# New Tradeoffs for Decremental Approximate All-Pairs Shortest Paths

We provide new tradeoffs between approximation and running time for the decremental all-pairs shortest paths (APSP) problem. For undirected graphs with m edges and n nodes undergoing edge deletions, we provide two new approximate decremental APSP algorithms, one for weighted and one for unweighted graphs. Our first result is an algorithm that supports (2+ ϵ)-approximate all-pairs constant-time distance queries with total update time Õ(m^1/2n^3/2) when m= O(n^5/3) (and m= n^1+c for any constant c >0), or Õ(mn^2/3) when m = Ω(n^5/3). Prior to our work the fastest algorithm for weighted graphs with approximation at most 3 had total Õ(mn) update time providing a (1+ϵ)-approximation [Bernstein, SICOMP 2016]. Our technique also yields a decremental algorithm with total update time Õ(nm^3/4) supporting (2+ϵ, W_u,v)-approximate queries where the second term is an additional additive term and W_u,v is the maximum weight on the shortest path from u to v. Our second result is a decremental algorithm that given an unweighted graph and a constant integer k ≥ 2, supports (1+ϵ, 2(k-1))-approximate queries and has Õ(n^2-1/km^1/k) total update time (when m=n^1+c for any constant c >0). For comparison, in the special case of (1+ϵ, 2)-approximation, this improves over the state-of-the-art algorithm by [Henzinger, Krinninger, Nanongkai, SICOMP 2016] with total update time of Õ(n^2.5). All of our results are randomized and work against an oblivious adversary.

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