Perfect Matching with Few Link Activations

TL;DR

This paper presents randomized and deterministic algorithms for perfect matching in bipartite networks with limited link activations, achieving near-optimal time and pulse complexities under various assumptions.

Contribution

It introduces new algorithms that significantly improve the efficiency of perfect matching computations in distributed networks with restricted communication.

Findings

Randomized algorithm terminates in O(log n) rounds with O(n log n) pulses.

Deterministic algorithms under KT_1 assumption reduce pulses to O(n) and time to O(log* n log log n).

All bounds also apply in the CONGEST model with single-bit messages.

Abstract

We consider the problem of computing a perfect matching problem in a synchronous distributed network, where the network topology corresponds to a complete bipartite graph. The communication between nodes is restricted to activating communication links, which means that instead of sending messages containing a number of bits, each node can only send a pulse over some of its incident links in each round. In the port numbering model, where nodes are unaware of their neighbor's IDs, we give a randomized algorithm that terminates in $O( \log n )$ rounds and has a pulse complexity of $O( n\log n )$, which corresponds to the number of pulses sent over all links. We also show that randomness is crucial in the port numbering model, as any deterministic algorithm must send at least $\Omega( n^2 )$ messages in the standard LOCAL model, where the messages can be of unbounded size. Then, we turn our attention to the KT_1 assumption, where each node starts out knowing its neighbors' IDs. We show that this additional knowledge enables significantly improved bounds even for deterministic algorithms. First, we give an $O( \log n )$ time deterministic algorithm that sends only $O( n )$ pulses. Finally, we apply this algorithm recursively to obtain an exponential reduction in the time complexity to $O( \log^*n\log\log n )$, while slightly increasing the pulse complexity to $O( n\log^*n )$. All our bounds also hold in the standard CONGEST model with single-bit messages.