Results 1  10
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20
A general approximation technique for constrained forest problems
 in Proceedings of the 3rd Annual ACMSIAM Symposium on Discrete Algorithms
, 1992
"... Abstract. We present a general approximation technique for a large class of graph problems. Our technique mostly applies to problems of covering, at minimum cost, the vertices of a graph with trees, cycles, or paths satisfying certain requirements. In particular, many basic combinatorial optimizatio ..."
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Cited by 355 (21 self)
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Abstract. We present a general approximation technique for a large class of graph problems. Our technique mostly applies to problems of covering, at minimum cost, the vertices of a graph with trees, cycles, or paths satisfying certain requirements. In particular, many basic combinatorial optimization problems fit in this framework, including the shortest path, minimumcost spanning tree, minimumweight perfect matching, traveling salesman, and Steiner tree problems. Our technique produces approximation algorithms that run in O(n log n) time and come within a factor of 2 of optimal for most of these problems. For instance, we obtain a 2approximation algorithm for the minimumweight perfect matching problem under the triangle inequality. Our running time of O(n log n) time compares favorably with the best strongly polynomial exact algorithms running in O(n 3) time for dense graphs. A similar result is obtained for the 2matching problem and its variants. We also derive the first approximation algorithms for many NPcomplete problems, including the nonfixed pointtopoint connection problem, the exact path partitioning problem, and complex locationdesign problems. Moreover, for the prizecollecting traveling salesman or Steiner tree problems, we obtain 2approximation algorithms, therefore improving the previously bestknown performance guarantees of 2.5 and 3, respectively [Math. Programming, 59 (1993), pp. 413420].
Nonlinearity of DavenportSchinzel sequences and of generalized path compression schemes
 Combinatorica
, 1986
"... DavenportSchinzel sequences are sequences that do not contain forbidden subsequences of alternating symbols. They arise in the computation of the envelope of a set of functions. We show that the maximal length of a DavenportSchinzel sequence composed of n symbols is 6(noc(n»), where t1.(n)is the f ..."
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Cited by 108 (17 self)
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DavenportSchinzel sequences are sequences that do not contain forbidden subsequences of alternating symbols. They arise in the computation of the envelope of a set of functions. We show that the maximal length of a DavenportSchinzel sequence composed of n symbols is 6(noc(n»), where t1.(n)is the functional inverse of Ackermann's function, and is thus very slowly increasing to infinity. This is achieved by establishing an equivalence between such sequences and generalized path compression schemes on rooted trees, and then by analyzing these schemes. 1.
A THEORY OF ALTERNATING PATHS AND BLOSSOMS FOR PROVING CORRECTNESS OF THE O(√VE) GENERAL GRAPH MAXIMUM MATCHING ALGORITHM
, 1994
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Improved shortest paths on the word RAM
 in: 27th Colloquium on Automata, Languages and Programming (ICALP), in: Lecture Notes in Comput. Sci
"... Abstract. Thorup recently showed that singlesource shortestpaths problems in undirected networks with n vertices, m edges, and edge weights drawn from {0,...,2 w − 1} can be solved in O(n + m) time and space on a unitcost randomaccess machine with a word length of w bits. His algorithm works by ..."
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Cited by 24 (0 self)
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Abstract. Thorup recently showed that singlesource shortestpaths problems in undirected networks with n vertices, m edges, and edge weights drawn from {0,...,2 w − 1} can be solved in O(n + m) time and space on a unitcost randomaccess machine with a word length of w bits. His algorithm works by traversing a socalled component tree. Two new related results are provided here. First, and most importantly, Thorup’s approach is generalized from undirected to directed networks. The resulting time bound, O(n + m log w), is the best deterministic linearspace bound known for sparse networks unless w is superpolynomial in log n. As an application, allpairs shortestpaths problems in directed networks with n vertices, m edges, and edge weights in {−2 w,...,2 w} can be solved in O(nm + n 2 log log n) time and O(n + m) space (not counting the output space). Second, it is shown that the component tree for an undirected network can be constructed in deterministic linear time and space with a simple algorithm, to be contrasted with a complicated and impractical solution suggested by Thorup. Another contribution of the present paper is a greatly simplified view of the principles underlying algorithms based on component trees. 1
Fast Congruence Closure and Extensions
, 2006
"... Congruence closure algorithms for deduction in ground equational theories are ubiquitous in many (semi)decision procedures used for verification and automated deduction. In many of these applications one needs an incremental algorithm that is moreover capable of recovering, among the thousands of i ..."
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Cited by 18 (1 self)
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Congruence closure algorithms for deduction in ground equational theories are ubiquitous in many (semi)decision procedures used for verification and automated deduction. In many of these applications one needs an incremental algorithm that is moreover capable of recovering, among the thousands of input equations, the small subset that explains the equivalence of a given pair of terms. In this paper we present an algorithm satisfying all these requirements. First, building on ideas from abstract congruence closure algorithms [Kapur (1997,RTA), Bachmair & Tiwari (2000,CADE)], we present a very simple and clean incremental congruence closure algorithm and show that it runs in the best known time O(n log n). After that, we introduce a proofproducing unionfind data structure that is then used for extending our congruence closure algorithm, without increasing the overall O(n log n) time, in order to produce a kstep explanation for a given equation in almost optimal time (quasilinear in k). Finally, we show that the previous algorithms can be smoothly extended, while still obtaining the same asymptotic time bounds, in order to support the interpreted functions symbols successor and predecessor, which have been shown to be very useful in applications such as microprocessor verification.
