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36
Improved Algorithms for Dynamic Shortest Paths
, 2000
"... 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 ca ..."
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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. 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.
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 9 (8 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...
EFFICIENT PARALLEL ALGORITHMS FOR SHORTEST PATHS IN PLANAR DIGRAPHS
 BIT 32 (1992),215236
, 1992
"... Efficient parallel algorithms are presented, on the CREW PRAM model, for generating a succinct encoding of all pairs shortest path information in a directed planar graph G with realvalued edge costs but no negative cycles. We assume that a planar embedding of G is given, togetber with a set of q fa ..."
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Cited by 7 (4 self)
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Efficient parallel algorithms are presented, on the CREW PRAM model, for generating a succinct encoding of all pairs shortest path information in a directed planar graph G with realvalued edge costs but no negative cycles. We assume that a planar embedding of G is given, togetber with a set of q faces that cover all the vertices. Then our algorithm runs in O(log 2 n) time and employs O{nq + M(q)) processors (where M(t) is the number of processors required to multiply two t x t matrices in O(log t) time). Let us note here that whenever q < n then our processor bound is better than the best previous one (M(n)). O(log 2 n) time, nprocessor algorithms are presented for various subproblems, including that of generating all pairs shortest path information in a directed outerplanar graph. Our work is based on the fundamental hammockdecomposition technique ofG. Frederickson. We achieve this decomposition in O(log n log * n) parallel time by using O(n) processors. The hammockdecomposition seems to be a fundamental operation that may help in improving efficiency of many parallel (and sequential) graph algorithms.
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.
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 5 (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.
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 5 (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
Efficient Sequential and Parallel Algorithms for the Negative Cycle Problem
 TO APPEAR IN ISAAC’94
, 1994
"... 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 imp ..."
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Cited by 4 (3 self)
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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).
On the computation of fast data transmissions in networks with capacities and delays
 In Proc. Workshop on Algorithms and Data Structures
, 1995
"... Abstract. 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 non–zero propagation delays. In the absence of delays, the problem has been shown to be solvable in polynomial time. In this pape ..."
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Abstract. 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 non–zero 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 NP–hard. In addition, we examine transmissions along a single path, called the quickest path, and present algorithms for general and sparse networks that outperform previous approaches. The first dynamic algorithm for the quickest path problem is also given. 1