42,784 research outputs found

    Improved Bounds for Shortest Paths in Dense Distance Graphs

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    We study the problem of computing shortest paths in so-called dense distance graphs, a basic building block for designing efficient planar graph algorithms. Let G be a plane graph with a distinguished set partial{G} of boundary vertices lying on a constant number of faces of G. A distance clique of G is a complete graph on partial{G} encoding all-pairs distances between these vertices. A dense distance graph is a union of possibly many unrelated distance cliques. Fakcharoenphol and Rao [Fakcharoenphol and Rao, 2006] proposed an efficient implementation of Dijkstra\u27s algorithm (later called FR-Dijkstra) computing single-source shortest paths in a dense distance graph. Their algorithm spends O(b log^2{n}) time per distance clique with b vertices, even though a clique has b^2 edges. Here, n is the total number of vertices of the dense distance graph. The invention of FR-Dijkstra was instrumental in obtaining such results for planar graphs as nearly-linear time algorithms for multiple-source-multiple-sink maximum flow and dynamic distance oracles with sublinear update and query bounds. At the heart of FR-Dijkstra lies a data structure updating distance labels and extracting minimum labeled vertices in O(log^2{n}) amortized time per vertex. We show an improved data structure with O((log^2{n})/(log^2 log n)) amortized bounds. This is the first improvement over the data structure of Fakcharoenphol and Rao in more than 15 years. It yields improved bounds for all problems on planar graphs, for which computing shortest paths in dense distance graphs is currently a bottleneck

    Reliable Hubs for Partially-Dynamic All-Pairs Shortest Paths in Directed Graphs

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    We give new partially-dynamic algorithms for the all-pairs shortest paths problem in weighted directed graphs. Most importantly, we give a new deterministic incremental algorithm for the problem that handles updates in O~(mn^(4/3) log{W}/epsilon) total time (where the edge weights are from [1,W]) and explicitly maintains a (1+epsilon)-approximate distance matrix. For a fixed epsilon>0, this is the first deterministic partially dynamic algorithm for all-pairs shortest paths in directed graphs, whose update time is o(n^2) regardless of the number of edges. Furthermore, we also show how to improve the state-of-the-art partially dynamic randomized algorithms for all-pairs shortest paths [Baswana et al. STOC\u2702, Bernstein STOC\u2713] from Monte Carlo randomized to Las Vegas randomized without increasing the running time bounds (with respect to the O~(*) notation). Our results are obtained by giving new algorithms for the problem of dynamically maintaining hubs, that is a set of O~(n/d) vertices which hit a shortest path between each pair of vertices, provided it has hop-length Omega(d). We give new subquadratic deterministic and Las Vegas algorithms for maintenance of hubs under either edge insertions or deletions

    Fully dynamic all-pairs shortest paths with worst-case update-time revisited

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    We revisit the classic problem of dynamically maintaining shortest paths between all pairs of nodes of a directed weighted graph. The allowed updates are insertions and deletions of nodes and their incident edges. We give worst-case guarantees on the time needed to process a single update (in contrast to related results, the update time is not amortized over a sequence of updates). Our main result is a simple randomized algorithm that for any parameter c>1c>1 has a worst-case update time of O(cn2+2/3log4/3n)O(cn^{2+2/3} \log^{4/3}{n}) and answers distance queries correctly with probability 11/nc1-1/n^c, against an adaptive online adversary if the graph contains no negative cycle. The best deterministic algorithm is by Thorup [STOC 2005] with a worst-case update time of O~(n2+3/4)\tilde O(n^{2+3/4}) and assumes non-negative weights. This is the first improvement for this problem for more than a decade. Conceptually, our algorithm shows that randomization along with a more direct approach can provide better bounds.Comment: To be presented at the Symposium on Discrete Algorithms (SODA) 201

    Dynamic Approximate All-Pairs Shortest Paths: Breaking the O(mn) Barrier and Derandomization

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    We study dynamic (1+ϵ)(1+\epsilon)-approximation algorithms for the all-pairs shortest paths problem in unweighted undirected nn-node mm-edge graphs under edge deletions. The fastest algorithm for this problem is a randomized algorithm with a total update time of O~(mn/ϵ)\tilde O(mn/\epsilon) and constant query time by Roditty and Zwick [FOCS 2004]. The fastest deterministic algorithm is from a 1981 paper by Even and Shiloach [JACM 1981]; it has a total update time of O(mn2)O(mn^2) and constant query time. We improve these results as follows: (1) We present an algorithm with a total update time of O~(n5/2/ϵ)\tilde O(n^{5/2}/\epsilon) and constant query time that has an additive error of 22 in addition to the 1+ϵ1+\epsilon multiplicative error. This beats the previous O~(mn/ϵ)\tilde O(mn/\epsilon) time when m=Ω(n3/2)m=\Omega(n^{3/2}). Note that the additive error is unavoidable since, even in the static case, an O(n3δ)O(n^{3-\delta})-time (a so-called truly subcubic) combinatorial algorithm with 1+ϵ1+\epsilon multiplicative error cannot have an additive error less than 2ϵ2-\epsilon, unless we make a major breakthrough for Boolean matrix multiplication [Dor et al. FOCS 1996] and many other long-standing problems [Vassilevska Williams and Williams FOCS 2010]. The algorithm can also be turned into a (2+ϵ)(2+\epsilon)-approximation algorithm (without an additive error) with the same time guarantees, improving the recent (3+ϵ)(3+\epsilon)-approximation algorithm with O~(n5/2+O(log(1/ϵ)/logn))\tilde O(n^{5/2+O(\sqrt{\log{(1/\epsilon)}/\log n})}) running time of Bernstein and Roditty [SODA 2011] in terms of both approximation and time guarantees. (2) We present a deterministic algorithm with a total update time of O~(mn/ϵ)\tilde O(mn/\epsilon) and a query time of O(loglogn)O(\log\log n). The algorithm has a multiplicative error of 1+ϵ1+\epsilon and gives the first improved deterministic algorithm since 1981. It also answers an open question raised by Bernstein [STOC 2013].Comment: A preliminary version was presented at the 2013 IEEE 54th Annual Symposium on Foundations of Computer Science (FOCS 2013
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