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• W2962916904 endingPage "1363" @default.
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• W2962916904 abstract "Depth first search (DFS) tree is a fundamental data structure for solving various problems in graphs. It is well known that it takes $O(m+n)$ time to build a DFS tree for a given undirected graph $G=(V,E)$ on $n$ vertices and $m$ edges. We address the problem of maintaining a DFS tree when the graph is undergoing updates (insertion and deletion of vertices or edges). We present the following results for this problem: (1) Fault tolerant DFS tree: There exists a data structure of size $tilde{O}(m)$ (where $tilde{O}$ hides the polylogarithmic factors) which can be preprocessed in $tilde{O}(m)$ time such that given any set ${cal F}$ of failed vertices or edges, a DFS tree of the graph $Gsetminus {cal F}$ can be reported in $tilde{O}(n|{cal F}|)$ time. (2) Fully dynamic DFS tree: There exists a fully dynamic algorithm for maintaining a DFS tree that takes $tilde{O}(m)$ time for preprocessing and worst case $tilde{O}(sqrt{mn})$ time per update for any arbitrary online sequence of updates. (3) Incremental DFS tree: There exists an incremental algorithm for maintaining a DFS tree that takes $tilde{O}(m)$ time for preprocessing and worst case $tilde{O}(n)$ time per update for any arbitrary online sequence of edge insertion. These are the first $o(m)$ worst case time results for maintaining a DFS tree of a dense graph in a dynamic environment. Moreover, our fully dynamic algorithm provides, in a seamless manner, the first deterministic algorithm for dense graphs with $O(1)$ query time and $o(m)$ worst case update time for connectivity, biconnectivity, and 2-edge connectivity in the dynamic subgraph model." @default.
• W2962916904 created "2019-07-30" @default.
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• W2962916904 date "2019-01-01" @default.
• W2962916904 modified "2023-10-02" @default.
• W2962916904 title "Dynamic DFS in Undirected Graphs: Breaking the $O(m)$ Barrier" @default.
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• W2962916904 doi "https://doi.org/10.1137/17m114306x" @default.
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