 
Summary: Inferring Graphs from Walks
(Extended Abstract)
Javed A. Aslam # Ronald L. Rivest +
Laboratory for Computer Science
Massachusetts Institute of Technology
Cambridge, MA 02139
Abstract
We consider the problem of inferring an undirected, degreebounded, edgecolored
graph from the sequence of edge colors seen in a walk of that graph. This problem can
be viewed as reconstructing the structure of a Markov chain from its output. (That is,
we are not concerned with inferring the transition probabilities, but only the underlying
graph structure of the Markov chain.) We present polynomialtime algorithms for the
inference of underlying graphs of degreebound 2 (linear chains and cycles), based on
some surprising properties about the confluence of various sets of rewrite rules.
1 Introduction
Consider an undirected, edgecolored graph G = (V, E, c) with vertex set V , edge set E,
and edge coloring c : E ## #. A walk of G starts at some vertex v i and makes transitions
from vertex to vertex by arbitrarily selecting some edge incident on the current vertex and
traversing it. The output of such a walk is the sequence of colors of the edges traversed.
We ask: given the output of a walk and a degreebound k, what is the smallest undirected,
