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 been studied in a long line of research [ACM81, FOCS99, FOCS01, STOC02, STOC03, SWAT04, STOC13] and the problem of $(1+epsilon)$-approximate, weighted APSP was solved to near-optimal update time $tilde{O}(mn)$ by Bernstein [STOC13], the problem has mainly been studied in the context of oblivious adversaries, which assumes that the adversary fixes the update sequence before the algorithm is started. In this paper, we make significant progress on the problem in the setting where the adversary is adaptive, i.e. can base the update sequence on the output of the data structure queries. We present three new data structures that fit different settings: We first present a deterministic data structure that maintains exact distances with total update time $tilde{O}(n^3)$. We also present a deterministic data structure that maintains $(1+epsilon)$-approximate distance estimates with total update time $tilde O(sqrt{m} n^2/epsilon)$ which for sparse graphs is $tilde O(n^{2+1/2}/epsilon)$. Finally, we present a randomized $(1+epsilon)$-approximate data structure which works against an adaptive adversary; its total update time is $tilde O(m^{2/3}n^{5/3} + n^{8/3}/(m^{1/3}epsilon^2))$ which for sparse graphs is $tilde O(n^{2+1/3})$. Our exact data structure matches the total update time of the best randomized data structure by Baswana et al. [STOC02] and maintains the distance matrix in near-optimal time. Our approximate data structures improve upon the best data structures against an adaptive adversary which have $tilde{O}(mn^2)$ total update time [JACM81, STOC03].
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