Title: (Min,+)-Convolutions, 3SUM and Additive Combinatorics 

Speaker: Moshe Lewenstein, Bar-Ilan University

Time/Date: Thursday, Feb 18, 3-4 pm
Location: Room 5510

Abstract:

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

The first truly subquadratic algorithm for

1. computing the (min,+) convolution for monotone increasing sequences
with integer values bounded by $O(n)$, 

2. solving 3SUM for monotone sets in 2D with integer coordinates
bounded by $O(n)$, and

3. preprocessing a binary string for histogram indexing (also called
jumbled indexing).

The running time is O(n^{1.859})$ with randomization, or
$O(n^{1.864})$ deterministically.

This greatly improves the previous $n^2/2^{\Omega(\sqrt{\log n})}$
time bound obtained from Williams' recent result on all-pairs shortest
paths [STOC'14], and answers an open question raised by several
researchers studying the histogram indexing problem.

The first algorithm for histogram indexing for any constant alphabet
size that achieves truly subquadratic preprocessing time and truly
sublinear query time.

1.  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 $\delta>0$.

2. An algorithm to preprocess any set of $n$ integers so that
subsequently 3SUM on any given subset can be solved in $\OO(n^{13/7})$
time.

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

Joint work with Timothy Chan