SOME ASPECTS OF

1Yohanna Tella (M.Sc, Ph.D),

2 D. Singh (M.Sc, Ph.D) & 3J.N. Singh (M.Sc, Ph.D)

1 (Corresponding Author) Department of Mathematical Sciences Kaduna State University, Kaduna, Nigeria. Email: yot2002ng @kasu.edu.ng

2 (Former Professor, Indian Institute of Technology Bombay). Department of Ahmadu Bello University, Zaria, Nigeria. Email: mathdss @yahoo.com

3Department of Mathematics and Computer Science, Barry University, 11300 North East Second Avenue Miami Shores, FL 33161, USA

Abstract: In this paper, we provide an outline of most of the known techniques and principal results pertaining to computing and counting topological sorts, realizers and dimension of a finite , and identify some new directions.

Key words: Partial Ordering, topological sorting, realizer, dimension.

1. Introduction and principal results pertaining to Topological sorting has been found computing and counting topological particularly useful in sorting and sorts, realizers and dimension of a scheduling problems such as PERT partially ordered set (Poset), and charts used to determine an ordering identify some new directions. of tasks, graphics to render objects 2. Definitions of some terms and from back to front to obscure hidden related results pertaining to surfaces, painting when applying partially ordered sets paints on a surface with various We borrow these definitions from parts and identifying errors in DNA various sources (Brualdi et al., fragment assembly (Knuth, 1973; 1992; Jung, 1992; Trotter 1991). Skiena, 1997; Rosen, 1999). In this Let denote a finite partially paper, we provide systematically an ordered set along with an implicit outline of most of the techniques assumption that denotes the 28 Covenant Journal of Informatics and Communication Technology (CJICT) Vol. 2, No. 1, June, 2014. underlying set and denotes its The incidence of a poset , order relation. Moreover, stands denoted , is defined as the set for reflexive partial order and for . strict partial order. A pair is called a

For an element , the set critical pair if is called an in implies in open upset of . The set and is called a in implies in closed upset of . For any nonempty for all . subset , U(A) = {x∈P│a<_P x Also, the set of all critical pairs is a∈A} = U(a∈A)a∈A U(a) and denoted . U[A]={x∈P│a≤x, a∈A}=Ua∈AU[a]. A subset of a poset is

Similarly, the open and closed down called a subposet if the suborder sets can be defined. Note that the defined on , is the restriction closed upset and the closed down set of on pairs of elements of . In are also called filter and other words, a subposet is a subset respectively. of with the induced order. An element is said to cover A chain in is a subposet of an element if and which is a linear order. The length with no element such of a chain of is . An that . Sometimes we antichain in is a subset of also say that is an immediate containing elements that are successor of or is an immediate mutually incomparable. predecessor of . For Two posets and are every , if then the Isomorphic, , if there exists pair is said to be a order preserving bijection comparability of . Two elements such that are called comparable, . denoted or if either A poset of the type or ; and incomparable denoted , if both is called an -poset as its Hasse and . Also, , diagram looks like the letter : iff and .

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c

d

b a

Figure 1

A poset is called N-free if A crown on elements there exists no subposet of is the isomorphic to an -poset. partial order defined: Let and be two For all indices , the elements disjoint posets. The disjoint and are incomparable, the (cardinal) sum is the poset elements and are such that if incomparable, but ; and for and only if and or each , the elements and are and . The linear incomparable. (ordinal) sum is the poset A 4-crown ( or a crown with four such that elements) poset is isomorphic to 2 if and only if or 2 (where 2 is a two elements antichain) and is a series parallel and with preceding . In poset. The N-poset can be described other words, is obtained from as by adding (or 2 2 with one comparability preceding ) for all and missing. For example let . and be the two elements A - antichain is defined as the antichains. It is clear that disjoint union of singletons. A poset is called series- for the poset (by definition). parallel if it can be constructed from The Hasse diagram follows: singletons by using disjoint union and linear sum.

