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29
The NPcompleteness column: an ongoing guide
 JOURNAL OF ALGORITHMS
, 1987
"... This is the nineteenth edition of a (usually) quarterly column that covers new developments in the theory of NPcompleteness. The presentation is modeled on that used by M. R. Garey and myself in our book "Computers and Intractability: A Guide to the Theory of NPCompleteness," W. H. Freem ..."
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This is the nineteenth edition of a (usually) quarterly column that covers new developments in the theory of NPcompleteness. The presentation is modeled on that used by M. R. Garey and myself in our book "Computers and Intractability: A Guide to the Theory of NPCompleteness," W. H. Freeman & Co., New York, 1979 (hereinafter referred to as "[G&J]"; previous columns will be referred to by their dates). A background equivalent to that provided by [G&J] is assumed, and, when appropriate, crossreferences will be given to that book and the list of problems (NPcomplete and harder) presented there. Readers who have results they would like mentioned (NPhardness, PSPACEhardness, polynomialtimesolvability, etc.) or open problems they would like publicized, should
Minimum Cuts in NearLinear Time
, 1999
"... We significantly improve known time bounds for solving the minimum cut problem on undirected graphs. We use a "semiduality" between minimum cuts and maximum spanning tree packings combined with our previously developed random sampling techniques. We give a randomized (Monte Carlo) algorit ..."
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Cited by 96 (12 self)
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We significantly improve known time bounds for solving the minimum cut problem on undirected graphs. We use a "semiduality" between minimum cuts and maximum spanning tree packings combined with our previously developed random sampling techniques. We give a randomized (Monte Carlo) algorithm that finds a minimum cut in an medge, nvertex graph with high probability in O(m log³ n) time. We also give a simpler randomized algorithm that finds all minimum cuts with high probability in O(n² log n) time. This variant has an optimal RNC parallelization. Both variants improve on the previous best time bound of O(n² log³ n). Other applications of the treepacking approach are new, nearly tight bounds on the number of near minimum cuts a graph may have and a new data structure for representing them in a spaceefficient manner.
Linear Assignment Problems and Extensions
"... This paper aims at describing the state of the art on linear assignment problems (LAPs). Besides sum LAPs it discusses also problems with other objective functions like the bottleneck LAP, the lexicographic LAP, and the more general algebraic LAP. We consider different aspects of assignment problems ..."
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Cited by 66 (0 self)
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This paper aims at describing the state of the art on linear assignment problems (LAPs). Besides sum LAPs it discusses also problems with other objective functions like the bottleneck LAP, the lexicographic LAP, and the more general algebraic LAP. We consider different aspects of assignment problems, starting with the assignment polytope and the relationship between assignment and matching problems, and focusing then on deterministic and randomized algorithms, parallel approaches, and the asymptotic behaviour. Further, we describe different applications of assignment problems, ranging from the well know personnel assignment or assignment of jobs to parallel machines, to less known applications, e.g. tracking of moving objects in the space. Finally, planar and axial threedimensional assignment problems are considered, and polyhedral results, as well as algorithms for these problems or their special cases are discussed. The paper will appear in the Handbook of Combinatorial Optimization to be published
An Efficient Algorithm to Compute Row and Column Counts for Sparse Cholesky Factorization
 SIAM J. MATRIX ANAL. APPL
, 1994
"... Let an undirected graph G be given, along with a specified depthfirst spanning tree T . We give almostlineartime algorithms to solve the following two problems: First, for every vertex v, compute the number of descendants w of v for which some descendant of w is adjacent (in G) to v. Second, ..."
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Cited by 31 (6 self)
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Let an undirected graph G be given, along with a specified depthfirst spanning tree T . We give almostlineartime algorithms to solve the following two problems: First, for every vertex v, compute the number of descendants w of v for which some descendant of w is adjacent (in G) to v. Second, for every vertex v, compute the number of ancestors of v that are adjacent (in G) to at least one descendant of v. These problems arise in Cholesky and QR factorizations of sparse matrices. Our algorithms can be used to determine the number of nonzero entries in each row and column of the triangular factor of a matrix from the zero/nonzero structure of the matrix. Such a prediction makes storage allocation for sparse matrix factorizations more efficient. Our algorithms run in time linear in the size of the input times a slowlygrowing inverse of Ackermann's function. The best previously known algorithms for these problems ran in time linear in the sum of the nonzero counts, which is...
Rectilinear and Polygonal pPiercing and pCenter Problems
 In Proc. 12th Annu. ACM Sympos. Comput. Geom
, 1996
"... We consider the ppiercing problem, in which we are given a collection of regions, and wish to determine whether there exists a set of p points that intersects each of the given regions. We give linear or nearlinear algorithms for small values of p in cases where the given regions are either axispa ..."
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We consider the ppiercing problem, in which we are given a collection of regions, and wish to determine whether there exists a set of p points that intersects each of the given regions. We give linear or nearlinear algorithms for small values of p in cases where the given regions are either axisparallel rectangles or convex coriented polygons in the plane (i.e., convex polygons with sides from a fixed finite set of directions) . We also investigate the planar rectilinear (and polygonal) pcenter problem, in which we are given a set S of n points in the plane, and wish to find p axisparallel congruent squares (isothetic copies of some given convex polygon, respectively) of smallest possible size whose union covers S. We also study several generalizations of these problems. New results are a lineartime solution for the rectilinear 3center problem (by showing that this problem can be formulated as an LPtype problem and by exhibiting a relation to Helly numbers). We give O(n log n...
