used for expressing the complexity of parallel algorithms are intro duced in Section 3. Sections 4 and 5 are devoted to survey ing the existing parallel search algorithms for trees and graphs, respectively. Finally, Section 6 provides a sum mary of this survey work highlighting the problems which deserve further investigation.
2 MODELS OF COMPUTATION Following Flynn’s taxonomy (Flynn, 1966) parallel com puters can broadly be classified into two categories viz., Copyright® 1992, Australian Computer Society Inc. General permission single-instruction stream, multiple-data stream (SIMD) to republish, but not for profit, all or part of this material is granted, provided that the ACJ’s copyright notice is given and that reference is and multiple-instruction stream, multiple-data stream made to the publication, to its date of issue, and to the fact that reprint (MIMD) computers. An SIMD machine such as ILLIAC- ing privileges were granted by permission of the Australian Computer IV (Barnes et al., 1968), typically consists of a number (say Society Inc. n) of processors each having an associated memory Manuscript received November 1991. module together with an interconnection network, and a
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memory has a significant impact on time complexity. Depending on the restrictions imposed on shared memory access, there are three variants of the PRAM models. In the most restrictive PRAM model, no two processors are allowed to access the same shared memory location at the same time. Such a model of computation is known as Interconnec tion Network Exclusive-Read, Exclusive-Write PRAM or EREW PRAM. The Concurrent-Read, Exclusive-Write (CREW) PRAM model allows two or more processors to access
CU: Control Unit simultaneously the same location of the shared memory CP: Control Processor CM: Control Memory for reading, but not for writing. Finally, the Concurrent- PE: Processing Element P: Processor Read, Concurrent-Write (CRCW) PRAM model allows MM: Memory Module any number of processors to access simultaneously the same location of the shared memory during both reading Figure 1. Structure of an SIMD computer. and writing. In the case of concurrent writing, different assumptions are made about the content of the memory control unit. The control unit broadcasts instructions to the location after the write operation. Most common write- processors, and all the processors execute the same conflict resolution rules are the priority-resolution and the instruction at the same time. Each processor executes equality-resolution (e.g. see Chaudhuri, 1987a; Chaudhuri instructions using data from the memory module with and Ghosh, 1986; Kucera, 1982; Shiloach and Vishkin, which it is associated. The interconnection network allows 1981; Vishkin, 1984). interprocessor communication. The block diagram of an SIMD model is shown Figure 1. Some additional examples In case of priority-resolution the processors are linearly of the SIMD machines include the Connection Machine ordered according to their priorities (or serial numbers). (Hillis, 1986; Hockney and Jessope, 1988), Goodyear When a simultaneous writing into the same shared Aerospace MPP (Batcher, 1980; Hwang and Briggs, memory location is attempted, the memory location 1985), and ICL DAP (Flanders et al., 1977). receives the value written by the processor with the highest An MIMD machine also typically consists of n proces priority (which may be defined based on their serial sors and associated memory modules, but each processor numbers). On the other hand, in equality-resolution simul can follow an independent instruction stream. Examples of taneous writing into the same shared memory location is MIMD machines are C.mmp (Mashburn, 1979; Wulf and allowed, provided all processors involved attempt to write Harbison, 1978) and Cm* (Haynes et al., 1982; Swan et al., the same value. 1977). 2.2 Mesh Connected Model In the survey, only various SIMD models are consi In the mesh connected SIMD (MC-SIMD) model proces dered. There are many variations of the SIMD models sors are thought of as arranged logically in the form of a depending on whether the processors communicate with k-dimensional array A[ 1: nk_b 1: nk. '2,__,1: n0j, where each other via a shared memory or using various intercon q is (he size of the ith dimension, i = 0,1,..., k—1. The nection schemes. A comparative study of various inter processor with index A(ik_,,ik_2,..., i0) is connected to the connection networks for SIMD machines is given in (Sie processors at locations A(ik_b ik_2,..., ij ± 1,..., i0), j = 0,1, gel, 1979). The following is a brief deggription of the . . . , k—1, provided these processors exist (Figure 2). various SIMD models used in computational graph theory. Clearly, the MC-SIMD model requires at most 2k connec-
2.1 Shared Memory Klodel This model assumes a system of n identical synchronous A(0,0) A(0,1) A(0,2) A(0,3) processors, each having access to a common or shared storage. Interprocessor communication is established through the shared memory. That is the block representing the interconnection network in Figure 1 is to be replaced A(1,0) A( 1,2) A( 1,3) with a shared memory to obtain the shared memory SIMD computer|This model is a direct extension of the standard Random Access Model (RAM) used for sequential compu A(2,0) A(2,1) A(2,2) A(2,3) tation and is more commonly referred to as the parallel- RAM or PRAM model. The case in which the number of processors n is considered arbitrarily large is called A(3,0) A(3,1) A( 3,2) A(3,3) unbounddd parallelism. The mechanism through which processors in a PRAM model access data in shared Figure 2. A mesh connected network with 16 processors.
62 THE AUSTRALIAN COMPUTER JOURNAL, VOL 24, No. 2, MAY 1992 ALGORITHMS FOR TREES AND GRAPHS tions per processor. MC-SIMD has been used by several 2.5 Other Models of SIMD Machines researchers for solving various graph theoretic problems Besides the above mentioned models many more models (Atallah and Kosaraju, 1984; Dekel et al, 1981; Nassimi of SIMD machines have been suggested. These include and Sahni, 1980). The ILLIAC-IV, Burroughs PEPE, and cub-connected cycles (Preparata and Vuillemin, 1981), Goodyear Aerospace MPP belong to the MC-SIMD class orthogonal tree cycles (Nath et al., 1983), the tree struc (see e.g., Hwang and Briggs, 1985, Quinn, 1987). tured computer of Bentley and Kung (1979), and recon- figurable computers (Siegel, 1979). 2.3 Cube Connected Model Assume that n = 2N and let (iN_,, iN_2,..., i0) be the binary 3 TERMINOLOGY representation of i (0 < i < n—1). Let i(b) be the integer The worst-case time complexity (or simply the time com whose binary representation is (iN_i, iN_2, • • • > ib+i> ib’> L-i • • plexity) of a parallel algorithm is defined to be the number ., i0), where ib’ is the complement of ib, for b = 0, 1,..., of parallel steps required by the algorithm. Processor com N—1. In a cube connected SIMD (CC-SIMD) model the plexity, in an unbounded model, is the number of proces ith processor is connected to the i(b)th processor, for b = 0, sors required. The cost of a parallel algorithm is the pro 1,. . . , N—1. Figure 3 shows the interconnection pattern duct of the parallel running time and the number of of the CC-SIMD computer with processors. Thus CC- processors used. A parallel algorithm solving a problem P SIMD model requires log n connections per processor (all is said to be cost-optimal (or optimal) if its cost is of the logarithms in this paper are to the base 2 unless otherwise order of the time complexity of the fastest known sequen specified). Dekel et al. (1981) used CC-SIMD for design tial algorithm solving P. The speedup of a parallel algo ing some parallel graph algorithms which include comput rithm solving a problem P is the ratio between the execu ing various spanning trees. The flip network used in STA- tion time of the fastest known sequential algorithm solving RAN and Pease’s binary n-cube networks are examples of P and the execution time of that parallel algorithm. cube networks (see Hwang and Briggs, 1985; Pease, Throughout this paper a tree rooted at a node r is 1977). denoted by T(r) =
Figure 3. A cube connected network with eight processors. 4 SEARCHING TREES IN PARALLEL The most common operation on trees is search or traver sal, i.e., visiting each node of the tree exactly once in some 2.4 Perfect Shuffle Model order. Three frequently used searching methods are Let n, N, and i(b) be as in the CC-SIMD model. In a perfect preorder (root first, then the subtrees), postorder (subtrees shuffle SIMD (PS-SIMD) model the ith processor is con first, then the root), and inorder (only for binary trees: first nected to the i(0)th processor and the processors with indi left subtree, then root, then right subtree). This section
ces given by the binary integers (iN_2, iN-3> • ■ • > io> >n-i) concentrates on the first two techniques. A lower bound to and(i0,iN_i,. . . ,i2,ii). These three connections are called the sequential computation time for searching a tree con the exchange, shuffle, and unshuffle, respectively. In PS- sisting of n nodes is D(n), since by definition every node SIMD, only three connections are required per processor has to be examined once. There exist optimal sequential (Figure 4). PS-SIMD is also used for designing some graph algorithms for these problems (e.g., see Horowitz and algorithms in Dekel et al. (1981). The shuffle and Sahni, 1977). exchange functions have been implemented with the mul There are two ways to search a general ordered tree. tistage Omega network (Lawrie, 1975). One is by converting it into the equivalent binary tree and then applying the corresponding search algorithm for the binary tree. The other avoids the conversion.
4.1 Transforming a General Ordered Tree into its Equivalent Binary Tree Assume that the given general ordered tree T(r) =
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given. It is assumed that p(r) = r. To obtain an equivalent binary tree representation, the leftmost-son-next-right- sibling relationship may be used. Assume, without loss of generality, that the leftmost-son and the next-right-sibling of every node of the general ordered tree as the leftson and the rightson in the equivalent binary tree.
Definition 4.1.1 Three functions leftson, rightson, and father on the set of nodes of a tree are defined as follows: (i) leftson® = j> if j = min{k | p(k) = i} = 0, otherwise; (ii) rightson(i) = j> if j> min{k | k > i AND p(k) = p(i)} Figure 6. (a) The binary tree T(l) of Figure 5 and the nonempty father = 0, otherwise; relations shown by labelled arrows; (b) The effect of folding on the (iii) father(i) = P® if rightson® = 0 father relations. = 0, otherwise. A general ordered tree and the corresponding equival- appears as the immediate next node to i. If node i has no ent binary tree, using the leftson and rightson relations, are sons, then to obtain the next node in the search list the nearest ancestor j of i having a rightson is identified. The rightson of j is then the immediate next node of i in the search list. Thus, if a node has either a leftson or a rightson, then the next node in the search list is easily computed. Otherwise, the required nearest ancestor that has a rightson may be readily found after propagating father relations as shown by Ghosh and Bhattacharjee (1984a). This is referred to as Figure 5. (a) A general ordered tree T(l); (b) The equivalent binary tree the folding on father relations. The father relation on the representation of T(l). The leftson, rightson, and father relations are nodes of the binary tree on Figure 5(b) and the effect of shown by labelled arrows H, r—, and f—■, respectively. folding on the father relations are illustrated in Figure 6. A parallel preorder traversal algorithm for the binary shown in Figure 5. Following the above definition a paral trees based on the above idea can be found in Ghosh and lel algorithm that computes the leftson, rightson, and Bhattacharjee (1984a) which works on the CREW PRAM father for each node of a tree can be obtained by parallelis model. The time complexity of Ghosh andBhattacharjee’s ing the corresponding sequential algorithm (Horowitz and algorithm is 0(k log n) using 0(n1+1/k),processors, where k Sahni, 1977). This algorithm first sorts the nodes of the > 2 is an integer. However, in Chaudhuri (1992a) it is tree in such a manner that the nodes having the same shown that the same algorithm can also be implemented in parent are stored in consecutive locations. For this pur 0(fog n) time with 0(n) processors if the required sorting pose, the sorting algorithm due to Cole (19J8) which runs of nodes is performed by using the parallel sorting algo in 0(log n) time with O(n) processors on a CREW PRAM rithm due to Cole (19^8). Similar algorithms can be may be used. Finally, fqr every node i, its leftson/rightson, designed for both inorder and postorder searching of and the father (as defined above) are computed in constant binary trees to-run in OQog n) time with O(n) processors on time using n processors. Thus the time complexity of this tfeq CREW PRAM model. parallel algorithm is 0(log n) using O(n) processors on the Kalra and Bhatt (1985) considered the same problem CREW PRAM model. The detailed descriptions of such a for binary trees on the EREW PRAM model. They showed parallel algorithm may be found in Ghosh and Bhattachar- that it is possible to search binary trees in preorder, inorder, jee (1984a), or in Chaudhuri (1992a). and postorder in 0(log n) time with O(n) processors. How ever, they assumed that the input of their algorithm was in 4.2 Searching the Equivalent Binary Tree the form of the leftmost-son and the next-right-sibling Searching1 the original general ordered tree in preorder relations. reduces to computing the successor of each node i (but the last) in preorder. The ideas behind this algorithm are as 4.3 Searching General Ordered Trees follows. The leftson of a node i, if it exists, should appear as In this section, parallel tree searching algorithms are con the immediate next node to i in the preorder list. Alterna sidered which directly search the given general ordered tively, if there is no leftson, then the rightson of i, if it exists, tree without converting it into its equivalent binary tree.
