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15
More algorithms for allpairs shortest paths in weighted graphs
 In Proceedings of 39th Annual ACM Symposium on Theory of Computing
, 2007
"... In the first part of the paper, we reexamine the allpairs shortest paths (APSP) problem and present a new algorithm with running time O(n 3 log 3 log n / log 2 n), which improves all known algorithms for general realweighted dense graphs. In the second part of the paper, we use fast matrix multipl ..."
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Cited by 56 (3 self)
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In the first part of the paper, we reexamine the allpairs shortest paths (APSP) problem and present a new algorithm with running time O(n 3 log 3 log n / log 2 n), which improves all known algorithms for general realweighted dense graphs. In the second part of the paper, we use fast matrix multiplication to obtain truly subcubic APSP algorithms for a large class of “geometrically weighted ” graphs, where the weight of an edge is a function of the coordinates of its vertices. For example, for graphs embedded in Euclidean space of a constant dimension d, we obtain a time bound near O(n 3−(3−ω)/(2d+4)), where ω < 2.376; in two dimensions, this is O(n 2.922). Our framework greatly extends the previously considered case of smallintegerweighted graphs, and incidentally also yields the first truly subcubic result (near O(n 3−(3−ω)/4) = O(n 2.844) time) for APSP in realvertexweighted graphs, as well as an improved result (near O(n (3+ω)/2) = O(n 2.688) time) for the allpairs lightest shortest path problem for smallintegerweighted graphs. 1
Allpairs shortest paths with real weights in O(n³ / log n) time
 PROC. OF THE 9TH WADS, LECTURE NOTES IN COMPUTER SCIENCE 3608
, 2005
"... We describe an O(n³ / log n) ..."
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 29 (2 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.
Networks Cannot Compute Their Diameter in Sublinear Time preliminary version please check for updates
, 2011
"... We study the problem of computing the diameter of a network in a distributed way. The model of distributed computation we consider is: in each synchronous round, each node can transmit a different (but short) message to each of its neighbors. We provide an ˜ Ω(n) lower bound for the number of commun ..."
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Cited by 11 (2 self)
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We study the problem of computing the diameter of a network in a distributed way. The model of distributed computation we consider is: in each synchronous round, each node can transmit a different (but short) message to each of its neighbors. We provide an ˜ Ω(n) lower bound for the number of communication rounds needed, where n denotes the number of nodes in the network. This lower bound is valid even if the diameter of the network is a small constant. We also show that a (3/2 − ε)approximation of the diameter requires ˜ Ω ( √ n) rounds. Furthermore we use our new technique to prove an ˜ Ω ( √ n) lower bound on approximating the girth of a graph by a factor 2 − ε. Contact author:
An InverseAckermann Type Lower Bound for Online Minimum Spanning Tree Verification
 Combinatorica
"... Given a spanning tree T of some graph G, the problem of minimum spanning tree verication is to decide whether T = MST(G). A celebrated result of Komlos shows that this problem can be solved in linear time. Somewhat unexpectedly, MST verication turns out to be useful in actually computing minimum spa ..."
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Cited by 2 (1 self)
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Given a spanning tree T of some graph G, the problem of minimum spanning tree verication is to decide whether T = MST(G). A celebrated result of Komlos shows that this problem can be solved in linear time. Somewhat unexpectedly, MST verication turns out to be useful in actually computing minimum spanning trees from scratch. It is this application that has led some to wonder whether a more flexible version of MST Verification could be used to derive a faster deterministic minimum spanning tree algorithm.
Subcubic Equivalences Between Graph Centrality Problems, APSP and Diameter
"... Measuring the importance of a node in a network is a major goal in the analysis of social networks, biological systems, transportation networks etc. Different centrality measures have been proposed to capture the notion of node importance. For example, the center of a graph is a node that minimizes ..."
