Free access
Proceedings
Proceedings of the 2022 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA)

Deterministic and Las Vegas Algorithms for Sparse Nonnegative Convolution

Abstract

Computing the convolution A∗B of two length-n integer vectors A, B is a core problem in several disciplines. It frequently comes up as a subroutine in various problem domains, e.g. in algorithms for Knapsack, k-SUM, All-Pairs Shortest Paths, and string pattern matching problems. For these applications it typically suffices to compute convolutions of nonnegative vectors. This problem can be classically solved in time O(n log n) using the Fast Fourier Transform.
However, in many applications the involved vectors are sparse and hence one could hope for output-sensitive algorithms to compute nonnegative convolutions. This question was raised by Muthukrishnan and solved by Cole and Hariharan (STOC '02) by a randomized algorithm running in near-linear time in the (unknown) output-size t and recently improved by Bringmann, Fischer and Nakos (STOC '21) in O(k log k) Monte Carlo time. Chan and Lewenstein (STOC '15) presented a deterministic algorithm with a overhead in running time and the additional assumption that a small superset of the output is given; this assumption was later removed by Bringmann and Nakos (ICALP '21).
In this paper we present the first deterministic near-linear-time algorithm for computing sparse nonnegative convolutions. This immediately gives improved deterministic algorithms for the state-of-the-art of output-sensitive Subset Sum, block-mass pattern matching, N-fold Boolean convolution, and others, matching up to log-factors the fastest known randomized algorithms for these problems. Our algorithm is a blend of algebraic and combinatorial ideas and techniques.
Additionally, we provide two fast Las Vegas algorithms for computing sparse nonnegative convolutions. In particular, we present a simple O(t log2 t) time algorithm, which is an accessible alternative to Cole and Hariharan's algorithm. Subsequently, we further refine this new algorithm to run in Las Vegas time O(t log t · log log t), which matches the running time of the dense case apart from the log log t factor.

Formats available

You can view the full content in the following formats:

Information & Authors

Information

Published In

cover image Proceedings
Proceedings of the 2022 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA)
Pages: 3069 - 3090
Editors: Joseph (Seffi) Naor, Technion Israel Institute of Technology, Israel and Niv Buchbinder, Tel Aviv University, Israel
ISBN (Online): 978-1-61197-707-3

History

Published online: 5 January 2022

Authors

Affiliations

Notes

This work is part of the project TIPEA that has received funding from the European Research Council (ERC) under the European Unions Horizon 2020 research and innovation programme (grant agreement No. 850979).
The full version of the paper can be accessed at https://arxiv.org/abs/2107.07625

Metrics & Citations

Metrics

Citations

If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.

Cited By

There are no citations for this item

View Options

View options

PDF

View PDF

Get Access

Media

Figures

Other

Tables

Share

Share

Copy the content Link

Share with email

Email a colleague

Share on social media

The SIAM Publications Library now uses SIAM Single Sign-On for individuals. If you do not have existing SIAM credentials, create your SIAM account https://my.siam.org.