Linear-Time Approximation Algorithms for Computing Numerical Summation with Provably Small Errors

TL;DR

This paper introduces the first polynomial-time algorithms for approximating numerical summation with small, provable error bounds, even for arbitrary data sets, improving upon previous methods in efficiency and accuracy.

Contribution

It presents new linear-time approximation algorithms for floating-point summation with provably small errors, including the first such algorithms for arbitrary data sets and improved bounds for sorted data.

Findings

Approximation error at most 2 times the minimal worst-case error for arbitrary data.

Error at most log log n times the minimal error for sorted data.

Algorithms run in linear time, significantly faster than previous methods.

Abstract

Given a multiset $X=\{x_1,..., x_n\}$ of real numbers, the {\it floating-point set summation} problem asks for $S_n=x_1+...+x_n$. Let $E^*_n$ denote the minimum worst-case error over all possible orderings of evaluating $S_n$. We prove that if $X$ has both positive and negative numbers, it is NP-hard to compute $S_n$ with the worst-case error equal to $E^*_n$. We then give the first known polynomial-time approximation algorithm that has a provably small error for arbitrary $X$. Our algorithm incurs a worst-case error at most $2(\mix)E^*_n$.\footnote{All logarithms $\log$ in this paper are base 2.} After $X$ is sorted, it runs in O(n) time. For the case where $X$ is either all positive or all negative, we give another approximation algorithm with a worst-case error at most $\lceil\log\log n\rceil E^*_n$. Even for unsorted $X$, this algorithm runs in O(n) time. Previously, the best linear-time approximation algorithm had a worst-case error at most $\lceil\log n\rceil E^*_n$, while $E^*_n$ was known to be attainable in $O(n \log n)$ time using Huffman coding.