Bisimplicial edges in bipartite graphs are closely related to pivots in Gaussian elimination that avoid turning zeroes into non-zeroes. We present a new deterministic algorithm to nd such edges in bipartite graphs. The expected time complexity of our new algorithm is $O(n^2 \log n)$ on random bipartite graphs in which each edge is present with a fixed probability p, a polynomial improvement over the fastest algorithm found in the existing literature

Bomhoff, Matthijs

Manthey, Bodo

English

Bomhoff, M.J.

University of Twente Research Information

Bisimplicial Edges in Bipartite Graphs ?Matthijs Bomhoff ∗ Bodo MantheyFaculty of Electrical Engineering, Mathematics and Computer ScienceUniversity of Twente, P.O. Box 217, 7500 AE Enschede, The NetherlandsAbstractBisimplicial edges in bipartite graphs are closely related to pivots in Gaussian elim-ination that avoid turning zeroes into non-zeroes. We present a new deterministicalgorithm to find such edges in bipartite graphs. The expected time complexity ofour new algorithm is O(n2 log n)on random bipartite graphs in which each edgeis present with a fixed probability p, a polynomial improvement over the fastestalgorithm found in the existing literature.Key words: bipartite graphs, random graphs, algorithms, Gaussian elimination1 IntroductionWhen applying Gaussian elimination to a square n× n matrix M containingsome elements with value zero, the choice of pivots can often determine theamount of zeroes turned into non-zeroes during the process, the so called fill-in.Some matrices even allow Gaussian elimination without any fill-in. Avoidingfill-in has the nice property of bounding the required space for intermediateresults of the Gaussian elimination to the space required for storing the inputmatrix M . This can be important for processing very large sparse matrices.Even when fill-in cannot be completely avoided, it may still be worthwhileto avoid it for several iterations, motivating the search for pivots that avoidfill-in.? This work was supported by the Dutch Innovation Oriented Research Program“Integral Product Creation and Realisation (IOP-IPCR)” of the Dutch Ministry ofEconomic Affairs.∗ Corresponding AuthorEmail addresses: m.j.bomhoff@utwente.nl (Matthijs Bomhoff),b.manthey@utwente.nl (Bodo Manthey).Preprint submitted to CTW 2010 3 March 2010If we assume subtracting a multiple of one row of M from another turns atmost one non-zero into a zero, we may restrict ourselves to considering only{0, 1} matrices. Given such a square matrix M , we can construct the bipartitegraph G[M ] with vertices corresponding to the rows and columns in M , wherevertices i and j are adjacent iff Mi,j is nonzero. The {0, 1} matrices thatallow Gaussian elimination without fill-in correspond to the class of perfectelimination bipartite graphs [1]. Central to the recognition of this class ofgraphs is the notion of a bisimplicial edge that corresponds to an element ofM that can be used as a pivot without causing fill-in. The fastest algorithmfor finding bisimplicial edges in the existing literature has a time complexityequal to that of matrix multiplication [2,3], i.e., O (n2.376) [4]. We present a newdeterministic algorithm for finding a bisimplicial edge in a bipartite graph, ifone exists, and show that its expected time complexity on random bipartitegraphs where each edge is present with some fixed probability p is O (n2 log n).2 Bisimplicial EdgesWe denote by Γ (u) the neighbors of a vertex u.Definition 1 An edge uv of a bipartite graph G = (U, V,E) is called bisim-plicial, if the induced subgraph G[Γ (u) ∪ Γ (v)] is a complete bipartite graph.Clearly, for a given edge uv we can determine in O (|E|) time if it is a bisim-plicial edge by simply checking all edges adjacent to it. So a simple algorithmto find a bisimplicial edge in a bipartite graph G, if one exists, takes O (|E|2)time.Goh and Rotem [2] present a faster algorithm based on the following: A rowMa,∗ is said to majorize a row Mb,∗ if for each 1 ≤ j ≤ n we have Ma,j ≥Mb,j.According to this definition, every row majorizes itself.Theorem 2 [Goh and Rotem [2]] Let M be an n×n {0, 1} matrix representinga bipartite graph G = (U, V,E). Let `i be the number of rows in M thatmajorize row i and let sj be the sum of the entries in column j of M . ThenMi,j = 1 and `i = sj iff the edge uivj is a bisimplicial edge of G.Let Q = MMT , this