Optimal Binary Search Trees with Near Minimal Height

Suppose we have n keys, n access probabilities for the keys, and n+1 access probabilities for the gaps between the keys. Let hmin(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 ≤ hmin(n) +∆ for some fixed ∆. It is shown, that for any fixed∆ optimal binary search trees with near minimal height can be constructed in time O(n). 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(Ln), whereby L is the maximal permitted height. Compared to these algorithms our algorithm is at least faster by a factor of log 2 n, because L is lower bounded by log 2 n.