Minimax Trees in Linear Time

TL;DR

This paper presents a linear-time algorithm for constructing minimax trees, improving upon previous algorithms with (n \, ext{log} \, n) complexity, and discusses applications in data compression, group testing, and circuit design.

Contribution

The paper introduces a linear-time algorithm for minimax tree construction on a word RAM, enhancing previous (n \, ext{log} \, n) solutions.

Findings

Linear-time minimax tree construction algorithm.

Applicable to data compression, group testing, and circuit design.

Improves efficiency over previous algorithms.

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, $\Oh{n \log n}$-time algorithm for building them. Drmota and Szpankowski (2002) gave another $\Oh{n \log n}$-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 (\Omega (\log n))-bit words. We also discuss how our solution applies to problems in data compression, group testing and circuit design.