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

Abstract

The Bottleneck Tower of Hanoi (BTH) problem, posed in 1981 by Wood, 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; when k = 1 we arrive at the classic TH problem. 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 the second author established tight asymptotic bounds for the worst-case complexity of the BTH problem, for all n and k. Moreover, they proved that the average-case complexity is asymptotically the same as the worst-case complexity, for all 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 from SOFSEM'07 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 n and k. To this end we provide a new proof that applies to all values of n > k. That is, our proof is not a patch over the previous proof of Dinitz and Solomon that is tailored only for the regime k < n ≤ 3k, but is rather a stronger proof that is based on different principles and deeper ideas. We also show that there are natural connections between the BTH problem, the problem of sorting with complete networks of stacks using a forklift [Albert and Atkinson 2002, Konig and Lubbecke 2008] and the well-studied pancake problem [Gates and Papadimitriou 1979].