Deciding a Graph Property by a Single Mobile Agent: One-Bit Memory Suffices

TL;DR

This paper demonstrates that a single mobile agent with just one bit of memory can decide certain graph properties efficiently, matching the power of agents with much larger memory, through novel algorithms and technical tools.

Contribution

It establishes the equivalence in computational power between one-bit memory agents and larger memory agents for graph decision tasks, introducing new algorithms for path maintenance and DFS.

Findings

One-bit memory suffices for certain graph decision problems.

Algorithms with polynomial overhead simulate larger memory agents.

One-bit memory agents can solve tasks impossible for oblivious agents.

Abstract

We investigate the computational power of the deterministic single-agent model where the agent and each node are equipped with a limited amount of persistent memory. Tasks are formalized as decision problems on properties of input graphs, i.e., the task is defined as a subset $\mathcal{T}$ of all possible input graphs, and the agent must decide if the network belongs to $\mathcal{T}$ or not. We focus on the class of the decision problems which are solvable in a polynomial number of movements, and polynomial-time local computation. The contribution of this paper is the computational power of the very weak system with one-bit agent memory and $O(1)$-bit storage (i.e. node memory) is equivalent to the one with $O(n)$-bit agent memory and $O(1)$-bit storage. We also show that the one-bit agent memory is crucial to lead this equivalence: There exists a decision task which can be solved by the one-bit memory agent but cannot be solved by the zero-bit memory (i.e., oblivious) agent. Our result is deduced by the algorithm of simulating the $O(n)$-bit memory agent by the one-bit memory agent with polynomial-time overhead, which is developed by two novel technical tools. The first one is a dynamic $s$-$t$ path maintenance mechanism which uses only $O(1)$-bit storage per node. The second one is a new lexicographically-ordered DFS algorithm for the mobile agent system with $O(1)$-bit memory and $O(1)$-bit storage per node. These tools are of independent interest.