Optimal Prefix-Free Codes for Unequal Letter Costs: Dynamic Programming with the Monge Property

In this paper we discuss a variation of the classical Huffman coding problem: finding optimal prefix-free codes for unequal letter costs. Our problem consists of finding a minimal cost prefix-free code in which the encoding alphabet consists of unequal cost (length) letters, with lengths α and β. The most efficient algorithm known previously required O(n) time to construct such a minimal-cost set of n codewords. In this paper we provide an O(n) time algorithm. Our improvement comes from the use of a more sophisticated modeling of the problem combined with the observation that the problem possesses a “Monge property” and that the SMAWK algorithm on monotone matrices can therefore be applied.