On an Optimal Split Tree Problem

TL;DR: It is shown that if all weights are equal and the optimal split tree is of depth O(log n), then the greedy algorithm guarantees O( log n/log log n) approximation ratio.

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 achieves Ω(log n/log log n) 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 O(log n/log log n) approximation ratio. We also extend our approximation algorithm to the construction of a search tree for partially ordered sets.