Topology recognition with advice

TL;DR

This paper explores the tradeoffs between the time required and the size of advice needed for topology recognition in anonymous networks, providing bounds that are often tight for graphs of size n and diameter D.

Contribution

It establishes upper and lower bounds on advice size for topology recognition within specific timeframes, advancing understanding of the time-advice tradeoff in anonymous networks.

Findings

Bounds on advice size are tight for many cases.

Advice size decreases as recognition time increases.

Results apply to graphs with size n and diameter D ≤ αn.

Abstract

In topology recognition, each node of an anonymous network has to deterministically produce an isomorphic copy of the underlying graph, with all ports correctly marked. This task is usually unfeasible without any a priori information. Such information can be provided to nodes as advice. An oracle knowing the network can give a (possibly different) string of bits to each node, and all nodes must reconstruct the network using this advice, after a given number of rounds of communication. During each round each node can exchange arbitrary messages with all its neighbors and perform arbitrary local computations. The time of completing topology recognition is the number of rounds it takes, and the size of advice is the maximum length of a string given to nodes. We investigate tradeoffs between the time in which topology recognition is accomplished and the minimum size of advice that has to be given to nodes. We provide upper and lower bounds on the minimum size of advice that is sufficient to perform topology recognition in a given time, in the class of all graphs of size $n$ and diameter $D\le \alpha n$, for any constant $\alpha< 1$. In most cases, our bounds are asymptotically tight.