Optimal Binary Search Trees with Near Minimal Height

TL;DR

This paper presents an efficient algorithm for constructing near minimal height optimal binary search trees in quadratic time, improving over previous methods by reducing the dependency on height restrictions.

Contribution

The authors introduce a quadratic-time algorithm for near minimal height optimal binary search trees, outperforming previous algorithms with higher complexity.

Findings

Constructed near minimal height trees in O(n^2) time.

Algorithm is as fast as unrestricted case, independent of height constraints.

Improves efficiency over previous height-restricted algorithms.

Abstract

Suppose we have n keys, n access probabilities for the keys, and n+1 access probabilities for the gaps between the keys. Let h_min(n) be the minimal height of a binary search tree for n keys. We consider the problem to construct an optimal binary search tree with near minimal height, i.e.\ with height h <= h_min(n) + Delta for some fixed Delta. It is shown, that for any fixed Delta optimal binary search trees with near minimal height can be constructed in time O(n^2). This is as fast as in the unrestricted case. So far, the best known algorithms for the construction of height-restricted optimal binary search trees have running time O(L n^2), whereby L is the maximal permitted height. Compared to these algorithms our algorithm is at least faster by a factor of log n, because L is lower bounded by log n.