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.

- In Proc. 3rd SIAM Symposium on Simplicity in Algorithms (SOSA), pages 1-7, 2020 (SOSA best paper)

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.

- PDF file
- ACM Transactions on Algorithms, 16(1): 7:1-7:23, 2020 (SODA special issue)
- In Proc. 29th ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 881-897, 2018

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

- The first truly subquadratic algorithm for
- computing the (min,+) convolution for monotone increasing sequences with integer values bounded by O(n),
- solving 3SUM for monotone sets in 2D with integer coordinates bounded by O(n), and
- preprocessing a binary string for histogram indexing (also called jumbled indexing).

- The first algorithm for histogram indexing for any constant alphabet size that achieves truly subquadratic preprocessing time and truly sublinear query time.
- A truly subquadratic algorithm for integer 3SUM in the case when the given set can be partitioned into n^{1-delta} clusters each covered by an interval of length n, for any constant delta > 0.
- An algorithm to preprocess any set of n integers so that subsequently 3SUM on any given subset can be solved in O(n^{13/7} polylog n) time.

- Preliminary arXiv version
- PDF talk slides
- In Proc. 47th ACM Symposium on Theory of Computing (STOC), pages 31-40, 2015

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.

- PDF file | arXiv version
- In Proc. 41st International Colloquium on Automata, Languages, and Programming (ICALP), Lecture Notes in Computer Science, volume 8572, pages 114-125, 2014

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Timothy Chan (Last updated Oct 2020)