On an optimal split tree problem

Abstract

We introduce and study a problem that we refer to as the optimal split tree problem. The problem generalizes a number of problems including two classical tree construction problems including the Huffman tree problem and the optimal alphabetic tree. We show that the general split tree problem is NP-complete and analyze a greedy algorithm for its solution. We show that a simple modification of the greedy algorithm guarantees O(log n) approximation ratio. We construct an example for which this algorithm achieves0(Formula presented.) approximation ratio. We show that if all weights are equal and the optimal split tree is of depth O(log n), then the greedy algorithm guarantees (Formula presented.) approximation ratio. We also extend our approximation algorithm to the construction of a search tree for partially ordered sets.