The Huffman Tree Problem with Upper-Bounded Linear Functions

Abstract

The construction of a Huffman code can be understood as the problem of finding a full binary tree such that each leaf is associated with a linear function of the depth of the leaf and the sum of the function values is minimized. Fujiwara and Jacobs extended this to a general function and proved the extended problem to be NP-hard. The authors also showed the case where the functions associated with leaves are each non-decreasing and convex is solvable in polynomial time. However, the complexity of the case of non-decreasing non-convex functions remains unknown. In this paper we try to reveal the complexity by considering non-decreasing non-convex functions each of which takes the smaller value of either a linear function or a constant. As a result, we provide a polynomial-time algorithm for two subclasses of such functions.