Observability of Lattice Graphs

TL;DR

This paper investigates the minimum number of edge colors needed for an agent to determine its position in a lattice graph based on observed color sequences, revealing bounds related to lattice and walk dimensions.

Contribution

It establishes tight bounds on the number of colors required for t-observability in lattice graphs, highlighting the dependence on lattice and walk dimensions.

Findings

Theta(n^(d/t)) colors are necessary and sufficient for t-observability.

Full-dimensional walks require Theta(n^(1/2)) colors in directed lattices.

Results extend to non-square lattices with size N, requiring Theta(N^(1/t)) colors.

Abstract

We consider a graph observability problem: how many edge colors are needed for an unlabeled graph so that an agent, walking from node to node, can uniquely determine its location from just the observed color sequence of the walk? Specifically, let G(n,d) be an edge-colored subgraph of d-dimensional (directed or undirected) lattice of size n^d = n * n * ... * n. We say that G(n,d) is t-observable if an agent can uniquely determine its current position in the graph from the color sequence of any t-dimensional walk, where the dimension is the number of different directions spanned by the edges of the walk. A walk in an undirected lattice G(n,d) has dimension between 1 and d, but a directed walk can have dimension between 1 and 2d because of two different orientations for each axis. We derive bounds on the number of colors needed for t-observability. Our main result is that Theta(n^(d/t)) colors are both necessary and sufficient for t-observability of G(n,d), where d is considered a constant. This shows an interesting dependence of graph observability on the ratio between the dimension of the lattice and that of the walk. In particular, the number of colors for full-dimensional walks is Theta(n^(1/2)) in the directed case, and Theta(n) in the undirected case, independent of the lattice dimension. All of our results extend easily to non-square lattices: given a lattice graph of size N = n_1 * n_2 * ... * n_d, the number of colors for t-observability is Theta (N^(1/t)).