Optimal binary search trees

Abstract

We consider the problem of building optimal binary search trees. The binary search tree is a widely used data structure for information storage and retrieval. A binary search tree T for a set of keys from a total order is a binary tree in which each node has a key value and all the keys of the left subtree are less than the key at the root and all the keys of the right subtree are greater than the key at the root, this property holding recursively for the left and right subtrees of the tree T. Suppose we are given n keys and the probabilities of accessing each key and those occurring in the gap between two successive keys. The optimal binary search tree problem is to construct a binary search tree on these n keys that minimizes the expected access time. One variant of this problem is when only the gaps have nonzero access probabilities, and is called the optimal alphabetic tree problem. Another related problem is when there is no order between the keys and there are probabilities associated only with the gaps and the objective is to build a binary tree with minimum expected weighted path length from the root. This is called the Huffman tree problem. In this survey, we assess known results on the structural properties of the optimal trees, algorithms and lower bounds to construct and to verify optimal trees and heuristics to construct nearly optimal trees and other related results.