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Some Aspects of Topological Sorting SOME ASPECTS OF TOPOLOGICAL SORTING 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 Mathematics 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 partially ordered set, 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 ideal 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 . 29 Covenant Journal of Informatics and Communication Technology (CJICT) Vol. 2, No. 1, June, 2014. 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. 30 Covenant Journal of Informatics and Communication Technology (CJICT) Vol. 2, No. 1, June, 2014. 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: 31 Covenant Journal of Informatics and Communication Technology (CJICT) Vol. 2, No. 1, June, 2014. 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 . 32 Covenant Journal of Informatics and Communication Technology (CJICT) Vol. 2, No. 1, June, 2014. 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.
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