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41
A Linear Algorithm for Analysis of Minimum Spanning and Shortest Path Trees of Planar Graphs
 Algorithmica
, 1992
"... We give a linear time and space algorithm for analyzing trees in planar graphs. The algorithm can be used to analyze the sensitivity of a minimum spanning tree to changes in edge costs, to find its replacement edges, and to verify its minimality. It can also be used to analyze the sensitivity of a s ..."
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Cited by 16 (0 self)
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We give a linear time and space algorithm for analyzing trees in planar graphs. The algorithm can be used to analyze the sensitivity of a minimum spanning tree to changes in edge costs, to find its replacement edges, and to verify its minimality. It can also be used to analyze the sensitivity of a singlesource shortest path tree to changes in edge costs, and to analyze the sensitivity of a minimum cost network flow. The algorithm is simple and practical. It uses the properties of a planar embedding, combined with a heapordered queue data structure. Let G = (V; E) be a planar graph, either directed or undirected, with n vertices and m = O(n) edges. Each edge e 2 E has a realvalued cost cost(e). A minimum spanning tree of a connected, undirected planar graph G is a spanning tree of minimum total edge cost. If G is directed and r is a vertex from which all other vertices are reachable, then a shortest path tree from r is a spanning tree that contains a minimumcost path from r to every...
Random Sampling in Matroids, with Applications to Graph Connectivity and Minimum Spanning Trees
, 1993
"... Random sampling is a powerful way to gather information about a group by considering only a small part of it. We give a paradigm for applying this technique to optimization problems, and demonstrate its effectiveness on matroids. Matroids abstractly model many optimization problems that can be solve ..."
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Cited by 14 (4 self)
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Random sampling is a powerful way to gather information about a group by considering only a small part of it. We give a paradigm for applying this technique to optimization problems, and demonstrate its effectiveness on matroids. Matroids abstractly model many optimization problems that can be solved by greedy methods, such as the minimum spanning tree (MST) problem. Our results have several applications. We give an algorithm that uses simple data structures to construct an MST in O(m+n logn) time (Klein and Tarjan [21] have recently shown that a better choice of parameters makes this algorithm run in O(m + n) time). We give bounds on the connectivity (minimum cut) of a graph suffering random edge failures. We give fast algorithms for packing matroid bases, with particular attention to packing spanning trees in graphs.
Finding All the Best Swaps of a Minimum Diameter Spanning Tree Under Transient Edge Failures
 Journal of Graph Algorithms and Applications
, 1998
"... Abstract. In network communication systems, frequently messages are routed along a minimum diameter spanning tree (MDST) of the network, to minimize the maximum delay in delivering a message. When a transient edge failure occurs, it is important to choose a temporary replacement edge which minimizes ..."
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Cited by 14 (6 self)
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Abstract. In network communication systems, frequently messages are routed along a minimum diameter spanning tree (MDST) of the network, to minimize the maximum delay in delivering a message. When a transient edge failure occurs, it is important to choose a temporary replacement edge which minimizes the diameter of the new spanning tree. Such an optimal replacement is called the best swap. As a natural extension, the allbestswaps (ABS) problem is the problem of finding the best swap for every edge of the MDST. Given a weighted graph G =(V, E), where V  = n and E  = m,wesolvetheABSprobleminO(n √ m)time and O(m + n) space, thus improving previous bounds for m = o(n 2). 1
Randomized Approximation Schemes for Cuts and Flows in Capacitated Graphs
, 2011
"... We describe random sampling techniques for approximately solving problems that involve cuts and flows in graphs. We give a nearlineartime randomized combinatorial construction that transforms any graph on n vertices into an O(n log n)edge graph on the same vertices whose cuts have approximately t ..."
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Cited by 9 (0 self)
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We describe random sampling techniques for approximately solving problems that involve cuts and flows in graphs. We give a nearlineartime randomized combinatorial construction that transforms any graph on n vertices into an O(n log n)edge graph on the same vertices whose cuts have approximately the same value as the original graph’s. In this new graph, for example, we can run the Õ(m3/2)time maximum flow algorithm of Goldberg and Rao to find an s– t minimum cut in Õ(n3/2) time. This corresponds to a (1 + ɛ)times minimum s–t cut in the original graph. A related approach leads to a randomized divide and conquer algorithm producing an approximately maximum flow in Õ(m √ n) time. Our algorithm is also used to improve the running time of sparsest cut algorithms from Õ(mn) to Õ(n²). Our approach also accelerates several other recent cut and flow algorithms. Our algorithms are based on a general theorem analyzing the concentration of cut values near their expectation in random graphs.
Random Sampling and Greedy Sparsification for Matroid Optimization Problems.
 Mathematical Programming
, 1998
"... Random sampling is a powerful tool for gathering information about a group by considering only a small part of it. We discuss some broadly applicable paradigms for using random sampling in combinatorial optimization, and demonstrate the effectiveness of these paradigms for two optimization problems ..."
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Cited by 9 (2 self)
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Random sampling is a powerful tool for gathering information about a group by considering only a small part of it. We discuss some broadly applicable paradigms for using random sampling in combinatorial optimization, and demonstrate the effectiveness of these paradigms for two optimization problems on matroids: finding an optimum matroid basis and packing disjoint matroid bases. Applications of these ideas to the graphic matroid led to fast algorithms for minimum spanning trees and minimum cuts. An optimum matroid basis is typically found by a greedy algorithm that grows an independent set into an the optimum basis one element at a time. This continuous change in the independent set can make it hard to perform the independence tests needed by the greedy algorithm. We simplify matters by using sampling to reduce the problem of finding an optimum matroid basis to the problem of verifying that a given fixed basis is optimum, showing that the two problems can be solved in roughly the same ...
