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An NC Algorithm for Minimum Cuts
 IN PROCEEDINGS OF THE 25TH ANNUAL ACM SYMPOSIUM ON THEORY OF COMPUTING
"... We show that the minimum cut problem for weighted undirected graphs can be solved in NC using three separate and independently interesting results. The first is an (m 2 =n)processor NC algorithm for finding a (2 + ffl)approximation to the minimum cut. The second is a randomized reduction from ..."
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Cited by 46 (3 self)
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We show that the minimum cut problem for weighted undirected graphs can be solved in NC using three separate and independently interesting results. The first is an (m 2 =n)processor NC algorithm for finding a (2 + ffl)approximation to the minimum cut. The second is a randomized reduction from the minimum cut problem to the problem of obtaining a (2 + ffl)approximation to the minimum cut. This reduction involves a natural combinatorial SetIsolation Problem that can be solved easily in RNC. The third result is a derandomization of this RNC solution that requires a combination of two widely used tools: pairwise independence and random walks on expanders. We believe that the setisolation approach will prove useful in other derandomization problems. The techniques extend to two related problems: we describe NC algorithms finding minimum kway cuts for any constant k and finding all cuts of value within any constant factor of the minimum. Another application of these techni...
Approximating MinimumSize kConnected Spanning Subgraphs via Matching
 SIAM J. Comput
, 1998
"... Abstract: An efficient heuristic is presented for the problem of finding a minimumsize k connected spanning subgraph of an (undirected or directed) simple graph G =(V#E). There are four versions of the problem, and the approximation guarantees are as follows: minimumsize knode connected spann ..."
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Cited by 35 (3 self)
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Abstract: An efficient heuristic is presented for the problem of finding a minimumsize k connected spanning subgraph of an (undirected or directed) simple graph G =(V#E). There are four versions of the problem, and the approximation guarantees are as follows: minimumsize knode connected spanning subgraph of an undirected graph 1+[1=k], minimumsize knode connected spanning subgraph of a directed graph 1+[1=k], minimumsize kedge connected spanning subgraph of an undirected graph 1+[2=(k + 1)], and minimumsize kedge connected spanning subgraph of a directed graph 1+[4= p k].
Algorithms for dense graphs and networks on the random access computer, Algorithmica 15
, 1996
"... Abstract. We improve upon the running time of several graph and network algorithms when applied to dense graphs. In particular, we show how to compute on a machine with word size L = f2 (log n) a maximal matching in an nvertex bipartite graph in time O (n 2 + n2"5/~.) = O (n2"5/log n), how to comp ..."
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Cited by 18 (5 self)
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Abstract. We improve upon the running time of several graph and network algorithms when applied to dense graphs. In particular, we show how to compute on a machine with word size L = f2 (log n) a maximal matching in an nvertex bipartite graph in time O (n 2 + n2"5/~.) = O (n2"5/log n), how to compute the transitive closure of a digraph with n vertices and m edges in time O(n 2 + nm/,k), how to solve the uncapacitated transportation problem with integer costs in the range [0..C] and integer demands in the range [U..U] in time O ((n 3 (log log / log n) 1/2 + n 2 log U) log nC), and how to solve the assignment problem with integer costs in the range [0..C] in time O(n 2"5 lognC/(logn/loglogn)l/4). Assuming a suitably compressed input, we also show how to do depthfirst and breadthfirst search and how to compute strongly connected components and biconnected components in time O(n~. + n2/L), and how to solve the single source shortestpath problem with integer costs in the range [0..C] in time O (n 2 (log C)/log n). For the transitive closure algorithm we also report on the experiences with an implementation. Key Words.
SubLinear Distributed Algorithms for Sparse Certificates and Biconnected Components.
, 1995
"... A certificate for the k connectivity y of a graph G = (V; E) is a subset E 0 of E such that (V; E 0 ) is k connected iff G is k connected. Let n = jV j and m = jEj. A certificate is called sparse if it has size O(kn). We present a distributed algorithm for computing sparse certificate for k co ..."
