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29
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
An ExternalMemory Data Structure for Shortest Path Queries
, 1999
"... In this paper, we present results related to satisfying shortest path queries on a planar graph stored in external memory. N denotes the total number of vertices and edges in the graph and sort(N) denotes the number of input/output (I/O) operations required to sort an array of length N . 1) We desc ..."
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Cited by 11 (1 self)
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In this paper, we present results related to satisfying shortest path queries on a planar graph stored in external memory. N denotes the total number of vertices and edges in the graph and sort(N) denotes the number of input/output (I/O) operations required to sort an array of length N . 1) We describe a data structure for supporting bottomup traversal of rooted trees in external memory. A tree of size S is stored in O(S=B) blocks, and traversing a path of length K towards the root in this tree takes O(K=B) I/Os. 2) We give an algorithm for computing a separator for an embedded planar graph in O(sort(N)) I/Os, provided that a breadthfirst search (BFS) tree is given. 3) We describe an algorithm for triangulating an embedded planar graph in O(sort(N)) I/Os. Using these results, we can obtain a data structure for shortest path queries on graphs with separators of size O( p N) that uses O(N 3=2 =B) blocks of external memory and allows for answering shortest path queries in O(( p ...
Online and Dynamic Algorithms for Shortest Path Problems
 Proc. 12th Symp. on Theor. Aspects of Comp. Sc. (STACS'95), LNCS 900
, 1995
"... Abstract. We describe algorithms for finding shortest paths and distances in a planar digraph which exploit the particular topology of the input graph. An important feature of our algorithms is that they can work in a dynamic environment, where the cost of any edge can be changed or the edge can be ..."
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Cited by 10 (9 self)
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Abstract. We describe algorithms for finding shortest paths and distances in a planar digraph which exploit the particular topology of the input graph. An important feature of our algorithms is that they can work in a dynamic environment, where the cost of any edge can be changed or the edge can be deleted. Our data structures can be updated after any such change in only polylogarithmic time, while a singlepair query is answered in sublinear time. We also describe the first parallel algorithms for solving the dynamic version of the shortest path problem. 1
HammockonEars Decomposition: A Technique for the Efficient Parallel Solution of Shortest Paths and Other Problems
 Theoretical Computer Science
, 1996
"... We show how to decompose efficiently in parallel any graph into a number, ~ fl, of outerplanar subgraphs (called hammocks) satisfying certain separator properties. Our work combines and extends the sequential hammock decomposition technique introduced by G. Frederickson and the parallel ear decom ..."
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Cited by 8 (5 self)
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We show how to decompose efficiently in parallel any graph into a number, ~ fl, of outerplanar subgraphs (called hammocks) satisfying certain separator properties. Our work combines and extends the sequential hammock decomposition technique introduced by G. Frederickson and the parallel ear decomposition technique, thus we call it the hammockonears decomposition. We mention that hammockonears decomposition also draws from techniques in computational geometry and that an embedding of the graph does not need to be provided with the input. We achieve this decomposition in O(logn log log n) time using O(n + m) CREW PRAM processors, for an nvertex, medge graph or digraph. The hammockonears decomposition implies a general framework for solving graph problems efficiently. Its value is demonstrated by a variety of applications on a significant class of (di)graphs, namely that of sparse (di)graphs. This class consists of all (di)graphs which have a ~ fl between 1 and \Theta(n...
Computing the Girth of a Planar Graph
 In Proc. 27th International Colloquium on Automata, Languages and Programming ICALP 2000, volume 1853 of LNCS
, 2000
"... The girth of a graph G has been de ned as the length of a shortest cycle of G. We design an O(n 5=4 log n) algorithm for finding the girth of an undirected nvertex planar graph, giving the first o(n 2 ) algorithm for this problem. Our approach combines several techniques such as graph separation, h ..."
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Cited by 6 (0 self)
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The girth of a graph G has been de ned as the length of a shortest cycle of G. We design an O(n 5=4 log n) algorithm for finding the girth of an undirected nvertex planar graph, giving the first o(n 2 ) algorithm for this problem. Our approach combines several techniques such as graph separation, hammock decomposition, covering of a planar graph with graphs of small treewidth, and dynamic shortest path computation. We discuss extensions and generalizations of our result.
A Dynamic Separator Algorithm
 IN PROC. 3RD WORKSH. ALGORITHMS AND DATA STRUCTURES
, 1993
"... Our work is based on the pioneering work in sphere separators done by Miller, Teng, Vavasis et al, [8, 12], who gave efficient static (fixed input) algorithms for finding sphere separators of size s(n) = O(n d ) for a set of points in R . We present ..."
