Universal traversal sequences for paths and cycles

doi:10.1016/0196-6774(87)90019-8

Journal of Algorithms

Volume 8, Issue 3, September 1987, Pages 395-404

https://doi.org/10.1016/0196-6774(87)90019-8 Get rights and content

Abstract

One approach to the reachability problem for rooted undirected graphs G is to order the neighbors of each vertex and traverse all vertices of the connected component containing the root by means of a sequence of positive integers (s_i)_{i ≥ 1} interpreted as instructions of the form, “Move to the s_ith neighbor or of the current vertex.” A sequence that does this for all d-regular connected graphs with n vertices is called an (n, d)-universal traversal sequence. Although the existence of universal traversal sequences is easy to verify, known methods for their construction involve some sort of exhaustive search. In this paper, a recursive algorithm for the construction of (n, 2)-universal traversal sequences in space O(log²n) is described.

References (4)

R Aleliunas et al.
Random walks, universal traversal sequences, and the complexity of maze problems
B Bollobás
Random Graphs
(1986)

There are more references available in the full text version of this article.

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In this paper we introduce a new notion of traversal sequences that we call exploration sequences. Exploration sequences share many properties with the traversal sequences defined in Aleliunas et al. (Proceedings on the 20th Annual Symposium of Foundations of Computer Science, 1979, pp. 218–223), 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.
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^☆: This work was supported by the MITRE-Sponsored Research Program.

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Universal traversal sequences for paths and cycles☆

Abstract

Random walks, universal traversal sequences, and the complexity of maze problems

Random Graphs