We provide new algorithms for maintaining approximate distances in a weighted undirected graph G=(V,E ) subject to edge deletions. Our first result is an algorithm that maintains (1+\epsilon) -approximate distances from a set of s sources in (sm) total update time , assuming that s= n^{\Omega(1), \epsilon = \Omega(1) and |E|= n^{1+\Omega(1)}. This }  matches the best known static algorithm , up to polylogarithmic factors for a wide range of settings. The currently best known algorithm for the problem is obtained by running the single-source algorithm of Henzinger, Krinninger and Nanongkai, FOCS'14%DIFDELCMD < ] %%% independently from each source. Our result improves over the update time bound of this solution by removing a 2 ^{\tilde{O(\log^{3/4} n)} factor. Additionally, we can maintain a } (\log^{3/4} n)}[HKN, FOCS'14]. Our decremental hopset algorithm allows us to obtain the following improved decremental algorithms for maintaining shortest paths. -} (1+\epsilon)-approximate single-source shortest paths with amortized update time of 2^{\tilde{O}(\sqrt{\log n})} , when 0< \epsilon<1 is a constant and |E|= n2^{\tilde{\Omega(\sqrt{\log n})}} . This improves  over the best known  update time of 2^{\tilde{O}(\log^{3/4} n)} by [ Henzinger, Krinninger and Nanongkai, FOCS'14]. Furthermore, for any integer k \geq 1 we give an algorithmfor maintaining\tilde{O}, and |E|= n^{1+\Omega(1)}. In this regime, we give the first decremental algorithm, whose running time matches, up to polylogarithmic factors, the best known static algorithm. -} (2k-1)(1+\epsilon)-approximate all-pairs-shortest-paths , in \tilde{O}( mn^{1/k update time and O(k) query time \footnote{Throughout this paper we use the notation%DIFDELCMD < \tilde{O}%%% (f(n)) to hide factors of O(\text{polylog (f(n))).}}%DIFAUXCMD \addtocounter{footnote}{-1}%DIFAUXCMD . This improves over the result of\tilde{O})\cdot(1/\epsilon)^{O(\sqrt{\log n})} } [Chechik, FOCS'18] in a twofold way. Namely, we improve the total update time bound by removing an n^{o(1) reduce the query time from O(\log \log (nW)) to  O(k) . Our results are based on a new decremental hopset construction that may be of independent interest .
Given a weighted undirected graph G=(V,E ,w), a hopset H of hopbound \beta and stretch (1+\epsilon) is a set of edges such that for any pair of nodes u, v \in V, there is a path in G \cup H of at most \beta hops, whose length is within a (1+\epsilon) factor from the distance between u and v in G. We provide a decremental algorithm for maintaining hopsets with a polylogarithmic hopbound, with a total update time , \epsilon = \Omega(1) and |E|= n^{1+\Omega(1)}. This } that matches the best known static algorithm up to polylogarithmic factors %DIFDELCMD < ] %%% . Previously, the best known decremental hopset algorithm had a hopbound of 2 (\log^{3/4} n)} factor. Additionally, we can maintain a } ^{\tilde{O(\log^{3/4} n)}[HKN, FOCS'14]. Our decremental hopset algorithm allows us to obtain the following improved decremental algorithms for maintaining shortest paths. -} (1+\epsilon)-approximate single source shortest paths in amortized update time of 2^{\tilde{O}(\sqrt{\log n})} (\sqrt{\log n})}} . This improves super-polynomially over the best known amortized update time of 2^{\tilde{O}(\log^{3/4} n)} by [ HKN, FOCS'14]. -( 1 +\epsilon)-approximate shortest paths from a set of s sources in\tilde{O}(s) amortized update time, assuming that s= n^{\Omega(1), and |E|= n^{1+\Omega(1)}. In this regime, we give the first decremental algorithm, whose running time matches, up to polylogarithmic factors, the best known static algorithm. -} (2k-1)(1+\epsilon)-approximate all-pairs shortest paths (for any constant k \geq 2) , in \tilde{O}( n^{1/k update time and O(k) query time \footnote{%DIFDELCMD < \tilde{O}%%% (f(n))).}}%DIFAUXCMD \addtocounter{footnote}{-1}%DIFAUXCMD . This improves over the best-known amortized update time of\tilde{O}(n^{1/k)\cdot(1/\epsilon)^{O(\sqrt{\log n})} } [Chechik, FOCS'18] . Moreover, we reduce the query time from O(\log \log (nW)) to a constant O(k) , and hence eliminate the dependence on n and the aspect ratio W .
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