Shaving Logs via Large Sieve Inequality: Faster Algorithms for Sparse Convolution and More
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Author(s)
Jin, Ce
Xu, Yinzhan
Date Issued
June 10, 2024
Publisher
ACM|Proceedings of the 56th Annual ACM Symposium on Theory of Computing
Citation
Jin, Ce and Xu, Yinzhan. 2024. "Shaving Logs via Large Sieve Inequality: Faster Algorithms for Sparse Convolution and More."
Version
Final published version
Abstract
In sparse convolution-type problems, a common technique is to
hash the input integers modulo a random prime 𝑝 ∈ [𝑄/2, 𝑄] for
some parameter 𝑄, which reduces the range of the input integers
while preserving their additive structure. However, this hash family
suffers from two drawbacks, which led to bottlenecks in many
state-of-the-art algorithms: (1) The collision probability of two
elements from [𝑁] is 𝑂(
log 𝑁
𝑄
) rather than 𝑂(
1
𝑄
); (2) It is difficult
to derandomize the choice of 𝑝; known derandomization techniques
lead to super-logarithmic overhead [Chan, Lewenstein STOC’15].
In this paper, we partially overcome these drawbacks in certain
scenarios, via novel applications of the large sieve inequality from
analytic number theory. Consequently, we obtain the following
improved algorithms for various problems (in the standard word
RAM model):
Sparse Nonnegative Convolution: We obtain an 𝑂(𝑡 log 𝑡)-
time Las Vegas algorithm that computes the convolution 𝐴 ★ 𝐵 of
two nonnegative integer vectors 𝐴, 𝐵, where 𝑡 is the output sparsity ∥𝐴 ★ 𝐵∥0. Moreover, our algorithm terminates in 𝑂(𝑡 log 𝑡)
time with 1 − 1/poly(𝑡) probability. This simultaneously improves
the 𝑂(𝑡 log 𝑡 log log 𝑡)-time Las Vegas algorithm [Bringmann, Fischer, Nakos SODA’22] and the Monte Carlo 𝑂(𝑡 log 𝑡)-time algorithm with failure probability 2
−
√
log 𝑡
[Bringmann, Fischer, Nakos
STOC’21].
Text-to-Pattern Hamming Distances: Given a length-𝑚 pattern 𝑃 and a length-𝑛 text 𝑇 , we obtain an 𝑂(𝑛
√︁
𝑚 log log𝑚)-time
deterministic algorithm that exactly computes the Hamming distance between 𝑃 and every length-𝑚 substring of 𝑇 . This improves
the previous 𝑂(𝑛
√
𝑚(log𝑚 log log𝑚)
1/4
)-time deterministic algorithm [Chan, Jin, Vassilevska Williams, Xu FOCS’23] and nearly
matches their 𝑂(𝑛
√
𝑚)-time Las Vegas algorithm.
Sparse General Convolution: For sparse convolution with possibly negative input, all previous approaches required Ω(𝑡 log2
𝑡) time, where 𝑡 is the maximum of input and output sparsity, and
an important question left open by [Bringmann, Fischer, Nakos
STOC’21] is whether this can be improved. We make partial progress
towards solving this question by giving a Monte Carlo 𝑂(𝑡 log 𝑡)
time algorithm in the restricted case where the length 𝑁 of the
input vectors satisfies 𝑁 ≤ 𝑡
1.99
.
Description
STOC ’24, June 24–28, 2024, Vancouver, BC, Canada
MIT Department
Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science
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DOI of Published Version
https://doi.org/10.1145/3618260.3649605
