21 research outputs found
Hardness of Detecting Abelian and Additive Square Factors in Strings
We prove 3SUM-hardness (no strongly subquadratic-time algorithm, assuming the
3SUM conjecture) of several problems related to finding Abelian square and
additive square factors in a string. In particular, we conclude conditional
optimality of the state-of-the-art algorithms for finding such factors.
Overall, we show 3SUM-hardness of (a) detecting an Abelian square factor of
an odd half-length, (b) computing centers of all Abelian square factors, (c)
detecting an additive square factor in a length- string of integers of
magnitude , and (d) a problem of computing a double 3-term
arithmetic progression (i.e., finding indices such that
) in a sequence of integers of
magnitude .
Problem (d) is essentially a convolution version of the AVERAGE problem that
was proposed in a manuscript of Erickson. We obtain a conditional lower bound
for it with the aid of techniques recently developed by Dudek et al. [STOC
2020]. Problem (d) immediately reduces to problem (c) and is a step in
reductions to problems (a) and (b). In conditional lower bounds for problems
(a) and (b) we apply an encoding of Amir et al. [ICALP 2014] and extend it
using several string gadgets that include arbitrarily long Abelian-square-free
strings.
Our reductions also imply conditional lower bounds for detecting Abelian
squares in strings over a constant-sized alphabet. We also show a subquadratic
upper bound in this case, applying a result of Chan and Lewenstein [STOC 2015].Comment: Accepted to ESA 202
Clustered Integer 3SUM via Additive Combinatorics
We present a collection of new results on problems related to 3SUM,
including:
1. The first truly subquadratic algorithm for
1a. computing the (min,+) convolution for monotone increasing
sequences with integer values bounded by ,
1b. solving 3SUM for monotone sets in 2D with integer coordinates
bounded by , and
1c. preprocessing a binary string for histogram indexing (also
called jumbled indexing).
The running time is:
with
randomization, or deterministically. This greatly improves the
previous 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.
2. The first algorithm for histogram indexing for any constant alphabet size
that achieves truly subquadratic preprocessing time and truly sublinear query
time.
3. A truly subquadratic algorithm for integer 3SUM in the case when the given
set can be partitioned into clusters each covered by an interval
of length , for any constant .
4. An algorithm to preprocess any set of integers so that subsequently
3SUM on any given subset can be solved in
time.
All these results are obtained by a surprising new technique, based on the
Balog--Szemer\'edi--Gowers Theorem from additive combinatorics
String Periods in the Order-Preserving Model
The order-preserving model (op-model, in short) was introduced quite recently but has already attracted significant attention because of its applications in data analysis. We introduce several types of periods in this setting (op-periods). Then we give algorithms to compute these periods in time O(n), O(n log log n), O(n log^2 log n/log log log n), O(n log n) depending on the type of periodicity. In the most general variant the number of different periods can be as big as Omega(n^2), and a compact representation is needed. Our algorithms require novel combinatorial insight into the properties of such periods
String periods in the order-preserving model
In the order-preserving model, two strings match if they share the same relative order between the characters at the corresponding positions. This model is quite recent, but it has already attracted significant attention because of its applications in data analysis. We introduce several types of periods in this setting (op-periods). Then we give algorithms to compute these periods in time O(n), O(nlogâĄlogâĄn), O(nlog2âĄlogâĄn/logâĄlogâĄlogâĄn), O(nlogâĄn) depending on the type of periodicity. In the most general variant, the number of different op-periods can be as big as Ω(n2), and a compact representation is needed. Our algorithms require novel combinatorial insight into the properties of op-periods. In particular, we characterize the FineâWilf property for coprime op-periods. © 2019 Elsevier Inc.Supported by ISF grants no. 824/17 and 1278/16 and by an ERC grant MPM under the EU's Horizon 2020 Research and Innovation Programme (grant no. 683064).Supported by the Ministry of Science and Higher Education of the Russian Federation, project 1.3253.2017.A part of this work was done during the workshop StringMasters in Warsaw 2017 that was sponsored by the Warsaw Center of Mathematics and Computer Science. The authors thank the participants of the workshop, especially Hideo Bannai and Shunsuke Inenaga, for helpful discussions
Dagstuhl Reports : Volume 1, Issue 2, February 2011
Online Privacy: Towards Informational Self-Determination on the Internet (Dagstuhl Perspectives Workshop 11061) : Simone Fischer-HĂŒbner, Chris Hoofnagle, Kai Rannenberg, Michael Waidner, Ioannis Krontiris and Michael Marhöfer Self-Repairing Programs (Dagstuhl Seminar 11062) : Mauro PezzĂ©, Martin C. Rinard, Westley Weimer and Andreas Zeller Theory and Applications of Graph Searching Problems (Dagstuhl Seminar 11071) : Fedor V. Fomin, Pierre Fraigniaud, Stephan Kreutzer and Dimitrios M. Thilikos Combinatorial and Algorithmic Aspects of Sequence Processing (Dagstuhl Seminar 11081) : Maxime Crochemore, Lila Kari, Mehryar Mohri and Dirk Nowotka Packing and Scheduling Algorithms for Information and Communication Services (Dagstuhl Seminar 11091) Klaus Jansen, Claire Mathieu, Hadas Shachnai and Neal E. Youn
All Growth Rates of Abelian Exponents Are Attained by Infinite Binary Words
We consider repetitions in infinite words by making a novel inquiry to the maximum eventual growth rate of the exponents of abelian powers occurring in an infinite word. Given an increasing, unbounded function f: ? ? ?, we construct an infinite binary word whose abelian exponents have limit superior growth rate f. As a consequence, we obtain that every nonnegative real number is the critical abelian exponent of some infinite binary word
45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)
We consider repetitions in infinite words by making a novel inquiry to the maximum eventual growth rate of the exponents of abelian powers occurring in an infinite word. Given an increasing, unbounded function , we construct an infinite binary word whose abelian exponents have limit superior growth rate . As a consequence, we obtain that every nonnegative real number is the critical abelian exponent of some infinite binary word.</p
Fast -fold {B}oolean Convolution via Additive Combinatorics
We consider the problem of computing the Boolean convolution (with wraparound) of ~vectors of dimension , or, equivalently, the problem of computing the sumset for . Boolean convolution formalizes the frequent task of combining two subproblems, where the whole problem has a solution of size if for some the first subproblem has a solution of size~ and the second subproblem has a solution of size . Our problem formalizes a natural generalization, namely combining solutions of subproblems subject to a modular constraint. This simultaneously generalises Modular Subset Sum and Boolean Convolution (Sumset Computation). Although nearly optimal algorithms are known for special cases of this problem, not even tiny improvements are known for the general case. We almost resolve the computational complexity of this problem, shaving essentially a factor of from the running time of previous algorithms. Specifically, we present a \emph{deterministic} algorithm running in \emph{almost} linear time with respect to the input plus output size . We also present a \emph{Las Vegas} algorithm running in \emph{nearly} linear expected time with respect to the input plus output size . Previously, no deterministic or randomized algorithm was known. At the heart of our approach lies a careful usage of Kneser's theorem from Additive Combinatorics, and a new deterministic almost linear output-sensitive algorithm for non-negative sparse convolution. In total, our work builds a solid toolbox that could be of independent interest