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Given a dynamic digraph $G = (V,E)$ undergoing edge deletions and given $sin V$ and $epsilon>0$, we consider the problem of maintaining $(1+epsilon)$-approximate shortest path distances from $s$ to all vertices in $G$ over the sequence of deletions. Even and Shiloach (J.~ACM$81$) give a deterministic data structure for the exact version of the problem with total update time $O(mn)$. Henzinger et al. (STOC$14$, ICALP$15$) give a Monte Carlo data structure for the approximate version with improved total update time $ O(mn^{0.9 + o(1)}log W)$ where $W$ is the ratio between the largest and smallest edge weight. A drawback of their data structure is that they only work against an oblivious adversary, meaning that the sequence of deletions needs to be fixed in advance. This limits its application as a black box inside algorithms. We present the following $(1+epsilon)$-approximate data structures: (1) the first data structure is Las Vegas and works against an adaptive adversary; it has total expected update time $tilde O(m^{2/3}n^{4/3})$ for unweighted graphs and $tilde O(m^{3/4}n^{5/4}log W)$ for weighted graphs, (2) the second data structure is Las Vegas and assumes an oblivious adversary; it has total expected update time $tilde O(sqrt m n^{3/2})$ for unweighted graphs and $tilde O(m^{2/3}n^{4/3}log W)$ for weighted graphs, (3) the third data structure is Monte Carlo and is correct w.h.p.~against an oblivious adversary; it has total expected update time $tilde O((mn)^{7/8}log W) = tilde O(mn^{3/4}log W)$. Each of our data structures can be queried at any stage of $G$ in constant worst-case time; if the adversary is oblivious, a query can be extended to also report such a path in time proportional to its length. Our update times are faster than those of Henzinger et al.~for all graph densities.
In the decremental Single-Source Shortest Path problem (SSSP), we are given a weighted directed graph $G=(V,E,w)$ undergoing edge deletions and a source vertex $r in V$; let $n = |V|, m = |E|$ and $W$ be the aspect ratio of the graph. The goal is to
Given a directed graph $G = (V,E)$, undergoing an online sequence of edge deletions with $m$ edges in the initial version of $G$ and $n = |V|$, we consider the problem of maintaining all-pairs shortest paths (APSP) in $G$. Whilst this problem has b
We present a new dynamic matching sparsification scheme. From this scheme we derive a framework for dynamically rounding fractional matchings against emph{adaptive adversaries}. Plugging in known dynamic fractional matching algorithms into our framew
Designing dynamic graph algorithms against an adaptive adversary is a major goal in the field of dynamic graph algorithms. While a few such algorithms are known for spanning trees, matchings, and single-source shortest paths, very little was known fo
In the decremental single-source shortest paths problem, the goal is to maintain distances from a fixed source $s$ to every vertex $v$ in an $m$-edge graph undergoing edge deletions. In this paper, we conclude a long line of research on this problem