# On Romeo and Juliet Problems: Minimizing Distance-to-Sight

## TL;DR

This paper introduces and solves efficiently the quickest pair-visibility problem in polygons, optimizing the travel distances for two persons to see each other, with solutions for both fixed and query scenarios.

## Contribution

It presents the first linear-time algorithms for the min-max and min-sum variants of the pair-visibility problem in simple polygons.

## Key findings

- Optimal linear-time algorithms for min-max and min-sum variants.
- Preprocessing enables fast query responses in O(log^2 n) time.
- Solutions applicable to polygons with no holes.

## Abstract

We introduce a variant of the watchman route problem, which we call the quickest pair-visibility problem. Given two persons standing at points $s$ and $t$ in a simple polygon $P$ with no holes, we want to minimize the distance they travel in order to see each other in $P$. We solve two variants of this problem, one minimizing the longer distance the two persons travel (min-max) and one minimizing the total travel distance (min-sum), optimally in linear time. We also consider a query version of this problem for the min-max variant. We can preprocess a simple $n$-gon in linear time so that the minimum of the longer distance the two persons travel can be computed in $O(\log^2 n)$ time for any two query positions $s,t$ where the two persons start.