Prefix Codes: Equiprobable Words, Unequal Letter Costs

We consider the following variant of Huffman coding in which the costs of the letters, rather than the probabilities of the words, are non-uniform: “Given an alphabet of r letters of nonuniform length, find a minimum-average-length prefix-free set of n codewords over the alphabet;” equivalently, “Find an optimal r-ary search tree with n leaves, where each leaf is accessed with equal probability but the cost to descend from a parent to its ith child depends on i.” We show new structural properties of such codes, leading to an O(n log r)-time algorithm for finding them. This new algorithm is simpler and faster than the best previously known O(nr min{logn, r})-time algorithm due to Perl, Garey, and Even [7].