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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 ..."
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Cited by 86 (8 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.
On dynamic shortest paths problems
 In ESA: Annual European Symposium on Algorithms
, 2004
"... Abstract. We obtain the following results related to dynamic versions of the shortestpaths problem: (i) Reductions that show that the incremental and decremental singlesource shortestpaths problems, for weighted directed or undirected graphs, are, in a strong sense, at least as hard as the static ..."
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Cited by 39 (2 self)
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Abstract. We obtain the following results related to dynamic versions of the shortestpaths problem: (i) Reductions that show that the incremental and decremental singlesource shortestpaths problems, for weighted directed or undirected graphs, are, in a strong sense, at least as hard as the static allpairs shortestpaths problem. We also obtain slightly weaker results for the corresponding unweighted problems. (ii) A randomized fullydynamic algorithm for the allpairs shortestpaths problem in directed unweighted graphs with an amortized update time of Õ(m n) and a worst case query time is O(n3/4). (iii) A deterministic O(n2 log n) time algorithm for constructing a (log n)spanner with O(n) edges for any weighted undirected graph on n vertices. The algorithm uses a simple algorithm for incrementally maintaining singlesource shortestpaths tree up to a given distance. 1
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 34 (5 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.
Improved dynamic algorithms for maintaining approximate shortest paths under deletions
 In 22nd ACM Symp. on Discrete Algorithms (SODA
"... We present the first dynamic shortest paths algorithms that make any progress beyond a longstanding O(n) update time barrier (while maintaining a reasonable query time), although it is only progress for nottoosparse graphs. In particular, we obtain new decremental algorithms for two approximate sh ..."
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Cited by 5 (0 self)
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We present the first dynamic shortest paths algorithms that make any progress beyond a longstanding O(n) update time barrier (while maintaining a reasonable query time), although it is only progress for nottoosparse graphs. In particular, we obtain new decremental algorithms for two approximate shortestpath problems in unweighted, undirected graphs. Both algorithms are randomized (Las Vegas). • Given a source s, we present an algorithm that maintains (1 + ɛ)approximate shortest paths from s with an expected total update time of Õ(n 2+O(1/ √ log n)) over all deletions (so the amortized time is about Õ(n2 /m)). The worstcase
Fully Dynamic Approximate Distance Oracles for Planar Graphs via ForbiddenSet Distance Labels
, 2012
"... 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) · ( ..."
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Cited by 2 (1 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) dynamic distance oracle for planar graphs, which has worst case query time Õ(n2/3) and amortized update time 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 ..."
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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