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They are illustrated in the case of preorder searching only. with 0(n2) processors on the CREW PRAM model Similar ideas can be applied for other search techniques. (Chaudhuri, 1987b, 1988). Several researchers have studied this problem and as a A different approach has been taken by Taijan and result a number of parallel algorithms are available (e.g., Vishkin (1984) for searching trees by using the Euler tour see Chaudhuri, 1987b; Chen et al., 1991; Taijan and Vish- technique on trees. The key idea behind this technique is to kin, 1984; Tsin and Chin, 1984). In Chaudhuri (1987b) construct a list of edges of the tree that forms an Euler tour and Tsin and Chin (1984), search algorithms were deve of the directed version of the tree. The directed version of loped using the ancestor array Ak(i), l
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lei dfs algorithm for dags but is also veiy effective in designing updating algorithms to update or recompute the dfs spanning tree of a dag when a minor change is made to the dag. The basis of Chaudhuri’s algorithm is the charac terisation of the dfs property with reference to a dag. This characterisation is performed by introducing the notions of k-tree and k-dfs-tree, defined below.
Definition 5.1.1 Given a dag G =
Definition 5.1.2 Suppose that G =
66 THE AUSTRALIAN COMPUTER JOURNAL, VOL. 24, No. 2, MAY 1992 ALGORITHMS FOR TREES AND GRAPHS
Input to the dfs algorithm Tu(i), i=1.2. Al T T
End of first iteration (lc = 1) - T (i), i = 1, 2,
Q) © ®
End of second iteration (k = 2) - T“ (i), i = 1, 2...... 10.
Figure 10. The trees T'(i) and T2(i) obtained from T°(i), i = 1,2...... 10, corresponding to the dag G of Figure 9. merging process is repeated flog dl times producing the It can be verified that the tree-merging process can (log d|-dfs-trees T[l°8dl(i), V i e N. Out of these n flog easily be implemented in 0(log n) time with 0(n2| n/log n ]) dl-dfs-trees, T(l°sdl(s) is the desired dfs spanning tree of the processors. Considering [log d] such tree-merging steps, dag G. An algorithmic description of the tree-merging the overall time complexity of the dfs algorithm becomes process is given below. Note that the input to the main 0(log d log n) when 0(n2[n/log nl) processors are algorithm is parent p(y| T°(x)) for each nodey in the 0-tree a employed on a CREW PRAM. The complete algorithm T°(x), x, y = 1,2,.. . , n. Thus the tree-merging process is including its proof of correctness can be found in Chaud repeated for k = 0 through flog dl to obtain the desired dfs huri (1990,1992). The tree-merging process is illustrated spanning tree of the dag. with the help of an example in Figure 10. Step 1 [Compute postorder ranks] For each pair of nodes x, y = 5.2 Breadth-First Search of Graphs 1, 2,. . . , n compute post(y | Tk(x)). The bfs is a good search technique in the context of parallel Step 2 processing. The first parallel algorithm for the bfs problem [Initialise matrices] For each triple x, y, z = 1,2,. . . , n was proposed by Alton and Eckstein (1977), and Eckstein do the following: (1979). Their algorithm runs in 0(d log k + (e + n)/k + if (z = x)A(p(y | Tk(x)) A 0) then £x((degree(x))2/k) on a k-processor CREW PRAM, where Mxk(y, z) := post(y | Tk(x)) x ranges over the nodes of the graph and degree(x) is the else if (z e Tk(x)) A(p(y | Tk(z» A 0) then number of edges incident on x. This algorithm is not very Mxk(y, z) := post(z | Tk(x)) attractive from the speedup point of view. Later, several else Mxk(y, z) := °°. fast parallel algorithms for this problem were reported, Step 3 including Dekel et al. (1981), Ghosh and Bhattacharjee [Find the minimum of each row of each matrix] For (1984b), and Kim and Chwa (1986). All these algorithms each pair of nodes x, y = 1, 2,. . . , n find zm such that essentially run in 0(log2n) time with 0(n3) processors and Mxk(y, zj := min {Mxk(y, z) | z = 1, 2,. . . , n}. are based either on a modified matrix multiplication Step 4 approach or a tree-merging technique of the type'des- [Define new trees] For each pair of nodes x, y = 1,2, cribed in the previous section. . . . , n set In the modified matrix multiplication approach, the P(y I Tk(x)) := p(y | Tk-1(zm)). multiplication and addition operations are replaced with
THE AUSTRALIAN COMPUTER JOURNAL, VOL. 24, No. 2, MAY 1992 67 ALGORITHMS FOR TREES AND GRAPHS
addition and min (which finds the minimum of two deserve further investigation with a view to reducing their numbers) operations. This technique is used by Dekel et al. costs, perhaps by choosing more appropriate data struc to design bfs algorithms on both CC-SIMD and PS-SIMD tures and exploiting the properties of these search methods models. Later, Kim and Chwa (1986) used this method to with respect to different classes of graphs more effectively. design an algorithm for the same problem on the CREW PRAM model. In order to apply the tree-merging tech ACKNOWLEDGEMENT nique for the bfs of graphs, the bfs property is defined by I would like to thank the referees for their detailed and using the bfs traversal rank and level information. Then helpful comments on the presentation of this material. the tree-merging is carried out in such a manner that at any stage the trees produced preserve the bfs-property. The REFERENCES details of such an algorithm can be found in Ghosh and ALTON, D.A. and ECKSTEIN, D.M. (1977): Parallel breadth-first Bhattachaijee (1984b), and Chaudhuri (1992). search of p-sparse graphs, Proc. West Coast Conf. on Combinatorics, Graph Theory, and Comput, pp. 79-93, Calif., USA. ATTALAH, M.J., and KOSARAJU, S.R. (1984): Graph Problems on a 5.3 Breadth-Depth Search of Graphs mesh-connected processor array. J. ACM, 31, PP. 649-667. The early work on the bds problem was due to Reghbati AWERBUCH, B. (1985a): A new distributed depth-first search algo rithm, Inform. Process. Lett., 20, pp. 147-150. and Corneil (1978). Their algorithm runs on a CREW AWERBUCH, B. (1985b): Complexity of network synchronisation. J. PRAM with k processors and takes 0(n log k + (e/k)) time. ACM, 32, pp. 804-823. The time complexity of this algorithm cannot be reduced AWERBUCH, B. (1985a): Reducing complexities of the distributed beyond 0(n log n) and thus form the speedup point of view max-flow and breadth-first search by means of network synchronisa tion, Networks, 15, pp. 425-437. Reghbati and Comeil’s algorithm is not very attractive. BARNES, G.H., BROWN, RM„ KATO, M„ KUCK, D J., SLOTNICK, Since the bds is somewhat in between dfs and bfs, it seems D.L., and STOKES, R.A. (1968): The Illiac IV computer. IEEE Tran that weaknesses of both of these will also be manifested in sactions on Computers, C-17, pp. 746-757. the bds. The direct consequence of this is that, since the dfs BATCHER, K.E. (1980): Design of a Massively Parallel Processor. IEEE Transactions on Computers, C-29, pp. 836-840. is known to be inherently sequential, the same is expected BENTLEY, J.L., and KUNG, H.T. (1979): A tree machine for searching to be the case for the bds. However, it has been shown in problems. Proc. 1979 Internal Conf. on Parallel Processing, pp. Chaudhuri (1988) that, although the bds of general graphs 257-266. seems to be inherently sequential, it is possible to design BERGE, C. (1973): Graphs and Hypergraphs, North-Holland, NY. CARRE, B. (1979): Graphs and Networks, Oxford University Press, fast parallel algorithm for the bds of dags. The algorithm England. was designed by using the same tree-merging technique CHANDRA, A.K. (1976): Maximal parallelism in matrix multiplication, mentioned earlier; the k-bds-trees, in this case, are charac IBM Tech. Rep. RC6193, Thomas J. Watson Research Center, York- town Heights, NY. terised by using both the preorder and reverse postorder CHANG, E.J.H. (1982): Echo algorithms: Depth parallel operations on traversal ranks for the nodes of the k-trees. This algorithm general graphs, IEEE Transactions on Software Engg., SE-8, pp. runs in 0(log d log n) time with 0(n3) processors on a 391-401. CREW PRAM. CHAUDHURI, P. (1987a): An 0(log n) parallel algorithm for strong connectivity augmentation problem, Intern. J. Computer Math., 22, pp. 187-97. 6 SUMMARY CHAUDHURI, P. (1987b): Fast parallel algorithms for a class of graph A brief survey of the results reported in the literature for problems. J. Comb. Inf. Syst. ScL, 12, pp. 51-65. ' searching of trees and graphs in parallel has been pres CHAUDHURI, P. (1987c): Algorithms for some graph problems on a distributed computational model, Information Sciences, 43, pp. ented. Section 4 shows that for the tree traversal problem, 205-228. » fast as well as efficient algorithms exist on various PRAM CHAUDHURI, P. (1988): Fast parallel graph searching with applica models. On the other hand, Section 5 covered the graph tions, BIT, 28, pp. 2-18. searching problem and provided important results related CHAUDHURI, (1990): Finding and updating depth first spanning trees of acyclic diagraphs in parallel. Computer Journal, 33, pp. to the dfs, bfs and the bds of graphs in the context of 247-51. parallel processing. The most notable results presented in CHAUDHURI, P. (1992a): Parallel Algorithms: Designs and Analysis, this section are that: (i) the dfs is inherently sequential, (ii) Prentice-Hall, Australia. CHAUDHURI, P. (1992b): Distributed Processing of graphs: Funda for dags and embedded planar graphs fast parallel algo mental cycles algorithm, Information Sciences, 60, pp. 41-50. rithms for the dfs problem are possible, (iii) fast parallel CHAUDHURI, P. and GHOSH, R.K. (1986): Parallel algorithms for algorithms for the bfs of general graphs are available, (iv) analysing activity networks, BIT, 26, pp. 418-29. , the bds djf general graphs seems to be inherently sequen CHEN, C, C-Y., and DAS, S.K. (1989): Parallel Algorithms for level- order traversal of general trees, J. Comb. Inf. Syst. Sci, 14, pp. tial, and (v) a fast parallel algorithm for the bds of dags is 135-162. available. CHEN, C, C-Y., DAS, S.K., and AKL, S.G. (1991): A unified approach to Although the parallel algorithms for the dfs of both parallel depth-first traversals of general trees, Inform Process. Lett, 38, pp. 49-55. dags and embedded planar graphs, bfs of general graphs, CHEUNG, T.Y. (1983): Graph traversal techniques and maximum flow and bds of dags, mentioned in Section 5, are sufficiently problem in distributed computation, IEEE Transactions on Software fast, their costs are far from being optimal. These problems Engg., SE-9, pp. 504-512.
