Universal traversal sequences with backtracking

Abstract

We introduce a new notion of traversal sequences that we call exploration sequences. Exploration sequences share many properties with the traversal sequences defined in (AKL+), but they also exhibit some new properties. In particular, they have an ability to backtrack, and their random properties are robust under choice of the probability distribution on labels. Further, we present extremely simple constructions of polynomial length universal exploration sequences for some previously studied classes of graphs (e.g. 2-regular graphs, cliques, expanders), and we also present universal exploration sequences for trees. Our constructions beat previously known lower-bounds on the length of universal traversal sequences.