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58
Approximating the Minimum Spanning Tree Weight in Sublinear Time
 In Proceedings of the 28th Annual International Colloquium on Automata, Languages and Programming (ICALP
, 2001
"... We present a probabilistic algorithm that, given a connected graph G (represented by adjacency lists) of average degree d, with edge weights in the set {1,...,w}, and given a parameter 0 < ε < 1/2, estimates in time O(dwε−2 log dw ε) the weight of the minimum spanning tree of G with a relativ ..."
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Cited by 49 (6 self)
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We present a probabilistic algorithm that, given a connected graph G (represented by adjacency lists) of average degree d, with edge weights in the set {1,...,w}, and given a parameter 0 < ε < 1/2, estimates in time O(dwε−2 log dw ε) the weight of the minimum spanning tree of G with a relative error of at most ε. Note that the running time does not depend on the number of vertices in G. We also prove a nearly matching lower bound of Ω(dwε−2) on the probe and time complexity of any approximation algorithm for MST weight. The essential component of our algorithm is a procedure for estimating in time O(dε−2 log d ε) the number of connected components of an unweighted graph to within an additive error of εn. (This becomes O(ε−2 log 1 ε) for d = O(1).) The time bound is shown to be tight up to within the log d ε factor. Our connectedcomponents algorithm picks O(1/ε2) vertices in the graph and then grows “local spanning trees” whose sizes are specified by a stochastic process. From the local information collected in this way, the algorithm is able to infer, with high confidence, an estimate of the number of connected components. We then show how estimates on the number of components in various subgraphs of G can be used to estimate the weight of its MST. 1
A New Approach to AllPairs Shortest Paths on RealWeighted Graphs
 Theoretical Computer Science
, 2003
"... We present a new allpairs shortest path algorithm that works with realweighted graphs in the traditional comparisonaddition model. It runs in O(mn+n time, improving on the longstanding bound of O(mn + n log n) derived from an implementation of Dijkstra's algorithm with Fibonacci heaps ..."
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Cited by 41 (3 self)
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We present a new allpairs shortest path algorithm that works with realweighted graphs in the traditional comparisonaddition model. It runs in O(mn+n time, improving on the longstanding bound of O(mn + n log n) derived from an implementation of Dijkstra's algorithm with Fibonacci heaps. Here m and n are the number of edges and vertices, respectively.
Distributed Verification of Minimum Spanning Trees
 Proc. 25th Annual Symposium on Principles of Distributed Computing
, 2006
"... The problem of verifying a Minimum Spanning Tree (MST) was introduced by Tarjan in a sequential setting. Given a graph and a tree that spans it, the algorithm is required to check whether this tree is an MST. This paper investigates the problem in the distributed setting, where the input is given in ..."
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Cited by 32 (23 self)
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The problem of verifying a Minimum Spanning Tree (MST) was introduced by Tarjan in a sequential setting. Given a graph and a tree that spans it, the algorithm is required to check whether this tree is an MST. This paper investigates the problem in the distributed setting, where the input is given in a distributed manner, i.e., every node “knows ” which of its own emanating edges belong to the tree. Informally, the distributed MST verification problem is the following. Label the vertices of the graph in such a way that for every node, given (its own label and) the labels of its neighbors only, the node can detect whether these edges are indeed its MST edges. In this paper we present such a verification scheme with a maximum label size of O(log n log W), where n is the number of nodes and W is the largest weight of an edge. We also give a matching lower bound of Ω(log n log W) (except when W ≤ log n). Both our bounds improve previously known bounds for the problem. Our techniques (both for the lower bound and for the upper bound) may indicate a strong relation between the fields of proof labeling schemes and implicit labeling schemes. For the related problem of tree sensitivity also presented by Tarjan, our method yields rather efficient schemes for both the distributed and the sequential settings.
LinearTime PointerMachine Algorithms for Least Common Ancestors, MST Verification, and Dominators
 IN PROCEEDINGS OF THE THIRTIETH ANNUAL ACM SYMPOSIUM ON THEORY OF COMPUTING
, 1998
"... We present two new data structure toolsdisjoint set union with bottomup linking, and pointerbased radix sortand combine them with bottomlevel microtrees to devise the first lineartime pointermachine algorithms for offline least common ancestors, minimum spanning tree (MST) verification, ..."
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Cited by 29 (4 self)
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We present two new data structure toolsdisjoint set union with bottomup linking, and pointerbased radix sortand combine them with bottomlevel microtrees to devise the first lineartime pointermachine algorithms for offline least common ancestors, minimum spanning tree (MST) verification, randomized MST construction, and computing dominators in a flowgraph.
Concurrent Threads and Optimal Parallel Minimum Spanning Trees Algorithm
 J. ACM
, 2001
"... This paper resolves a longstanding open problem on whether the concurrent write capability of parallel random access machine (PRAM) is essential for solving fundamental graph problems like connected components and minimum spanning trees in O(log n) time. Specically, we present a new algorithm to so ..."
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Cited by 24 (2 self)
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This paper resolves a longstanding open problem on whether the concurrent write capability of parallel random access machine (PRAM) is essential for solving fundamental graph problems like connected components and minimum spanning trees in O(log n) time. Specically, we present a new algorithm to solve these problems in O(log n) time using a linear number of processors on the exclusiveread exclusivewrite PRAM. The logarithmic time bound is actually optimal since it is well known that even computing the \OR" of n bits
Estimating the weight of metric minimum spanning trees in sublineartime
 in Proceedings of the 36th Annual ACM Symposium on Theory of Computing (STOC
"... In this paper we present a sublinear time (1+ ɛ)approximation randomized algorithm to estimate the weight of the minimum spanning tree of an npoint metric space. The running time of the algorithm is Õ(n/ɛO(1)). Since the full description of an npoint metric space is of size Θ(n 2),the complexity ..."
