An algorithm for the weighted metric dimension of two-dimensional grids

TL;DR

This paper presents an efficient algorithm to find the minimum cost landmark set in two-dimensional grid graphs, which is crucial for applications like network navigation and robot localization.

Contribution

It introduces a novel algorithm specifically designed for the weighted metric dimension problem on 2D grids, optimizing landmark selection based on vertex costs.

Findings

Algorithm efficiently computes minimum landmark sets

Optimizes landmark placement considering vertex weights

Applicable to large grid graphs for practical scenarios

Abstract

A two-dimensional grid consists of vertices of the form (i,j) for 1 \leq i \leq m and 1 \leq j \leq n, for fixed m,n > 1. Two vertices are adjacent if the \ell_1 distance between their vectors is equal to 1. A landmark set is a subset of vertices L \subseteq V, such that for any distinct pair of vertices u,v \in V, there exists a vertex of L whose distances to u and v are not equal. We design an efficient algorithm for finding a minimum landmark set with respect to total cost in a grid graph with non-negative costs defined on the vertices.