Superconcentrators of Density 25.3

TL;DR

This paper proves the existence of superconcentrators with a density of 25.3, improving upon previous bounds and demonstrating more efficient graph constructions for network connectivity.

Contribution

It introduces superconcentrators with asymptotic density 25.3, advancing the known bounds and providing new constructions with lower edge-to-node ratios.

Findings

Existence of superconcentrators with density 25.3 proven.

Improvement over previous densities of 28 and 27.4136.

Supports more efficient network designs.

Abstract

An $N$-superconcentrator is a directed, acyclic graph with $N$ input nodes and $N$ output nodes such that every subset of the inputs and every subset of the outputs of same cardinality can be connected by node-disjoint paths. It is known that linear-size and bounded-degree superconcentrators exist. We prove the existence of such superconcentrators with asymptotic density $25.3$ (where the density is the number of edges divided by $N$). The previously best known densities were $28$ \cite{Scho2006} and $27.4136$ \cite{YuanK12}.