九州大学学術情報リポジトリ Kyushu University Institutional Repository
Parallel Complexity and P-Complete Problems
Miyano, Satoru Research Institute of Fundamental Information Science Kyushu University
http://hdl.handle.net/2324/3112
出版情報：RIFIS Technical Report. 6, 1988-10-24. Research Institute of Fundamental Information Science, Kyushu University バージョン： 権利関係： cal Report
Parallel Complexity and P-Complete Problems
Satoru Miyano
October 24, 1988
Research Institute of Fundamental Information Science Kyushu University 33 hkuoka812, Japan E-rnaii: [email protected] Phone: 092(641)1101 Ext.4471 PARALLEL GOnllFLEXfTY AND P-COMPLETE, FROBLEhllS
Reseascll Irlstitute of F~~ndamenltalInformation Science Kyushu University 33, Fuli~asla812, Japan
ABSTRACT The ciass NC is a subclass of P but, unfortunately, r~o!iody I:as succecdwl irl proving P#NC althougll it The class IVC consists of probiem solvable by par- is strongly believed that PfNG Like NPfP question. allel algorithms running in time O((log n)v using poly- If P#N@ is approved, a proof that a problem is P- nomial number of processors for some k 2 (3. It is complete iviil be a social proof of nonparallelizability. strongly believed that PfNC. If P#N@ is assumed, This paper gives very general P-complete theorems the P-completeness of a problem implies that no efi- concerning graph algorithms. These theorem yield a cient parallel algorithm exists for the problem. This neiai series of P-cornpiete problems arising from graph paper presents very general P-comp!eteness tileorems optixr~ization probienss including the lexicographically which yield a new series of P-complete psoblenls includ- first ~naxirrlalindependent set problem (Cook 1983). Po- ing the lexicographically first maxima! independent set te~~tially,inftnitely rnanj- nontrivial P-complete prob- problem (Cook 1983). We also give a rneliiod of firding 1c111scalr he derived, parallelisrn in sorrle kind of sequential aigoritfiil~s. This paper is also concerned with a method of find- ing parallelism in problems. Parallelism can be con- sidered in two ways. One is paralleiisrn in existing se- quential algorithms (Kuck 1973). Tlie other is inherent An important role of the parailel complexity theory parallelism in problerns themselves that requires more is to gain insights into inherent parallelism in various mathematical analysis. A method we present in this pa- types of computing problem. It is intended to provide per iocate between these two parallelisms. This method knowledges for answering the following basic questions: is very helpful io show that some problems are in PJC. We exemplify the method through some applications. 1. What kind of problems allow fast eflicient parallel algorithms? 2 EFFICENT PARALLEL ALGORITHhTS 2. What kind of problerns are inherently sequential? AND NC
Namely, it aims at understanding of the range of prob- NC is a mnemonics for Nick's Class (Nick Pippenger lems which allow fast emcient parallel algorithms. Si- 1979) named by S.A. Cook in recognition of his contribu- multaneously, it tries ts capture theoretical limits of tion. Tlre purpose of this section is to convince that NC parallel comput- t'ions. is a reasonable class which provides a yardstick mmsur- ing parallel complexity of problems. It has been observed that there are some probienls which can be solved by easy polynomial ti~riesecjnenlial 2.1 Efliciel~tParallel Algoritllms algorikhnls but do not seem to allow any fast parallel al- gorithms. Recent researches show that these probtems hiany formal parallel cornputation models have been are P-complete. On the otller hand, the class NC, idern- considered (Cook 1981). A broadly accepted model is tified by Pippenger (19791, is understood as the class of the paraliel RAhl (PItAhi) model in which a number problem which allow fast efficient parallel algorithm, of