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A New Approach to Dynamic All Pairs Shortest Paths
, 2002
"... We study novel combinatorial properties of graphs that allow us to devise a completely new approach to dynamic all pairs shortest paths problems. Our approach yields a fully dynamic algorithm for general directed graphs with nonnegative realvalued edge weights that supports any sequence of operatio ..."
Abstract

Cited by 73 (9 self)
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We study novel combinatorial properties of graphs that allow us to devise a completely new approach to dynamic all pairs shortest paths problems. Our approach yields a fully dynamic algorithm for general directed graphs with nonnegative realvalued edge weights that supports any sequence of operations in e O(n amortized time per update and unit worstcase time per distance query, where n is the number of vertices. We can also report shortest paths in optimal worstcase time. These bounds improve substantially over previous results and solve a longstanding open problem. Our algorithm is deterministic and uses simple data structures.
Improved Dynamic Reachability Algorithms for Directed Graphs
, 2002
"... We obtain several new dynamic algorithms for maintaining the transitive closure of a directed graph, and several other algorithms for answering reachability queries without explicitly maintaining a transitive closure matrix. Among our algorithms are: (i) A decremental algorithm for maintaining the ..."
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Cited by 29 (3 self)
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We obtain several new dynamic algorithms for maintaining the transitive closure of a directed graph, and several other algorithms for answering reachability queries without explicitly maintaining a transitive closure matrix. Among our algorithms are: (i) A decremental algorithm for maintaining the transitive closure of a directed graph, through an arbitrary sequence of edge deletions, in O(mn) total expected time, essentially the time needed for computing the transitive closure of the initial graph. Such a result was previously known only for acyclic graphs.
Fully Dynamic Approximate Distance Oracles for Planar Graphs via ForbiddenSet Distance Labels
"... This paper considers fully dynamic (1 + ε) distance oracles and (1 + ε) forbiddenset labeling schemes for planar graphs. For a given nvertex planar graph G with edge weights drawn from [1,M]andparameterε>0, our forbiddenset labeling scheme uses labels of length λ = O(ε −1 log 2 n log (nM) · (ε − ..."
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Cited by 1 (0 self)
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This paper considers fully dynamic (1 + ε) distance oracles and (1 + ε) forbiddenset labeling schemes for planar graphs. For a given nvertex planar graph G with edge weights drawn from [1,M]andparameterε>0, our forbiddenset labeling scheme uses labels of length λ = O(ε −1 log 2 n log (nM) · (ε −1 +logn)). Given the labels of two vertices s and t and of a set F of faulty vertices/edges, our scheme approximates the distance between s and t in G \ F with stretch (1 + ε), in O(F  2 λ)time. We then present a general method to transform (1 + ε) forbiddenset labeling schemas into a fully dynamic (1 + ε) distance oracle. Our fully dynamic (1 + ε) distanceoracle is of size O(n log n · (ε −1 +logn)) and has Õ(n1/2)query and update time, both the query and the update time are worst case. This improves on the best previously known (1+ε) dynamicdistanceoracleforplanargraphs,whichhas worst case query time Õ(n2/3)andamortizedupdatetime of Õ(n2/3). Our (1 + ε) forbiddenset labeling scheme can also be extended into a forbiddenset labeled routing scheme with stretch (1 + ε).
A New Approach to Dynamic All Pairs . . .
 IN PROCEEDINGS OF THE 35TH ANNUAL ACM SYMPOSIUM ON THEORY OF COMPUTING (STOC’03
, 2003
"... We study novel combinatorial properties of graphs that allow us to devise a completely new approach to dynamic all pairs shortest paths problems. Our approach yields a fully dynamic algorithm for general directed graphs with nonnegative realvalued edge weights that supports any sequence of opera ..."
Abstract
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We study novel combinatorial properties of graphs that allow us to devise a completely new approach to dynamic all pairs shortest paths problems. Our approach yields a fully dynamic algorithm for general directed graphs with nonnegative realvalued edge weights that supports any sequence of operations in O(n time per update and unit worstcase time per distance query, where n is the number of vertices. We can also report shortest paths in optimal worstcase time. These bounds improve substantially over previous results and solve a longstanding open problem. Our
Improved Deterministic Algorithms for Decremental Transitive Closure and Strongly Connected Components
"... This paper presents a new deterministic algorithm for decremental maintenance of the transitive closure in a directed graph. The algorithm processes any sequence of edge deletions in O(mn) time and answers queries in constant time. Until now such time bound has only been achieved by a randomized Las ..."
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This paper presents a new deterministic algorithm for decremental maintenance of the transitive closure in a directed graph. The algorithm processes any sequence of edge deletions in O(mn) time and answers queries in constant time. Until now such time bound has only been achieved by a randomized Las Vegas algorithm. In addition to that, a few decremental algorithms for maintaining strongly connected components are shown, whose time complexity is O(n 1.5) for planar graphs, O(n log n) for graphs with bounded treewidth and O(mn) for general digraphs. 1