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Figure 2

figure 2 above is an poset Hasse diagram with the comparability missing. Series-parallel posets can also be characterized as N-free posets (Valdes et al.,1982) A cycle is a poset with Hasse diagram in figure 3(a) where

(a) (b)

Figure 3 (a) Cycle (b) Crown

The young’s lattice , where are positive integers, is a poset defined on with the order relation:

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if called topological sorting and the and only if . outcome is called a topological sort The height of a poset , (or linear extension). In other denoted , is defined to be the words, a linear extension of is a cardinality of its longest chain. The linear order which contains . width of , denoted by , Let denote a linear extension of is defined to be the cardinality of its and denotes set of all largest antichain. It is easy, though linear extensions of . is not trivial, to see that the following nonempty for any (Szpilrajn, results hold 1930). That is, every order can be ( Dilworth, 1950; Brualdi et al., extended to a linear order. In fact, a 1992): stronger result has been proved: Let equals the minimum number be an order on and let of chains in a partition of into such that . Then there exist two chains, equals the minimum linear extensions and of number of antichains in a partition such that and . of into antichains, A linear extension is said to reverse the incomparable pair and when in . A family of . linear extensions of reverses A Poset is called width-critical if for every , if and height- there exists some such that critical in . if , for all The dual of a linear extension of a . poset denoted , is a linear It follows that a poset is width- order obtained by reversing the critical if and only if is an order of the linear extension . The antichain and height-critical if and dual of a poset , denoted , only if is totally ordered. is the poset obtained by reversing its

3. Algorithms for constructing order. linear extensions A consecutive pair of 3.1 Definitions of some basic elements in is called a jump or terms and related results setup of in if and pertaining to ordered sets are incomparable in . Let be a finite nonempty We denote the number of jumps of poset. A total ordering is said to in by . The jump be compatible with the partial number of is the ordering if whenever minimum of over all linear . The scheme for extensions of . constructing a compatible total A Poset is called jump critical if ordering from a partial ordering is for each .

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A jump-critical Poset with order , , … , of elements of jump number has atmost such that whenever elements ( El-Zahar & in i.e; precedes in the partial Schmerl, 1984), and there are ordering implies precedes in the precisely jump-critical posets linear extension (Knuth, 1973). The with jump number atmost idea is to pick a minimal element ( El-Zahar & Rival, 1985). It is and then to remove it from the recognized that characterizing jump- poset, and continue the process with critical posets turns out to be a the truncated poset until it gets considerably complicated problem. exhausted. A very fast algorithm Pulleyblank proved that jump and its implementation for number problem viz. schedule the computing a topological sort of a tasks to minimize the number of poset is presented in (Knuth, 1973) jumps is -hard ( Bouchitte & . As a matter of fact, this is a well- Habib, 1987). documented work on sorting. In its It follows from Dilworth’s theorem simplest form, the algorithm for that . If , constructing a total ordering in the then is called a Dilworth finite poset can be depicted poset or simply a - poset . It is as below: shown that a poset which does not Since is finite and have a subposet isomorphic to a nonempty, it has minimal elements. cycle in figure 2(a) is a - poset Let be a minimal element i.e. (Duffus et al., 1982). is not preceded by any other object Syslo (( Bouchitte & Habib, 1987). in the ordering which is put forward a polynomial algorithm chosen first. Again , is to characterize Dilworth posets in also a poset. If it is non-empty, let the case where the antichain of be one of its minimal elements, maximal elements is a maximal- which is chosen next and continue sized antichain. It is observed that the process until no element remains the class of Dilworth posets does not to be further chosen. Since is seem to be nice with respect to finite, this process must terminate computational complexity. and give rise to a sequence of If , then is called elements , , …, along with an optimal linear extension of the desired total ordering defined by . We denote the set of all . The optimal linear extensions of compatibility of the total ordering by . with the original partial ordering

can easily be verified i.e. by the 3.2 Knuth’s (Bucket) sorting definition given above, algorithm Essentially, the topological sort of a for all in the ordering. It needs to be constantly finite partial order is a linear