processors work together synchronously and cornmu- nicate with a common random access memory. The 2.2 Uniform Circuits and Parallel RAM PRAM model is further classified according to read and The class NC is defined by the uniform circuit model write access abilities as CREW PRAM (Concurrent Read that is suited for classifying problems precisely. Exclusive Write), EREW PRAM and CRCW PRAM. a. n n Problems for which we consider parailel complexity Defirlition A circuit wit11 inputs and outputs is a finite labelled directed acyclic graph satisfying the fol- are formulated as follows: lowings: There are exactly n nodes of indegree 0 called Definition A problem (or search problem) S with a size inputs which are labelled with XI,..., zn, respectively. parameter h(n) is a family (Sn),ll of relations S, C The nodes of indegree 2 (resp. indegree 1) are labelled (0,l)" x (0,13"") for n 2 1. For n 2 1, s E (0, 11, is with either V (or) or i\ (and) (resp. 7 (not)). Exactly called an instance and an object y E (0, I)~(")satisfying m nodes labelled with yl,..., y, are specified as outputs. S,(x, y), if any, is called a solution for z. We denote by size(a) (resp. depi'h(a)) the number of node in rr (resp. the length of the longest path from Example 2.1 A nasirna! independent set (XIIS) of an some input to some output). undirected graph is a maximal set U of vertices such that no two vertices in CT are adjacent. The problem of find- Definition We say that a circuit family {a,),>l- with ing a MIS is formulated in the following way: .A graph output size h(n) compute a function J = (Jn)nll, if with n vertices is represented by an n x n-adjacency ma- each a, computes fn, where a, is a circuit, with n inputs trix and a subset of vertices is represented by an n-bit and h(n) outputs and j, : (O,1jn -+ {O, l)'In). For a vector, Then MIS=(MTS,) is defined only for Integers search problem S = (S,),>1, wesay that solves of the form n2 as S if the function {fn],zl- computed by {an),LI satisfies MIS,? (0,1)"' x (0, l)", S,(s,f,fx)) for all n 2 1. where for (s,yj EMIS,=, 3; is a symmetric matrix repre- Several kinds of uniformities of circuits have been senting an undirected graph with n vertices and y a bit proposed (Ruzzo 1981, Cook 1951, 1983, 1985). One of vector representing a MIS in the graph.
Definition A parallel algorithm on a PRAM solving a problem is efficient if, given an input of size n, it runs
(1) in time O((log njk) for some constant k 2 0,
(2) with a polynomial number of processors.
This definition is based on the observation that the time Ofjlog n)k)is very fast and a polynomial number of pro- cessors is feasible.
The PRAM model may not seem very realistic in the practical sense. A more realistic model is, for example, a network model. But there are some works showing that PRAM is a good model. For example, Alt et al. Fig. 2.1 Overview of Complexity Classes (1987) showed that the shared memory of any EREW PRAM with n processors and m cells of shared mem- them defines that a circuit family (a,),2l is "uniform" ory can be emulated by an n-processor module parallel (called log uniform) if, given 12 in unary, the description computer with a slow down of O(log nz). Ranade (1987) of the nth circuit a, is log space computable" But NC proved that one step of an n-processor CRCW PRAM defined below does not change by the choice of unifor- can be emulated on an n-processor FFT network in time mity (Ruzzo 1981)- The location of NC is shown in O(Log n) with high probability, using an FIFO queue of Fig. 2.1 together with its neighbors. size 0(1) at each node. Hence the gap is only O(lognj 'A function f is log space computable if it is computed by while keeping the number of processors. a deterministic off-line Turing machine using O(logn3 worktape space. Definition the hardness of parallelization.
