Alphabetic Minimax Trees in Linear Time

Abstract

We develop a linear time algorithm for the following problem: given an ordered sequence of n real weights, construct a binary tree on n leaves labelled with those weights when read from left to right minimizing the maximum value of w _i plus the depth of the corresponding leaf. This improves the previously known O(nlogn) time solutions [3,10,12]. Assuming that the integer and the fractional part of each weight is given separately, our solution works in the linear decision tree model, i.e., we use only the basic arithmetical operations on the input numbers. To decide (efficiently) which operations to perform we need the word RAM model, though. We provide a simplified $\mathcal{O}(nd)$ version of the algorithm, where d is the number of distinct integer parts, which does not require the full power of the word RAM model in order to decide which operations to perform. Nevertheless, it improves the previously known $\mathcal{O}(nd\log\log n)$ solution of Gagie [5].