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Cited by 22 (5 self)
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In this paper we present a sublinear time (1+ ɛ)approximation randomized algorithm to estimate the weight of the minimum spanning tree of an npoint metric space. The running time of the algorithm is Õ(n/ɛO(1)). Since the full description of an npoint metric space is of size Θ(n 2),the complexity of our algorithm is sublinear with respect to the input size. Our algorithm is almost optimal as it is not possible to approximate in o(n) time the weight of the minimum spanning tree to within any factor. Furthermore,it has been previously shown that no o(n 2) algorithm exists that returns a spanning tree whose weight is within a constant times the optimum.
Lineartime compression of boundedgenus graphs into informationtheoretically optimal number of bits
 In: 13th Symposium on Discrete Algorithms (SODA
, 2002
"... 1 I n t roduct ion This extended abstract summarizes a new result for the graph compression problem, addressing how to compress a graph G into a binary string Z with the requirement that Z can be decoded to recover G. Graph compression finds important applications in 3D model compression of Computer ..."
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Cited by 16 (1 self)
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1 I n t roduct ion This extended abstract summarizes a new result for the graph compression problem, addressing how to compress a graph G into a binary string Z with the requirement that Z can be decoded to recover G. Graph compression finds important applications in 3D model compression of Computer Graphics [12, 1720] and compact routing table of Computer Networks [7}. For brevity, let a ~rgraph stand for a graph with property n. The informationtheoretically optimal number of bits required to represent an nnode ngraph is [log 2 N~(n)], where N,~(n) is the number of distinct nnode *rgraphs. Although determining or approximating the close forms of N ~ (n) for nontrivial classes of n is challenging, we provide a lineartime methodology for graph compression schemes that are informationtheoretically optimal with respect to continuous uperadditive functions (abbreviated as optimal for the rest of the extended abstract). 1 Specifically, if 7r satisfies certain properties, then we can compress any nnode medge 1rgraph G into a binary string Z such that G and Z can be computed from each other in O(m + n) time, and that the bit count of Z is at most fl(n) + o(fl(n)) for any continuous uperadditive function fl(n) with log 2 N~(n) < fl(n) + o(fl(n)). Our methodology is applicable to general classes of graphs; this extended abstract focuses on graphs with sublinear genus. 2 For example, if the input nnode,rgraph G is equipped with an embedding on its genus surface, which is a reasonable assumption for graphs arising from 3D model compression, then our methodology is applicable to any 7r satisfying the following statements:
A shortest path algorithm for realweighted undirected graphs
 in 13th ACMSIAM Symp. on Discrete Algs
, 1985
"... Abstract. We present a new scheme for computing shortest paths on realweighted undirected graphs in the fundamental comparisonaddition model. In an efficient preprocessing phase our algorithm creates a linearsize structure that facilitates singlesource shortest path computations in O(m log α) ti ..."
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Cited by 16 (4 self)
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Abstract. We present a new scheme for computing shortest paths on realweighted undirected graphs in the fundamental comparisonaddition model. In an efficient preprocessing phase our algorithm creates a linearsize structure that facilitates singlesource shortest path computations in O(m log α) time, where α = α(m, n) is the very slowly growing inverseAckermann function, m the number of edges, and n the number of vertices. As special cases our algorithm implies new bounds on both the allpairs and singlesource shortest paths problems. We solve the allpairs problem in O(mnlog α(m, n)) time and, if the ratio between the maximum and minimum edge lengths is bounded by n (log n)O(1) , we can solve the singlesource problem in O(m + nlog log n) time. Both these results are theoretical improvements over Dijkstra’s algorithm, which was the previous best for real weighted undirected graphs. Our algorithm takes the hierarchybased approach invented by Thorup. Key words. singlesource shortest paths, allpairs shortest paths, undirected graphs, Dijkstra’s
Fast Euclidean minimum spanning tree: algorithm, analysis, and applications.
 In Proceedings of the 16th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining (KDD ’10),
, 2010
"... ABSTRACT The Euclidean Minimum Spanning Tree problem has applications in a wide range of fields, and many efficient algorithms have been developed to solve it. We present a new, fast, general EMST algorithm, motivated by the clustering and analysis of astronomical data. Largescale astronomical sur ..."
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Cited by 13 (4 self)
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ABSTRACT The Euclidean Minimum Spanning Tree problem has applications in a wide range of fields, and many efficient algorithms have been developed to solve it. We present a new, fast, general EMST algorithm, motivated by the clustering and analysis of astronomical data. Largescale astronomical surveys, including the Sloan Digital Sky Survey, and large simulations of the early universe, such as the Millennium Simulation, can contain millions of points and fill terabytes of storage. Traditional EMST methods scale quadratically, and more advanced methods lack rigorous runtime guarantees. We present a new dualtree algorithm for efficiently computing the EMST, use adaptive algorithm analysis to prove the tightest (and possibly optimal) runtime bound for the EMST problem todate, and demonstrate the scalability of our method on astronomical data sets.