(1) NCk=(S 1 S is solvable by a uniform circuit The knowledge that a problem is P-complete pro- family such that sizejru,) vides algorithm designers valuable information about is boilnded by some polynomiai the approaches they should choose. It would relieve and simultaneously depih(ccn) = wasting efforts for devising drastically fast paraliel ai- ?zIk)). Q(Qog gorithms and, instead, direct toward ways which lead to useful algorithnw. Proofs of P-completeness may also tell us which parts of problems are hard to parallelize. From the above arguments and the following result, we see that NC is a reasonable class. The P-completeness due to Cook (1985) adopted the NC1-reducibility that uses NC1-computable O(1og n) Theorem 2.1 (Stockmeyer and Vishkin 1984) depth uniform circuits with oracle gates. However, we use an extended version of the many-one log space re- ducibility (Hopcroft and Ull~nan1979) instead of the where CRCW-PRAM(~~(",O((log n)O("))) is the class NC1-reducibility because of the following reasons. It of problems solvable on a CRCW PRAM with polyno- is easier to find required log space computable functions mial number of processors in time O(jlogn)k) for some which are also computable in NC2. Above all, all known constant k 2 O . P-completeness of natural problems can also be showr, via log space reductions. In the arguments above, we have ignored constants appearing in 0-notation and the degree of poiynomi- Definition Let S and T be problems of size param- als. In parallel computations, constants affect very seri- eters t~(123 and h(nj, respectively. We say that S is ously and we should he more careful about these things. Zog space reducible to T, denoted S ~"gT, if there For example, the Batcher's sorting network is of depth are log space computable functions f = (fn),>l and 1/2iognjl+ logn). On the other hand, the AKS sort- g = {,on)nL1with f, : (0,I)" --, {0,1)~(~)and g, : ing network inlproved by Cole and 6'~~nlaing(1986) is (0, lIn 4 (0,I]~(") such that for every x E (0, the still of depth about 200. log n. It should be noted that following statement holds: 200 > log n even if n is the total number of atoms in the solar system.
However, the class NC and the hierarchy within it If S No mathematical proof has been given showing that (1) The relation "g is transitive. P#NC. Even a proof separating NP from NC1 has not (2) IfS siUg T and T ENC, then S ENC. been known, The best result known is that NCO#NC1 (Furst et al. 4981). There is, however, a strong belief Definition A problem 3 is said to be P-complete if that P#NC. the following conditions are satisfied: Assuming that PfNC, we can prove that no P- (1) 3 is in P. complete problem allows any parailel algorithm running (2) For each problem S in P, S In this section we prove very general P-completeness For each such property T, LFMsp(71.1 can be com- theorems that cover many problems solvable by polyno- puted in polynomial time by the greedy algorithm in mial time greedy algorithms. These are the first results Algorithm 4.1 since 7i is polynomial time testable. in this direction. Our main results are the following theorems begin u + &i; for i t 1 to n if G[U 9 (i)]satisfies a then U +- U U (i] end for end Algorithm 4.1 Algorithm for LFMSP(a) Fig, 4.1 Theorem 4.3 Let a be a polynorrial time testable nontrivial property on undirected graphs which is on sorted lists of positive integers. Let c~ be any cut- hereditary on induced su bgraphs. Then LFhfSP(x) is point with cr~= acH,~. If N is biconnected, we defined P-complete. a~ = (!HI)and CH be any vertex. For a graph G with connected components GI, ...; Gt, the ,&sequence of Theorem 4,2 Let a be a polynomial time testable non- PG trivial property- on directed graphs which is hereditary G is ja~,, ..., a~,),where aa, > . . . > a~,.It should be noticed that any set of P-sequences has a minimum on induced