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CIDON, I. (1987/88): Yet another distributed depth-first search algo PEASE, M.C. (1977): The indirect binary n-cube microprocessor array. rithm, Inform. Process. Lett, 26, pp. 301-305. IEEE Transactions on Computers, C-25, pp. 252-264. COLE, R. (1988): Parallel merge sort. SIAM J. Computing, 17, pp. PREPARATA, F.P., and VUILLEMIN, J. (1981): The cube-connected 770-785. cycles: A versatile network for parallel computation, Comm. ACM, DEKEL, E„ NASSIMI, D„ and SAHNI, S. (1981): Parallel matrix and 24, pp. 300-309. graph algorithms, SIAMJ. Computing, 10, pp. 657-675. QUINN, M. J. (1987): Designing efficient algorithmsfor parallel computers, DEO, N. (1974): Graph Theory with Applications to Engineering and McGraw-Hill, NY. Computer Science. Prentice-Hall, Englewood Cliffs, NJ. REGHBATI, E„ and CORNEIL, D.G. (1978): Parallel computations in ECKSTEIN, D.M. (1977): BFS and biconnectivity, Tech. Rep. TR-79- graph theory, SIAMJ. Computing, 2, pp. 230-237. 19, Iowa State University, 1979. REIF, J. (1985): Depth-first search is inherently sequential. Inform. Pro ECKSTEIN, D.M., and ALTON, D.A. (1977): Parallel graph processing cess. Lett., 20, pp. 229-234. using depth-first search. Proc. Conf. on Theoretical Computer Set, SAVAGE, C., and JA’ JA’, J. (1981): Fast, efficient parallel algorithms for Waterloo, Ont., pp. 21-29. some graph problems, SIAM J. Computing, 10, pp. 682-691. FLANDERS, P.M., HUNT, D.J., REDDAWAY, S.F., and PARKINSON, SHARMA, M.B., IYENGAR, S., and MANDYAM, N.K. (1989): An D. (1977): Efficient high speed computing with the distributed array efficient distributed depth-first search algorithm, Inform. Process. processor. In High Speed Computer and Algorithm Organisation. Aca Lett, 32, pp. 183-186. demic Press, pp. 113-128. SHILO ACH, Y„ and VISHKIN, U. (19 81): Finding the maximum, merg FLYNN, M.J. (1966): Very high speed computing systems. Proc. IEEE ing, and sorting in a parallel computation model, J. Algorithms, 2, pp. 54, pp. 1901-1909. 88-102. GALIL, Z., and PAUL, W.J. (1983): An efficient general purpose parallel SIEGEL, H.J. (1979): A model of SIMD machines and a comparison of computer. J. ACM, 30, pp. 360-387. various interconnection networks, IEEE Transactions on Computers, GHOSH, R.K., and BHATTACHARJEE, G.P. (1984a): A parallel search C-28, pp. 907-917. algorithm for directed graphs, BIT, 24, pp. 34-50. SWAN, R.J., BECHTOLSHEIM, A., LAI, K.-W., and OUSTERHOUT, GHOSH, R.K. and BHATTACHARJEE, G.P. (1984b): Parallel breadth- J.K. (1977): The implementation of the Cm* multi-microprocessor, first search algorithms for trees and graphs, Intern. J. Computer Math., Proc. National Computer Conf, pp. 645-655. 15, pp. 255-268. TARJAN, R.E., and VISHKIN, U. (1984): Finding biconnected compo HAGERUP, T. (1990): Planar depth-first search in 0(log n) parallel nents and computing tree functions in logarithmic parallel time, Proc. time, SIAMJ. Computing, 19, pp. 678-704. 23rd Annual Symposium on Foundations of Computer Set, pp. 12-20. HARARY, F. (1972): Graph Theory, Addison-Wesley, Reading, MA. TSIN, Y.H., and CHIN, F.Y. (1984): Efficient parallel algorithms for a HAYNES, L.S., LAU, R.L., SIEWIOREK, D.P., and MIZELL, D. (1982): class of graph theoretic problems, SIAM J. Computing, 13, pp. A survey of highly parallel computing, Computer, 14, pp. 9-24. 580-599. HILLIS, W.D. (1986): The Connection Machine, MIT Press, MA. VISHKIN, U. (1984): An optimal parallel connectivity algorithm, Dis HOCKNEY, R.W. and JESSHOPE, C.R. (1988): Parallel Computers 2, crete App. Math., 9, pp. 197-207. IOP Publishing Ltd., UK. WULF, W„ and H ARBISON, S.P. (1978): Reflections in a pool of proces HOROWITZ, E. and SAHNI, S. (1977): Fundamentals of Data Structures, sors. Tech. Rep., Dept, of Computer Science, Camegie-Mellon Uni Computer Science Press, Potomac, Md. versity, Pittsburgh, Pa. HWANG, K„ and BRIGGS, F.A. (1985): Computer Architecture and ZHANG, Y. (1986): A note on parallel depth first search, BIT, 26, pp. Parallel Processing, McGraw-Hill, NY. 195-198. KALRA, N.C., and BHATT, P.C.P. (1985): Parallel algorithms for tree traversals, Parallel Computing, 2,pp. 163-171. BIOGRAPHICAL NOTE KIM, T„ and CHWA, K. (1986): Parallel algorithms for a depth first Pranay Chaudhuri received his BSc(Hons) andBTech(Elec- search and a breadth first search, Intern. J. Computer Math., 19, pp. , 39-54. tronics) from Calcutta University. His ME and PhD both in KUCERA, L. (1982): Parallel computation and conflicts in memory Computer Science and Engineering came from Jadavpur access, Inform. Process. Lett., 14, pp. 93-96. University, Calcutta. At present he is a Senior Lecturer in the LAWRIE, D.H. (1975): Access and alignment of data in an array proces sor. IEEE Transactions on Computers, C-24, pp. 1145-1155. School of Computer Science and Engineering at the Univer MANBER, U. (1989): Introduction to Algorithms: A Creative Approach, sity of New South Wales. Dr Chaudhuri’s research interests Addison-Wesley, Reading, MA. include Parallel and Distributed Algorithms, Graph theory, MASHBURN, H.H. (1979): The C.mmplHydra project: An architectural and Distributed O/S. In these areas he has published more overview, Tech. Rep., Dept, of Computer Science, Camegie-Mellon University, Pittsburgh, Pa. than 35 technical papers in 15 different International Jour NASSIMI, D., and SAHNI, S. (1980): Finding connected components and nals and Conference Proceedings. He is also the author of a connected ones on a mesh-connected parallel computer, SIAM J. book entitled Parallel Algorithms: Design and Analysis Computing, 9, pp. 744-757. (Prentice Hall, 1992). Dr Chaudhuri is the recipient of the NATH, D„ MAHESWARI, S.N., and BHATT, P.C.P. (1983): Efficient VLSI networks for parallel processing based on orthogonal trees, 1987 M.N. Saha Memorial Award by the Institution of IEEE Transactions on Computers, C-32, pp. 569-581. Electronics and Telecommunication Engineers, India.
BACK ISSUES OF AUSTRALIAN COMPUTER JOURNAL A member has a near complete collection of Australian Computer Journals. As he has no further use of these issues, he would be willing to give them to a person or organisation who has , use for them. Please contact the editor Mr Rob Cook on (07) 365 4321
THE AUSTRALIAN COMPUTER JOURNAL, VOL. 24, No. 2, MAY 1992 69 1 INTRODUCTION The problem of sorting is one of the most fundamental in An Overview of computing, and has been intensively studied for many years. Moreover, it is also a very practical problem, and Adaptive Sorting even trivial interactions with a computer system often involve sorting. For example, obtaining a directory listing or querying the current users of a system will probably Alistair Moffat and Ola Petersson* involve sorted output. Department of Computer Science Here we concentrate on adaptive sorting algorithms. An The University of Melbourne algorithm is said to be adaptive if ‘easy’ problem instances Victoria, Australia 3052 are solved faster than ‘hard’. For example, the ‘easiest’ *Department of Computer Science possible instance of the sorting problem is probably to Lund University ‘sort’ an already sorted list: Box 118, S-221 00 Lund, Sweden Xi = (10,11,16,18,21,27,45,64,65,67,84,85,86). As ‘human computers’ we have no difficulty in recognising the special nature of list Xu and can quickly output —write An algorithm is adaptive if ‘easy’problem instances are down — the sorted equivalent. Not so ‘easy’ to process is solved faster than ‘hard’ instances. Here we give a tutorial list X2: overview of the field of adaptive sorting, considering in X2 = (67,10,65,16,64,21,85,18,84,27,45,86,11), turn each of the three main paradigms for the design of and somewhere in between, ‘nearly sorted’, is list Xy. sorting algorithms. We show that each of these paradigms leads to a corresponding family of adaptive algorithms. X3 = (10,11,18,21,27,45,16,64,85,65,67,84,86). Categories and Subject Descriptors: F.2.2 (Analysis of An adaptive algorithm attempts to capture and act Algorithms and Problem Complexity): Nonnumerical upon in some analytic manner this intuitive notion of Algorithms and Problems — sorting and searching. ‘nearly sorted’. To be of interest, an adaptive sorting algo General Terms: Algorithms, Theory. rithm should, as a minimum specification, consume1 0(ri) Keywords: adaptive sorting, measures of presortedness, time on a sorted list, and never more than CKn log n) time nearly sorted on any list, where it is supposed that the list contains n items and the only operation permitted of the algorithm is pairwise comparison of items. We would also hope that the transition between these two extremes of permitted behav iour should be ‘smooth’ in some sense. The fl(n) require ment comes from the straightforward observation that no algorithm can claim to sort its inpift without' evfery item being involved in at least one comparison; and the fl(n log n) worst case requirement is developed from an argument based upon the minimum depth of a binary decision tree j* with n\ leaves, where eaph leaf of the decision tree repres ents one possible input permutation (Knuth, 1973). The interest in adaptive sorting is motivated by two concerns: firsf, a purely academic curiosity, to see (asymp- tdmatically) just how fast certain categories of lists can be sorted; and second, by an empirical observation that many lists to be sorted in practice are already nearly sorted, and applications requiring such sorting can be made to execute faster. For example, if a large sorted file of records is to be )— edited (old records altered or deleted, new records I Copyright ® 1992, Australian Computer Society Inc. General permission appended) and then re-sorted, it is likely that the number of to republish, but not for profit, all or part of this material is granted, records in the file that must be moved to regain sorted provided that the ACJ's copyright notice is given and that reference is order is small, and the use of a @(n log n) sort might be made to the publication, to its date of issue, and to the fact that reprint ing privileges were granted by permission of the Australian Computer Society Inc. 1 A function A«) is said to be 0(g(n)) if An) < c.gfn) for some positive constant c and all sufficiently large values of n; is said to be ft(g(n)) if gin) is 70 THE AUSTRALIAN COMPUTER JOURNAL, VOL 24, No. 2, MAY 1992 ADAPTIVE SORTING unnecessarily expensive. An understanding of adaptive ‘better’ than a Rem value of s/n? More generally, a list of k sorting and the requirements of a particular application runs must have k — 1 pairs of items XyXi+i such that x( > might lead to significant performance improvements. xi+l. In each of these pairs at least one of the items must contribute to Rem, since they cannot both remain in the 2 MEASURES OF PRESORTEDNESS sorted subsequence. Hence Runs(X) < Rem{X) + 1 for all X, To formalise the notion of ‘nearly sorted’ several different and we might be tempted to conclude that Runs is, in some measures of presortedness (Mannila, 1985) have been pro sense, a ‘more accurate’ measure than Rem. posed. Perhaps the most intuitive of these is the number of However it is important to note that simple relation ascending sequences or Runs in the input. For example, if ships such as Runs{X) < Rem(X) + 1, and indeed, Runs(X) < Runs(X) is the number of ascending runs in list X, RunsfXi) Inv{X) + 1, do not necessarily imply that Runs is a better = 1; RumiX2) = 7; and Runs(Xf) = 3. The maximum number measure than Rem or Inv (or vice versa). In the latter case it of runs appears in a reverse sorted list: is perhaps obvious that we are comparing apples and X4 = (n,n — 1, n — 2,...,2,1). oranges, since 1 < Runs(X) < n and 0 < Inv(X) < n(n —1)/2; that is, the measures have different ranges. Sim An algorithm that is adaptive with respect to Runs has a ilarly, even though /ferns and Rem have almost identical running time that is a function both of the number of items ranges, it is erroneous to compare their values. It is not the n in the list and the number of runs in the list. As we shall numeric value that tells how close to sorted a measure see below, the best bound that can be achieved in terms of judges a list, but the number of lists that are regarded by the adaptivity with respect to Runs is2 ®(n log Runs(X)); an measure to be at least as sorted as this particular list. algorithm which attains this bound is said to be optimally This notion was formalised by Mannila (1985). Let M adaptive (with respect to Runs). be some measure of presortedness. Using a slightly differ Another obvious measure of presortedness is the ent notation he defined number of inversions — pairs of items that are out of order. If xi is the fth item of the list X, then the number of belowpfn,k) = {tt t Sn : M{ir) < k], inversions Inv(X) in X is defined as where Sn is the set of all permutations of {1,2,.That is, Inv(X) = | {(if) :i THE AUSTRALIAN COMPUTER JOURNAL, VOL 24, No. 2, MAY 1992 71 ADAPTIVE SORTING that k evenly divides n. Let ir be the permutation corres that have been discussed in the literature (see, for example, ponding to the concatenation of the sorted sets, taken in Knuth, 1973; Mannila, 1985; Petersson, 1990). order. Then Runs(ir) < k, and so 7re belowRum(n,k). It is The measure Exc(X) counts the minimum number of easily seen that each partitioning gives rise to one and only exchanges of arbitrary items necessary to sort the input list; one permutation ir. Counting the number of different ways the measure Block(X) counts the number of contiguous of performing the initial partitioning gives blocks of items in X that remain together in the sorted output (i.e., one more than the minimum number of ‘cuts’ \belowRuJ,n,k)\>n\/(ii)\ (1) that must be made if we were sorting the list using scissors which after taking the logarithm yields and paper); and Max(X) is the maximum distance any item CRuns(n,k) = fXn log k). in X is from its correct sorted position. For example, Exc(X3) = 7; Block(X3)=7; and Max(X3) = 4. The relation Mannila (1985) also gave this result, but with a more ships between these measures and the measures already complex combinatorial construction. described will be discussed in Section 7. Hence, an algorithm taking 0(n log k) time to sort a list Given these definitions, and the notions of optimal of n items and k runs is optimal with respect to Runs. As we adaptivity and superiority there are several problems to be shall see in Section 4, such algorithms do exist. addressed: Let us now establish a similar bound for Rem. Let ir be — We should attemptto order (usings) existing measures, any permutation obtained by permuting the first k + 1 so that algorithm designers can know which measures items of the identity permutation in Sn. By construction, or combinations of measures pose ‘challenges’; removing any k of the first k + 1 items in 7r leaves a sorted — We should attemptto categorise current adaptive algo list, and hence, Rem(ir) < k. As there are (k + 1)! different rithms in terms of this ordering of measures; it’s that can be constructed this way, we have — We should attempt to develop new measures that are CRem(n,k) = Ct(n + log((k + 1)!)) = tt(n + k log k). strictly superior to current measures or combinations of Cook and Kim (1980) described an adaptive sorting current measures, to better capture the behaviour of algorithm that runs in time 0(n + Rem(X)\ogRem(X)), and particular algorithms. (We would also, however, like so it is Rem-optimal. , any new measures to still be ‘natural’.) We now return to the question of which is better, Rem or In the sections that follow we give examples of each of Runs. Consider the list A8, a slight variant of Xy. these lines of investigation. We start by giving an overview of the main paradigms for the construction of sorting Xg = (\Jn + \,\Jn + 2r..,n, \/n, \jn— 1 ,...,2,1). algorithms, and then show how each paradigm can also be We have Rem(Xg) = \/n and Runs(Xg) = \Jn + 1. But applied to the development of adaptive sorting algorithms. according to the results above, a /fans-optimal algorithm 3 SORTING PARADIGMS may spend ®(n log Runs(X^)') = 0(« log n) time, while a General purpose comparison based sorting algorithms fall /tem-optimal algorithm must complete the sorting of Xg in into three broad categories. 