Randomized Tree-Intersection Leader Election

2026-08-09

AI Summary

This work proposes a randomized leader election algorithm in the synchronous CONGEST model on a complete graph, achieving a continuous trade-off between time and communication costs through an adjustable branching parameter ℓ. By integrating dynamically truncated ℓ-ary tree expansion, a silent pulse verification mechanism, and topology-aware weight aggregation, the algorithm attains a round complexity of O(log_ℓ√(n log n)) and per-node message complexity of O(ℓ), while maintaining a sublinear total message complexity of O(√n log^{1.5} n). This design enables flexible configurations ranging from O(1) messages per node to O(1) time rounds, establishing for the first time a fine-grained balance between time and message complexity under sublinear total communication.

Abstract

We present a randomized leader election algorithm for synchronous complete $n$-node graphs in the \textsf{CONGEST} model that introduces a highly tunable trade-off between time complexity and the per-node message complexity. By adjusting a single branching parameter, $\ell$, system designers can smoothly shift the algorithmic burden from execution time to per-node message complexity, all while maintaining a strictly sublinear total message complexity of $O(\sqrt{n} \log^{1.5} n)$. We achieve this by utilizing dynamically truncated $\ell$-ary tree expansions coupled with a novel ``silent pulse'' verification mechanism. By forcing the expansions to form exact-volume almost-complete trees, nodes can safely aggregate topological weights without overshooting the sublinear message bounds. Specifically, our algorithm achieves $O(\log_\ell \sqrt{n \log n})$ time (round) complexity and $O(\ell)$ per-node message complexity. This flexibility allows networks with tight bandwidth constraints to operate with a minimal $O(1)$ per-node burden, while high-bandwidth environments can collapse the election into $O(1)$ time units.

Problem

• leader election
• CONGEST model
• time-message trade-off
• sublinear message complexity
• synchronous networks

Innovation

• leader election
• randomized algorithm
• CONGEST model
• message complexity
• tree expansion