subgraphs. Then LFMSP(a) is P-complete. since iists are sorted. Theorem 4.3 Let 7~ be a hereditary polynomial time testabie property which is nontrivial on undirected p4a- Notation For subsets t;,& of a linearly ordered set vl vz vl nar bipartite graphs and satisfied by all independent V,we denote VI < 'ti if < for any 6 Vland any vz edges. Then LFAISP(r) restricted to planar bipartite E t;. graphs is P-complete. Proof of Theorem 4.1 As shown in Lewis and Yan- nakakis (1980), if a property r is nontrivial and hered- 4.3 Reducing LFMIS itary, it follows from Ramsey's Theorem that either a is satisfied by all independent sets of vertices or a is As a basis of reduction we use the following lemma satisfied by ail cliques. (Miyano 1987). Since a is nontrivial, there exists an undirected graph Lemma 4.4 The following problems are P-complete. J such that pJ = rnin{,& 1 G is an undirected graph vi- (1) LFMIS restricted to planar graphs of degree 3. olating T). Let J1,..., Jt be the connected components of J that are sorted as aj, . . . 2 a~,, Hence the j3- (2) LFMIS restricted to bipartite graphs of degree 3. > sequence of J is (aJ,, ..., CYJ~).Let c be c~ and ;lei be the (3) The lfm subgraph of maximum degree one prob- largest connected component of J1 relative to c. lem restricted to planar bipartite graphs of degree 3. Case 1. x is satisfied by all independent sets: Since Let H be a connected undirected graph. A vertex c x is hereditary and satisfied by all independent sets, lo is called a cutpoint of H if deletion of c from N separates must contain an edge. Therefore there is a vertex d of the graph into at least two connected components. A lowith d # c. Let Il be the graph obtained by deleting subgraph consisting of a resulting connected component except c from J1 (Fig 4.1 (a)). together with c and the edges joining c and the compo- nent is called a component relative to c. A connected We reduce LFMIS to the problem. For an undi- graph without any cutpoint is called biconnected. rected graph G = (V,/,E), we construct a graph = (T/, 6) as follows (Fig, 4.1 (b)): For each vertex u of G, For a connected graph PI, we define the a-sequence a copy of Il is attached by identifying u with c. Then CYHof N in the following way. If N is not biconnected, let each edge {u; u) in E is replaced by a copy of do by iden- c be any cutpoint of H and let HI, ..., Hjlc)be connecked tifying u (resp. vj with c (resp. d). Finally, independent components relative to c. Then acv~= (iN11, ..., INj(,)\),graphs J2,..., Jt are added. it foliows from the choice of where lH;I represents the number of vertices in N;and J that for any independent set U of G the induced sub- we assume /HI/> 0.. > IfIj-ij(ct!.Then cr~is defined graph of (V - V) U U has a P-sequence smaller than ,BJ by a~ = rnin(aCvHj c is a cutpoint of N), where rnin and for each edge (u, v) in E the induced subgraph of is the minimum with respect to the lexicographic order (p-V)il(u, v) violates a. Therefore we define an order on so that - V < V, the order on V is the same as G and the order on - V is arbitrary. With this or- Let C7 be the Ifm independent set of G and be der, first all verticm in - V,which were newly added, the lfrn subset of VDwhose induced subgraph satisfies K. are chosen. Then the vertices in the lfm independent Then by induction we shall show the following reiation. set of G are cfzosen according to the order on V. Thus the ifm independent set U of G and the Ifm subset 0 (I) Eio = fu~,..., u,) U (up;I i E U, 1 I p < k). of V whose induced subgraph satisfies sr are related as 0 = (6,- V) u U. Since the induced subgraph of (vll, %I, ..., vkl) is the c.a.t. directed graph of size k, it follows from the choice Case 2. sr is satisfied by all cliques: This case can be of s and C that the vertices ul , ..., u, and v13, v21, ..., ulil solved in the same way as Case 1 by