33 Covenant Journal of Informatics and Communication Technology (CJICT) Vol. 2, No. 1, June, 2014. observed that only if is At this stage, 2 is the only minimal chosen before . element. Next we need to select any minimal element of , . Alternatively, for a subset of , Here, 4 is the only minimal element. we denote the set of minimal Next, as both 8 and 12 are minimal elements of restricted to by elements of , , we select . The algorithm for 12. Finally, 8 is left as the last computing a linear extension element. The outcome is the total of the poset ordering ( Kierstead et al., 1987) is defined: A linear representation of the above can be depicted as follows:

Another compatible total ordering for the same partial ordering may be constructed as follows: . Hence it follows that a compatible total ordering for a given partial ordering For any sequence of choices of the may not be unique. In fact, the size points , the algorithm of the family of linear extensions of produces a linear extension of a poset varies from (if is a ; and every linear extension chain) to ( if is an -element of is obtained from by antichain). Note, however, that it a suitable sequence of choices of may not be even possible to . construct such a total ordering if loops were present. Example 1: Let and the 3.3 Depth first traversal algorithm partial ordering relation be “divides” Another algorithm used for the denoted by . The scheme to find a computation of a topological sort is compatible total ordering for the totally based on depth first traversal poset , may be ( Papamanthou, 2004): outlined as follows: At first stage, 1 is the only minimal (i) Perform DFS to compute the element and hence gets selected. discovery/finishing times Next, we need to select a minimal for each vertex element of , . At representing an element in the this stage, 2 and 3 are the two Hasse diagram of the poset. minimal elements, we select 3. (ii) As each vertex is finished, insert Next, we need to select a minimal it to the front of a linked list element of , .

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(iii) Return the linked list of Example 2: Below is an original vertices graph of the (iv) Output the vertices in reverse poset , and order of finishing time to get the its Depth first search (DFS) forest. topological sort of the poset (Skiena, 1997).

1/12

8.9 10/11 1 1

2 3 2 3

4 6/7 4 8 2/3 12 4/5 12 8

Depth First Search (DFS) forest of poset

Original graph of poset P Figure 4

Final order: .

Note that the DFS could generate other distinct topological sorts using the same vertex .

4. Algorithm for constructing Brualdi et al., 1992) is given as greedy linear extensions follows: A more restrictive class of linear (i) Choose a minimal element of extensions of a poset is obtained by further restricting the (ii) Suppose have been choice of to generate chosen. topological sorts called greedy linear extensions. The algorithm for : If there is at least one minimal computing greedy linear extensions element of of a poset ( Cogis & Habib, 1979;

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which covers then choose to be any such minimal

element; otherwise, choose to be any minimal element of . More precisely, a linear extension of

is greedy if and only if it is obtained from the following algorithm by a suitable sequence of choices of the points ( Kierstead et al., Example 3: Hasse diagram of the 1987): poset , and its corresponding linear extensions:

8 12 12 8 12 8 8 12 12 12 8 3 8 1 12 8 2 3 4 2 3 3 8 4 4 4 2 4 2 2 4 2 4 32 2 1 2 2 12 4 23 2 22 3 2 3 2 2 12 2 1 1 1 1 2 1 2 2 1 2 L7

Poset P Linear extensions of the poset ( )

Figure 5

By definition, L2, L6 and L7 are , but L1 , L3 , L4 and L5 are not greedy linear extensions of the poset greedy.

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Let denote the set of all greedy (i) Choose for any minimal linear extensions of the poset . A element of poset is greedy if (ii) If is super greedy, ; that is, every greedy then choose for any minimal linear extension is optimal. element of Every greedy linear extension is optimal for the jump number on the covering , class of series parallel posets where is the greatest (Cogis & Habib, 1979 ) . Every - subscript, if there exists one; free poset is greedy (Rival, 1986). otherwise choose any minimal An optimal linear extension of element of . Dilworth poset is necessarily greedy Alternatively, a linear extension (Bouchitte &Habib, 1987). of is super greedy if and The Young’s Lattice is only if it is obtained by applying greedy if and only if one of (1) the following algorithm or and (2) and by a suitable is satisfied. Every poset sequence of choices of the points , containing no subposet (Kierstead et al., 1987): isomorphic to figure 2(b) given in section 2, satisfies (El-Zahar & Rival, 1985). A poset is reversible if whenever . A poset is reversible if and only if (Rival &

Zaguia, 1986; Jung, 1992).

