On the Complexity of Approximate Sum of Sorted List

TL;DR

This paper presents an efficient algorithm for approximating the sum of sorted nonnegative numbers with a tight lower bound, and shows the difficulty of approximation when negative numbers are involved.

Contribution

It introduces a near-optimal algorithm for approximate sum of sorted nonnegative lists and establishes fundamental lower bounds, highlighting the complexity differences with negative elements.

Findings

Algorithm computes (1+ε)-approximation in near-linear time.

Lower bound matches the upper bound, proving optimality.

No sublinear algorithm exists for lists with negative numbers.

Abstract

We consider the complexity for computing the approximate sum $a_1+a_2+...+a_n$ of a sorted list of numbers $a_1\le a_2\le ...\le a_n$. We show an algorithm that computes an $(1+\epsilon)$-approximation for the sum of a sorted list of nonnegative numbers in an $O({1\over \epsilon}\min(\log n, {\log ({x_{max}\over x_{min}})})\cdot (\log {1\over \epsilon}+\log\log n))$ time, where $x_{max}$ and $x_{min}$ are the largest and the least positive elements of the input list, respectively. We prove a lower bound $\Omega(\min(\log n,\log ({x_{max}\over x_{min}}))$ time for every O(1)-approximation algorithm for the sum of a sorted list of nonnegative elements. We also show that there is no sublinear time approximation algorithm for the sum of a sorted list that contains at least one negative number.