considering com- can be chosen into UD. Let ~2'= br~ I? ((ul, ..., u,) U plement graphs.U (up;11 < i< j, 1 'A collection of disjoint edges is called rndeyendeni edges, begin N(i)= (j 1 (i, j) E E and j < i). The rules are defined for k = l to 7 log(size(&j) by pardo for each pi,..., ,On+ a E a (I) j + -i for each j E Nji), if {PI, ..., Pn) CTH(Q) then TN(Q) +TM($) i_J {a] (2) 7j1,..., ~j,+ i, where N(i)= (jl,..., ji). end pardo It is easy to see that vertex i is in the lfm indepen- end for dent set if and only if i is a theorem of the inference end system, Further, the resulting inference system has the Algorithm 5.1 unique path property, Hence the problern is solvable on CREW PRAM in time O(iog n). Froposidions 5.1, 5.2 guarantee that Algorithm 5.1 on a CRClV PRAM computes TH(Q) with polyno- Example 5.2 The lexicographically first maximal tri- mial number of processors although the polynomial is angle free edge-induced subgraph problem is, given a of rather large degree. Possibly, size(Q) is exponential graph G = (If,E) with a linear order on E, to find the with respect to \Xi. However, if size(&>is polynomially Ifm edge set F 2 E such thak the graph formed by F bounded, it runs in O(logn) time. This observation is contains no cycle of length three. This problem is P- also found in Ullman and van Gelder (1986). complete for graphs with degree 6 (Miyano 1987). Wow- ever, it can be shown by constructing an inference sys- Definition We say that a family (Q, = (X,,R,)) of tem from a graph that the problem restricted to outer- inference systems has polynomzai saze proof trees if there planar graphs is parallelized by Theorem 5.4 (1) (Miyano is a polynomial p(n) such that every theorem 7 of Q, has 1988a). a proof tree T,(T)for 7 in Q, with szze(T(-y)) 5 p(iX,/). Example 5.3 2-satisfiability problem (2SAT) is also For Q = (X,R), we define a directed graph DG = solved on a CRCW PRAhf in O(lognj time. Let S = (X,A) by setting A = ((z, y) 1 (5,y) E TN(~)).We ((~~I~PI),(~Z,P~I,..., (~m9Pm)) be a set of clauses of say that Q has tEle unique paih property if DG is uni- size two, where a,,p, are in {xi,-al, ..., x,, ~e,].The connected, that is, there is at most one directed path problem is to decide whether S is satisfiable. For S,we between any pair of distinct vertices. construct an inference system Qs = (Xs, Rs) as foilows: Theorem 5.4 (Rytter 1985) Let {Qi = (Xi,R;)) be a family of inference systems with polynomial size proof trees. Let n = jX;/. The rules of Qs are (1) THjQ,) can be computed on a CRC3V PRAM j (oi, for i = 1, m with polj~norniainumber of processors in O(iog n) time. a,) ..., 1,Y (a, r) (2) If the unique path property is satisfied, then {PI>f7/3>7) =+ 1~1 TH(Qi) can be computed on a CREW PRAM r.iith {PI,(-PI =+ polynomiai number of processors in O(1og n) time. where cr,@,r are literals in (el, 7x1, ..., z,, 7~~). 5.3 Applications It is known (Jones et al. 1976) that S is nod satisfi- able if and only if there is a sequence of literals %, ..., 7;; An interesting feature of Theorem 5.4 is that it such that (1) holds, and either (2) or (3) holds. gives a method of transforming sequentiai algorithms to parallel algorithms. We show some applications of the (1) {-7i,-ti+l) is in S for a11 i = 1, ~..?k - 1. theorem. Example 5.1 The Ifm independent set problem is (3) yl = ~yj= yk for some 1 < j < k known P-complete but the problem can be parallelized by Theorem 5.4 (2) if the instances are restricted to Then it can be shown that D is in theorem(Qs) if and forests. For a forest G = (V3E) with V = (1, ...: n), we only if S is not satisfiable. Moreover, it is not hard to construct an inference system as follows: For i E V,Let show that size(Qs) is linear with respect to m, Exanlpie 5.4 The class A~x'DPDA(~'('),log n) of by an example. Let G be an unambiguous context-free functions computable by deterministic auxiliary