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Cited by 16 (1 self)
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A certificate for the k connectivity y of a graph G = (V; E) is a subset E 0 of E such that (V; E 0 ) is k connected iff G is k connected. Let n = jV j and m = jEj. A certificate is called sparse if it has size O(kn). We present a distributed algorithm for computing sparse certificate for k connectivity whose time complexity is O(k(D+n 0:614 )) where D is the diameter of the network. A new algorithm for identifying biconnected components is also presented. This algorithm is significantly simpler than many existing algorithms and can be implemented in a distributed environment to run in O(D+n 0:614 ) time. Both algorithms improve on the previous best known time bounds. Our main focus in this paper is the time complexity. However, no more than a polynomial number of messages, each of size O(log n), are generated by the algorithm. 1 Introduction Connectivity is an important property of graphs with many applications in computer science. We study the distributed time complexity o...
Fast Algorithms for kShredders and kNode Connectivity Augmentation
, 1996
"... A kseparator (kshredder) of an undirected graph is a set of k nodes whose removal results in two or more (three or more) connected components. Let the given (undirected) graph be knode connected, and let n denote the number of nodes. Solving an open question, we show that the problem of counti ..."
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Cited by 15 (0 self)
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A kseparator (kshredder) of an undirected graph is a set of k nodes whose removal results in two or more (three or more) connected components. Let the given (undirected) graph be knode connected, and let n denote the number of nodes. Solving an open question, we show that the problem of counting the number of kseparators is #Pcomplete. However, we present an O(k )time (deterministic) algorithm for finding all the kshredders. This solves an open question: efficiently find a kseparator whose removal maximizes the number of connected 4, our running time is within a factor of k of the fastest algorithm known for testing knode connectivity. One application of shredders is in increasing the node connectivity from k to (k +1)by effi tly adding an (approximately) minimum number of new edges. Jord'an [JCT(B) 1995] gaveanO(n )time augmentation algorithm such that the number of new edges is within an additive term of (k 2) from a lower bound. We improve the running time to ), while achieving the same performance guarantee. For k 4, the running time compares favorably with the running time for testing knode connectivity.
Dynamic Graph Algorithms
, 2000
"... INTRODUCTION Dynamic graph algorithms are algorithms that maintain properties of a (possibly edgeweighted) graph while the graph is changing. These algorithms are potentially useful in a number of application areas, including communication networks, VLSI design, distributed computing, and graphics, ..."
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Cited by 12 (0 self)
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INTRODUCTION Dynamic graph algorithms are algorithms that maintain properties of a (possibly edgeweighted) graph while the graph is changing. These algorithms are potentially useful in a number of application areas, including communication networks, VLSI design, distributed computing, and graphics, where the underlying graphs are subject to dynamic changes. Efficient dynamic graph algorithms are also used as subroutines in algorithms that build and modify graphs as part of larger tasks, e.g., the algorithm for constructing Voronoi diagrams by building planar subdivisions. GLOSSARY Update: an operation that changes the graph. The primitive updates considered in the literature are edge insertions and deletions and, in the case of edgeweighted graphs, changes in edge weights. Query: a request for information about the property being maintained. For example, if the property is planarity, a query simply asks whether the graph is currently
Directed st Numberings, Rubber Bands, and Testing Digraph kVertex Connectivity
"... Let G = (V, E) be a directed graph and n denote V. We show that G is kvertex connected iff for every subset X of V with IX I = k, there is an embedding of G in the (k I)dimensional space Rkl, ~ : V ~Rkl, such that no hyperplane contains k points of {~(v) \ v G V}, and for each v E V – X, f( ..."
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Cited by 10 (2 self)
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Let G = (V, E) be a directed graph and n denote V. We show that G is kvertex connected iff for every subset X of V with IX I = k, there is an embedding of G in the (k I)dimensional space Rkl, ~ : V ~Rkl, such that no hyperplane contains k points of {~(v) \ v G V}, and for each v E V – X, f(v) is in the convex hull of {~(w) I (v, W) G E}. This result generalizes to directed graphs the notion of convex embedding of undirected graphs introduced by Linial, LOV6SZ and Wigderson in ‘Rubber bands, convex embedding and graph connectivity, ” Combinatorics 8 (1988), 91102. Using this characterization, a directed graph can be tested for kvertex connectivity by a Monte Carlo algorithm in time O((M(n) + nkf(k)). (log n)) with error probability < l/n, and by a Las Vegas algorithm in expected time O((lf(n)+nM(k)).k), where M(n) denotes the number of arithmetic steps for multiplying two n x n matrices (Al(n) = 0(n2.3755)). Our Monte Carlo algorithm improves on the best previous deterministic and randomized time complexities for k> no. *9; e.g., for k = @, the factor of improvement is> n0.G2. Both algorithms have processor efficient parallel versions that run in O((log n)2) time on the EREW PRAM model of computation, using a number of processors equal to (logn) times the respective sequential time complexities. Our Monte Carlo parallel algorithm improves on the number of processors used by the best previous (Monte Carlo) parallel algorithm by a factor of at least (n2/(log n)3) while having the same running time. Generalizing the notion of st numberings, we give a combinatorial construction of a directed st nulmberiug for any 2vertex connected directed graph.
