We show how to build an alphabetic minimax tree for a sequence W = w1,...,wn of real weights in O(nd log log n) time, where d is the number of distinct integers ⌈wi⌉. We apply this algorithm to building an alphabetic prefix code given a sample.

Abstract. A minimax tree is similar to a Huffman tree except that, instead of minimizing the weighted average of the leaves ’ depths, it minimizes the maximum of any leaf’s weight plus its depth. Golumbic (1976) introduced minimax trees and gave a Huffman-like, O(nlog n)-time algorithm for building them. Drmota and Szpankowski (2002) gave another O(nlog n)-time algorithm, which checks the Kraft Inequality in each step of a binary search. In this paper we show how Drmota and Szpankowski’s algorithm can be made to run in linear time on a word RAM with Ω(log n)-bit words. We also discuss how our solution applies to problems in data compression, group testing and circuit design. 1