First Near-Additive All-Pairs Approximate Shortest Paths in Decremental Weighted Graphs
We present the first four All-Pairs Approximate Shortest Paths (APASP) algorithms for decremental weighted undirected graphs. We extend the hierarchy induced edge sets of Baswana and Kavitha [FOCS'06 and SICOMP'10] to the decremental setting. At any point in time, for any $u,v$, $W_{i}$ denotes the weight of the $i$-th heaviest edge on any shortest path between them in the current graph. For any $\varepsilon>0$, our first algorithm maintains $(1+\varepsilon,2W_{2})$-APASP in $\tilde{O}(n^{\frac{3}{2}+o(1)}m^{\frac{1}{2}})$ total update time, matching a single static computation of Cohen and Zwick [SODA'97 and JALG'01], up to a subpolynomial factor. It also improves the stretch of the $(2+\varepsilon)$-APASP algorithm of Dory, Forster, Nazari and de Vos [ICALP'24]. Our second algorithm maintains $(1+\varepsilon,2W_{1}+2W_{2})$-APASP in $\tilde{O}(n^{\frac{7}{3}})$ total update time, improving upon Chechik's $(3+\varepsilon)$-APASP algorithm [FOCS'18] for $m=Ω(n^{\frac{11}{6}})$. We also obtain two $(1+\varepsilon,2\cdot(k-1)W_{1})$-APASP algorithms with total update times $\tilde{O}(n^{2-\frac{1}{k}+o(1)}m^{\frac{1}{k}})$ and $\tilde{O}(n^{2+\frac{2}{3k-2}+o(1)})$, adapting the static $+2\sum_{i=1}^{k-1}W_{i}$-APASP algorithms of Cohen and Zwick [SODA'97 and JALG'01] and Roditty and Sapir [FSTTCS'25]. For unweighted graphs, we present $(1+\varepsilon,2)$-APASP in $\tilde{O}(n^{\frac{7}{3}})$ total update time and $(1+\varepsilon,2\cdot(k-1))$-APASP in $\tilde{O}(n^{2+\frac{1}{3k-4}+o(1)})$. Combined with the results of Dory, Forster, Nazari and de Vos [ICALP'24], these match the fastest combinatorial static bounds of Dor, Halperin and Zwick [FOCS'96 and SICOMP'00] up to subpolynomial factors.