Optimal-Length Labeling Schemes and Fast Algorithms for k-gathering and k-broadcasting

TL;DR

The paper tackles k-gathering and k-broadcasting in radio networks under the advice-labeling framework, aiming for labeling lengths that are asymptotically optimal and for fast, provably time-efficient algorithms. It introduces scheduling tools (children-parents assignments) and constructs phase-based, leaf-to-root strategies on BFS trees to achieve $D+k$ rounds for k-gathering, with a complementary $DΔ$-time algorithm for large k. The authors prove tight lower bounds on both time ($D+k$) and label size ($Θ(min(log Δ, log k))$) and show that the labeling-comparable, advice-based approach yields fast, nearly optimal solutions, including corollaries for unit and multi-message k-broadcasting. A key contribution is the demonstration of a separation between advice-enabled and fully-labeled models, and the work provides a cohesive framework for extending efficiency results from gathering to broadcasting in radio networks. The results have implications for designing compact labeling schemes that support fast distributed communication under interference-prone wireless settings.

Abstract

We consider basic communication tasks in arbitrary radio networks: $k$-broadcasting and $k$-gathering. In the case of $k$-broadcasting messages from $k$ sources have to get to all nodes in the network. The goal of $k$-gathering is to collect messages from $k$ source nodes in a designated sink node. We consider these problems in the framework of distributed algorithms with advice. Krisko and Miller showed in 2021 that the optimal size of advice for $k$-broadcasting is $Θ(\min(\log Δ,$ $ \log k))$, where $Δ$ is equal to the maximum degree of a vertex of the input communication graph. We show that the same bound $Θ(\min(\log Δ, \log k))$ on the size of optimal labeling scheme holds also for the $k$-gathering problems. Moreover, we design fast algorithms for both problems with asymptotically optimal size of advice. For $k$-gathering our algorithm works in at most $D+k$ rounds, where $D$ is the diameter of the communication graph. This time bound is optimal even for centralized algorithms. We apply the $k$-gathering algorithm for $k$-broadcasting to achieve an algorithm working in time $O(D+\log^2 n+k)$ rounds. We also exhibit a logarithmic time complexity gap between distributed algorithms with advice of optimal size and distributed algorithms with distinct arbitrary labels.

Figures (3)

• Figure 1: Inductive step in the proof of Lemma \ref{['l:level']}, Case 1. Segments denote edges, fat blue segments correspond to edges connecting elements of $X'$ with their parents. There are no edges connecting nodes from $X'$ with $y_k$ in $G(V,E)$.
• Figure 2: Illustration for the lower bounds on $k$-gathering.
• Figure 3: An example for $(X_0,Y_0,\text{par}_0)$ and $(X_0,Y'_0,\text{par}'_0)$. Fat edges with arrows connect children with their parents. Dashed edges do not connect parents and children. The schedule $S_0$ might be such that $x_1$ and $x_2$ transmit simultaneously in the first round, $x_3$ and $x_4$ transmit simultaneously in the second round. (Note that the parents of $x_3$ and $x_4$ in $\text{par}'_0$ are $y_1$ and $y_2$, respectively.) If messages are originally located only in $x_1,\ldots,x_4$ then $x_1,\ldots,x_4$ as well as $y_3$ are removed from $T_0$ in order to obtain $T_1$.

Theorems & Definitions (18)

• theorem thmcountertheorem
• definition thmcounterdefinition: Children-parents assignment
• definition thmcounterdefinition
• lemma thmcounterlemma
• proof
• lemma thmcounterlemma
• proof
• lemma thmcounterlemma
• theorem thmcountertheorem
• lemma thmcounterlemma