Exact minimum number of input reads for sum-exclude-self under sublinear working memory

Determine the exact minimum total number of input-element reads required by any correct algorithm that, given a read-only input array In of length n with d-bit unsigned integers and at most t = o(nd) bits of working memory and write-only access to an output array, computes Out[i] = sum_{j ≠ i} In[j] over the integers. Ascertain whether the lower bound (n − 1) + (n − ⌊t/d⌋) element reads is tight for all n, d, and sublinear t, or whether strictly more reads are inherently necessary.

Background

The paper proves two lower bounds for computing the sum-exclude-self array under a streaming model with sublinear working memory: at least n−1 input elements must be read before any output cell receives its final value (the first pass), and at least n − ⌊t/d⌋ additional elements must be read after that point (the second pass), where t = o(nd) bits is the working memory bound.

They show that a straightforward two-pass algorithm reads 2n elements, and an optimized variant reads 2n − 1 elements by avoiding one final read. However, when analyzing the second pass under the space used to store the total sum S (t = d + ⌈log2 n⌉), the lower bound indicates that as few as n − 1 − ⌊⌈log2 n⌉/d⌋ second-pass reads might suffice, leaving a persistent gap of ⌊⌈log2 n⌉/d⌋ reads compared to the optimized algorithm’s n − 1 reads. The authors note that better summaries or summary-free strategies might close this gap, and explicitly state that the exact minimum number of reads is not yet determined.