Finding the minimum number of elements with sum above a threshold

Abstract

Motivated by the wavelet compression techniques and their applications, we consider the following problem: Given an unsorted array of numerical values and a threshold, what is the minimum number of elements chosen from the array, such that the sum of these elements is not less than the threshold value. In this article, we first provide two linear time algorithms for the problem. We then demonstrate the efficacy of these algorithms through experiments. Lastly, as an application of this research, we indicate that the construction of wavelet synopses on a prescribed error bound (in L₂ metric) can be solved in linear time.

Introduction

Given an unsorted list of numerical values A with n elements and a threshold ρ > 0, the error-bound sum selection (SelectSum_ρ) problem¹ termed in this article is to find S ⊆ A of minimized cardinality such that ${\sum }_{s\in S}s⩾\rho$. That is, acquiring the top most valued elements in an array with the sum equal to or greater than the threshold value. One obvious solution for this problem is to sort the sequence into decreasing numerical order and then output the current largest elements one by one until their sum is equal to or greater than the threshold value. Clearly, this will take O(nlog n) time in the worst case. As the array A becomes very large, this naive approach can be inefficient in time and expensive for online stream data processing.

The SelectSum_ρ problem is slightly related to the Selection problem. Solved by Blum et al. [2] in 1972, the Selection problem aims at finding the kth largest element in an unsorted array. Using the divide-and-conquer technique, Blum et al. derived a linear time algorithm² through wisely selecting pivot elements for partitions. This was an important complexity result because till then the selection problem was assumed to be as difficult as sorting. A lower bound of 2n comparisons (in the worst-case) for the median selection problem was due to Bent and John [1] in 1985. Refer to [3], [4] for the detailed research on the Selection problem.

The SelectSum_ρ Problem is a fundamental problem for many practical applications. In Section 4, we will use an example to illustrate its application on streaming data compression.

In this article, we explore the SelectSum_ρ problem and provide linear time algorithms for the problem. We first indicate that the SelectSum_ρ problem is solvable in linear time by using the Selection algorithm in a straight way. We then provide a more involved algorithm for the problem. With this article’s result, we conclude that the error-bound wavelet compression on square error (L₂) can be computed in linear time.

The rest of the paper is organized as follows. Section 2 presents the relevant concepts and the two algorithms for the SelectSum_ρ problem. Section 3 reports the experiment results of these proposed algorithms. Section 4 is an application of the SelectSum_ρ problem on streaming data compression and Section 5 concludes this article.