8,177 research outputs found

    Near Optimal Parallel Algorithms for Dynamic DFS in Undirected Graphs

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    Depth first search (DFS) tree is a fundamental data structure for solving graph problems. The classical algorithm [SiComp74] for building a DFS tree requires O(m+n)O(m+n) time for a given graph GG having nn vertices and mm edges. Recently, Baswana et al. [SODA16] presented a simple algorithm for updating DFS tree of an undirected graph after an edge/vertex update in O~(n)\tilde{O}(n) time. However, their algorithm is strictly sequential. We present an algorithm achieving similar bounds, that can be adopted easily to the parallel environment. In the parallel model, a DFS tree can be computed from scratch using mm processors in expected O~(1)\tilde{O}(1) time [SiComp90] on an EREW PRAM, whereas the best deterministic algorithm takes O~(n)\tilde{O}(\sqrt{n}) time [SiComp90,JAlg93] on a CRCW PRAM. Our algorithm can be used to develop optimal (upto polylog n factors deterministic algorithms for maintaining fully dynamic DFS and fault tolerant DFS, of an undirected graph. 1- Parallel Fully Dynamic DFS: Given an arbitrary online sequence of vertex/edge updates, we can maintain a DFS tree of an undirected graph in O~(1)\tilde{O}(1) time per update using mm processors on an EREW PRAM. 2- Parallel Fault tolerant DFS: An undirected graph can be preprocessed to build a data structure of size O(m) such that for a set of kk updates (where kk is constant) in the graph, the updated DFS tree can be computed in O~(1)\tilde{O}(1) time using nn processors on an EREW PRAM. Moreover, our fully dynamic DFS algorithm provides, in a seamless manner, nearly optimal (upto polylog n factors) algorithms for maintaining a DFS tree in semi-streaming model and a restricted distributed model. These are the first parallel, semi-streaming and distributed algorithms for maintaining a DFS tree in the dynamic setting.Comment: Accepted to appear in SPAA'17, 32 Pages, 5 Figure

    Anytime Hierarchical Clustering

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    We propose a new anytime hierarchical clustering method that iteratively transforms an arbitrary initial hierarchy on the configuration of measurements along a sequence of trees we prove for a fixed data set must terminate in a chain of nested partitions that satisfies a natural homogeneity requirement. Each recursive step re-edits the tree so as to improve a local measure of cluster homogeneity that is compatible with a number of commonly used (e.g., single, average, complete) linkage functions. As an alternative to the standard batch algorithms, we present numerical evidence to suggest that appropriate adaptations of this method can yield decentralized, scalable algorithms suitable for distributed/parallel computation of clustering hierarchies and online tracking of clustering trees applicable to large, dynamically changing databases and anomaly detection.Comment: 13 pages, 6 figures, 5 tables, in preparation for submission to a conferenc
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