On The Average-Case Complexity of the Bottleneck Tower of Hanoi Problem

Noam Solomon, Shay Solomon
2013 2014 Proceedings of the Eleventh Workshop on Analytic Algorithmics and Combinatorics (ANALCO)  
The Bottleneck Tower of Hanoi (BTH) problem, posed in 1981 by Wood [29], is a natural generalization of the classic Tower of Hanoi (TH) problem. There, a generalized placement rule allows a larger disk to be placed higher than a smaller one if their size difference is less than a given parameter k ≥ 1. The objective is to compute a shortest move-sequence transferring a legal (under the above rule) configuration of n disks on three pegs to another legal configuration. In SOFSEM'07, Dinitz and
more » ... M'07, Dinitz and Solomon [7] established tight asymptotic bounds for the worst-case complexity of the BTH problem, for all values of n and k. Moreover, they proved that the average-case complexity is asymptotically the same as the worst-case complexity, for all values of n > 3k and n ≤ k, and conjectured that the same phenomenon also occurs in the complementary range k < n ≤ 3k. In this paper we settle the conjecture of Dinitz and Solomon [7] in the affirmative, and show that the average-case complexity of the BTH problem is asymptotically the same as the worst-case complexity, for all values of n and k. We also discuss some connections between the BTH problem, the problem of sorting with complete networks of stacks using a forklift [1, 19] , and the pancake problem [11] . ⋆
doi:10.1137/1.9781611973204.10 dblp:conf/analco/SolomonS14 fatcat:ha454kbay5a4homi7l7f6yalhq