The Rectilinear Marco Polo Problem

2025-08-20

AI Summary

This paper studies the Rectilinear Marco Polo problem in rectangular geometric spaces: a mobile searcher Δ, starting from the origin, must locate a point within unit L₁- or L∞-distance of any point of interest (POI) using minimal probes and shortest traversal distance. It is the first work to generalize the classical Marco Polo problem to high-dimensional rectilinear metric spaces, proposing a unified framework for adaptive probing strategies. Leveraging geometric analysis and asymptotic complexity modeling, we design several efficient algorithms that, for a search domain of size n, achieve worst-case probe count and traversal distance bounded by O(log n) or O(polylog n)—a substantial improvement over naive linear approaches. Our key contributions are: (i) a rigorous theoretical model for geometric localization under L₁ and L∞ metrics; and (ii) a scalable, provably correct search paradigm with guaranteed performance bounds.

Technology Category

• Search and Optimization: Learning to Search
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• Constraint Satisfaction and Optimization: Search

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines
• Security and Privacy: Large-scale security measurements

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, $Δ$, from the origin to a location within distance $1$ of a POI. We periodically issue probes from $Δ$ out a given distance (in either the $L_1$ or $L_infty$ metric) and if a POI is within the specified distance of $Δ$, then we learn this (but no other location information). Optimization goals are to minimize the number of probes and the distance traveled by $Δ$. 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.

Problem

• Generalizing Marco Polo problem to rectilinear geometries and higher dimensions
• Locating points of interest using limited probe information in urban-like settings
• Minimizing probes and travel distance for search-and-rescue applications

Innovation

• Rectilinear search strategies for geometric localization
• Probing method using L1 or Linfinity distance metrics
• Minimizing probes and distance in search domain