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Cited by 4 (0 self)
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Our work is based on the pioneering work in sphere separators done by Miller, Teng, Vavasis et al, [8, 12], who gave efficient static (fixed input) algorithms for finding sphere separators of size s(n) = O(n d ) for a set of points in R . We present
Algorithms for Approximate Shortest Path Queries on Weighted Polyhedral Surfaces
, 2008
"... We consider the well known geometric problem of determining shortest paths between pairs of points on a polyhedral surface P, where P consists of triangular faces with positive weights assigned to them. The cost of a path in P is defined to be the weighted sum of Euclidean lengths of the subpaths w ..."
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Cited by 3 (0 self)
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We consider the well known geometric problem of determining shortest paths between pairs of points on a polyhedral surface P, where P consists of triangular faces with positive weights assigned to them. The cost of a path in P is defined to be the weighted sum of Euclidean lengths of the subpaths within each face of P. We present query algorithms that compute approximate distances and/or approximate shortest paths on P. Our allpairs query algorithms take as input an approximation parameter ε ∈ (0,1) and a query time parameter q, in a certain range, and builds a data structure APQ(P,ε;q), which is then used for answering εapproximate distance queries in O(q) time. As a building block of the APQ(P,ε;q) data structure, we develop a single source query data structure SSQ(a;P,ε) that can answer εapproximate distance queries from a fixed point a to any query point on P in logarithmic time. Our algorithms answer shortest path queries in weighted surfaces, which is an important extension, both theoretically and practically, to the extensively studied Euclidean distance case. In addition, our algorithms improve upon previously known query algorithms for shortest paths on surfaces. The algorithms are based on a novel graph separator algorithm introduced and analyzed here, which extends and generalizes previously known separator algorithms.
Efficient Sequential and Parallel Algorithms for the Negative Cycle Problem ⋆ (to appear in ISAAC’94)
"... Abstract. We present here an algorithm for detecting (and outputting, if exists) a negative cycle in an nvertex planar digraph G with real edge weights. Its running time ranges from O(n) up to O(n 1.5 log n) as a certain topological measure of G varies from 1 up to Θ(n). Moreover, an efficient CREW ..."
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Cited by 2 (1 self)
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Abstract. We present here an algorithm for detecting (and outputting, if exists) a negative cycle in an nvertex planar digraph G with real edge weights. Its running time ranges from O(n) up to O(n 1.5 log n) as a certain topological measure of G varies from 1 up to Θ(n). Moreover, an efficient CREW PRAM implementation is given. Our algorithm applies also to digraphs whose genus γ is o(n). 1
Transmissions in a Network with Capacities and Delays
, 1996
"... We examine the problem of transmitting in minimum time a given amount of data between a source and a destination in a network with finite channel capacities and nonzero propagation delays. In the absence of delays, the problem has been shown to be solvable in polynomial time. In this paper, we sho ..."
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Cited by 1 (0 self)
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We examine the problem of transmitting in minimum time a given amount of data between a source and a destination in a network with finite channel capacities and nonzero propagation delays. In the absence of delays, the problem has been shown to be solvable in polynomial time. In this paper, we show that the general problem is NPcomplete. In addition, we examine transmissions along a single path, called the quickest path, and present algorithms for general and special classes of networks that improve upon previous approaches. The first dynamic algorithm for the quickest path problem is also given. Keywords: Data transmission, sparse network, transmission time, quickest path, dynamic algorithm. 1 Introduction Consider an nnode, medge network N = (V; E; c; l), where G = (V; E) is a directed graph, c : E ! IR + is the capacity function and l : E ! IR is the delay function. The nodes represent transmitters/receivers without data memories and the edges represent communication ...
Edge disjoint paths and multicut problems in graphs generalizing the trees
, 2005
"... We generalize all the results obtained for maximum integer multiflow and minimum multicut problems in trees by Garg et al. [Primaldual approximation algorithms for integral flow and multicut in trees. Algorithmica 18 (1997) 3–20] to graphs with a fixed cyclomatic number, while this cannot be achieve ..."
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Cited by 1 (0 self)
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We generalize all the results obtained for maximum integer multiflow and minimum multicut problems in trees by Garg et al. [Primaldual approximation algorithms for integral flow and multicut in trees. Algorithmica 18 (1997) 3–20] to graphs with a fixed cyclomatic number, while this cannot be achieved for other classical generalizations of the trees. Moreover, we prove that the minimum multicut problem with a fixed number of sourcesink pairs is polynomialtime solvable in planar and in bounded treewidth graphs. Eventually, we introduce the class of kedgeouterplanar graphs and show that the integrality gap of the maximum edgedisjoint paths problem is bounded in these graphs. We also provide stronger results for cacti (k = 1).