Optimal Search Trees with 2-Way Comparisons

TL;DR

This paper presents the first polynomial-time algorithms for constructing optimal binary search trees using only 2-way comparisons, addressing a long-standing open problem in search tree optimization.

Contribution

It introduces an $O(n^4)$-time exact algorithm and an $O(n \, log \, n)$-time approximation for 2-way comparison search trees, solving the general case.

Findings

Provided an $O(n^4)$-time exact algorithm.

Developed an $O(n \, log \, n)$-time approximation algorithm.

Proved previous methods for binary split trees incorrect.

Abstract

In 1971, Knuth gave an $O(n^2)$-time algorithm for the classic problem of finding an optimal binary search tree. Knuth's algorithm works only for search trees based on 3-way comparisons, while most modern computers support only 2-way comparisons (e.g., $<, \le, =, \ge$, and $>$). Until this paper, the problem of finding an optimal search tree using 2-way comparisons remained open -- poly-time algorithms were known only for restricted variants. We solve the general case, giving (i) an $O(n^4)$-time algorithm and (ii) an $O(n \log n)$-time additive-3 approximation algorithm. Also, for finding optimal binary split trees, we (iii) obtain a linear speedup and (iv) prove some previous work incorrect.