Faster Planar Diameter using Pseudoline Envelopes

We give an improved algorithm for weighted directed planar diameter that runs in $\widetilde{O}(n^{3/2})$ time. This improves upon the $\widetilde{O}(n^{5/3})$ time algorithm for weighted directed planar diameter of Gawrychowski, Kaplan, Mozes, Sharir, and Weimann [SIAM J. Comput. 2021] and the recent $\widetilde{O}(n^{8/5})$ time algorithm for unweighted undirected planar diameter of Fredslund-Hansen, Mozes, and Weimann [SOSA 2027]. The main new result is an improved data structure for computing the farthest vertex when given feasible planar distances to the vertices on the outer face. The critical insight is the fact that normalized distances to the vertices of a face behave like pseudolines, for which we can leverage range searching and upper envelope data structures. Using the farthest vertex data structure, we are able to compute all vertex eccentricities and diameter in $\widetilde{O}(n^{3/2})$.