Random $k$-out subgraph leaves only $O(n/k)$ inter-component edges

TL;DR

The paper proves that sampling each vertex with $k$ edges in a graph leaves only $O(n/k)$ inter-component edges when $k \\ge c \\log n$, and applies this to develop an efficient distributed spanning forest protocol.

Contribution

It establishes an $O(n/k)$ bound on inter-component edges after random $k$-out sampling for large $k$, and introduces a new communication protocol for spanning forest.

Findings

Expected inter-component edges are $O(n/k)$ for $k \\ge c \\log n$

Sampling preserves connectivity structure with high probability

Efficient one-way communication protocol for spanning forest

Abstract

Each vertex of an arbitrary simple graph on $n$ vertices chooses $k$ random incident edges. What is the expected number of edges in the original graph that connect different connected components of the sampled subgraph? We prove that the answer is $O(n/k)$, when $k\ge c\log n$, for some large enough $c$. We conjecture that the same holds for smaller values of $k$, possibly for any $k\ge 2$. Such a result is best possible for any $k\ge 2$. As an application, we use this sampling result to obtain a one-way communication protocol with \emph{private} randomness for finding a spanning forest of a graph in which each vertex sends only ${O}(\sqrt{n}\log n)$ bits to a referee.