2021/11/17 by Aaman, Anders, Karczmarz, Adam, Łącki, Jakub +3
#Data Structures and Algorithms (cs.DS) #FOS: Computer and information sciences
paper · doi:10.48550/arxiv.2111.09376
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 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 Ω(n 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 (n2 / m) + n polylog n)-time algorithm of Thorup [J.Alg. 1999], which runs in linear time only for graphs with Ω(n2) edges. Our constant amortized cost for edge deletions in decremental connectivity in non-sparse graphs should be contrasted with an Ω(log n/loglog n) worst-case time lower bound in the decremental setting [Alstrup, Thore Husfeldt, FOCS'98] as well as an Ω(log n) amortized time lower-bound in the fully-dynamic setting [Patrascu and Demaine STOC'04].