push- grammar with productions down machines ffiopcroft and Ullrnan 1979) which run s 4 ss, s -+ fs), s -+ (1, in polynomial time using O(log n) worktape space con- tains many important problems, for example, sorting, where 'Y'' and lL)" are terminal symbols. For a string pattern matching, parsing, etc. UP = 21 . . zn of terminal symbols, we define an inference system Q, = (X,, R,) by setting X, = (S[i, jj 1 0 < We show by constructing inference systems that prob- i < j < n, j - i is even), R, consists of the following lem in this class are computable on a GREW PRAM rules: with polynomial number of processors in O(1og n) time. This class is known to be contained in NC2 (Ruzzo 1980, 1981). (2) S[i,j] =+ S[i - 1, j + 11 if xi = ( and xj+l = ). Assume that an auxiliary pushdown machine 114 runs in polynomial time, say p(tz), using O(1ognj worktape space. A su7;face configuration C = (],A)of Al on an Then SI0, n] E TN(Q,) 5' J* w and the family input 3: of length n consists of the top symbol A of the (Q,) satisfies the conditions of Theorem 5.4 (2). pushdown store and a configuration d except the push- down store describing the current state, the input head position, and the contents of the wrorktape together with 6 CONCLUSION the worktape head location. For each z,we construct an inference system as fol- Given a problem in P, there may be two ways to lows: Let 6,C', D, I)', E be surface configurations of 114 go. The first reaches P-completeness. The other falls on 2. For an indexed pair (C,D); of surface configu- into NC. In these respects, this paper contributed in rations, it means that M on a: can move from C to D the following points: keeping the stack height at least h. The rules are defined as follows: 1. Very general P-cornpIeteness theorem are found that derives a new series of P-complete problems that are solvable by simple greedy algorithms. 2. By using inference systems, a method of finding parallelism in sequential algorithm is introduced. where it is assumed in 3 that, 6" C by push(i4) and 1) t- I?' by pp(A) for some A, and the lower index h is While the search for a really efficient and exact al- between i and p(nj and the upper index k is between O gorithm is always expected, the P-completeness would and p(n), light up the way to go. Also, a variety of parallelizing If the pair (Co,Do): of the initial and fina! surface techniques should be searched in detail. configurations is a theorem of the inference system for some k, then M accepts x, and vice versa. We can show that the above family of inference systems indexed by REFERENCES inputs has polynomial size proof trees and the unique path property. Alt, H.: Kagerup, T., Mehlhorn, X. and Freparata, F.F. (19871, Deterministic simulation of idealized parallel com- hlloreover, if M is nondeterministic, the unique path puters on more realistic ones, SAM J. Cornput. 16, 808- property may be lost but it still has polynomial size 835. proof trees. Therefore Theorem 5.4 (1) is applicable. Anderson, R. and h'iayr, E.W. (19841, Parallelism and greedy algoritlirns, Report No. STAN-CS-84-1003, Department of Example 5.5 Unambiguous context-free languages are Computer Science, Stanford University. recognizable In time O(log n) on CREW PRAM (Rytter 1985). This result can be also proved by directly con- Anderson, It. and hfayr, E.W. (1987), Parallelism and the structing inference systems. We show the construction maximal path problem, Inf. Process. Lett. 24, 121-126. Asano, T. and Ilirata, T. (19821, Edge-deletion and edge- contraction problems, Proc. 14th ACAl STOC, 245-254. mal independent set problem, Proc, 17th ACAf STOC, 1-10. 270-295. Miyano, S. (1987), The lexicographically first maximal sub- Cole, R.J. and i"'l)bnlaing, C. (19861, Note on the AKS graph problems: P-completeness and NC algorithms, Proc, sorting network, Uitracomputer Note No. 109, Courant In- 13th ICALP (Lecture Notes in Computer Science 2871, 425- stitute of hlatliernatical Sciences, New York University. 