The Travelling Salesman Problem and Minimum Matching in the Unit Square
, 1983
"... We show that the cost (length) Of the shortest traveling salesman tour through n points in the unit square is, in the worst case, aopt v/n + o (x/n), where 1.075 atsPopt < = 1.414. The cost of the minimum matching of n points in the unit square is shown to be, in the worst case, a opt 4 + O(4), wh ..."
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Cited by 12 (0 self)
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We show that the cost (length) Of the shortest traveling salesman tour through n points in the unit square is, in the worst case, aopt v/n + o (x/n), where 1.075 atsPopt < = 1.414. The cost of the minimum matching of n points in the unit square is shown to be, in the worst case, a opt 4 + O(4), where mat 0.537 opt <0.707 Furthermore, for each of these two problems there is an almost linear time heuristic algorithm whose worst case cost is, neglecting lower order terms, as low as possible.
Efficient Algorithms for the Domination Problems on Interval and CircularArc Graphs
 SIAM J. Comput
, 1998
"... Abstract. This paper first presents a unified approach to design efficient algorithms for the weighted domination problem and its three variants, i.e., the weighted independent, connected, and total domination problems, on interval graphs. Given an interval model with endpoints sorted, these algorit ..."
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Cited by 9 (1 self)
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Abstract. This paper first presents a unified approach to design efficient algorithms for the weighted domination problem and its three variants, i.e., the weighted independent, connected, and total domination problems, on interval graphs. Given an interval model with endpoints sorted, these algorithms run in time O(n) orO(n log log n) where n is the number of vertices. The results are then extended to solve the same problems on circulararc graphs in O(n + m) time where m is the number of edges of the input graph.
Efficient Sequential and Parallel Algorithms for Maximal Bipartite Sets
 Journal of Algorithms
, 1993
"... A maximal bipartite set (MBS) in an undirected graph G = (V; E) is a maximal collection of vertices B ` V whose induced subgraph is bipartite. In this paper we present efficient sequential (linear time) and parallel (NC) algorithms for constructing an MBS. 1 Introduction In the last few years sever ..."
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Cited by 7 (0 self)
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A maximal bipartite set (MBS) in an undirected graph G = (V; E) is a maximal collection of vertices B ` V whose induced subgraph is bipartite. In this paper we present efficient sequential (linear time) and parallel (NC) algorithms for constructing an MBS. 1 Introduction In the last few years several efficient parallel algorithms have appeared for the maximal independent set (MIS) problem [GS, KW, Luby]. This may be thought of as the problem of finding a maximal 1colorable set in a given graph. In this paper we consider the natural extension to finding a maximal 2colorable set. Supported by a General Electric Foundation Graduate Fellowship y Supported in part by NSF Grant DCR 8552938 and PYI matching funds from AT&T Bell Laboratories and Sun Microsystems Inc. at Cornell University. A maximal bipartite set (MBS) in an undirected graph G = (V; E) is a maximal collection of vertices B such that the subgraph induced on B is bipartite. We give a lineartime sequential algorithm f...
Optimizing Connected Component Labeling Algorithms
"... This paper presents two new strategies that can be used to greatly improve the speed of connected component labeling algorithms. To assign a label to a new object, most labeling algorithms use a scanning step that examines some of its neighbors. The first strategy exploits the dependencies among the ..."
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Cited by 7 (1 self)
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This paper presents two new strategies that can be used to greatly improve the speed of connected component labeling algorithms. To assign a label to a new object, most labeling algorithms use a scanning step that examines some of its neighbors. The first strategy exploits the dependencies among the neighbors to reduce the number of neighbors examined. When considering 8connected components in a 2D image, this can reduce the number of neighbors examined from four to one in many cases. The second strategy uses an array to store the equivalence information among the labels. This replaces the pointer based rooted trees used to store the same equivalence information. It reduces the memory required and also produces consecutive final labels. Using an array based instead of the pointer based rooted trees speeds up the connected component labeling algorithms by a factor of 5 # 100 in our tests on random binary images.
Private communication
, 1995
"... Given a graph G and positive integers b and w, the blackandwhite coloring problem asks about the existence of a partial vertexcoloring of G, with b vertices colored black and w white, such that there is no edge between a black and a white vertex. We suggest an improved algorithm for solving this ..."
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Cited by 5 (0 self)
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Given a graph G and positive integers b and w, the blackandwhite coloring problem asks about the existence of a partial vertexcoloring of G, with b vertices colored black and w white, such that there is no edge between a black and a white vertex. We suggest an improved algorithm for solving this problem on trees. Submitted: