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Optimal-Length Labeling Schemes for Fast Deterministic Communication in Radio Networks

Adam Gańczorz, Tomasz Jurdziński, Andrzej Pelc

TL;DR

It is shown that appropriately chosen node labels of constant length permit to achieve the optimal centralized broadcasting time, which is, perhaps, the most surprising finding of this paper.

Abstract

We consider two fundamental communication tasks in arbitrary radio networks: broadcasting (information from one source has to reach all nodes) and gossiping (every node has a message and all messages have to reach all nodes). Nodes are assigned labels that are (not necessarily different) binary strings. Each node knows its own label and can use it as a parameter in the same deterministic algorithm. The length of a labeling scheme is the largest length of a label. The goal is to find labeling schemes of asymptotically optimal length for the above tasks, and to design fast deterministic distributed algorithms for each of them, using labels of optimal length. Our main result concerns broadcasting. We show the existence of a labeling scheme of constant length that supports broadcasting in time $O(D+\log^2 n)$, where $D$ is the diameter of the network and $n$ is the number of nodes. This broadcasting time is an improvement over the best currently known $O(D\log n + \log^2 n)$ time of broadcasting with constant-length labels, due to Ellen and Gilbert (SPAA 2020). It also matches the optimal broadcasting time in radio networks of known topology. Hence, we show that appropriately chosen node labels of constant length permit to achieve, in a distributed way, the optimal centralized broadcasting time. This is, perhaps, the most surprising finding of this paper. We are able to obtain our result thanks to a novel methodological tool of propagating information in radio networks, that we call a 2-height respecting tree. Next, we apply our broadcasting algorithm to solve the gossiping problem. We get a gossiping algorithm working in time $O(D + Δ\log n + \log^2 n)$, using a labeling scheme of optimal length $O(\log Δ)$, where $Δ$ is the maximum degree. Our time is the same as the best known gossiping time in radio networks of known topology.

Optimal-Length Labeling Schemes for Fast Deterministic Communication in Radio Networks

TL;DR

It is shown that appropriately chosen node labels of constant length permit to achieve the optimal centralized broadcasting time, which is, perhaps, the most surprising finding of this paper.

Abstract

We consider two fundamental communication tasks in arbitrary radio networks: broadcasting (information from one source has to reach all nodes) and gossiping (every node has a message and all messages have to reach all nodes). Nodes are assigned labels that are (not necessarily different) binary strings. Each node knows its own label and can use it as a parameter in the same deterministic algorithm. The length of a labeling scheme is the largest length of a label. The goal is to find labeling schemes of asymptotically optimal length for the above tasks, and to design fast deterministic distributed algorithms for each of them, using labels of optimal length. Our main result concerns broadcasting. We show the existence of a labeling scheme of constant length that supports broadcasting in time , where is the diameter of the network and is the number of nodes. This broadcasting time is an improvement over the best currently known time of broadcasting with constant-length labels, due to Ellen and Gilbert (SPAA 2020). It also matches the optimal broadcasting time in radio networks of known topology. Hence, we show that appropriately chosen node labels of constant length permit to achieve, in a distributed way, the optimal centralized broadcasting time. This is, perhaps, the most surprising finding of this paper. We are able to obtain our result thanks to a novel methodological tool of propagating information in radio networks, that we call a 2-height respecting tree. Next, we apply our broadcasting algorithm to solve the gossiping problem. We get a gossiping algorithm working in time , using a labeling scheme of optimal length , where is the maximum degree. Our time is the same as the best known gossiping time in radio networks of known topology.

Paper Structure

This paper contains 20 sections, 19 theorems, 8 equations, 3 figures, 3 tables.

Table of Contents

  1. Introduction
  2. The model and the problem
  3. Our results
  4. Related work
  5. Organization of the paper
  6. High-level Description of our Results
  7. High-level description of broadcasting
  8. High-level description of gossiping
  9. $2$-height Respecting Trees
  10. Broadcasting Algorithms
  11. Levelled Executor Algorithm
  12. Constructive Variant of Levelled Executor Algorithm
  13. Fast Broadcast Algorithm: Linear Dependency on $D$
  14. Express Broadcast Algorithm -- Optimization of the polylog Additive Summand
  15. Acknowledged broadcasting
  16. ...and 5 more sections

Key Result

Lemma 1

For every graph $G=(V,E)$ and each $r\in V$, one can construct a BFS spanning tree $T$ of $G$ rooted at $r$ such that $T$ is a 2-HRT. Moreover, the tree $T$ can be constructed in time $\text{poly}(n)$.

Figures (3)

  • Figure 1: Illustration for type (a) violation. Fat edges connect nodes with their parents. Blue edges are fast, i.e., connect children with their parents such that the value of $h_2$ of the child and the parent are equal. Red edges might be fast, but it is not determined. Non-fat edges are not fast.
  • Figure 2: Illustration for type (b) violation. Fat edges connect nodes with their parents. Blue edges are fast, i.e., connect children with their parents such that the value of $h_2$ of the child and the parent are equal. Red edges might be fast, but it is not determined. Non-fat edges are not fast.
  • Figure 4: Illustration for the lower bounds on $k$-gathering.

Theorems & Definitions (29)

  • Definition 1: 2-height, fast edge, slow edge
  • Definition 2: $2$-height respecting tree
  • Lemma 1
  • proof
  • Theorem 1
  • Lemma 2
  • Lemma 3
  • proof
  • Corollary 1
  • Lemma 4
  • ...and 19 more