WHILE M⋂U(a_k )=∅ AND k≠0 DO 5. Algorithm for constructing SET k=k-1 super greedy ( depth-first greedy (dfgreedy)) linear extensions. , A further restrictive class of linear extensions of a poset is the class of super greedy ( depth-first greedy (dfgreedy)) linear extensions. In example 3, L5, L6 and L7 are super greedy linear extensions. A greedy linear extension of is super greedy if it is The notion of super greedy linear obtained by applying the following extension was introduced in scheme (Bouchitte et al.,1985; (Pretzel, 1985), and studied some of Ducournau & Habib, 1987) : its algorithmic properties studied (Bouchitte et al., 1985) . Every super greedy linear extension is

37 Covenant Journal of Informatics and Communication Technology (CJICT) Vol. 2, No. 1, June, 2014. greedy i.e; , where (v) , denotes the set of all super where denotes the number of greedy linear extensions of a poset all linear extensions of a given Poset (Bouchitte & Habib, 1987). . The for various linear Computational complexity aspect of extensions outlined above can be greedy and super greedy linear computed using the formulae viz; extension construction associated with the jump number has been and studied (Kierstead, 1986).

6. Counting topological sorts,

Dimension, and Realizers of a poset where The following are some established facts in this regard (Trotter, 1991; denote the number of linear Brualdi et al., 1992; Skiena , 1997; extensions, greedy linear extensions Schroder, 2003; Kloch, 2007): and super greedy linear extensions

of a given Poset 6.1 Counting topological sorts respectively. The expressions (i) Posets with no elements have are exactly one linear extension, the null set. also useful for enumerating all the (ii) A Poset that is a chain has just linear extensions of each kind. one linear extension which is itself. Example 4. We enumerate (iii) A Poset that is an antichain of elements has linear extensions of the poset given by the (iv) The number of linear Hasse diagram below: extensions of all other Posets with elements lies between two bounds mentioned in (ii) and (iii).

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: d

c

b y

x a

Figure 6

Let denote a shorthand notation for . We have the following:

.

Corresponding to the linear extensions ,

, respectively.

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corresponding to the greedy linear extensions , , and , respectively.

=1 corresponding to the super geedy linear extensions

and , Poset turns out to be sufficient to respectively. give rise to the original Poset. That is, for any poset , there exists a 6.2 Realizers of a poset finite set of its linear extensions which realizes . This leads to the Szpilrajn [33] proved that any order following definition: relation is the intersection of its linear extensions. In fact, not very If is a family of linear extensions infrequently, the intersection of only of the poset whose a few linear extensions of a given intersection is the order relation ,

40 Covenant Journal of Informatics and Communication Technology (CJICT) Vol. 2, No. 1, June, 2014. then is called a realizer of . If upper bound is given by is a realizer of . Moreover, if , the poset , then for any then . , the set A Poset (of dimension is is a called dimension-critical ( - realizer of the subposet . irreducible) provided The set of all greedy linear . In extensions of a poset is a other words, is -irreducible if it realizer called the greedy realizer has a dimension and the and the family of greedy realizers is removal of any element lowers its nonempty (Bouchitte et al., 1985). dimension. The -irreducible Posets There exists a super greedy realizer have been characterized, but no for every ordered set (Kierstead et characterization is known for the - al.,1987). irreducible posets for . 6.3 Dimension of a poset However, it is known that for each The dimension of a poset , , there exist infinitely many denoted , is defined as the dimension-critical posets of minimum cardinality of a realizer dimension ( Kelley, 1977; Trotter for the poset ( Dushnik & & Moore, 1976). It is shown that Miller, 1941) . In other words, the the computation of dimension itself dimension of a poset is the minimal is an NP-complete problem. In number of its linear extensions particular, it is polynomial time whose intersection is the original solvable if a partial order has poset. That is, is the least dimension atmost , but the case for positive integer for which there having dimension atmost is NP- exists a family complete (Yannakakis, 1982) . of linear extensions of such that However, whether the jump number . It follows is NP-complete for the particular from Dilworth’s theorem that the class of two dimensional posets is dimension of an order never exceeds still an open problem (Bouchitte & Habib, 1987). its width i.e., . The notion of greedy dimension of a Also, it follows from the definition, poset is studied in (Bouchitte et al., that the removal of a point from 1985). It is observed that the cannot increase its dimension but it existence of a greedy realizer and can decrease by atmost one thus of the greedy dimension (Hiraguchi, 1951). Thus we have immediately follows from a result