0(n + Rem(Xs) log RemCXg)) = O(n) time. List A5 suffices to Merge-based sorts subdivide the input list into a set of complete the demonstration that Runs and Rem are two or more sorted sublists, and by repeated merging of independent. theseflists reduce the set to one sorted list A recursive The considerations outlined in the preceding para Jjy foWnulation is often particularly convenient for describing graphs led us to make these further definitions (Petersson the sequence of merges ^that take place. The standard and Moffat, 1991). Suppose that My and Mfare two mea example of this paradigmls Straight Mergesort, which sorts sures of presortedness. Then * a list by recursively splitting the input list into two sublists — M\ is superior to Mj, denoted My 2 M2, if of equal size, and then merges the two sorted halves using a CM] (|X|, My(X)) = 0(CMJ\ X\, M2(X))); straightforward linear time merge. Straight Mergesort requires at most n log2 n — n + 1 comparisons and @(n log — is strictly superior to M2, denoted M, d M2, if My 2 M2 n) time. and M2 2 My, Selection sorts repeatedly select an item from its input — My andM2 are equivalent, denoted My =M2, if My 2 M2 and place it into its correct location. Most selection sorts and M2 2. M\, store the input list in a priority queue, and select the maxi mum item remaining. The simplest such algorithm is Lin — Mx andiM2 are independent if M\ 2 M2 and M22 My. ear Selection Sort, which uses the input array as priority There are two important consequences of these defini queue, and takes @(«2) time. Slightly more complex, but tions. Firstly, if algorithm S is My-optimal, and My 2 M2, much more efficient, is Heapsort, which stores the items in then S is M2-optimal. Secondly, if algorithm S is not M2- a heap. As a heap can be built in linear time and supports optimal arid My 2 M2 then S cannot be My-optimal. extraction of the maximum item in logarithmic time, We now briefly describe a number of other measures Heapsort runs in ©(« log n) time. An algorithm that does 72 THE AUSTRALIAN COMPUTER JOURNAL, VOL 24, No. 2, MAY 1992 ADAPTIVE SORTING not select the maximum item, but any item, is Quicksort, cribes a number of experiments in which Natural Merge which partitions the input about one of the items and then sort is compared empirically with other sorting algorithms. recursively sorts the two partitions. Quicksort consumes Carlsson, Levcopoulos, and Petersson (1990) demon S(n2) time in the worst case, but is @(n log n) on average, strated that the adaptivity of Natural Mergesort can be and is widely held to be the best general purpose sorting extended significantly by applying an adaptive merging method, because of its small implicit constant factor and algorithm. This merging algorithm runs in sublinear time the fact that it requires only 0(log ri) extra space. if the lists taking part in any particular merge are disjoint, The third of the three standard paradigms is sorting by or ‘easily’ merged in some other way. Their algorithm insertion. These algorithms maintain the first i — 1 items in achieves optimality with respect to Runs, Bbck and Max, some kind of sorted list, and repeatedly insert the i’th item, but is not /nv-optimal. with i growing from 1 ton. The sorted list is initially empty, Moffat (1991) described an adaptive mergesort — corresponding to i = 1; when i reaches n all of the items Margesort — that attains Runs- optimality without pre have been inserted into the sorted list and the algorithm is scanning the input identifying runs, instead making use of done. A well known example of this paradigm is Linear an adaptive merging algorithm similar to that of Carlsson, Insertion Sort, where the sorted component of the list is Levcopoulos and Petersson. Moffat, Petersson and Wor- simply maintained in the first i — 1 positions in the array mald (1992a) further analysed the behaviour of Marge- being sorted. Linear Insertion Sort is normally regarded as sort, and showed it to be optimal with respect to Inv and being ‘an 0(n2) sorting algorithm’. More accurate is to say Rem, and also gave a counter-example that showed it to be that Linear Insertion Sort requires ®(n + Inv(X)) time, and not optimal with respect to Block. so on a sorted list will take linear time. However, in the The merging paradigm also leads to adaptive algo worst case — a reverse sorted list — Linear Insertion Sort rithms for other measures. Levcopoulos and Petersson spends @(n2) time. (1990) and Skiena (1988) have developed algorithms that Descriptions of these algorithms and further examples identify ascending subsequences where the items in each of the three paradigms can be found in Knuth (1973). subsequence are not necessarily adjacent in the input. The behaviour of these algorithms is captured by a number of 4 ADAPTIVE MERGE SORTING new measures. One such measure is SUS(X), defined as the The essence of the merging paradigm is loosely captured minimum number of shuffled upsequences into which X in the following pseudo-code: can be decomposed. For example the lists X5 and X6 of Section 2 can both be formed by the shuffle of two upse procedure Adaptive Merge Sort (X-.list) quences. In the case of Xs there is no shuffling required; Divide X into ascending lists X\,Xi,... ,Xk and in the case of Xb the shuffle is ‘perfect’, with items in X6 Let Q be a queue containing the k lists being drawn alternately from each of the two shuffles. while Q contains more than 1 list do Since a list with k ascending runs is, de facto, a member of Remove the first two lists from Q, and merge them Append the resultant list at the tail of Q , belowsus(n,k), we have CSu^n,k) = D(« log k) by eq. (1). endwhile Here we briefly describe an SOS-optimal algorithm that return the single list in Q attains this bound. end The algorithm operates in two phases as follows. In the first phase each item in the input list is considered and The adaptivity of a merge sort depends on how the appended to the/th of a set of sorted lists Xh chosen to be division and the merging is performed. One of the oldest the smallest j such thatjc, is greater than or equal to the item adaptive sorting algorithms, Knuth’s (1973) Natural Mer most recently appended to Xj. Ifxt is less than all of the tail gesort, differs from Straight Mergesort only in how the items, a new list is started, and xt becomes the first item in division is carried out. Rather than merging a set of lists that list. Item x, will always be1 the first item into Xh each initially of length 1, as is done by Straight Mergesort, Then, in the second phase, the lists are merged, exactly it first scans the input list locating the ascending runs. This as for Natural Mergesort. pre-scan requires linear time, and will identify the starting Suppose that at the end of the first phase there are k lists. point of each of the k = RunsQC) ascending sublists in the Each of the n items was appended to one of the lists; this input. Then the first run is merged with the second, the takes 0(1) time per item. To identify the list takes longer. third with the fourth, and so on, resulting in [k/2] longer However the set of tail items in the lists always form a runs. These longer runs are then repeatedly merged pair decreasing sequence, and if pointers to the tails are kept in wise until there is just one run left, which is the sorted list. an array, the correct list for x,- can be identified with a Since the number of runs is approximately halved in each binary search in 0(log k) time. Over all items, this totals pass, each item takes part in [log2 k] merges. Moreover, 0(n log k) time. each item causes no more than constant cost per merge, The merging of the second phase will also require 0(n and hence, Natural Mergesort runs in 0(n log k) time, log k) time. To demonstrate that this algorithm is SOS- which is optimal with respect to Runs. Harris (1981) des optimal all that remains is to show that k = SUS(X). Con THE AUSTRALIAN COMPUTER JOURNAL, VOL 24, No. 2, MAY 1992 73 ADAPTIVE SORTING sider the last item in list X*. At the time it was appended to without mentioning with respect to what. The adaptivity of Xk the last item in list X*_i was larger, and appeared Smoothsort was later investigated by Hertel (1983), who earlier in the input list JX. Similarly, there must be an item in proved that it was not optimal with respect to the number X/c_2 both larger than this item (in X*_i) and earlier in X. of inversions. In fact, to date it is not known whether Continuing in this fashion, there must be an item in Xj Smoothsort is optimal with respect to any known measure larger than all of these items, and appearing before any of of presortedness. Recently, however, Chen and Carlsson them. These k items, one in each list Xk, form a decreasing (1991) demonstrated that /nv-optimality can be obtained subsequence in the original list X, and so must of necessity by spending linear time rearranging the input before build appear in different shuffles. Hence SUS(X) > k. Since the ing the forest of heaps. lists Xt are a decomposition of X into upsequences, we The second way of achieving adaptivity is motivated by must have SUS(X) = k the observation that instead of storing dl items in the This algorithm is thus optimal with respect to SUS. priority queue, only the ones that can possibly be the Furthermore, since SUS(X) < Runs(X) for any list X, we maximum of the remaining items, the max-candidates, have need to be stored. To support this scheme we employ some CSU^\X\, SUS(X)) = 0(|X| log SUS(X)) data structure that provides max-candidates and interacts = 0(\X\ log Runs(X)) with the priority queue. The following describes a generic = 0(CRuJ\X\,Runs(X))). adaptive selection sort algorithm based on this approach: That is, we have shown that SUS 2. Runs. It is not difficult procedure Adaptive Selection Sort (X:list) to show that SUS o Runs. Consider the list X6 described Construct a data structure 74 THE AUSTRALIAN COMPUTER JOURNAL, VOL. 24, No. 2, MAY 1992 ADAPTIVE SORTING (1989) devised a more sophisticated adaptive selection d) time, where d is the number of keys between the finger sort, called Adaptive Heapsort. This algorithm uses a Car and the accessed item. Finger search trees also provide tesian tree (Vuillemin, 1980) to support the heap. The insertion adjacent to a fingered node in 0(1) amortised (max-)C artesian tree for a list X of length n is the binary time (Mehlhorn, 1984a). One possible implementation of tree with root x,- = max{xi,...,x„}. Its left subtree is the finger search trees using level-linked 2-3 trees was des Cartesian tree for (xi,...^c,_i) and its right subtree is the cribed by Brown and Taijan (1980). Cartesian tree for (xfrl,...;<:„). Given the operations possible on a finger search tree, a After building the Cartesian tree in linear time, Adap number of adaptive sorting algorithms follow. Mehlhorn’s tive Heapsort inserts its root into a heap. Then, each extrac A-Sort fingers the largest item in the tree, and then, using tion of the maximum item is followed by the insertion into this finger, searches for the insertion point of the next item the heap of the children of the extracted item in the Carte to be inserted. The cost of the i’th search will be 0(log hf sian tree; that is, these are the max-candidates. Levcopou where ht is the number of items preceding x; that are larger los and Petersson (1989) proved that Adaptive Heapsort than x,-. Since Inv(X) = XU ht, the total cost is proportional runs in 0(n log(Osc(X)/ri)) time, where (Mehlhorn, 1984a) to n n .« >l/« Osc(X) = X | {/': 1 THE AUSTRALIAN COMPUTER JOURNAL, VOL. 24, No. 2, MAY 1992 75 ADAPTIVE SORTING since each item after the second is inserted adjacent to an of Figure 1, where an upward edge from M2 to Mi indi item that was itself inserted just two operations previously. cates that MpM2.For example, there is an upward edge to This definition leads naturally to the notion of historical show that SUS ^ Runs. Details of the relationships between searching, where we require fast access to items that have measures, and, where there is no relationship and the recently been accessed. For example, a move-to-front list measures are independent, of the lists that are counter provides reaccess to recently accessed items in 0(f) time, examples, may be found in Petersson (1990) and Petersson where t is the number of distinct items that have been and Moffat (1991). accessed since the most recent access to this item. In Since d is transitive, each edge in the diagram is a Moffat and Petersson (1991) we introduced a data struc containment relation on optimality, with all optimal algo ture we call a Historical Search Tree that provides 0(log t) rithms for the higher measure automatically inheriting reaccess to recently accessed items, and allows insertion of optimality for any connected lower measures. For exam new items in <9(log t) comparisons, where t in this case is ple, the edge from Rem to Block reflects the fact that every the number of items that have been accessed since the R/ocfc-optimal algorithm must be Rem-optimal. Conver most recent access to either of the neighbours of the new sely, the presence of upward paths from Block to Loc, Hist, item. By repeated insertion into a Historical Search Tree and Reg means that an algorithm that is notB/ock-optimal we arrived at a Historical Insertion Sort that is sensitive to cannot be optimal with respect to any of the higher temporal locality. We also introduced two measures Loc measures. and Hist to capture the notions of spatial and temporal Each adaptive sorting algorithm corresponds to a des presortedness, and to precisely model the running times of criptor line across the diagram. For example, the descriptor Local Insertion Sort and Historical Insertion Sort for Local Insertion Sort crosses (immediately) below Hist, respectively. Reg, and SUS — it is not optimal with respect to any of It is also possible to combine these two assessments, these three, but is optimal with respect to everything below and to accept as nearly sorted any list in which most of the (Mannila, 1985; Moffat and Petersson, 1991; Petersson insertions are not too far away (in space) from an item that and Moffat, 1991). was itself inserted not too long ago (in time). This is a very To establish lower bounds on the location of the des natural definition. For example, if we lose our car keys, we criptor corresponding to some algorithm we need proofs of search not only in the immediate vicinity of where we first optimality , such as the example proof given here for Natu notice that we no longer have them, but also in the vicinity ral Mergesort. To establish upper bounds on the location of of where we have been recently, both tracing our steps the descriptor for an algorithm we need to describe lists further and further backward in time and also searching in that are nearly sorted according to the measure, but for a wider and wider radius from each intervening point. which the algorithm is not optimal. Our Regional Insertion Sort (Moffat and Petersson, The partial order on measures of presortedness can thus 1991) does exactly that. Perhaps in part because this is a be used as a yardstick to measure the importance (or particularly ‘intuitive’ searching strategy the associated otherwise) of new adaptive algorithms; and as well serves measure of presortedness — Reg — can be shown to be as a framework within which new measures of presorted superior to all previous measures (Petersson and Moffat, ness can be evaluated. The goal of the. algorithm designer 1991), and so a Reg-optimal algorithm will be optimal must be to show that any new algorithm is optimal with with respect to all of the measures of presortedness des cribed here. Both Historical Insertion Sort and Regional Insertion Sort require efficient historical searching. The Historical Search Tree (Moffat and Petersson, 1991) supports fast Hist -t Loc SMS reaccess to items recently inserted, but to date we have been unable to implement all of the desired operations in the necessary time bounds, although we can meet the requirements in terms of comparisons. The development Block of such a data structure remains an important open prob lem, and to date Mst-optimal and Reg-optimal (in terms of running Jtime) algorithms have not been completely described. Inv=DS 7 A PARTIAL ORDER ON MEASURES OF PRESORTEDNESS The relation 2. allows the construction of a partial order on the set of measures of presortedness. The relationships in the partial order are best illustrated with the Hasse diagram Figure 1: Partial order on measures of presortedness. 