An incremental algorithm for a generalization of the shortest-path problem
Journal of Algorithms
An On-Line Edge-Deletion Problem
Journal of the ACM (JACM)
Improved decremental algorithms for maintaining transitive closure and all-pairs shortest paths
STOC '02 Proceedings of the thiry-fourth annual ACM symposium on Theory of computing
Fully dynamic biconnectivity and transitive closure
FOCS '95 Proceedings of the 36th Annual Symposium on Foundations of Computer Science
Fully Dynamic Algorithms for Maintaining All-Pairs Shortest Paths and Transitive Closure in Digraphs
FOCS '99 Proceedings of the 40th Annual Symposium on Foundations of Computer Science
Dynamic Approximate All-Pairs Shortest Paths in Undirected Graphs
FOCS '04 Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science
A new approach to dynamic all pairs shortest paths
Journal of the ACM (JACM)
Compact oracles for reachability and approximate distances in planar digraphs
Journal of the ACM (JACM)
Mathematics for the Analysis of Algorithms: Modern Birkhuser Classics
Mathematics for the Analysis of Algorithms: Modern Birkhuser Classics
FOCS '09 Proceedings of the 2009 50th Annual IEEE Symposium on Foundations of Computer Science
Highway dimension, shortest paths, and provably efficient algorithms
SODA '10 Proceedings of the twenty-first annual ACM-SIAM symposium on Discrete Algorithms
Near linear time (1 + ε)-approximation for restricted shortest paths in undirected graphs
Proceedings of the twenty-third annual ACM-SIAM symposium on Discrete Algorithms
Subquadratic algorithm for dynamic shortest distances
COCOON'05 Proceedings of the 11th annual international conference on Computing and Combinatorics
Improved dynamic algorithms for maintaining approximate shortest paths under deletions
Proceedings of the twenty-second annual ACM-SIAM symposium on Discrete Algorithms
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We present an improved algorithm for maintaining all-pairs 1 + ε approximate shortest paths under deletions and weight-increases. The previous state of the art for this problem was total update time ~O (n2√m/ε) for directed, unweighted graphs [2], and ~O(mn/ε) for undirected, unweighted graphs [12]. Both algorithms were randomized and had constant query time. Note that ~O(mn) is a natural barrier because even with a (1 + ε) approximation, there is no o(mn) combinatorial algorithm for the static all-pairs shortest path problem. Our algorithm works on directed, weighted graphs and has total (randomized) update time ~O (mn log(R)/ε) where R is the ratio of the largest edge weight ever seen in the graph, to the smallest such weight (our query time is constant). Note that log(R) = O(log(n)) as long as weights are polynomial in n. Although ~O(mn log(R)/ε) is the total time over all updates, our algorithm also requires a clearly unavoidable constant time per update. Thus, we effectively expand the ~O(mn) total update time bound from undirected, unweighted graphs to directed graphs with polynomial weights. This is in fact the first non-trivial algorithm for decremental all-pairs shortest paths that works on weighted graphs (previous algorithms could only handle small integer weights). By a well known reduction from decremental algorithms to fully dynamic ones [9], our improved decremental algorithm leads to improved query-update tradeoffs for fully dynamic (1 + ε) approximate APSP algorithm in directed graphs.