Constrained Routing Between Non-Visible Vertices

TL;DR

This paper introduces a deterministic local routing algorithm for geometric graphs with constraints, guaranteeing pathfinding in visibility graphs and establishing bounds on competitiveness in triangulations.

Contribution

It presents the first deterministic 1-local $O(1)$-memory routing algorithm for constrained visibility graphs and analyzes competitiveness limits in triangulations.

Findings

A 1-local $O(1)$-memory routing algorithm guarantees paths in constrained visibility graphs.

No $o(n)$-competitive routing exists using triangle-intersected segments in certain triangulations.

An $O(n)$-competitive routing algorithm is provided for triangulations, matching the lower bound.

Abstract

In this paper we study local routing strategies on geometric graphs. Such strategies use geometric properties of the graph like the coordinates of the current and target nodes to route. Specifically, we study routing strategies in the presence of constraints which are obstacles that edges of the graph are not allowed to cross. Let $P$ be a set of $n$ points in the plane and let $S$ be a set of line segments whose endpoints are in $P$, with no two line segments intersecting properly. We present the first deterministic 1-local $O(1)$-memory routing algorithm that is guaranteed to find a path between two vertices in the visibility graph of $P$ with respect to a set of constraints $S$. The strategy never looks beyond the direct neighbors of the current node and does not store more than $O(1)$-information to reach the target. We then turn our attention to finding competitive routing strategies. We show that when routing on any triangulation $T$ of $P$ such that $S\subseteq T$, no $o(n)$-competitive routing algorithm exists when the routing strategy restricts its attention to the triangles intersected by the line segment from the source to the target (a technique commonly used in the unconstrained setting). Finally, we provide an $O(n)$-competitive deterministic 1-local $O(1)$-memory routing algorithm on any such $T$, which is optimal in the worst case, given the lower bound.