The Rectilinear Marco Polo Problem

TL;DR

This paper investigates the rectilinear Marco Polo problem, extending geometric localization to urban-like environments in higher dimensions, proposing efficient search strategies to locate points of interest with minimal probes and travel.

Contribution

It introduces and analyzes new search strategies specifically designed for rectilinear environments and higher dimensions, generalizing the classical Marco Polo problem.

Findings

Proposed efficient search algorithms for rectilinear environments.

Analyzed strategies in terms of domain size and search efficiency.

Provided bounds on the number of probes and travel distance.

Abstract

We study the rectilinear Marco Polo problem, which generalizes the Euclidean version of the Marco Polo problem for performing geometric localization to rectilinear search environments, such as in geometries motivated from urban settings, and to higher dimensions. In the rectilinear Marco Polo problem, there is at least one point of interest (POI) within distance $n$, in either the $L_1$ or $L_\infty$ metric, from the origin. Motivated from a search-and-rescue application, our goal is to move a search point, $\Delta$, from the origin to a location within distance $1$ of a POI. We periodically issue probes from $\Delta$ out a given distance (in either the $L_1$ or $L_\infty$ metric) and if a POI is within the specified distance of $\Delta$, then we learn this (but no other location information). Optimization goals are to minimize the number of probes and the distance traveled by $\Delta$. We describe a number of efficient search strategies for rectilinear Marco Polo problems and we analyze each one in terms of the size, $n$, of the search domain, as defined by the maximum distance to a POI.