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.
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