Patrolling Grids with a Bit of Memory

TL;DR

The paper investigates the space complexity of patrolling $d$-dimensional grid graphs with a single agent that has $b$ bits of memory and sensing range $V$. It proves a tight 0-bit characterization: patrolling is possible with no memory iff at most one dimension exceeds $2V+1$ and $\prod_{i=1}^{d} \min(n_i,2V+1)$ is even or equals 1, leveraging a sensing-region partition and a Hamiltonian-cycle argument. It then delivers a surprising universal result: a 1-bit, $V=1$ patrolling algorithm exists for all $d$-dimensional grids, built recursively across floor-structures and subroutines for low dimensions, achieving patrolling in at most $2|\mathcal{Q}|$ steps. The work demonstrates that grid environments admit far smaller memory footprints for patrolling than general graphs, and the techniques—especially sensing-region partitioning and higher-dimensional decomposition—offer a path to analyze space complexity in other regular environments with practical implications for robotics and traffic management.

Abstract

This work addresses the challenge of patrolling regular grid graphs of any dimension using a single mobile agent with minimal memory and limited sensing range. We show that it is impossible to patrol some grid graphs with $0$ bits of memory, regardless of sensing range, and give an exact characterization of those grid graphs that can be patrolled with $0$ bits of memory and sensing range $V$. On the other hand, we show that an algorithm exists using $1$ bit of memory and $V=1$ that patrols any $d$-dimensional grid graph. This result is surprising given that the agent must be able to move in $2d$ distinct directions to patrol, while $1$ bit of memory allows specifying only two directions per sensory input. Our $1$-bit patrolling algorithm handles this by carefully exploiting a small state-space to access all the needed directions while avoiding getting stuck. Overall, our results give concrete evidence that extremely little memory is needed for patrolling highly regular environments like grid graphs compared to arbitrary graphs. The techniques we use, such as partitioning the environment into sensing regions and exploiting distinct coordinates resulting from higher-dimensionality, may be applicable to analyzing the space complexity of patrolling in other types of regular environments as well.

Figures (8)

• Figure 1: Gray-colored vertices illustrate what our mobile agent can see with sensing range $V=1$ and $V=2$, respectively, in a $2$-dimensional grid graph.
• Figure 2: The sensing regions on $2$-dimensional grid graphs given sensing range $V = 1$, $2$, or $3$. Regions are labelled from $1$ to $(2V + 1)^2$.
• Figure 3: The sensing regions of $3$-dimensional grid graphs given sensing range $V = 1$, $2$, or $3$.
• Figure 4: The behavior of \ref{['alg:V=1 b=0-Patrolling Algorithm']} over $\mathcal{Q} = [5] \times [3] \times [3] \times [2]$. Arrows denote the next step of $\mathcal{R}$ from a given location.
• Figure 5: The behavior of \ref{['alg:Vmorethan1 m=0-Patrolling Algorithm']} over $\mathcal{Q} = [7] \times [5] \times [5] \times [2]$.

Theorems & Definitions (26)

• Definition 2.1
• Theorem 3.1
• Definition 3.2
• Lemma 3.3
• proof
• Lemma 3.4
• proof
• Corollary 3.5: Special case of \ref{['theorem:grid graphsthatcanbepatrolledwithVvisibility0bits']}
• proof
• Definition 3.6