Sorting Networks On Restricted Topologies

TL;DR

This paper investigates the minimum depth of sorting networks constrained to graph topologies, providing bounds based on graph properties like path length and degree, and relating sorting number to other graph parameters.

Contribution

The paper introduces general bounds on sorting numbers for graphs based on properties like path length and degree, and connects sorting number to routing number and matching size.

Findings

Graphs with a path of length d have sorting networks of depth O(n log(n/d))

Graphs with maximum degree Δ have sorting networks of depth O(Δ n)

New bounds on sorting numbers for specific graph classes

Abstract

The sorting number of a graph with $n$ vertices is the minimum depth of a sorting network with $n$ inputs and outputs that uses only the edges of the graph to perform comparisons. Many known results on sorting networks can be stated in terms of sorting numbers of different classes of graphs. In this paper we show the following general results about the sorting number of graphs. Any $n$-vertex graph that contains a simple path of length $d$ has a sorting network of depth $O(n \log(n/d))$. Any $n$-vertex graph with maximal degree $\Delta$ has a sorting network of depth $O(\Delta n)$. We also provide several results that relate the sorting number of a graph with its routing number, size of its maximal matching, and other well known graph properties. Additionally, we give some new bounds on the sorting number for some typical graphs.