76 THE AUSTRALIAN COMPUTER JOURNAL, VOL. 24, No. 2, MAY 1992 ADAPTIVE SORTING respect to the highest possible combination of measures; monotone sequences. Information and Computation. To appear. Prel. version in Proc. 2nd Scandinavian Workshop on Algorithm Theory, and an algorithm will only be of interest if its descriptor lies LNCS 447, Springer-Verlag,pp. 181-191. above all other algorithm descriptors at at least one point LEVCOPOULOS, C. and PETERSSON, O. (1991a): An optimal adap in the partial order. tive in-place sorting algorithm. In Proc. 8th International Conference on Fundamentals of Computation Theory, LNCS 529, Springer- Verlag, pp. 329-338. 8 SUMMARY LEVCOPOULOS, C. and PETERSSON, O. (1991b): Splitsort — an We have shown that each of the three main sorting para adaptive sorting algorithm. Information Processing Letters, Vol. 39, No. 4, pp. 205-211. digms leads naturally to corresponding adaptive algo MANNILA, H. (1985): Measures of presortedness and optimal sorting rithms. We have also surveyed a wide range of measures of algorithms. IEEE Transactions on Computers, Vol. C-34, No. 4, pp. presortedness, and shown how measures of presortedness 318-325. MEHLHORN, K. (1984a): Data Structures and Algorithms, Vol L.Sorting can, in some cases, be compared. This relationship leads to and Searching, Springer-Verlag, Berlin, Germany. a partial ordering on measures of presortedness, and pro MEHLHORN, K. (1984b): Data Structures and Algorithms, Vol 3:Multi- vides a framework not only for the evaluation of new dimensional Searching and Computational Geometry, Springer- Verlag, Berlin/Heidelberg, Germany. measures of presortedness, but also for the evaluation of all MOFFAT, A.M. (1991): Adaptive merging and a naturally Natural adaptive sorting algorithms. Mergesort. In Proc. 14th Australian Computer Science Conference, pages 08.1-08.8. University of New South Wales, February. MOFFAT, A.M. and PETERSSON, O. (1991): Historical searching and ACKNOWLEDGEMENTS sorting. In Proc. ISA'91 International Symposium on Algorithms, We thank the referees for their helpful comments and LNCS 557, Springer-Verlag, December, pp. 263-272. suggestions. This work was supported by the Australian MOFFAT, A.M., PETERSSON, O. and WORMALD, N.C. (1992a): Further analysis of an adaptive sorting algorithm. In Proc. 15th Research Council. Australian Computer Science Conference, University of Tasmania, January, pp. 603-613. REFERENCES MOFFAT, A.M., PETERSSON, O. and WORMALD, N.C. (1992b): Sorting and/by merging finger trees. Technical Report 92/4, BENTLEY, J.L. (1984): Programming pearls: How to sort., Communica Department of Computer Science, The University of Melbourne, tions of the ACM, Vol. 27, pp. 287-291. May. BROWN, M.R. and TARJAN, R.E. (1980): Design and analysis of a data PETERSSON, O. (1990): Adaptive Sorting. PhD thesis, Department of structure for representing sorted lists. SIAM Journal on Computing, Computer Science, Lund University, Lund, Sweden. Vol. 9, pp. 594-614. PETERSSON, O. (1991): Adaptive selection sorts. Tech. Report LU-CS- CARLSSON, S„ LEVCOPOULOS, C. and PETERSSON, O. (1990): TR:91-82, Department of Computer Science, Lund University, Lund, Sublinear merging and Natural Mergesort. Algorithmica. To appear. Sweden, October. Prel. version in Proc. Internal Symp. on Algorithms, LNCS 450, PETERSSON, O. and MOFFAT, A.M. (1991): A framework for adap Springer-Verlag, pp. 251-260. tive sorting. In Proc. 3rd Scandinavian Workshop on Algorithm The CHEN, J. and CARLSSON,S. (1991): On partitions and presortedness of ory, July 1992. To appear. Also available as Technical Report LU- sequences. Acta Information. To appear. Prel. version in Proc. 2nd CS-TR:91-81, Department of Computer Science, Lund University, ACM-SIAM Symposium on Discrete Algorithms, pp. 62-71. Sweden, October. COOK, C.R. and KIM, D.J. (1980): Best sorting algorithms for nearly SEDGEWICK, R. (1978): Implementing quicksort programs. Communi sorted lists. Communications of the ACM, Vol. 23, No. 11, pp. cations of the ACM, Vol. 21, pp. 847-857. 620-624. SEDGEWICK, R. (1980): Quicksort. Garland Publishing Inc., New York DIJKSTRA, E.W. (1982): Smoothsort, an alternative to sorting in situ. » and London. Science of Computer Programming, Vol. 1, pp. 223-233. SKIENA, S.S. (1988): Encroaching lists as a measure of presortedness. DROMEY, G.R. (1984): Exploiting partial order with Quicksort. Soft BIT, Vol. 28, No. 4, pp. 775-784. ware — Practice and Experience, Vol. 14, No. 6, pp. 509-518. VUILLEMIN, J. (19 80): A unifying look at data structures. Communica ESTIVILL-CASTRO, V. and WOOD, D. (1989): A new measure on tions of the ACM, Vol. 23, No. 4, pp. 229-237. presortedness. Information and Computation, Vol. 83, No. 1, pp. WAINWRIGHT, R.L. (1985): A class of sorting algorithms based on 111-119. Quicksort. Communications of the ACM, Vol. 28, No. 4, pp. 396-402. ESTIVILL-CASTRO, V. and WOOD, D. (1991): Practical adaptive sorting. In Proc. International Conference on Computing and Informa tion, LNCS 497, Springer-Verlag, pp. 47-54. BIOGRAPHICAL NOTES GUIBAS, L.J., McCREIGHT, E.M., PLASS, M.F. and ROBERTS, J.R. (1977): A new representation of linear lists. In Proc. 9th Annual ACM Alistair Moffat is a Senior Lecturer at The University of Symposium on Theory of Computing, pp. 49-60. Melbourne, having completed his PhD at the University of HARRIS, J.D. (1981): Sorting unsorted and partially sorted lists using the Natural Merge Sort. Software — Practice and Experience, Vol. 11, No. Canterbury in 1986. His research interests cover the broad 12, pp. 1339-1340. areas of data structures and algorithms, and he is currently HERTEL, S. (1983): Smoothsort’s behaviour on presorted sequences. involved in two research projects: data compression for full- Information Processing Letters, Vol. 16, No. 4, pp. 165-170. IGARASHI, Y. and WOOD, D. (1991): Roughly sorting: a generalisation text retrieval systems; and an investigation of algorithms for of sorting. Journal of Information Processing, Vol. 14, No. 1, pp. adaptive sorting and merging. He is a member of the A CM 36-42. , and the IEEE Computer Society. KNUTH, D.E. (1973): The Art of Computer Programming, Vol 3: Sorting and Searching. Addison-Wesley, Reading, Mass. Ola Petersson received his BS degree in Mathematics and LEVCOPOULOS, C. and PETERSSON, O. (1989): Adaptive Heapsort. Computer Science in 1987atVdxjd University,, Sweden and Journal of Algorithms. To appear. Prel. version in Proc. 1989 Work his PhD degree in Computer Science in 1990 at Lund fJni- shop on Algorithms and Data Structures, LNCS 382, Springer-Verlag, pp. 499-509. versity, Sweden. His research interests include design and LEVCOPOULOS, C. and PETERSSON, O. (1990): Sorting shuffled analysis of algorithms. He is a member of EATCS. THE AUSTRALIAN COMPUTER JOURNAL, VOL. 24, No. 2, MAY 1992 77 BOOK REVIEWS Editor: Dr Chris Andrews McDERMID, D. C. (1990): Software Engineering for Information Sys interfaces. Unfortunately the authors appear to have specialisations tems, Blackwell Scientific Publications, Oxford, 384pp., $54.00 which are in one of the two camps and so it is left to the reader to try and (paperback). integrate them. However, each component in the this section introduces pertinent information and is generally well written to give the reader a This book is part of an Information Systems Series by Blackwell Scien basic understanding of simulation in its various aspects and how user tific Publications and is intended to be an introductory text for students of interfaces play a role in presenting that simulation in a realistic way. The systems analysis and design. However, new students should not be dis second part of the book entitled ‘Embedded Simulations’ was found to be couraged from attempting this book, because it assumes no prior knowl fairly heavy going in that each chapter concentrates on fairly specific edge except for previous exposure to general programming concepts and topics. Despite that, the second part does manage to cover quite a wide basic computer terminology. Degree students in computing science as range of applications from speech, eye input and output, sound and well as industry practitioners (engaged in technical or user roles) will cognitive modelling as well as a more traditional software engineering equally be able to gain something from McDermid’s book which, approach to computer-based help facilities. although a British publication, benefits from his stay at Curtin University The final two parts of the book essentially constitute a number of case of Technology (WA). studies presented in reasonable detail. These case studies do cover a wide It was particularly interesting to see how McDermid treats Data Flow range of topics from task analysis through simulations for cockpits and Diagrams, Structure Charts and Pseudocode which are the three tools various information systems. I found this the most useful part of the book that I personally use. In fact, the chapters on these subjects turned out to with its emphasis on specific tools for building both the simulation be the best parts of the book. McDermid’s brief style is sufficiently algorithm and the user interface. It is interesting to note how frequently explanatory that the concepts can be easily understood and applied the Macintosh HyperCard application is used in many case studies. Hyp further as most chapters conclude with several exercises that test the erCard appears to have become the de facto standard for prototyping and reader’s comprehension. (Solutions to the exercises are not included). simulating user interfaces particularly in the area of simulation. Examples from a detailed case study are also used throughout out the On the whole, this book will be useful to practitioners in user interface book to illustrate the concepts of most chapters. The presentation mode design generally, as well as those professionals working in the specific of McDermid’s Data Flow Diagrams appears to follow Gane and Sarson area of simulation, both for command and control applications and (1979). McDermid’s book may be thought of as a reasonably thorough simple cognitive and mathematical models. Both ergonomists and pro update on this well known text with the main concepts abbreviated for fessionals in the HCI area should find something of use. The book was those who do not wish to read all the details of the older book. originally composed of papers from a conference entitled ‘Simulation in Extra concepts covered by McDermid include chapters on entity the development of user interfaces’ organised by the Ergonomic Society modelling, normalisation and testing procedures. In these cases, special and the BCS Human Computer Interaction Specialist Group. The editors ist publications will provide a more in-depth coverage, but as an introduc have subsequently re-ordered this material in a logical way and have tory text, this book is more than satisfactory. McDermid places emphasis, smoothed the boundaries between the individual contributions. On the not only on explaining the basic techniques, but also on how to use them. whole, standard terminology is used except that we see the use of MMI in This is refreshing and forms a methodology which students can adopt to one chapter which is an obsolete use of this term despite its Alvey history. perform requirements analysis as well as systems analysis and design. Michael J, Rees Project management and its relation to Case tools are also covered and, Bond University in the former, McDermid draws upon his considerable practical experience. WECHSLER, H (1990): Computational Vision, Academic Press, xvii, In regard to Case tools, McDermid bases his discussion on ‘Automate 558pp., SUS64.50. Plus’, which may date this chapter unnecessarily. The final chapters stress the need to quantify work estimates for software development and opera In the opening paragraph, Computational Vision promises to provide “an tion of systems. McDermid also provides some simple financial//statisti integrated theory on the workings and implementation of computational cal algorithms which can be used during cost//benefit analysis of compu vision”. The word “computational” is intended to include human (non ter systems. However, the treatment is probably too sketchy to be used in computer) vision! Unfortunately, the book presents a smorgasbord of practice. mathematics lacking anything but superficial structure. It is poorly orga In short, this book is interesting reading, covering all the major parts of nised and lacks focus. software engineering in a technical, but not too demanding way, that is By means of illustrations and philosophical remarks, the author briefer than Gane and Sarson (1979) and useful to university students and attqnfpts to cast the study of the processes of vision into thfe mould of practitioners alike. As an introductory text, I would certainly regard it as a Reiaissance Man. He claims to present a synergistic view of the world of useful reference tool that would also enable more detailed material to be computational vision. Unfortunately, the book remains caught in the located. fourteenth century of yet to lje connected facts, and is certainly not L synergistic. A reader faced with this smorgasbord senses that there is stuff Reference worth knowing, but he will do better to find out about it elsewhere. Gane, C. and Sarson, T. (1979): Structured Systems Analysis: Tools and Contextual discussion about, for example, the importance and physical Techniques, Prentice