Timothy M. Chan's Publications: 3SUM

Faster algorithms for text-to-pattern Hamming distances

Ce Jin, Virginia Vassilevska Williams, and Yinzhan Xu)

Fredman's trick meets dominance product: Fine-grained complexity of unweighted APSP, 3SUM counting, and more

Virginia Vassilevska Williams and Yinzhan Xu)

In this paper we carefully combine Fredman's trick [SICOMP'76] and Matousek's approach for dominance product [IPL'91] to obtain powerful results in fine-grained complexity:

Hardness for triangle problems under even more believable hypotheses: Reductions from Real APSP, Real 3SUM, and OV

Virginia Vassilevska Williams and Yinzhan Xu)

The 3SUM hypothesis, the APSP hypothesis and SETH are the three main hypotheses in fine-grained complexity. So far, within the area, the first two hypotheses have mainly been about integer inputs in the Word RAM model of computation. The "Real APSP" and "Real 3SUM" hypotheses, which assert that the APSP and 3SUM hypotheses hold for real-valued inputs in a reasonable version of the Real RAM model, are even more believable than their integer counterparts.

Under the very believable hypothesis that at least one of the Integer 3SUM hypothesis, Integer APSP hypothesis or SETH is true, Abboud, Vassilevska W. and Yu [STOC 2015] showed that a problem called Triangle Collection requires n^{3-o(1)} time on an n-node graph.

Our main result is a nontrivial lower bound for a slight generalization of Triangle Collection, called All-Color-Pairs Triangle Collection, under the even more believable hypothesis that at least one of the Real 3SUM, the Real APSP, and the OV hypotheses is true. Combined with slight modifications of prior reductions, we obtain polynomial conditional lower bounds for problems such as the (static) ST-Max Flow problem and dynamic Max Flow, now under the new weaker hypothesis.

Our main result is built on the following two lines of reductions.

Along the way we show that Triangle Collection is equivalent to a simpler restricted version of the problem, simplifying prior work. Our techniques also have other interesting implications, such as a super-linear lower bound of Integer All-Numbers 3SUM based on the Real 3SUM hypothesis, and a tight lower bound for a string matching problem based on the OV hypothesis.

Reducing 3SUM to Convolution-3SUM

(with Qizheng He)

Given a set S of n numbers, the 3SUM problem asks to determine whether there exist three elements a,b,c in S such that a+b+c = 0. The related Convolution-3SUM problem asks to determine whether there exist a pair of indices i,j such that A[i]+A[j] = A[i+j], where A is a given array of nnumbers.

When the numbers are integers, a randomized reduction from 3SUM to Convolution-3SUM was given in a seminal paper by Patrascu [STOC 2010], which was later improved by Kopelowitz, Pettie, and Porat [SODA 2016] with an O(log n) factor slowdown. In this paper, we present a simple deterministic reduction from 3SUM to Convolution-3SUM for integers bounded by U. We also describe additional ideas to obtaining further improved reductions, with only a (loglog n)^{O(1)} factor slowdown in the randomized case, and a (log U)^{O(1)} factor slowdown in the deterministic case.

More logarithmic-factor speedups for 3SUM, (median,+)-convolution, and some geometric 3SUM-hard problems

We present an algorithm that solves the 3SUM problem for n real numbers in O((n^2/log^2n) (loglog n)^{O(1)}) time, improving previous solutions by about a logarithmic factor. Our framework for shaving off two logarithmic factors can be applied to other problems, such as (median,+)-convolution/matrix multiplication and algebraic generalizations of 3SUM. We also obtain the first subquadratic results on some 3SUM-hard problems in computational geometry, for example, deciding whether (the interiors of) a constant number of simple polygons have a common intersection.

Clustered integer 3SUM via additive combinatorics

Moshe Lewenstein)

We present a collection of new results on problems related to 3SUM, including:

All these results are obtained by a surprising new technique, based on the Balog-Szemeredi-Gowers Theorem from additive combinatorics.

On hardness of jumbled indexing

Amihood Amir, Moshe Lewenstein, and Noa Lewenstein)

Jumbled indexing is the problem of indexing a text T for queries that ask whether there is a substring of T matching a pattern represented as a Parikh vector, i.e., the vector of frequency counts for each character. Jumbled indexing has garnered a lot of interest in the last four years. There is a naive algorithm that preprocesses all answers in O(n^2 |Sigma|) time allowing quick queries afterwards, and there is another naive algorithm that requires no preprocessing but has O(n log |Sigma|) query time. Despite a tremendous amount of effort there has been little improvement over these running times.

In this paper we provide good reason for this. We show that, under a 3SUM-hardness assumption, jumbled indexing for alphabets of size omega(1) requires Omega(n^{2-epsilon}) preprocessing time or Omega(n^{1-delta}) query time for any epsilon,delta>0. In fact, under a stronger 3SUM-hardness assumption, for any constant alphabet size r >= 3 there exist describable fixed constant epsilon_r and delta_r such that jumbled indexing requires Omega(n^{2-epsilon_r}) preprocessing time or Omega(n^{1-delta_r}) query time.

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Timothy Chan (Last updated Aug 2023)