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Measuring the importance of a node in a network is a major goal in the analysis of social networks, biological systems, transportation networks etc. Different centrality measures have been proposed to capture the notion of node importance. For example, the center of a graph is a node that minimizes the maximum distance to any other node (the latter distance is the radius of the graph). The median of a graph is a node that minimizes the sum of the distances to all other nodes. Informally, the betweenness centrality of a node w measures the fraction of shortest paths that have w as an intermediate node. Finally, the reach centrality of a node w is the smallest distance r such that any st shortest path passing through w has either s or t in the ball of radius r around w. The fastest known algorithms to compute the center and the median of a graph, and to compute the betweenness or reach centrality even of a single node take roughly cubic time in the number n of nodes in the input graph. It is open whether these problems admit truly subcubic algorithms, i.e. algorithms with running time Õ(n3−δ) for some constant δ> 01. We relate the complexity of the mentioned centrality problems to two classical problems for which no truly subcubic algorithm is known, namely All Pairs Shortest Paths (APSP) and Diameter. We show that Radius, Median and Betweenness Centrality are equivalent under subcubic reductions to APSP, i.e. that a truly subcubic algorithm for any of these problems implies a truly subcubic algorithm for all of them. We then show that Reach Centrality is equivalent to Diameter under subcubic reductions. The same holds for the problem of approximating Betweenness Centrality within any constant factor. Thus the latter two centrality problems could potentially be solved in truly subcubic time, even if APSP requires essentially cubic time. 1
Priority Queues and Dijkstra’s Algorithm
, 2007
"... We study the impact of using different priority queues in the performance of Dijkstra’s SSSP algorithm. We consider only general priority queues that can handle any type of keys (integer, floating point, etc.); the only exception is that we use as a benchmark the DIMACS Challenge SSSP code [1] which ..."
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Cited by 1 (0 self)
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We study the impact of using different priority queues in the performance of Dijkstra’s SSSP algorithm. We consider only general priority queues that can handle any type of keys (integer, floating point, etc.); the only exception is that we use as a benchmark the DIMACS Challenge SSSP code [1] which can handle only integer values for distances. Our experiments were focussed on the following: 1. We study the performance of two variants of Dijkstra’s algorithm: the wellknown version that uses a priority queue that supports the DecreaseKey operation, and another that uses a basic priority queue that supports only Insert and DeleteMin. For the latter type of priority queue we include several for which highperformance code is available such as bottomup binary heap, aligned 4ary heap, and sequence heap [33]. 2. We study the performance of Dijkstra’s algorithm designed for flat memory relative to versions that try to be cacheefficient. For this, in main part, we study the difference in performance of Dijkstra’s algorithm relative to the cacheefficiency of the priority queue used, both incore and outofcore. We also study the performance of an implementation
An O(n 3 log log n/ log 2 n) time algorithm for all pairs shortest paths
, 2009
"... Abstract. We present an O(n 3 log log n / log 2 n) time algorithm for all pairs shortest paths. This algorithm improves on the best previous result of O(n 3 (log log n) 3 / log 2 n) time. ..."
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Abstract. We present an O(n 3 log log n / log 2 n) time algorithm for all pairs shortest paths. This algorithm improves on the best previous result of O(n 3 (log log n) 3 / log 2 n) time.
Hybrid BellmanFordDijkstra Algorithm
"... We consider the singlesource cheapest paths problem in a digraph with negative edge costs allowed. A hybrid of BellmanFord and Dijkstra algorithms is suggested, improving the running time bound upon BellmanFord for graphs with a sparse distribution of negative cost edges. The algorithm iterates D ..."
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We consider the singlesource cheapest paths problem in a digraph with negative edge costs allowed. A hybrid of BellmanFord and Dijkstra algorithms is suggested, improving the running time bound upon BellmanFord for graphs with a sparse distribution of negative cost edges. The algorithm iterates Dijkstra several times without reinitializing the tentative value d(v) at vertices. At most k + 2 executions of Dijkstra solve the problem, if for any vertex reachable from the source, there exists a cheapest path to it with at most k negative cost edges. In addition, a new, straightforward proof is suggested that the BellmanFord algorithm results in a cheapest path tree from the source. 1
Proximity Graphs inside Large Weighted Graphs
, 2010
"... Given a large weighted graph G = (V, E) and a subset U of V, we define several graphs with vertex set U in which two vertices are adjacent if they satisfy some prescribed proximity rule. These rules use the shortest path distance in G and generalize the proximity rules that generate some of the most ..."
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Given a large weighted graph G = (V, E) and a subset U of V, we define several graphs with vertex set U in which two vertices are adjacent if they satisfy some prescribed proximity rule. These rules use the shortest path distance in G and generalize the proximity rules that generate some of the most common proximity graphs in Euclidean spaces. We prove basic properties of the defined graphs and provide algorithms for their computation.