## The Matching Model for Routing Permutations on Graphs

Abstract

A model for routing on connected graphs is investigated. On each vertex of
an undirected, connected graph, `G=(V,E)`, is placed a unique "pebble".
Each pebble has a destination which a unique vertex in `V`, so that the map
from a vertex `v` to the destination of the pebble on `v` forms a
permutation of the vertex set. At each step in the routing, a matching of
the edges of `G` is selected, and the pebbles on the pair of vertices
incident with each edge in the matching are interchanged. Our goal is to
route all pebbles to their destinations in a minimal number of steps and we
define the routing number of a graph, `rt(G)`, to be the maximum
number of steps ever required to route a permutation on `G` with a minimal
length routing. The matching model is compared to parallel sorting
algorithms and an algorithm for routing on cycles with an even number of
vertices is presented. This algorithm is used to determine the routing
number for even cycles. Bounds on the routing number for several classes of
graphs are determined. Finally, a new lower bound for any graph based upon
diameter is considered.

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