434. Cook, S.A. (19741, An observation on time-storage trade off, Miyano, S. (1988a), A parallelizable iexicographicaiiy first J. Cornput. System Scz. 9, 308-316. maximal edge-induced stibgraph problem, Inf. Process. Lett. 27, 75-78, Cook. S.A, (1981), Towards a complexity theory of syn- chronous parallel computation, Enseign. Math. 27, 99-124. Miyano, S. (1988b1, A:-complete lexicographically first max- imal subgraph problems, Proc. 13th MFCS (Lecture Notes Cook, S.A. (19831, The classification of problems which have in Computer Science 317),454-462. fast parallel algorithms, Proc. 1983 International FCT Con- ference (Lecture Notes in Computer Science 758), 78-93. Miyano, S., Shiraishi, S. and Shoudai, T. (19881, A list of P-complete problems, in preparation, Cook, S.A. (1985), A taxonomy of problems with fast par- allel algorithms, Inform. and Control 64, 2-22. Pippenger, N. (19791, On simultaneous resource bounds, Proc. 20th IEEE FOCS, 307-311. Dobkin, D., Lipton, R.J. and Reiss, S. (19791, Linear pre gramming is log-space hard for P, Inf. Process, Lett. 9, Ranade, A.G. (1987), Wow to emulate shared memory, Prm. 96-97. 28th IEEE FOCS, 185.194. Dwork, D., ICa~ellakis,P.C. and Michel!, J.C. (1984). On Reif, J.11. (1985), Depth-first search is inherently sequential, the sequential nature of unification, J. Logic Progmrnming In$ PI-ocess. Lett. 20, 229-234. 1, 35-50. Ruzzo, W.L. (19801, Tree-size bounded alternation, J. Corn- Furst, M., Saxe. J.B. and Sipser, hi. (1981). Parity, circuits, put. System Sci. 21, 218-235. and polynomial-time hierarchy, Proc. d2nd IEEE FOCS, 260-270. Ruzzo, W.L. (1981), On uniform circuit complexity, 9. Corn- put. System Sci. 22, 365-383. Goldschlager, L.hil, (1977), The monotone and planar circuit value problems are log space complete for P, SIGACT News Rytter, W, (49851, Parallel time O(logn) recognition of un- 8,25-29. ambiguous cfi's, Proc. Fundamentals oJ Comp?mtation The- ory (Lecture Notes in Computer Science 1991, 380-389. Goldschlager, L.M., Shaw, R.A. and Staples, 3. (19821, The maximum flow problem is log-space complete for P, Theorei. Shiloach, Y. and Vishkin, TJ. (1982), An O(nZlogn) parallel Compul. Sci. 21, 105-111. LIAX-FLOW algorithm, J. Algorithms 3, 128-146. IIopcroft, J.E. and Ullman, J.D. (19791, Intrductz'on to Stockmeyer, L. and Vishkin, U. (19841, Simulation of par- Automata Theory, Languages, and Computation, Addison- allel random access machines by circuits, SIAM J. Comput. Wesley Publishing Company. 43, 409-422. Jones, N.D., Lien, Y.E. and Laaser, W.T. (19761, New prob- Ullman, J.D. and van Gelder, A. (1986), Parallel complexity lems complete for nondeterministic log space, Math. Sys- of logical query programs, Prm. 27th IFEE FOGS, 438-454. terns Theory 10, 1-17. Vaiiant, L.G. (1982), Parallel computation, Proc. 7th IBM Jones, N.D. and Laaser, W.T. (1977), Complete problems Symp. on Mathematical Foundations of Computer Science, for deterministic polynomial time, Theoxt. Cornput. Sei. 172-189. 3, 105-117. Watanabe, T., Ae, T. and Nakamura, A. (1081), On the Icuck, D.J. (1977), A survey of parallel machine organization removal of forbidden graphs by edge-deletion or by edge- and programming, Comput. Survey 9, 29-59. contraction, Discrete Appl. Math. 3, 151-153. Ladner, R.E. (1977), The circuit value problem is log space Yannakakis, M. (19811, Node-deletion problems on bipartite complete for P, SlGACT News 7, 583-.590. graphs, SIAM 9. Comput. 10, 310-327. Lewis, J.M. and Yannakakis, hl, (1980), The node deletion Yasnura, Ei. (19841, On parallel computational complexity problems for hereditary properties is NP-complete, J. Corn- of unification, Proc. FGCS'84, 235-243. put. System Sci. 20, 219-230. Luhy, M. (19851, A simple parallel algorithm for the maxi-