# Subquadratic Algorithms for the Diameter and the Sum of Pairwise Distances in Planar Graphs

## TL;DR

This paper introduces subquadratic algorithms for efficiently computing the diameter, sum of pairwise distances, and counting pairs within a threshold in planar graphs, improving over previous quadratic time solutions.

## Contribution

It presents the first subquadratic algorithms for these problems in planar graphs, applicable to directed, weighted, and unweighted cases, with improved expected time complexities.

## Key findings

- Algorithms run in $O(n^{11/6}{
m polylog}(n))$ expected time for diameter and sum of distances.
- Expected time $O(n^{15/8}{
m polylog}(n))$ for counting pairs below a threshold.
- First algorithms with $O(n^c)$ time for some $c<2$ for these problems in planar graphs.

## Abstract

We show how to compute for $n$-vertex planar graphs in $O(n^{11/6}{\rm polylog}(n))$ expected time the diameter and the sum of the pairwise distances. The algorithms work for directed graphs with real weights and no negative cycles. In $O(n^{15/8}{\rm polylog}(n))$ expected time we can also compute the number of pairs of vertices at distance smaller than a given threshold. These are the first algorithms for these problems using time $O(n^c)$ for some constant $c<2$, even when restricted to undirected, unweighted planar graphs.