kconnectivity in the semistreaming model
, 2006
"... We present the first semistreaming algorithms to determine kconnectivity of an undirected graph with k being any constant. The semistreaming model for graph algorithms was introduced by Muthukrishnan in 2003 and turns out to be useful when dealing with massive graphs streamed in from an external ..."
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Cited by 4 (0 self)
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We present the first semistreaming algorithms to determine kconnectivity of an undirected graph with k being any constant. The semistreaming model for graph algorithms was introduced by Muthukrishnan in 2003 and turns out to be useful when dealing with massive graphs streamed in from an external storage device. Our two semistreaming algorithms each compute a sparse subgraph of an input graph G and can use this subgraph in a postprocessing step to decide kconnectivity of G. To this end the first algorithm reads the input stream only once and uses timeO(k 2 n) to process each input edge. The second algorithm reads the input k +1 times and needs timeO(k +α(n)) per input edge. Using its constructed subgraph the second algorithm can also generate all lseparators of the input graph for all l < k.
SeparatorBased Sparsification II: Edge And Vertex Connectivity
 SIAM J. Comput
, 1998
"... . We consider the problem of maintaining a dynamic planar graph subject to edge insertions and edge deletions that preserve planarity but that can change the embedding. We describe algorithms and data structures for maintaining information about 2 and 3vertexconnectivity, and 3 and 4edgeconnec ..."
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Cited by 3 (0 self)
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. We consider the problem of maintaining a dynamic planar graph subject to edge insertions and edge deletions that preserve planarity but that can change the embedding. We describe algorithms and data structures for maintaining information about 2 and 3vertexconnectivity, and 3 and 4edgeconnectivity in a planar graph in O(n 1/2 ) amortized time per insertion, deletion, or connectivity query. All of the data structures handle insertions that keep the graph planar without regard to any particular embedding of the graph. Our algorithms are based on a new type of sparsification combined with several properties of separators in planar graphs. Key words. analysis of algorithms, dynamic data structures, edge connectivity, vertex connectivity, planar graphs AMS subject classifications. 68P05, 68Q20, 68R10 PII. S0097539794269072 1. Introduction. Sparse certificates, small graphs that retain some property of a larger graph, appear often in graph theory, especially in problems of edge and...
A Simple Randomized Scheme for Constructing LowWeight kConnected Spanning Subgraphs with Applications to Distributed Algorithms
"... The main focus of this paper is the analysis of a simple randomized scheme for constructing lowweight kconnected spanning subgraphs. We first show that our scheme gives a simple approximation algorithm to construct a minimumweight kconnected spanning subgraph in a weighted complete graph, a NPh ..."
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Cited by 3 (1 self)
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The main focus of this paper is the analysis of a simple randomized scheme for constructing lowweight kconnected spanning subgraphs. We first show that our scheme gives a simple approximation algorithm to construct a minimumweight kconnected spanning subgraph in a weighted complete graph, a NPhard problem even if the weights satisfy the triangle inequality. We show that our algorithm gives an approximation ratio of O(k log n) for a metric graph, O(k) for a random graph with nodes uniformly randomly distributed in [0, 1] 2 and O(log n k) for a complete graph with random edge weights U(0, 1). We show that our scheme is optimal with respect to the amount of “local information ” needed to compute any connected spanning subgraph. We then show that our scheme can be applied to design an efficient distributed algorithm for constructing such an approximate kconnected spanning subgraph (for any k ≥ 1) in a pointtopoint distributed model, where the processors form a complete network. Our algorithm takes O(log n n k) time and expected O(nk log