A coefficient related to splay-to-root traversal, correct to thousands of decimal places

08/12/2021

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by   Colm Ó Dúnlaing, et al.

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This paper takes another look at the cost of traversing a binary tree using repeated splay-to-root. This was shown to cost O(n) (in rotations) by Tarjan and later, in different ways, by Elmasry and others. It would be interesting to know the minimal possible coefficient implied by the O(n) cost; call this coefficient β. In this paper we define a related coefficient α describing the cost of splay-to-root traversal on maximal (i.e., complete) binary trees, and show that β≥ 2 + α. We give the first 3009 digits of α, including the decimal point, and show that every digit is correct. We make two conjectures: first, that β = 2 + α, and second, that α is irrational.