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Planar spanners and approximate shortest path queries among obstacles
 in the plane, Proc. 4th European Sympos. Algorithms
, 1996
"... Abstract. We consider the problem of finding an obstacleavoiding path between two points s and t in the plane, amidst a set of disjoint polygonal obstacles with a total of n vertices. The length of this path should be within a small constant factor c of the length of the shortest possible obstacle ..."
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Cited by 40 (14 self)
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Abstract. We consider the problem of finding an obstacleavoiding path between two points s and t in the plane, amidst a set of disjoint polygonal obstacles with a total of n vertices. The length of this path should be within a small constant factor c of the length of the shortest possible obstacleavoiding st path measured in the Lvmetric. Such an approximate shortest path is called a cshort path, or a short path with stretch]actor c. The goal is to preprocess the obstaclescattered plane by creating an efficient data structure that enables fast reporting of a cshort path (or its length). In this paper, we give a family of algorithms for the above problem that achieve an interesting tradeoff between the stretch factor, the query time and the preprocessing bounds. Our main results are algorithms that achieve logarithmic length query time, after subquadratic time and space preprocessing. 1
Shortest Paths in Digraphs of Small Treewidth. Part I: Sequential Algorithms
, 1995
"... We consider the problem of preprocessing an nvertex digraph with real edge weights so that subsequent queries for the shortest path or distance between any two vertices can be efficiently answered. We give algorithms that depend on the treewidth of the input graph. When the treewidth is a consta ..."
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Cited by 35 (4 self)
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We consider the problem of preprocessing an nvertex digraph with real edge weights so that subsequent queries for the shortest path or distance between any two vertices can be efficiently answered. We give algorithms that depend on the treewidth of the input graph. When the treewidth is a constant, our algorithms can answer distance queries in O(ff(n)) time after O(n) preprocessing. This improves upon previously known results for the same problem. We also give a dynamic algorithm which, after a change in an edge weight, updates the data structure in time O(n fi ), for any constant 0 ! fi ! 1. Furthermore, an algorithm of independent interest is given: computing a shortest path tree, or finding a negative cycle in linear time.
Improved Algorithms for Dynamic Shortest Paths
 Algorithmica
, 1996
"... We describe algorithms for finding shortest paths and distances in outerplanar and planar digraphs that 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 ..."
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Cited by 15 (3 self)
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We describe algorithms for finding shortest paths and distances in outerplanar and planar digraphs that 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. In the case of outerplanar digraphs, our data structures can be updated after any such change in only logarithmic time and a distance query is answered also in logarithmic time. In the case of planar digraphs, we give an interesting tradeoff between preprocessing, query and update times depending on the value of a certain topological parameter of the graph. Our results can be extended to nvertex digraphs of genus O(n 1\Gamma" ) for any " ? 0. Keywords: Shortest path, dynamic algorithm, planar digraph, outerplanar digraph. This work was partially supported by the NSF grant No. CCR9409191 and by the EU ESPRIT LTR Project No. 20244 (ALCOMIT). 1 Introduction 1.1 The prob...
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...
AllPairs MinCut in Sparse Networks
, 1996
"... Algorithms are presented for the allpairs mincut problem in bounded treewidth, planar and sparse networks. The approach used is to preprocess the input nvertex network so that, afterwards, the value of a mincut between any two vertices can be efficiently computed. A tradeoff is shown between th ..."
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Cited by 5 (1 self)
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Algorithms are presented for the allpairs mincut problem in bounded treewidth, planar and sparse networks. The approach used is to preprocess the input nvertex network so that, afterwards, the value of a mincut between any two vertices can be efficiently computed. A tradeoff is shown between the preprocessing time and the time taken to compute mincuts subsequently. In particular, after an O(n log n) preprocessing of a bounded treewidth network, it is possible to find the value of a mincut between any two vertices in constant time. This implies that for such networks the allpairs mincut problem can be solved in time O(n 2 ). This algorithm is used in conjunction with a graph decomposition technique of Frederickson to obtain algorithms for sparse and planar networks. The running times depend upon a topological property, fl, of the input network. The parameter fl varies between 1 and \Theta(n); the algorithms perform well when fl = o(n). The value of a mincut can be found in t...
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 ...
© 2000 SpringerVerlag New York Inc. Shortest Paths in Digraphs of Small Treewidth. Part I: Sequential Algorithms 1
"... Abstract. We consider the problem of preprocessing an nvertex digraph with real edge weights so that subsequent queries for the shortest path or distance between any two vertices can be efficiently answered. We give algorithms that depend on the treewidth of the input graph. When the treewidth is a ..."
Abstract
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Abstract. We consider the problem of preprocessing an nvertex digraph with real edge weights so that subsequent queries for the shortest path or distance between any two vertices can be efficiently answered. We give algorithms that depend on the treewidth of the input graph. When the treewidth is a constant, our algorithms can answer distance queries in O(α(n)) time after O(n) preprocessing. This improves upon previously known results for the same problem. We also give a dynamic algorithm which, after a change in an edge weight, updates the data structure in time O(nβ), for any constant 0 <β<1. Furthermore, an algorithm of independent interest is given: computing a shortest path tree, or finding a negative cycle in linear time. Key Words. Shortest path, Graph theory, Treewidth, Dynamic algorithm.
© 2000 SpringerVerlag New York Inc. Improved Algorithms for Dynamic Shortest Paths 1
"... Abstract. We describe algorithms for finding shortest paths and distances in outerplanar and planar digraphs that 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 th ..."
Abstract
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Abstract. We describe algorithms for finding shortest paths and distances in outerplanar and planar digraphs that 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. In the case of outerplanar digraphs, our data structures can be updated after any such change in only logarithmic time. A distance query is also answered in logarithmic time. In the case of planar digraphs, we give an interesting tradeoff between preprocessing, query, and update times depending on the value of a certain topological parameter of the graph. Our results can be extended to nvertex digraphs of genus O(n1−ε) for any ε>0. Key Words.
AllPairs MinCut in Sparse Networks? (FSTTCS'95)
"... Abstract. Algorithms for the allpairs mincut problem in bounded treewidth and sparse networks are presented. The approach usedisto preprocess the input network so that, afterwards, the value of a mincut between any two vertices can be e ciently computed. A tradeo between the preprocessing time a ..."
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Abstract. Algorithms for the allpairs mincut problem in bounded treewidth and sparse networks are presented. The approach usedisto preprocess the input network so that, afterwards, the value of a mincut between any two vertices can be e ciently computed. A tradeo between the preprocessing time and the time taken to compute mincuts subsequently is shown. In particular, after O(n log n) preprocessing of a bounded treewidth network, it is possible to nd the value of a mincut between any two vertices in constant time. This implies that for such networks the allpairs mincut problem can be solved in time O(n 2). This algorithm is used in conjunction with a graph decomposition technique of Frederickson to obtain algorithms for sparse networks. The running times depend upon a topological property of the input network. The parameter varies between 1 and (n) � the algorithms perform well when = o(n). The value of a mincut can be found in time O(n + 2 log) and allpairs mincut can be solved in time O(n 2 + 4 log). 1