Optimal Decremental Connectivity in Non-Sparse Graphs

Anders Aaman; Adam Karczmarz; Jakub Łącki; Nikos Parotsidis; Peter M. R. Rasmussen; Mikkel Thorup

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arXiv · 2111.09376

Optimal Decremental Connectivity in Non-Sparse Graphs

Anders Aaman·Adam Karczmarz·Jakub Łącki·Nikos Parotsidis·Peter M. R. Rasmussen·Mikkel Thorup

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Abstract

We present a dynamic algorithm for maintaining the connected and 2-edge-connected components in an undirected graph subject to edge deletions. The algorithm is Monte-Carlo randomized and processes any sequence of edge deletions in $O(m + n \operatorname{polylog} n)$ total time. Interspersed with the deletions, it can answer queries to whether any two given vertices currently belong to the same (2-edge-)connected component in constant time. Our result is based on a general Monte-Carlo randomized reduction from decremental $c$-edge-connectivity to a variant of fully-dynamic $c$-edge-connectivity on a sparse graph. While being Monte-Carlo, our reduction supports a certain final self-check that can be used in Las Vegas algorithms for static problems such as Unique Perfect Matching. For non-sparse graphs with $\Omega(n \operatorname{polylog} n)$ edges, our connectivity and $2$-edge-connectivity algorithms handle all deletions in optimal linear total time, using existing algorithms for the respective fully-dynamic problems. This improves upon an $O(m \log (n^2 / m) + n \operatorname{polylog} n)$-time algorithm of Thorup [J.Alg. 1999], which runs in linear time only for graphs with $\Omega(n^2)$ edges. Our constant amortized cost for edge deletions in decremental connectivity in non-sparse graphs should be contrasted with an $\Omega(\log n/\log\log n)$ worst-case time lower bound in the decremental setting [Alstrup, Thore Husfeldt, FOCS'98] as well as an $\Omega(\log n)$ amortized time lower-bound in the fully-dynamic setting [Patrascu and Demaine STOC'04].

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Anders Aaman, Adam Karczmarz, Jakub Łącki, Nikos Parotsidis, Peter M. R. Rasmussen, Mikkel Thorup. 2021-11-17. Optimal Decremental Connectivity in Non-Sparse Graphs. https://arxiv.org/abs/2111.09376

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