Fast and Efficient Distributed Computation of Hamiltonian Cycles in Random Graphs

TL;DR

This paper introduces a fast, distributed randomized algorithm for finding Hamiltonian cycles in certain random graphs, operating efficiently in the CONGEST model with high probability and improved performance over previous methods.

Contribution

It presents the first distributed algorithm for Hamiltonian cycles in $G(n,p)$ graphs with near-optimal running time and low message complexity, applicable to denser graphs.

Findings

Finds Hamiltonian cycles in $G(n,p)$ with high probability

Operates in $ ilde{O}(n^{ ext{delta}})$ rounds in the CONGEST model

Works efficiently on denser random graphs

Abstract

We present fast and efficient randomized distributed algorithms to find Hamiltonian cycles in random graphs. In particular, we present a randomized distributed algorithm for the $G(n,p)$ random graph model, with number of nodes $n$ and $p=\frac{c\ln n}{n^{\delta}}$ (for any constant $0 < \delta \leq 1$ and for a suitably large constant $c > 0$), that finds a Hamiltonian cycle with high probability in $\tilde{O}(n^{\delta})$ rounds (the notation $\tilde{O}$ hides a $\text{polylog}(n)$ factor). Our algorithm works in the (synchronous) CONGEST model (i.e., only $O(\log n)$-sized messages are communicated per edge per round) and its computational cost per node is sublinear (in $n$) per round and is fully-distributed (each node uses only $o(n)$ memory and all nodes' computations are essentially balanced). Our algorithm improves over the previous best known result in terms of both the running time as well as the edge sparsity of the graphs where it can succeed; in particular, the denser the random graph, the smaller is the running time.