Maintaining and updating shortest paths information in a graph is a fundamental problem with many applications. As computations on dense graphs can be prohibitively expensive, and it is preferable to perform the computations on a sparse skeleton of the given graph that roughly preserves the shortest paths information. Spanners and emulators serve this purpose. This paper develops fast dynamic algorithms for sparse spanner and emulator maintenance and provides evidence from fine-grained complexity that these algorithms are tight. Under the popular OMv conjecture, we show that there can be no decremental or incremental algorithm that maintains an $n^{1+o(1)}$ edge (purely additive) $+n^{delta}$-emulator for any $delta<1/2$ with arbitrary polynomial preprocessing time and total update time $m^{1+o(1)}$. Also, under the Combinatorial $k$-Clique hypothesis, any fully dynamic combinatorial algorithm that maintains an $n^{1+o(1)}$ edge $(1+epsilon,n^{o(1)})$-spanner or emulator must either have preprocessing time $mn^{1-o(1)}$ or amortized update time $m^{1-o(1)}$. Both of our conditional lower bounds are tight. As the above fully dynamic lower bound only applies to combinatorial algorithms, we also develop an algebraic spanner algorithm that improves over the $m^{1-o(1)}$ update time for dense graphs. For any constant $epsilonin (0,1]$, there is a fully dynamic algorithm with worst-case update time $O(n^{1.529})$ that whp maintains an $n^{1+o(1)}$ edge $(1+epsilon,n^{o(1)})$-spanner. Our new algebraic techniques and spanner algorithms allow us to also obtain (1) a new fully dynamic algorithm for All-Pairs Shortest Paths (APSP) with update and path query time $O(n^{1.9})$; (2) a fully dynamic $(1+epsilon)$-approximate APSP algorithm with update time $O(n^{1.529})$; (3) a fully dynamic algorithm for near-$2$-approximate Steiner tree maintenance.
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