In the decremental single-source shortest paths problem, the goal is to maintain distances from a fixed source <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">s</tex> to every vertex <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">v</tex> in an m-edge graph undergoing edge deletions. In this paper, we conclude a long line of research on this problem by showing a near-optimal deterministic data structure that maintains (1 + E) -approximate distance estimates and runs in m 1+o(1) total update time. Our result, in particular, removes the oblivious adversary assumption required by the previous breakthrough result by Henzinger et al. [FOCS'14], which leads to our second result: the first almost-linear time algorithm for (1 - E) -approximate min-cost flow in undirected graphs where capacities and costs can be taken over edges and vertices. Previously, algorithms for max flow with vertex capacities, or min-cost flow with any capacities required super-linear time. Our result essentially completes the picture for approximate flow in undirected graphs. The key technique of the first result is a novel framework that allows us to treat low-diameter graphs like expanders. This allows us to harness expander properties while bypassing shortcomings of expander decomposition, which almost all previous expander-based algorithms needed to deal with. For the second result, we break the notorious flow-decomposition barrier from the multiplicative-weight-update framework using randomization.
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Bernstein et al. (2022) studied this question.