implies `i is equal to the number of elements in the rowQi,∗ that are equal to Qi,i (including Qi,i itself). Clearly, the time complexityof finding a bisimplicial edge in G[M ] is bounded by the time complexity ofthe matrix multiplication. The fastest currently known algorithm for this hasa time complexity of O (n2.376) [4].2To improve on this, our new approach first selects a set of candidate edges.If a bisimplicial edge exists, then one of our candidates is bisimplicial. Thuswe can restrict ourselves to checking the candidate edges for bisimpliciality.By bounding the number of candidates we achieve an improved expected timecomplexity. The following observation is the basis of our candidate selectionprocedure.Lemma 3 If an edge uv of a bipartite graph G = (U, V,E) is bisimplicial, wemust have δ (u) = minu′∈Γ(v) δ (u′) and δ (v) = minv′∈Γ(u) δ (v′).Translated to the matrix M , this means that if Mi,j = 1, it can only correspondto a bisimplicial edge if row i has a minimal number of ones over all the rowsthat have a 1 in column j and column j has a minimal number of ones over allthe columns having a 1 in row i. In what follows, we will call the row (column)in M with the minimal number of ones over all the rows (columns) in M thesmallest row (column). Using this observation, we construct an algorithm topick candidate edges that may be bisimplicial:Algorithm 1 (1) Determine the row and column sums (ai and bj) for eachrow and column of M .(2) Determine for each row i the index ci of the smallest column with Mi,ci =1 (breaking ties by favoring the lowest index); or ci = 0 if row i has noone.(3) Determine for each column j the index rj of the smallest row with Mrj ,j =1 (breaking ties by favoring the lowest index); or rj = 0 if column j hasno one.(4) Mark Mi,j as a candidate edge if ci = j and rj = i.Clearly, all steps in the algorithm can be performed in O (n2) time. Further-more, the last step will mark at most n candidate edges (and at least 1).Theorem 4 If G = (U, V,E) contains a bisimplicial edge, at least one of thecandidates marked by the algorithm will be bisimplicial.As each of the candidates can subsequently be checked for bisimpliciality inO (n2), we obtain a O (n3) algorithm that finds a single bisimplicial edge in abipartite graph G, if one exists, without using matrix multiplication. By itself,this is not really interesting, as the worst case time complexity is not an im-provement over previously known algorithms. However, for random bipartitegraphs, our new algorithm performs significantly better.33 Asymptotic Expected Behavior on Random GraphsFor a fixed value of p ∈ (0, 1), we consider random bipartite graphs from theGn,n,p model: i.e., we have n vertices in each vertex class, and each edge ispresent with probability p. Such a random graph corresponds to a stochasticn × n {0, 1} matrix M with P [Mi,j = 1] = p. We denote by random variableXi the {0, 1} vector that forms row i of M and use |Xi| to denote the sum ofits elements. If we order the Xi vectors according to the number of ones theycontain (breaking ties by favoring lower values of i), we denote by X(1) therow with the least number of ones, by X(2) the row with the second-to-leastnumber etc.Lemma 5 For any ε > 0 and sufficiently large n, we haveP[|X(1)| < (1− ε)pn]≤ 1n.Lemma 6 For any p′, k with 0 < p′ < p and k ≥ 1 and sufficiently large n,we haveP [Algorithm 1 selects more than k candidates] ≤ n(1− p′)k + 1n.From this, we immediately get a bound on the expected number of candidatesthat our algorithm selects.Theorem 7 For any p′ with 0 < p′ < p and sufficiently large n, we haveE [number of candidates selected by Algorithm 1] ≤ 2 + 2 log(1−p′)1n.Corollary 8 For any fixed p, the expected time complexity of finding a bisim-plicial edge or deciding there is none using Algorithm 1 is O (n2 log n).References[1] Martin Charles Golumbic and Clinton F. Goss. Perfect elimination and chordalbipartite graphs. J. Graph Theory, 2(2):155–163, 1978.[2] L. Goh and D. Rotem. Recognition of perfect elimination bipartite graphs.Inform. Process. Lett., 15(4):179–182, 1982.[3] Jeremy P. Spinrad. Recognizing quasi-triangulated graphs. Discrete Appl. Math.,138(1-2):203–213, 2004.[4] Don Coppersmith and Shmuel Winograd. Matrix multiplication via arithmeticprogressions. J. Symbolic Comput., 9(3):251–280, 1990.4

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