obtained in (El-Zahar & Rival, for all . -shaped posets are 1985) that for every incomparable -dimensional. In general, for a pair , there exists a greedy poset with , its linear extension with .

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This is proved by demonstrating Removing one point from a poset algorithmically that such a greedy does not increase any of the linear extension exists for every parameters: width, height, jump critical pair. Further, in course of number and dimension. However, it studying the relationship between can decrease each of them by atmost the greedy dimension and the one. ordinary dimension of a poset, the If one comparability pair is removed existence of equality between them from a poset, its result is a poset in for a wide range of posets, including general. However, if only a the N-free posets, two dimensional comparability pair which cannot be posets and distributive lattices has recovered by transitivity is removed, been proved. the result is still a poset. Thus, Following the definition of removing the comparability for a poset , a family of results in a poset if and only if linear extensions of is a realizer covers . of if and only if for every Similarly, the addition of one , there exists some comparability pair does not in such that in , and general results in a poset. hence the dimension of is just the However,if only a comparability least integer for which there exists which does not force other a family of linear extension of comparabilities is added, the result which reverses (Trotter, is again a poset. In other words, the 1991). comparability can be added Furthermore, if , then to with the result being a poset every minimal realizer of is (with exactly one more greedy. comparability) if and only if Since , we note here in implies , and in that implies . Such a pair is called an nonforcing ordered pair holds of ( Rabinovitch & Rival, 1979). (Kierstead & Trotter, 1985). Despite the emergence of consequences that posets exist with Besides a wealth of results related bounded height but arbitrary large to bounds for ordinary and greedy dimension (Trotter, 1991), dimensions of a poset, the best numerous significant contributions possible upper bounds for the super towards characterizing the greedy dimension of a poset dimension parameter of a poset are in terms of and width around (Kelley & Trotter, 1982). , where is a maximal 7. Some future directions antichain has been proved (i) In face of the fact that every (Kierstead et al., 1987). Summarily, poset has a greedy optimal linear we have the following: extension, the characterization for

42 Covenant Journal of Informatics and Communication Technology (CJICT) Vol. 2, No. 1, June, 2014. the existence of non-greedy optimal (v) Studies related to various linear extension of a poset need to concepts described in this paper on a be investigated. finite multiset are yet to be (ii) The choice of some useful conducted ( Anderson, 1987; Girish characterization (say, stability, etc.) & Sunil, 2009). of a poset interms of the size of its (vi) Multiset as a model for multi- realizers versus the size of the class attribute objects used in discovering of its all linear extensions could be intelligent systems, control of Non- investigated further. linear mechanical systems etc. may (iii) Many nice properties of get simplified by using topological realizers are known, but how to sorting (Petrovsky, 1997). compute them effectively needs (vii) Topological sorting can also be further vindication. used in discovering computing (iv) A number of optimization simulators for biological systems problems need to be addressed; for (Krishnamurthy, 2005). example, constructing an efficient (viii) Last but not the least, some algorithm to compute a linear open problems, like extension that minimizes the problem need our attention number of jumps (Bouchitte & (Felsner & Trotter, 1993). Habib, 1987).

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