Routing by matching on convex pieces of grid graphs

TL;DR

This paper extends the understanding of routing numbers in grid graphs by showing that convex polygon subgraphs of the infinite square lattice have routing numbers proportional to their width and height, generalizing previous results for rectangular grids.

Contribution

It proves that the routing number for convex polygon subgraphs of the infinite square lattice is in O(w(P)+h(P)), broadening the class of graphs with known routing bounds.

Findings

Routing number of convex polygon subgraphs is in O(w(P)+h(P))

Generalizes previous bounds from rectangular grids to convex polygons

Provides a broader understanding of routing complexity in grid-based graphs

Abstract

The routing number is a graph invariant introduced by Alon, Chung, and Graham in 1994, and it has been studied for trees and other classes of graphs such as hypercubes. It gives the minimum number of routing steps needed to sort a set of distinct tokens, placed one on each vertex, where each routing step swaps a set of disjoint pairs of adjacent tokens. Our main theorem generalizes the known estimate that a rectangular grid graph R with width w(R) and height h(R) has routing number rt(R) in O(w(R)+h(R)). We show that for the subgraph P of the infinite square lattice enclosed by any convex polygon, its routing number rt(P) is in O(w(P)+h(P)).