Hall. meaning of the Wigner Distribution is absent. Perhaps it is not important Lindsay P. Whitehead because there are few references in the index. Similarly, a section on HCS Australia fractals is interpolated without explanation and is not referenced later. Another fault of the book is its failure to develop discussion about its material. To take one example, Chapter 7 deals with object recognition. ANDREW-LIFE, M., NARBOROUGH-HALL, C, and HAMILTON, The discussion on Human Recognition is incomplete because it merely I., (eds.) (1990): Simulation and the User Interface, Taylor and works through a list of literature without drawing the material together at Francis, London, 269 pp., Stg.32.00 (hardback). the end of the section. This reader is left dissatisfied. Later, the author’s object recognition system developed using parallel distributed comput This book Iattempts to combine the areas of simulation and human ing is presented. However, what related work is there? None presented. computer interfaces. These two areas have been reasonably distinct in the What choices were made and why? The reader is in the dark. The system past and their combination is tenuous and not an easy task. The editors is presented and analysed piece by piece, then shown to work with one have done & reasonable job of combining contributions from 38 authors example! It’s not science. in a readable fashion. These are some particular shortcomings of the book. However, there The book is split into four parts which come together quite well. In the are also several errors in Chapter 2, “Distributed and Multiscale Repres first part entitled ‘General Issues’ we see overview contributions which entations”. For example on page 23, “F(s) F(a>)” should read “F(s) attempt to define the boundaries of simulation and how it relates to user F(joi)”. Indeed, there is no need to mention the LaPlace Transform in the 78 THE AUSTRALIAN COMPUTER JOURNAL, VOL 24, No. 2, MAY 1992 BOOK REVIEWS context of the mathematics of vision systems. Such signals are stable, not Several papers look at fuzzy reasoning. Buckley discusses design the one-sided (causal) signals of dynamic systems. Again, the phrase “the decisions relevant to an expert system development tool called FLOPS power spectrum is given as (amplitude/magnitude, phase)” is really while Novak and Pedrycz discuss aspects of many-valued fuzzy logic. describing the polar notation of the fourier transform (page 25). On page Both papers provide axioms which any reasonable fuzzy reasoning sys 28, the expression (j22pu, j2pv) is meaningless. Again on page 30, the tem must satisfy, and both leave the feeling that the fuzzy reasoning words explaining g(i, j ) do not relate to the expression or the matrix. framework is still too.flexible to be converted to a technology in the near There are other typographical errors throughout the text. future. In contrast Zwick & Wallsten consider the psychological validity This reviewer cannot recommend this book either to the student, the of several different models of fuzzy reasoning. The different models experienced practitioner, or the library. allow fuzziness at different points in the reasoning process. The paper P. Horan gives a clear exposition of these models with well-worked examples. Deakin University They each combine numeric and linguistic descriptions in some combi nation. Two groups of subjects were identified: those with probability// GAINES, B., and BOOSE, J., (eds) (1990): Machine Learning and Uncer statistics background and those without. A few experiments were tain Reasoning, Academic Press, 498pp., $83.70. included which tested a knowledge of (non-fuzzy) probability and these completely failed to distinguish between these groups. Apparently the This is volume 3 in the series Knowledge-Based Systems published by Einstelling effect, documented in the psychological literature can explain Academic Press. The two previous volumes are on knowledge acquisi this observation. Apart from this amusing aside it is refreshing to see tion and tools for knowledge acquisition. The three volumes have covers work on the psychological validity of aspects of fuzzy reasoning espe which identify them as belonging to a series. cially amongst the large body of turgid literature in this area. The present volume is similar to the others in origin. It is a compen In a paper that does not fit comfortably into the two themes of the book dium of papers previously published in the International Journal of Yager shows how logical inference of the form often implemented in a Man-Machine Studies with the exception of one paper from the journal (pure) production system may be represented as integer (linear) pro Knowledge Acquisition. The papers appeared in these journals between gramming problems. This has the advantage that the full logical content June 1988 and November 1989. Unlike the previous two volumes not all of a set of production rules is used, instead of being subservient to the way of the papers resulted from an AAAI Workshop on Knowledge Acquisi a production rule is written down. Using this framework it is possible to tion for Knowledge-Based Systems at Banff, Canada, although some determine easily whether a goal is certainly true, certainly false or simply papers originated from work presented at the 1987 Workshop. consistent with the logical axioms underlying a production system. The In the introductory paper Gaines and Boose describe how human paper goes on to show how default rules or non-monotonic reasoning) experts are not static repositories of knowledge rather they are able to may be handled using the integer programming idea. learn and even create new knowledge. Additionally all human beings are As usual with a collection of papers there are a large variety of subjects able to function in a world where much knowledge and data is uncertain. discussed. In the machine learning section there are other papers which Commercial tools for expert system development offer only primitive address exemplar-based learning and another on expert system learning features for learning and reasoning in the presence of uncertainty. The from natural language explanations. Other uncertain reasoning papers aim of this volume is collect together some current research ideas on discuss Depster-Shafer theory, INFERNO-style reasoning, non these topics. monotonic reasoning and possibilistic reasoning. Additional papers Two survey papers attempt to place machine learning systems into the which defy classification under the title of the book include one on the context of knowledge acquisition for expert systems integration of experiential and model-based reasoning. Another paper . Witten & McDonald, in a particularly readable paper, classify machine examines the performance of high and low-skill workers in the context of learning systems which are clearly defined in the literature. All of the a diagnostic expert system. classifications and associated algorithms are described briefly, but there This volume contains a reasonably high level of symbolism and is little discussion of how this classification relates to knowledge acquisi theory. Frequently this is tempered by examples and case studies. On the tion. Shalin, Wisniewski, Levi and Scott introduce a formal analysis of whole the papers are of a high standard with only the odd typographical machine learning systems for knowledge acquisition which is based on a> error. They represent a worthwhile insight into several aspects of rather different taxonomy (or classification) of learning systems. The research into advanced features for expert systems of the future. Addi paper describes aspects of expert systems which are amenable to help tionally the volume looks good on the book-shelf alongside its compan from machine learning methods. Two case studies are presented of a ion volumes! rational method for selecting machine learning methods for expert sys P.A. Collier tem tasks. The authors conclude that the work is a modest beginning in University of Tasmania attempting to understand the relationship between knowledge acquisi tion and machine learning. BIELAWSKI, L., and LEWAND, R. (1991.): Intelligent Systems Design: The paper by Crawford considers extensions to the inductive infer Integrating Expert Systems, Hypermedia and Database Technolo ence algorithm CART, which is similar to the well-known ID3 family of gies, John Wiley and Sons, 302pp., unstated price. algorithms. The problem of estimating the accuracy of an induced theory is addressed and analysed in terms of bias and variance in the estimator. A The two authors, from the case studies they present, are obviously highly new estimator is introduced and shown to perform better than a cross- effective users of software packages for PCs and compatible machines, validation estimate, especially for small data sets. This estimator can also and for the Apple Macintosh. The book is written for those who may wish be used as an aid to select good theories after adjustment to compensate to emulate the authors in selecting and using effectively such packages. for over-learning. Crawford also discusses an approach to incremental The book has been logically organised by grouping chapters into four inductive learning which only rebuilds a theory when it is significantly sections dealing with: Intelligent Systems Concepts; Intelligent Systems deficient. Development; Models for Intelligent Systems Design; and Intelligent The intriguingly styled frustration-based learning method introduced Systems Development Tools. The authors’ concept of an intelligent by Suwa and Motoda is used to learn methods of geometry theorem system, apparent from the title, is one which integrates two or all three of proving where one new point and associated lines have to be constructed. the component technologies. Intelligent systems are able to: The system is able to learn rules for these constructions from a frustration — behave logically, state, where further progress is stalled before a successful proof is com — solve complex problems, pleted. Even though several strategies are learnt these can result in a — be responsive and adaptive, • combinatorial explosion since ‘human beings perform effective reason — permit ‘non-linear program navigation’, ing by visually searching only promising ways of matching /(methods — effectively use existing information and with problems/). Such intellectual behaviour is beyond our learning — be user-friendly and interactive. method’. It is refreshing to see such honesty. It is not completely apparent why any well-written software package THE AUSTRALIAN COMPUTER JOURNAL, VOL. 24, No. 2, MAY 1992 79 BOOK REVIEWS should not deliver similar benefits. useful though superficial case studies. Reading the case studies, I feel that The first section of the book equips the reader with basic definitions the authors are justifying the access they have been given to information and a framework on which to base the development of ideas and the about the systems by making only positive statements; there is a singular evaluation of packages and applications. The architecture of an expert lack of critical comment. Many of the statements are those I would expect system is described, with the development of concepts relating to decision to glean from a glossy brochure. trees, object-oriented programming, procedural and relational constructs I gain the impression that linear navigation is related to a fixed and forward and backward chaining. There is an extremely superficial sequential data entry sequence! treatment of uncertainty, which relates more to its treatment in some of The fourth part of the book is devoted to a review of PC and Macin the available packages than to any notion of desirable methodology. tosh based intelligent system development tools. The discussion includes Hypermedia is defined as 1 .. an information management tool that details about pricing and vendors. The tools are subjected to scrutiny on a links text, graphics, sound or other types of media in an associative way’. common set of criteria. Once again, glossy brochure comments abound, Again, this puzzling notion of ‘non-linear navigation’ is raised — I was with a lack of any direct criticisms. left with the impression that this is the sort of thing offered by an on-line The book should be useful to anyone wanting to approach intelligent help facility, which allows one to choose individualised paths through systems at the package level. The structure is good, but at times the detail data. Finally, different classes of intelligent systems (to be explored later) is a little woolly. The most important contribution is the glimpse that it are defined, dependent on whether the dominant construct is the expert affords the reader into effectively using and combining standard soft system, hypermedia, or an integration of the two. ware. It seems to me that this management of data is what hypermedia is The second section in the book deals with intelligent systems devel all about, and if the book limits itself to hypertext, or fails to deliver opment, based on ‘a five step procedure that has evolved into a standard systems complete with images and sound, then this is at least realistic, and methdodology for intelligent system construction’. The steps are: offers a good view of what is achievable with minimal effort and devel — Problem identification opment time. I would recommend the book, but read it quickly before the — Knowledge acquisition and representation information it presents is outdated by new packages. — Tool selection Tim Hesketh — Prototyping and development University of New South Wales — Testing and maintenance Cognitive rather than computational capabilities are stressed. The authors point out that the modern tendency is to limit the scope of TURBAN, E (1990): Decision Support and Expert Systems, (2nd edn), projects, providing many specific systems, each with very short develop MacMillan Publishing Company, New York, 846 pp (paperback), ment time. It is claimed that nowadays, the experts are frequently the price unstated. developers of the expert systems, given suitable tools with which to work. Suggestions are made about evaluating the worthiness of a project, The purpose of this book is ‘to introduce the reader’ to EIS (executive though I found this section woolly. information system) and DSS (decision support system) technologies. It is The section on knowledge representation and acquisition is very clearly aimed at students, and contains questions, exercises and case representative of the book as a whole. There is a useful figure illustrating studies, which is unusual for books on these topics. An accompanying sources of knowledge, which encapsulates suggested approaches for instructional package (instructor’s manual, test bank, demo software, and anyone who is unsure about where to start. However, coverage is very free student software) is available. The second edition has new chapters general, and details about knowledge representation are totally inade on modelling, IFPS, knowledge acquisition, and user interfaces. The quate for anyone who is not relying on a software package. The discus material on DSS, EIS, and specific expert system products has also been sion covers representations of procedural and relational knowledge. expanded. The section on development tools is aimed at providing the reader This is a massive tome. One might wonder at