Linear Time Recognition of Optimal L-Restricted Prefix Codes

Abstract

Given an alphabet Σ = {a ₁,...,a _n} and a corresponding list of weights [w ₁,...,w _n], an optimal prefix code is a prefix code for Σ that minimizes the weighted length of a code string, defined to be $\sum_{i=1}^{n} w_i l_i$, where l _i is the length of the codeword assigned to a _i. This problem is equivalent to the following problem: given a list of weights [w ₁,...,w _n], find an optimal binary code tree, that is, a binary tree T that minimizes the weighted path length $\sum_{i=1}^{n} w_i l_i$, where l _i is the level of the i-th leaf of T from left to right. If the list of weights is sorted, this problem can be solved in O(n) by one of the efficient implementations of Huffman’s Algorithm [Huf52]. Any tree constructed by Huffman’s Algorithm is called a Huffman tree.