# Short Labeling Schemes for Topology Recognition in Wireless Tree Networks

## TL;DR

This paper investigates minimal labeling schemes and efficient algorithms for topology recognition in wireless tree networks, establishing bounds on label length and recognition time based on network diameter and maximum degree.

## Contribution

It determines the minimal label length for topology recognition in wireless trees and provides bounds on the recognition time using such schemes.

## Key findings

- Minimal label length is Θ(log log Δ) for trees with degree Δ ≥ 3.
- Recognition algorithms have an upper bound of O(DΔ) time.
- Recognition algorithms have a lower bound of Ω(DΔ^ε) time for any ε<1.

## Abstract

We consider the problem of topology recognition in wireless (radio) networks modeled as undirected graphs. Topology recognition is a fundamental task in which every node of the network has to output a map of the underlying graph i.e., an isomorphic copy of it, and situate itself in this map. In wireless networks, nodes communicate in synchronous rounds. In each round a node can either transmit a message to all its neighbors, or stay silent and listen. At the receiving end, a node $v$ hears a message from a neighbor $w$ in a given round, if $v$ listens in this round, and if $w$ is its only neighbor that transmits in this round. Nodes have labels which are (not necessarily different) binary strings. The length of a labeling scheme is the largest length of a label. We concentrate on wireless networks modeled by trees, and we investigate two problems. \begin{itemize} \item What is the shortest labeling scheme that permits topology recognition in all wireless tree networks of diameter $D$ and maximum degree $\Delta$? \item What is the fastest topology recognition algorithm working for all wireless tree networks of diameter $D$ and maximum degree $\Delta$, using such a short labeling scheme? \end{itemize} We are interested in deterministic topology recognition algorithms. For the first problem, we show that the minimum length of a labeling scheme allowing topology recognition in all trees of maximum degree $\Delta \geq 3$ is $\Theta(\log\log \Delta)$. For such short schemes, used by an algorithm working for the class of trees of diameter $D\geq 4$ and maximum degree $\Delta \geq 3$, we show almost matching bounds on the time of topology recognition: an upper bound $O(D\Delta)$, and a lower bound $\Omega(D\Delta^{\epsilon})$, for any constant $\epsilon<1$.