Tradeoffs Between Cost and Information for Rendezvous and Treasure Hunt

TL;DR

This paper investigates the minimal amount of initial advice needed for agents to efficiently perform rendezvous and treasure hunt tasks in networks, establishing bounds based on network size, initial distance, and cost savings.

Contribution

It provides tight bounds on advice size necessary for rendezvous and treasure hunt, relating advice to network parameters and task costs, especially for trees and arbitrary graphs.

Findings

Advice size depends on initial distance and cost ratio.

Bounds are tight for trees and nearly tight for general graphs.

Advice significantly reduces traversal costs in network tasks.

Abstract

In rendezvous, two agents traverse network edges in synchronous rounds and have to meet at some node. In treasure hunt, a single agent has to find a stationary target situated at an unknown node of the network. We study tradeoffs between the amount of information ($\mathit{advice}$) available $\mathit{a\ priori}$ to the agents and the cost (number of edge traversals) of rendezvous and treasure hunt. Our goal is to find the smallest size of advice which enables the agents to solve these tasks at some cost $C$ in a network with $e$ edges. This size turns out to depend on the initial distance $D$ and on the ratio $\frac{e}{C}$, which is the $\mathit{relative\ cost\ gain}$ due to advice. For arbitrary graphs, we give upper and lower bounds of $O(D\log(D\cdot \frac{e}{C}) +\log\log e)$ and $\Omega(D\log \frac{e}{C})$, respectively, on the optimal size of advice. For the class of trees, we give nearly tight upper and lower bounds of $O(D\log \frac{e}{C} + \log\log e)$ and $\Omega (D\log \frac{e}{C})$, respectively.