Tree exploration in dual-memory model

TL;DR

This paper investigates conditions under which a deterministic mobile agent can explore trees efficiently, demonstrating that dual-memory models with specific features enable linear exploration time, while others lead to quadratic time complexity.

Contribution

The paper introduces new algorithms for linear tree exploration using dual-memory models with fixed initial memory or a movable token, and establishes lower bounds showing limitations without these features.

Findings

Linear exploration is possible with constant agent memory and logarithmic node memory if certain features are present.

Presence of a movable token or fixed initial node memory enables linear exploration algorithms.

Without these features, exploration can require quadratic time, even on simple paths.

Abstract

We study the problem of online tree exploration by a deterministic mobile agent. Our main objective is to establish what features of the model of the mobile agent and the environment allow linear exploration time. We study agents that, upon entering to a node, do not receive as input the edge via which they entered. In such a model, deterministic memoryless exploration is infeasible, hence the agent needs to be allowed to use some memory. The memory can be located at the agent or at each node. The existing lower bounds show that if the memory is either only at the agent or only at the nodes, then the exploration needs superlinear time. We show that tree exploration in dual-memory model, with constant memory at the agent and logarithmic at each node is possible in linear time when one of two additional features is present: fixed initial state of the memory at each node (so called clean memory) or a single movable token. We present two algorithms working in linear time for arbitrary trees in these two models. On the other hand, in our lower bound we show that if the agent has a single bit of memory and one bit is present at each node, then exploration may require quadratic time on paths, if the initial memory at nodes could be set arbitrarily (so called dirty memory). This shows that having clean node memory or a token allows linear exploration of trees in the model with two types of memory, but having neither of those features may lead to quadratic exploration time even on a simple path.