Deterministic Graph Exploration with Advice
Barun Gorain, Andrzej Pelc

TL;DR
This paper investigates how much prior knowledge (advice) an agent needs to efficiently explore any graph, establishing precise bounds on advice size for different exploration time thresholds and oracle types.
Contribution
It precisely characterizes the minimum advice size needed for polynomial-time exploration and explores thresholds for large-time exploration in deterministic graph exploration.
Findings
Minimum advice size for polynomial-time exploration is $ ext{log} ext{log} ext{log} n - ext{Theta}(1)$.
Large advice enables exploration in $ ext{Theta}(n^2)$ or $ ext{Theta}(n)$ time depending on oracle type.
Advice size bounds relate to achievable exploration times, with thresholds at $n^ ext{delta}$ and $n ext{log} n$.
Abstract
We consider the task of graph exploration. An -node graph has unlabeled nodes, and all ports at any node of degree are arbitrarily numbered . A mobile agent has to visit all nodes and stop. The exploration time is the number of edge traversals. We consider the problem of how much knowledge the agent has to have a priori, in order to explore the graph in a given time, using a deterministic algorithm. This a priori information (advice) is provided to the agent by an oracle, in the form of a binary string, whose length is called the size of advice. We consider two types of oracles. The instance oracle knows the entire instance of the exploration problem, i.e., the port-numbered map of the graph and the starting node of the agent in this map. The map oracle knows the port-numbered map of the graph but does not know the starting node of the agent. We first consider…
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