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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 12 (3 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
Discrete sensor placement problems in distribution networks
 SIAM Conference on Mathematics for Industry
, 2003
"... Abstract—We consider the problem of placing sensors in a network to detect and identify the source of any contamination. We consider two variants of this problem: (1) sensorconstrained: we are allowed a fixed number of sensors and want to minimize contamination detection time; and (2) timeconstrai ..."
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Cited by 9 (1 self)
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Abstract—We consider the problem of placing sensors in a network to detect and identify the source of any contamination. We consider two variants of this problem: (1) sensorconstrained: we are allowed a fixed number of sensors and want to minimize contamination detection time; and (2) timeconstrained]: we must detect contamination within a given time limit and want to minimize the number of sensors required. Our main results are as follows. First, we give a necessary and sufficient condition for source identification. Second, we show that the sensor and time constrained versions of the problem are polynomially equivalent. Finally, we show that the sensorconstrained version of the problem is polynomially equivalent to the asymmetric kcenter problem and that the timeconstrained version of the problem is
On the ComparisonAddition Complexity of AllPairs Shortest Paths
 In Proc. 13th Int'l Symp. on Algorithms and Computation (ISAAC'02
, 2002
"... We present an allpairs shortest path algorithm for arbitrary graphs that performs O(mn log (m; n)) comparison and addition operations, where m and n are the number of edges and vertices, resp., and is Tarjan's inverseAckermann function. Our algorithm eliminates the sorting bottleneck inherent in a ..."
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Cited by 6 (5 self)
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We present an allpairs shortest path algorithm for arbitrary graphs that performs O(mn log (m; n)) comparison and addition operations, where m and n are the number of edges and vertices, resp., and is Tarjan's inverseAckermann function. Our algorithm eliminates the sorting bottleneck inherent in approaches based on Dijkstra's algorithm, and for graphs with O(n) edges our algorithm is within a tiny O(log (n; n)) factor of optimal. Our algorithm can be implemented to run in polynomial time (granted, a large polynomial). We leave open the problem of providing an efficient implementation.
An InverseAckermann Style Lower Bound for Online Minimum Spanning Tree Verification
 Combinatorica
"... 1 Introduction The minimum spanning tree (MST) problem has seen a flurry of activity lately, driven largely by the success of a new approach to the problem. The recent MST algorithms [20, 8, 29, 28], despite their superficial differences, are all based on the idea of progressively improving an appro ..."
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Cited by 3 (2 self)
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1 Introduction The minimum spanning tree (MST) problem has seen a flurry of activity lately, driven largely by the success of a new approach to the problem. The recent MST algorithms [20, 8, 29, 28], despite their superficial differences, are all based on the idea of progressively improving an approximately minimum solution, until the actual minimum spanning tree is found. It is still likely that this progressive improvement approach will bear fruit. However, the current
Ramachandran V. “Oracles for Distances Avoiding a NodeLink Failure
, 2002
"... We consider the problem of preprocessing an edgeweighted directed graph G to answer queries that ask for the shortest distance from any given node x to any other node y avoiding an arbitrary failed node or link. We describe an oracle (i.e, a simple data structure) for such queries that can be store ..."
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Cited by 2 (0 self)
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We consider the problem of preprocessing an edgeweighted directed graph G to answer queries that ask for the shortest distance from any given node x to any other node y avoiding an arbitrary failed node or link. We describe an oracle (i.e, a simple data structure) for such queries that can be stored in O(n 2 log n) space, and which allows queries to be answered in O(1) time, where n is the number of nodes in G. We also show that if we are willing to use Θ(n 2.5) space, we can reduce the preprocessing time by a factor of √ n while maintaining the constant query time. We can also keep track of the shortest path avoiding any failed node or link by maintaining for each node the outgoing edge that should be used to get on such a path.
Engineering Shortest Path Algorithms
"... In this paper, we report on our own experience in studying a fundamental problem on graphs: all pairs shortest paths. In particular, we discuss the interplay between theory and practice in engineering a simple variant of Dijkstra's shortest path algorithm. In this context, we show that studying ..."
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Cited by 1 (0 self)
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In this paper, we report on our own experience in studying a fundamental problem on graphs: all pairs shortest paths. In particular, we discuss the interplay between theory and practice in engineering a simple variant of Dijkstra's shortest path algorithm. In this context, we show that studying heuristics that are e#cient in practice can yield interesting clues to the combinatorial properties of the problem, and eventually lead to new theoretically e#cient algorithms.
Discrete Sensor Placement Problems in Distribution Networks
"... We consider the problem of placing sensors in a network to detect and identify the source of any contamination. We consider two variants of this problem: 1) sensorconstrained: we are allowed a fixed number of sensors and want to minimize contamination detection time; and 2) timeconstrained: we mu ..."
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We consider the problem of placing sensors in a network to detect and identify the source of any contamination. We consider two variants of this problem: 1) sensorconstrained: we are allowed a fixed number of sensors and want to minimize contamination detection time; and 2) timeconstrained: we must detect contamination within a given time limit and want to minimimize the number of sensors required. Our main results are as follows. First, we give a necessary and sufficient condition Preprint submitted to Elsevier Science 15 December 2004 for source identification. Second, we show that the sensor and time constrained versions of the problem are polynomially equivalent. Finally, we show that the sensorconstrained version of the problem is polynomially equivalent to the asymmetric kcenter problem and that the timeconstrained version of the problem is polynomially equivalent to the dominating set problem. 1
The 25th Workshop on Combinatorial Mathematics and Computation Theory The Weighted AllPairsShortestPathLength Problem on TwoTerminal SeriesParallel Graphs
"... Let G(V, E) be a nvertex graph with V = { v1 vn} and each edge e = ( i, ) is associated with two real weights, W ( → ) and W ( →). v v j v v v v Let P be a path from any source vertex s to any destination vertex d and assume that P is s = u → u2 ut i = d, t ≥ 2. The length of P, denoted by Len(P ..."
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Let G(V, E) be a nvertex graph with V = { v1 vn} and each edge e = ( i, ) is associated with two real weights, W ( → ) and W ( →). v v j v v v v Let P be a path from any source vertex s to any destination vertex d and assume that P is s = u → u2 ut i = d, t ≥ 2. The length of P, denoted by Len(P), is defined as ∑ →. ( W u u j t−1 j=