Data structures for range minimum queries in multidimensional arrays
 In Proceedings of the 20th Annual ACMSIAM Symposium on Discrete Algorithms
, 2010
"... Given a ddimensional array A with N entries, the Range Minimum Query (RMQ) asks for the minimum element within a contiguous subarray of A. The 1D RMQ problem has been studied intensively because of its relevance to the Nearest Common Ancestor problem and its important use in stringology. If constan ..."
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Cited by 9 (0 self)
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Given a ddimensional array A with N entries, the Range Minimum Query (RMQ) asks for the minimum element within a contiguous subarray of A. The 1D RMQ problem has been studied intensively because of its relevance to the Nearest Common Ancestor problem and its important use in stringology. If constanttime query answering is required, linear time and space preprocessing algorithms were known for the 1D case, but not for the higher dimensional cases. In this paper, we give the first lineartime preprocessing algorithm for arrays with fixed dimension, such that any range minimum query can be answered in constant time. 1
Representing all Minimum Spanning Trees with Applications to Counting and Generation
, 1995
"... We show that for any edgeweighted graph G there is an equivalent graph EG such that the minimum spanning trees of G correspond oneforone with the spanning trees of EG. The equivalent graph can be constructed in time O(m + n log n) given a single minimum spanning tree of G. As a consequence we can ..."
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Cited by 7 (0 self)
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We show that for any edgeweighted graph G there is an equivalent graph EG such that the minimum spanning trees of G correspond oneforone with the spanning trees of EG. The equivalent graph can be constructed in time O(m + n log n) given a single minimum spanning tree of G. As a consequence we can count the minimum spanning trees of G in time O(m + n 2.376 ), generate a random minimum spanning tree in time O(mn), and list all minimum spanning trees in time O(m+n log n+k) where k denotes the number of minimum spanning trees generated. We also discuss similar equivalent graph constructions for shortest paths, minimum cost flows, and bipartite matching.
StateoftheArt Algorithms for Minimum Spanning Trees  A Tutorial Discussion
, 1997
"... The classic “easy” optimization problem is to find the minimum spanning tree (MST) of a connected, undirected graph. Good polynomialtime algorithms have been known since 1930. Over the last 10 years, however, the standard O(mlogn) results of Kruskal and Prim have been improved to linear or nearli ..."
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Cited by 7 (0 self)
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The classic “easy” optimization problem is to find the minimum spanning tree (MST) of a connected, undirected graph. Good polynomialtime algorithms have been known since 1930. Over the last 10 years, however, the standard O(mlogn) results of Kruskal and Prim have been improved to linear or nearlinear time. The new methods use several tricks of general interest in order to reduce the number of edge weight comparisons and the amount of other work. This tutorial reviews those methods, building up strategies step by step so as to expose the insights behind the algorithms. Implementation details are clarified, and some generalizations are given. Specifically, the paper attempts to shed light on the classical algorithms of Kruskal, of Prim, and of Bor˙uvka; the improved approach of Gabow, Galil, and Spencer, which takes time only O(mlog(log*n−log * m n)); and the randomized O(m) algorithm of Karger, Klein, and Tarjan,
An Optimal EREW PRAM Algorithm For Minimum Spanning Tree Verification
, 1997
"... We present a deterministic parallel algorithm on the EREW PRAM model to verify a minimum spanning tree of a graph. The algorithm runs on a graph with n vertices and m edges in O(log n) time and O(m + n) work. The algorithm is a parallelization of King's linear time sequential algorithm for the prob ..."
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Cited by 7 (2 self)
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We present a deterministic parallel algorithm on the EREW PRAM model to verify a minimum spanning tree of a graph. The algorithm runs on a graph with n vertices and m edges in O(log n) time and O(m + n) work. The algorithm is a parallelization of King's linear time sequential algorithm for the problem. 1 Introduction The problem of verifying if a given spanning tree in an edgeweighted graph is a minimum spanning tree for the graph is an important problem which is closely related to the problem of finding a minimum spanning tree of a graph. Recently Dixon, Rauch and Tarjan ([DRT92]) and King ([Kin95]) have developed sequential linear time algorithms for the problem. Dixon and Tarjan have also given an optimal CREW algorithm ([DT94]). In this paper, we present a parallel algorithm which runs in optimal time and work bounds on the weaker EREW PRAM model. This resolves an open question posed in [DT94]. The highlevel structure of the algorithm has been adapted from [DT94] and [Kin95]. W...
Reconstructing a Minimum Spanning Tree after Deletion of Any Node
 Algorithmica
, 1999
"... Updating a minimum spanning tree (MST) is a basic problem for communication networks. In this paper, we consider single node deletions in MSTs. Let G = (V; E) be an undirected graph with n nodes and m edges, and let T be the MST of G. For each node v in V , the node replacement for v is the minim ..."
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Cited by 6 (0 self)
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Updating a minimum spanning tree (MST) is a basic problem for communication networks. In this paper, we consider single node deletions in MSTs. Let G = (V; E) be an undirected graph with n nodes and m edges, and let T be the MST of G. For each node v in V , the node replacement for v is the minimum weight set of edges R(v) that connect the components of T \Gamma v. We present a sequential algorithm and a parallel algorithm that find R(v) for all V simultaneously. The sequential algorithm takes O(m log n) time, but only O(mff(m; n)) time when the edges of E are presorted by weight. The parallel algorithm takes O(log 2 n) time using m processors on a CREW PRAM. 2 1 INTRODUCTION For communication networks, minimum spanning trees (MSTs) are used for basic network tasks such as broadcast, leader election, and synchronization. Updating the MST after changes in network topology is a fundamental problem. In this paper, we update MSTs after single node deletions. In a graph G with ...