the scope for applying with a framework for evaluating off-the-shelf systems. The coverage is the technology of the topic, which is all about turning masses of data into generic in nature, although it must be heavily influenced by the ideas intelligible information, but perhaps that will come later in the evolution encapsulated in existing packages. Useful suggestions about developing of books. Chapter 1 is an overview of management support systems semantic connections only for highly related information, maintaining which tries to cover’ managerial decision making’ and ‘management consistency of relational features, developing stylistic navigation aids, science’ in 2 pages. It goes on to define decisiofi support arid executive chunking and the preservation of the original document structures are information systems, then expert systems. ThiY breathtaking pace is provided. It is also good to see comments concerning prototyping during undertaken only at the expense of a structured, formal, and instructive development, and testing and maintenance following it, though these appjjjach to these large topics. Subsequent chapters move to modelling, sections are not memorable. development tools for DSSs and case studies. The information on pro Three broad model classes are offered for intelligent systems design. ducts is dated, some of it drawins on 1984 and 1985 publications. The first model is that of an expert system with Jjypermedia support. The Chapter 10 is on EISs, and chapters 11 — 15 on expert systems. It is role of the expert system for diagnosis, classification, predictigif schedul hard to know who is the audience for these latter chapters: The material is ing, decision support and configuration is highlighted. The discussion of a too elementary for the serious Al student, and too superficial for the hypermedia semantic network seems to imply a close relationship with non-specialist to emerge anything but confused. object oriented representations. There is a mention of decision tree or rule -This is a criticism of the style of the entire book: The range of topics is based expert system realisations. enormous, and the coverage correspondingly light. The result is an The second model is that of a hypermedia based intelligent system unsatisfying melange with no firm structure. As a reference book, it lacks with expert system support. At last I was hoping to get a clearer picture of depth and rigour, yet as a teaching tool it may confuse and fail to impart what hypermedia is, but was merely left with the impression that it is understanding. Its greatest value is as a storehouse of case studies. In the awfully like a database involving an organisational structure with infor expert system area, the examples are well known, although still likely to mation decomposed into screens, some functional realisation of links interest business studies students. The case studies for DSSs and EISs are between the screens, and a suitable user interface. wide-ranging and valuable for students of business information studies. Finally, tile third class of system is presented: that of an intelligent There is hardly, however, a good basis here for the beginning practioner. system integrated with a data base, or at least a data base with an Ronald Watts intelligent system front- or back-end. All three models are illustrated by Information Resources, Sydney 80 THE AUSTRALIAN COMPUTER JOURNAL, VOL. 24, No. 2, MAY 1992 NEWS BRIEFS ‘News Briefs’ is a regular feature which covers NEW COMMUNICATIONS CABLING deluged with government regulations and forms,” local and overseas developments in the computer SYSTEMS STANDARD Mr Bulluss said. industry including new products and other topical Standards Australia has recently published an im “They have struggled to keep up-to-date and events of interest. portant new Australian standard dealing with inte have allowed the management accounting tech grated communications cabling systems for com niques to sit on the shelf to the detriment of small NEW INFORMATION TECHNOLOGY mercial premises. businesses in Australia.” STANDARDS Thestandard,basedonthe current US standard, “And while computers have got cheaper, faster AS 4016 Information processing systems — will be of importance to building owners and man and more user-friendly, accountants have mainly Text communication — Reliable transfer agers, client organizations, equipment vendors and been performing everyday compliance work.” Parti: Model andservicedefinition provides carrier organizations. “The launch of this package enables account a conceptual and terminological framework for the It will help owners and managers in the areas of ants to use technology to improve the profit and other parts of this standard. The functional and marketability, ‘future proofing’, reduced disrup productivity of their clients.” organizational models define ways in which the tion, flexibility and reduced fit-out time. For clients, Mr Bulluss said the package —which took Directory can be distributed, both functionally and it will mean uniformity in communications wiring 2,000 hours to set up—had been designed to deal administratively, the security model defines the combined with flexibility, structured management, with eight key areas: accounting and calculators; framework within which security features, such as reduced costs associated with changes and moves cash management; costing; finance; investment; access control, are provided and the information and simplified problem diagnosis. management; practice management; and superan model describes the logical structure of the Direc Compliance with the new standard by the in nuation. tory Information Base. This standard is identical dustry will provide equipment vendors with known “This structure allows it to be used as a manage- with and has been reproduced from ISO/IEC9594- operating environments, connectivity conventions, ment tool providing decision makers with useful 2:1990. simplified fault diagnosis and wider market accept projections and analysis in a range of important Part 2: Protocol specification specifies the ance of their products. Carrier organizations will accounting areas,” Mr Bulluss said. protocol (abstract syntax) and procedures for the similarly be assisted. Ray Harrold, a Lecturer in the Faculty of Busi reliable transfer service element (RTSE) services. The standard provides a glossary of terms and ness and Technology at UWS — Macarthur, said These procedures are defined in terms of the inter abbreviations and information on planning and the new package was very versatile and had the actions between peer RTSE protocol machines design, as well as covering technical issues, earth potential to be an important asset to accountants. through the use of association control service ele ing, installation and testing and administration. “It seems to provide a support facility to many ment and the presentation-service and the interac There are many issues such as environment, of the main services, especially that of management tions between the RTSE protocol machines and its quality, compatibility, operation, installation, inter consultants, that accountants should be providing,” service-user. This standard is identical with and has ference, redundancy, and administration which are Mr Harrold said. been reproduced from ISO/IEC 9066-2:1989. not addressed by mandatory standards, and this new Mr Harrold also believes the package will be These documents may be purchased from any standard is intended to address these issues. The put to good use as a teaching aid that will provide Standards Australia sales office, and from the Na application of this standard should lead to the accounting students with valuable practical experi tional Sales Centre, tel (02) 746 4600. installation of cabling systems which satisfy user ence. requirements and provide a useful life of at least 10 “Using it will enhance their potential as future years. practitioners in the industry,” he said. IMPORTANT NEW SUBSET OF IGES FOR Adherence to this standard within premises will CAD DATA EXCHANGE SWITCHING ON TO THE ELECTRONIC lead to opportunities for cost savings both during A new computer graphics data exchange standard CAMPUS construction and throughout the life of the building, which will assist those in the architectural, engi Western Australia’s Edith Cowan University has and will provide for the use of multi-vendor prod neering and construction industries has recently opened an ‘electronic’ campus to improve commu ucts on existing or emerging standards. been published by Standards Australia. nication and educational opportunities for its exter In addition to this new standard, a further three This important project, a world first, has been nal students, some of whom are in remote areas of cabling systems standards are now under develop developed in Australia after extensive research Australia. ment by Standards Australia. When published, they both here and overseas. It will directly benefit The ‘electronic’ campus involves linking stu will cover, in detail, cabling pathways, administra dents and University staff via a computer network software developers, managers and users of com tion and light commeicial and domestic buildings, puter aided drafting (CAD) systems, and is a subset using Telecom Discovery software. Students and or ‘smart houses’. staff are able to communicate with each other to the more substantial and complex document, The standard ‘Integrated communications ca ‘Computer graphics — Initial graphics exchange through ‘noticeboards’, send messages to indi bling systems for commercial premises’ (AS 3080) vidual ‘mailboxes’ and ‘chat’ during open forum specification (IGES) for digital exchange of prod is available for purchase from any Standards Aus uct definition data’. sessions. The network will also provide access to tralia sales office, and from the National Sales course materials which may be downloaded to a The new standard covers definitions, subset Centre, tel (02) 746 4600. application levels, application protocol, 2-D draw student’s computer and allow students to upload ing entity subset, and testing, as well as guidelines assignments or documents. for data exchange users. It is aimed at those in NEW ACCOUNTANT’S PACKAGE To participate students require a modem, com volved in the planning, design, documentation and A new software package aimed at improving ac munications software on their computer and a construction of buildings, but is also applicable to 2- countant’s productivity and professional effective telephone line. Students in Edith Cowan’s Compu D drawing transfer in other sectors, such as the ness was launched at the University of Western ter Education and Computer Science units have automotive and aerospace industries and other com Sydney — Macarthur’s Campbelltown campus in been assisting with field testing of the network mercial CAD systems users. April. during semester one. This important document will make the rapid The package — ”50 Plus Ways to Provide The University is developing new communica and accurate transfer of drawings between different Additional Management Services” — has been tion software for second semester when 30 Library CAD systems possible, which typically happens developed by Frank Bulluss, a Certified Practising Technology students will be invited to join the when different organizations wish to share data or Accountant based at Chester Hill. The first copy of network as part of an equity project. The University when information is transferred from a CAD to a the software and accompanying manual was do will provide the Library students with a modem and CAM (computer aided manufacturing) system. nated to UWS—Macarthur’s Faculty of Business software and cover communication costs. Copiesof‘Computer graphics— Initial graph and Technology at the launch. Project leader Jan Ring, from the University's ics exchange specification (IGES) for digital ex While Mr Bulluss say s the package can be used Department of Computer Science, said the network change of product definition data, Part 2: Subset of to assist accountants to quickly and accurately will be expanded to provide access to the Universi AS 3643.1 —Two-dimensional drawings for ar assess their client’s taxation position through a ty’s Library catalogue, CD-ROM and the AARNet chitectural, engineering and construction (AEC) range of pre-programmed methods, he believes it information system which will provide interna industries’ (AS3643.2)may bepurchased from any has other more important benefits. tional links. Standards Australia sales office, and from the Na “For two decades public accountants, and to Each call costs the price of a local phone call to tional Sales Centre, tel (02) 746 4600. some extent accountants in commerce, have been establish the connection. Calls are then charged THE AUSTRALIAN COMPUTER JOURNAL, VOL. 24, No. 2, MAY 1992 81 INFORMATION MANAGEMENT & AN BA R TECHNOLOGY Three top international ABSTRACTS journals for experts in Anbar-Information Management & Technology Abstracts is one of the most recent additions to the abstracting journals, Information Technology providing you with a quick overview of key marketing issues appearing in current international management literature to ensure that nothing escapes your notice. THE ABSTRACTS Twelve monthly issues brings you concise summaries and appraisals of the most relevant and useful articles published in the field of Information Management & Technology. Over 300 international management journals are scanned by A experts who are practising managers and consultants Computer Control themselves, ensure that only articles which will make a positive contribution to the knowledge of the practising Quarterly manager are selected, abstracted and annotated. 