New techniques and fine-grained hardness for dynamic near-additive spanners

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. Unfortunately, very little is known about dynamically maintaining sparse spanners and emulators as the graph is modified by a sequence of edge insertions and deletions. This paper develops fast dynamic algorithms for spanner and emulator maintenance and provides evidence from fine-grained complexity that these algorithms are tight. For unweighted undirected m-edge n-node graphs we obtain the following results. Under the popular OMv conjecture, there can be no decremental or incremental algorithm that maintains an n1+o(1) edge (purely additive) +nδ-emulator for any δ < 1/2 with arbitrary polynomial preprocessing time and total update time m1+o(1). Also, under the Combinatorial k-Clique hypothesis, any fully dynamic combinatorial algorithm that maintains an n1+o(1) edge (1 + ∊, no(1))-spanner or emulator for small ∊ must either have preprocessing time mn1–o(1) or amortized update time m1–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 m1–o(1) update time for dense graphs. For any constant ∊ ∊ (0, 1], there is a fully dynamic algorithm with worst-case update time O(n1.529) that whp maintains an n1+o(1) edge (1 + ∊, no(1))-spanner. Our new algebraic techniques allow us to also obtain a new fully dynamic algorithm for All-Pairs Shortest Paths (APSP) that can perform both edge updates and can report shortest paths in worst-case time O(n1.9), which are correct whp. This is the first path-reporting fully dynamic APSP algorithm with a truly subquadratic query time that beats O(n2.5) update time. It works against an oblivious adversary. Finally, we give two applications of our new dynamic spanner algorithms: (1) a fully dynamic (1 + ∊)-approximate APSP algorithm with update time O(n1.529) that can report approximate shortest paths in n1+o(1) time per query; previous subquadratic update/query algorithms could only report the distance, but not obtain the paths; (2) a fully dynamic algorithm for near-2-approximate Steiner tree maintenance with both terminal and edge updates.