What Else Can Voronoi Diagrams Do for Diameter in Planar Graphs?

The Voronoi diagrams technique, introduced by Cabello [SODA’17] to compute the diameter of planar graphs in subquadratic time, has revolutionized the field of distance computations in planar graphs. We present novel applications of this technique in static, fault-tolerant, and partially-dynamic undirected unweighted planar graphs, as well as some new limitations. In the static case, we give n³⁺o⁽¹⁾/D² and Õ(n·D²) time algorithms for computing the diameter of a planar graph G with diameter D. These are faster than the state of the art Õ(n⁵/³) [SODA’18] when D < n¹/³ or D > n²/³. In the fault-tolerant setting, we give an n⁷/3+o⁽¹⁾ time algorithm for computing the diameter of G \ {e} for every edge e in G (the replacement diameter problem). This should be compared with the naive Õ(n⁸/³) time algorithm that runs the static algorithm for every edge. In the incremental setting, where we wish to maintain the diameter while adding edges, we present an algorithm with total running time n⁷/3+o⁽¹⁾. This should be compared with the naive Õ(n⁸/³) time algorithm that runs the static algorithm after every update. We give a lower bound (conditioned on the SETH) ruling out an amortized O(n^1−ε) update time for maintaining the diameter in weighted planar graph. The lower bound holds even for incremental or decremental updates. Our upper bounds are obtained by novel uses and manipulations of Voronoi diagrams. These include maintaining the Voronoi diagram when edges of the graph are deleted, allowing the sites of the Voronoi diagram to lie on a BFS tree level (rather than on boundaries of r-division), and a new reduction from incremental diameter to incremental distance oracles that could be of interest beyond planar graphs. Our lower bound is the first lower bound for a dynamic planar graph problem that is conditioned on the SETH.