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He established the Australian Computer Abuse subscribing to Computer Abstracts, you can be Research Bureau and is a Past President of the EDP certain nothing of value escapes your attention. Auditors’ Association. Vol 10 Nos 1-4 1992 $299 An international journal Vol 36 Nos 1-12 1992 $699 An international journal Publications Ltd. IDEAS YOU CAN USE ISSN 0813 7099 published in the UK ISSN 0010 4469 published in the UK The Latest Texts & References in Computer Science from <%) Addison-Wesley • Benjamiri/Curnmings 0 Publishing Company A.C.N. 000 663 203 Unit Al, 6 Byfield Street, North Ryde NSW 2113 Telephone (02) 878 5411 Fax (02) 878 5830 Distributed Operating University Revision The Magic of Modula-2 Systems Notes Series FORTRAN 77 By K. Hopper, G. Holmes, W. Rogers By Andrzej Goscinski Documentation & Style The Magic of Modula-2 introduces computer problem solving through This book examines the basic con modeling and data abstraction. The au By Stephen R Kessell cepts of distributed operating systems: thors present practical solutions to realis transparency, heterogeneity, interprocess This book provides detailed guidance tic problems that support the develop communication, the synchronization, on how best to incorporate good docu ment of abstractions and painlessly show naming, and process and resource man mentation, style, structure and portability users the features of Modula-2 agement. It is the first book to cover all into any FORTRAN program, regardless ISBN 50993,1991, Paper cover, $36.95 design issues of distributed operating of its size or complexity. systems. ISBN 55611,1992, Paper cover, $21.95 ISBN 41704,1991, Hard cover, $54.95 TtX for the Beginner By Wynter Snow UNIX: Productivity Tools ADA TgX is now widely used for computer For Teachers, Writers & A First Course in Programming typesetting in mathematics, science, and Researchers and Software Engineering engineering. This book is a carefully paced, tutorial introduction for people firstleamingthe system. Special emphasis By Gerald Tan By David Bover, Kevin Maciunas & Michael Oudshoorn is given to what can go wrong, and how This book provides a set of programs to fix it when it does. LaTgX notes are The book introduces fundamental for those who use UNIX systems. Using provided for use with a set of macros. programming and software engineer the programs set forth in this book, UNIX ISBN 54799,1992, Paper cover, $48.95 ing concepts through the use of a subset users will be able to use their computers of the programming language Ada. more productively. The programs in Emphasis is placed on the design and Designing the User Interface cluded alio w users to maintain a database development of algorithms, the con of notes on a number of topics, display Second Edition cept of abstraction, and the software clashes in a timetable and all free slots in engineering concepts of modularity and a timetable, create bibliographies, store Ben Shneiderman information hiding. These features are phone numbers, and much more. not peculiar to Ada and will aid the This substantial revision expands ISBN 55612,1992, Paper cover, $52.95 reader throughout their programming upon the first edition's broad coverage life, no matter what language might be of key topics in the field of user interface Introduction to Robotics used. design. Like the earlier edition, the ISBN 50992,1992, Paper cover, $39.95 second edition highlights major issues By Phillip McKerrow in human factors, and combines de scriptions of theoretical underpinnings Robotics is now a significant area of with practical applications. The book importance within computer science, es Relational Database points to solutions of key problems pecially in Al, and also electrical and Writings 1989 -1991 where possible, and to further work mechanical engineering courses: an area where necessary. The new edition which is emerging as a discipline in its C.J. Date with Hugh Darwen featrures much new and expanded own right. This major field draws on material, including recent information expertise from mechanical engineering Like its predecessors, this book on interaction devices, gestural imput, (kinematics and dynamics), electrical en presents a collection of papers - many of high-precision touch screens, user in gineering (control), computer science them previously unpublished - on vari terface management systems, compu (programming), and Al (vision). Every ous, cutting-edge aspects of relational ter-supported collaborative work, one of these aspects is covered in this database technology. hypermedia, telepresence, and graphi ludd, fully illustrated text whose unique ISBN 54303,1992, Hard cover, $49.95 cal user interfaces. format is wider than usual. ISBN 57286,1992, Hard cover, $48.95 ISBN 18240,1991, Hard cover, $48.95 Programming for Knowledge-Based I Artificial Intelligence Artificial Intelligence [anufacturing Management Third Edition Methods, Tools, & Applications By Roger Kerr Patrick Henry Winston This book provides a comprehensive in This is an eagerly awaited revision of the By W. Kreutzer & B. McKenzie troduction to the modem approach to plan single bestselling introduction to Artificial ning and controlling manufacturing opera Intelligence ever published. Artificial In This book details the tools, techniques, tions. In particular, it shows how new tech telligence, Third Edition retains the best and environments of artificial intelligence. nologies of software engineering database features of the earlier works including su Each programming paradigm is fully illus design, and artificial intelligence can be com perior readability, currency, and excellence trated by actual case studies, examples, and bined with modem manufacturing manage in the selection of the examples. Winston Al applications. A concise, technical intro ment techniques resulting ultimately in total emphasizes how artificial intelligence can duction to Scheme, Flavours, Prolog, and computer integrated manufacturing (CIM). be viewed from an engineering point of Smalltalk is provided within the book. ISBN 41622,1991, Hard cover, $42.95 view. The new edition offers comprehen ISBN 41621,1991, Paper cover, $42.95 sive coverage of more material and many of the ideas presented are enhanced with a UNIX for the Impatient variety of side pieces including application Mathematica examples such as the Westinghouse nuclear A System for Doing fuel plant optimizer. Mathematics by Computer P. Abrahams and B. Larson ISBN 53377,1992, Hard cover, $44.95 Second Edition UNIX for the Impatient is a UNIX handbook for people who need to learn UNIX quickly - people who are interested in UNIX By Stephen Wolfram as a tool rather than an object of study. It can This new edition maintains the format of serve both as a means of learning UNIX and the original book and is the single most im as a reference source once one has mastered a portant user guide and reference for topic. Concise and direct, this book condenses Mathematica. The Second Edition includes a a lot of information into a modest number of comprehensive review of the increased func pages. It is particulaiy suitable for people tionality that Version 2.0 affords users as well converting to UNIX from another operating as an enhanced reference section. Contains system. sixteen pages of full-colour graphics created ISBN 55703,1992, Paper cover, $42.95 wth Mathematica. ISBN 51507,1991, Paper cover, $48.95 ISBN 51502,1991, Hard cover, $54.95 rder Form Please send me a copy of (post free): □ Winston Artificial Intelligence 3E (ISBN 53377) ...$44.95 □ Abrahams UNIX for the Impatient (ISBN 55703) ....$42.95 □ Wolfram Mathematica 2EP/b (ISBN 51507)...... $48.95 □ Bover ADA (ISBN 50992)...... $39.95 □ Wolfram Mathematica 2EH/c (ISBN 51502)...... $54.95 □ Date Relational Database Writings 1989 -1991 □ Please deduct my 20% academic discount. (ISBN 54303)...... $49.95 I enclose a cheque for $______□ Goscinski Distributed Operating Systems OR Debit my: □ Bankcard □ Visa Card (ISBN 41704)...... $54.95 □ Mastercard □ American Express □ Hopper The Magic of Modula-2 (ISBN 50993) ....$36.95 Card No______Expiry Date______□ Kerr Knowledge-Based Manufacturing Signature______Management (ISBN 41622)...... $42.95 Name______□ Kessell FORTRAN 77 (ISBN 55611)...... $21.95 □ Kreutzer Programming for Artificial Address______Intelligence (ISBN 41621)...... $42.95 ______State______Postcode______□ McKerrow Introduction to Robotics (ISBN 18240) ...$48.95 Note: Prices are subject to change without notice □ Shneiderman Designing the User Interface (ISBN 57286)...... $48.95 ADDISON-WESLEY • BENJAMIN/CUMMINGS Q PUBLISHING COMPANY □ Snow T^ for the Beginner (ISBN 54799)...... $48.95 Unit A1,6 Byfield Street, North Ryde NSW 2113 □ Tan UNIX: Productivity Tools (ISBN 55612) .$52.95 Telephone (02) 878 5411 Fax (02) 878 5830 NEWS BRIEFS (Continued) between 10-20C a minute depending on the time of In line with previous years, Online Information • practical/manual activities; day. 92 seeks contributions covering a number of gen • persuasive/selling activities; Students will receive a training package and be eral information themes, and this year’s meeting • clerical/administrative activities; able to call a telephone hot-line to assist with invites contributions specifically to four major pro • outdoor/mobile activities; problems. fessional interest areas. • mathematical/numerate activities; Ms Ring said the aim of the ‘ electronic’ campus Conference Chairman, David Raitt explains, • creative/innovative activities; was to reduce the isolation experienced by external “We recognise that there are at least four major • scientific/analytic activities; and students. professional interest areas that should be addressed, • social/people-related activities. “External students are usually far removed and these are online use and applications, CD- DPA also indicates the degree to which the from the support of teachers and fellow students ROM acquisition and use, computer systems in person prefers working with others or working which is considered important in encouraging per libraries, and archival and documentation systems. with information. It maps the respondent’s ap sistence in difficult courses,” she said. “We are also seeking papers on a number of proach to problem solving and preferred work The network will be extended for use by other technical and practical issues which cut across these environment. external students next year. four areas, such as quality control, standards, edu The “TEAM” version of DPA may help man cation and training, technological aspects, and fu agers draw conclusions about the respondent in the PIONEERING VISION TECHNOLOGY ture roles,” he added. context of a work team. The DPA TEAM Profile LAUNCHED Last year’s Meeting once again exceeded the reveals the extent to which the respondent prefers to The first field trials of pioneering CSIRO research Organiser’s expectations, with over 7,000 partici work with others as a team member, and provides into vision technology, code-named Safe-T-Cam, pants from 38 countries over the three days. a base for recognizing the individual’s tendency to were launched in New South Wales by Deputy Online Information 92 — the 16th Interna assume one of a series of team roles. These are Premier and Minister for Roads Wal Murray. tional Online Information Meeting, takes place at described as: Team producer, Persuader, Adminis The Safe-T-Cam system will be used to moni Olympia 2, London on 8-10 December 1992. trator, Field Agent, Examiner, Innovator, Analyser, tor heavy vehicle movement at the Wilson Road Team leader, Private producer, Self-promoter, overpass on the Hume Highway as part of a NSW CSIRO SOFTWARE HELPS TAILOR JOBS Blocker, Absent member, Pedant, Potential inno Road Traffic Authority incentive aimed at improv TO WORKER’S INTERESTS, ABILITIES vator, Disputer, and Controller. ing road safety. CSIRO designed the software system and The unique dual-camera system was devel- the algorithm which implements the preference opedbyCSIRO’sDivisionofManufacturingTech- theory. The software is comprised of several nology with Telecom and local software developer modules: Iconix, and uses leading edge infra-red sensor • a data entry module supports the convenient technology to decipher heavy vehicles amongst input of test data based on the questionnaires; m - ordinary traffic on multi-lane highways. • an automatic data analysis module; According to CSIRO researcher Ian Macintyre, • a report generation module for representing the technology is a world first. “We carried out an analysis results on the screen and in printed international technology audit to find a sensor mad form; suitablefor demanding on-site conditions, but found • a research module; and nothingappropriate”hesaid.“Weultimately worked • a security device that limits the number of with a group of international researchers to develop users of the software. a whole new high resolution camera system.” For more information, call Mr Neville Smith, The Safe-T-Cam system incorporates unique NIS Associates, tel (02) 977 0366 or Judy Marcure, computer architectures, sensor devices and flash CSIRO Division of Information technology, tel systems specially designed for continuous auto (02) 887 9307. matic operation in all road conditions, and can process images in the time taken for a vehicle to COMPUTER POWER WINS $20 MILLION move in and out of a camera’s sight. R&D EXPORT CONTRACT The technology offers superior road monitor WA Deputy Premier, Mr Ian Taylor, welcomed the ing since it is portable, operates in all road condi win by the Western Australian operations of the tions, emits no electromagnetic signals and needs Computer Power Group of a large research and devel no human maintenance, unlike current traffic moni CSIRO has just completed delivery of a software opment contract with IBM Corporation in the US. toring systems which activate only when sensors or system which analyzes workers’ preferences. The The four year contract will generate more than radars are triggered and often require road modifi system,calledDPA(for Decision Preference Analy 30 new jobs and result in export revenues of ap cation. sis) was developed by CSERO’s Division of Infor proximately $20 million flowing into the State over According to Macintyre, the technology pro mation Technology for NIS Associates. The com this period. duces vastly superior picture resolutions that allow pany’s Managing Director, Mr Neville Smith, says Mr Tailor added that this achievement, against automatic, high quality monitoring useful for many that the package can help firms assess the suitability vigorous competition world-wide, should prove an other applications that require image processing, of employees for specific positions, help guide inspiration to local firms seeking international mar such as car parks, toll gates and industrial inspection those considering a new career, or help managers^ kets. systems. restructure jobs or work teams to motivate workers He congratulated Computer Power and IBM for The Safe-T-Cam system adds to the previous and make the best use of their skills. “Designing working together on their commitment to a long successes of the Division of Manufacturing Tech jobs to reflect human preferences is a vital step to term export strategy and said it confirmed the capa nology’s Vision Applications Group, who have maximising performance. And in the current eco bilities of local information technology specialists. created novel vision architecture for high speed nomic climate, this is clearly important.” Mr Taylor said that this was the latest and most image processing and feedback systems for robot DPA was developed by the Division’s significant in a string of successes for the two control. Knowledge-based Systems Research Program, companies stemming from the State Government’s For further information contact Mrs Rae using HyperCard and drawing from base career Associated Benefits Scheme. Established in 1985, Robinson, CSIRO Division ofManufacturingTech- preference research of Dr Jim Kable of the the Scheme ensured that companies supplying com nology, Locked Bag 9, Preston Vic 3072, tel (03) International Management Centre, Brisbane, and puter equipment to the WA Government generated 487 9217 or fax (03) 480 0757. Dr Richard Hicks, a psychologist at Queensland work for local firms. University of Technology. Users respond to a The four year contract will give Computer CALL FOR PAPERS PUBLISHED list of about 90 questions and the answers are Power responsibility for developing and maintain Learned Information, organiser and sponsor of automatically analyzed by DPA. For individu ing one of IBM’s existing software products and ONLINE Information 92—the 16th International als, DPA then produces a report which provides supporting its customers world-wide. The com Online Information Meeting, has announced the a profile of the respondent’s personal prefer mercial sensitivity of this work means that few publication of the Meeting’s Call for Papers. ences in the following areas: other details can be released.