Two Linear Passes Are Necessary for Sum-Exclude-Self Under Sublinear Space

TL;DR

This paper proves that computing sum-exclude-self in sublinear space requires exactly two linear passes over the input, establishing a fundamental lower bound and demonstrating the choke-point technique.

Contribution

It establishes a tight lower bound of two linear passes for sum-exclude-self under sublinear space and introduces the choke-point technique for such proofs.

Findings

Any sublinear-space algorithm must perform two linear passes.

A simple two-pass algorithm achieves the proven lower bound.

The proof illustrates the choke-point technique for lower bounds.

Abstract

We prove that any algorithm computing the sum-exclude-self of an unsigned $d$-bit integer array of length $n$ under sublinear space must perform two linear passes over the input. More precisely, the algorithm must read at least $n-1$ input elements before any output cell receives its final value, and at least $n - \lfloor t/d \rfloor$ additional elements thereafter, where $t = o(nd)$ bits is the working memory size. This gives a total of $2n - 1 - \lfloor t/d \rfloor$ element reads. A trivial modification of the standard two-pass algorithm achieves this bound exactly for all practical input sizes. The proof uses this toy problem as a worked example to demonstrate the choke-point technique for proving sublinear-space lower bounds.