On the Common Substring Alignment Problem
"... The Common Substring Alignment Problem is defined as follows: Given a set of one or more strings and a target string. is a common substring of all strings, that is. The goal is to compute the similarity of all strings with, without computing the part of again and again. Using the classical dynamic p ..."
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Cited by 25 (2 self)
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The Common Substring Alignment Problem is defined as follows: Given a set of one or more strings and a target string. is a common substring of all strings, that is. The goal is to compute the similarity of all strings with, without computing the part of again and again. Using the classical dynamic programming tables, each appearance of in a source string would require the computation of all the values in a dynamic programming table of size where is the size of. Here we describe an algorithm which is composed of an encoding stage and an alignment stage. During the first stage, a data structure is constructed which encodes the comparison of with. Then, during the alignment stage, for each comparison of a source with, the precompiled data structure is used to speed up the part of. We show how to reduce the alignment work, for each appearance of the common substring in a source string, to at the cost of encoding work, which is executed only once.
I/Oefficient batched unionfind and its applications to terrain analysis
 IN PROC. 22ND ANNUAL SYMPOSIUM ON COMPUTATIONAL GEOMETRY
, 2006
"... Despite extensive study over the last four decades and numerous applications, no I/Oefficient algorithm is known for the unionfind problem. In this paper we present an I/Oefficient algorithm for the batched (offline) version of the unionfind problem. Given any sequence of N union and find opera ..."
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Despite extensive study over the last four decades and numerous applications, no I/Oefficient algorithm is known for the unionfind problem. In this paper we present an I/Oefficient algorithm for the batched (offline) version of the unionfind problem. Given any sequence of N union and find operations, where each union operation joins two distinct sets, our algorithm uses O(SORT(N)) = O ( N B log M/B N I/Os, where M is the memory size and B is the disk block size. This bound is asymptotically optimal in the worst case. If there are union operations that join a set with itself, our algorithm uses O(SORT(N) + MST(N)) I/Os, where MST(N) is the number of I/Os needed to compute the minimum spanning tree of a graph with N edges. We also describe a simple and practical O(SORT(N) log ( N M))I/O algorithm for this problem, which we have implemented. We are interested in the unionfind problem because of its applications in terrain analysis. A terrain can be abstracted as a height function defined over R2, and many problems that deal with such functions require a unionfind data structure. With the emergence of modern mapping technologies, huge amount of elevation data is being generated that is too large to fit in memory, thus I/Oefficient algorithms are needed to process this data efficiently. In this paper, we study two terrainanalysis problems that benefit from a unionfind data structure: (i) computing topological persistence and (ii) constructing the contour tree. We give the first O(SORT(N))I/O algorithms for these two problems, assuming that the input terrain is represented as a triangular mesh with N vertices. Finally, we report some preliminary experimental results, showing that our algorithms give orderofmagnitude improvement over previous methods on large data sets that do not fit in memory.
Efficient and Practical Modular Decomposition
 EIGHTH ANNUAL ACMSIAM SYMPOSIUM ON DISCRETE ALGORITHMS
, 1997
"... We give a simple recursive algorithm for modular decomposition of undirected graphs that runs in O(n+ma(m;n)) time. Previous algorithms with this bound are of theoretical use only. By adding some data structure tricks, we get a much simpler proof of an O(n+m) bound than was previously available. Key ..."
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Cited by 18 (5 self)
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We give a simple recursive algorithm for modular decomposition of undirected graphs that runs in O(n+ma(m;n)) time. Previous algorithms with this bound are of theoretical use only. By adding some data structure tricks, we get a much simpler proof of an O(n+m) bound than was previously available. Key components of the algorithm are variations of a procedure for finding a depthfirst forest on the complement of a directed graph G in O(n+m) time. This is surprising, given that it takes Ω(n²) time to compute the complement explicitly.
Unique Maximum Matching Algorithms
, 2002
"... We consider the problem of testing the uniqueness of maximum matchings, both in the unweighted and in the weighted case. For the unweighted case, we have two results. First, given a graph with n vertices and m edges, we can test whether the graph has a unique perfect matching, and find it if it exis ..."
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Cited by 17 (0 self)
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We consider the problem of testing the uniqueness of maximum matchings, both in the unweighted and in the weighted case. For the unweighted case, we have two results. First, given a graph with n vertices and m edges, we can test whether the graph has a unique perfect matching, and find it if it exists, in O(m log^4 n) time. This algorithm uses a recent dynamic connectivity algorithm and an old result of Kotzig characterizing unique perfect matchings in terms of bridges. For the special case of...
Greedy Matching Algorithms, an Experimental Study
 Proceedings of the 1 st Workshop on Algorithm Engineering
, 1997
"... We conduct an experimental study of several greedytype algorithms for finding large matchings in graphs. Further we propose a new graph reduction, called kBlock Reduction, and present two novel algorithms using extra heuristics in the matching step and kBlock Reduction for k = 3. Greedy type mat ..."
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We conduct an experimental study of several greedytype algorithms for finding large matchings in graphs. Further we propose a new graph reduction, called kBlock Reduction, and present two novel algorithms using extra heuristics in the matching step and kBlock Reduction for k = 3. Greedy type matching algorithms can be used for finding a good approximation of the maximum matching in a graph G if no exact solution is required, or as a fast preprocessing step to some other matching algorithm. The studied greedytype algorithms run in O(m) and are easy to implement and to prove. Our experiments show that a good greedytype algorithm looses on average at most one edge on random graphs with up to 10,000 vertices. Furthermore the experiments show for which edge densities the maximum matching problem is difficult to solve. 1. Introduction Let G = (V; E) be a graph with vertex set V and edge set E and let n = jV j and m = jEj. A matching is a subset M of the edge set E such that no two ed...