Sublinear Time Approximate Sum via Uniform Random Sampling

TL;DR

This paper presents a randomized algorithm that approximates the sum of nonnegative numbers in sublinear time using uniform sampling, and establishes a near-matching lower bound for the problem.

Contribution

It introduces a novel sublinear time algorithm for approximate sum computation and proves a tight lower bound, advancing understanding of sampling-based approximation.

Findings

The algorithm achieves an $(1+ ext{epsilon})$-approximation in $O(n(\log\log n)/\sum a_i)$ time.

A lower bound of $\Omega(n/\sum a_i)$ is established, nearly matching the upper bound.

The method relies on uniform random sampling to efficiently approximate the sum.

Abstract

We investigate the approximation for computing the sum $a_1+...+a_n$ with an input of a list of nonnegative elements $a_1,..., a_n$. If all elements are in the range $[0,1]$, there is a randomized algorithm that can compute an $(1+\epsilon)$-approximation for the sum problem in time ${O({n(\log\log n)\over\sum_{i=1}^n a_i})}$, where $\epsilon$ is a constant in $(0,1)$. Our randomized algorithm is based on the uniform random sampling, which selects one element with equal probability from the input list each time. We also prove a lower bound $\Omega({n\over \sum_{i=1}^n a_i})$, which almost matches the upper bound, for this problem.