In this paper we consider the decremental single-source shortest paths (SSSP) problem, where given a graph $G$ and a source node $s$ the goal is to maintain shortest distances between $s$ and all other nodes in $G$ under a sequence of online adversarial edge deletions. In their seminal work, Even and Shiloach [JACM 1981] presented an exact solution to the problem in unweighted graphs with only $O(mn)$ total update time over all edge deletions. Their classic algorithm was the state of the art for the decremental SSSP problem for three decades, even when approximate shortest paths are allowed. A series of results showed how to improve upon $O(mn)$ if approximation is allowed, culminating in a recent breakthrough of Henzinger, Krinninger and Nanongkai [FOCS 14], who presented a $(1+epsilon)$-approximate algorithm for undirected weighted graphs whose total update time is near linear: $O(m^{1+o(1)}log(W))$, where $W$ is the ratio of the heaviest to the lightest edge weight in the graph. In this paper they posed as a major open problem the question of derandomizing their result. Until very recently, all known improvements over the Even-Shiloach algorithm were randomized and required the assumption of a non-adaptive adversary. In STOC 2016, Bernstein and Chechik showed the first emph{deterministic} algorithm to go beyond $O(mn)$ total update time: the algorithm is also $(1+epsilon)$-approximate, and has total update time $tilde{O}(n^2)$. In SODA 2017, the same authors presented an algorithm with total update time $tilde{O}(mn^{3/4})$. However, both algorithms are restricted to undirected, unweighted graphs. We present the emph{first} deterministic algorithm for emph{weighted} undirected graphs to go beyond the $O(mn)$ bound. The total update time is $tilde{O}(n^2 log(W))$.
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