On Dynamic Approximate Shortest Paths for Planar Graphs with Worst-Case Costs.

Given a base weighted planar graph Ginput on n nodes and parameters M, ε we present a dynamic distance oracle with 1 + ε stretch and worst case update and query costs of ε--3M4 · poly-log(n). We allow arbitrary edge weight updates as long as the shortest path metric induced by the updated graph has stretch of at most M relative to the shortest path metric of the base graph Ginput. For example, on a planar road network, we can support fast queries and dynamic traffic updates as long as the shortest path from any source to any target (including using arbitrary detours) is between, say, 80 and 3 miles-per-hour. As a warm-up we also prove that graphs of bounded treewidth have exact distance oracles in the dynamic edge model. To the best of our knowledge, this is the first dynamic distance oracle for a non-trivial family of dynamic changes to planar graphs with worst case costs of o(n1/2) both for query and for update operations.