144 research outputs found
Front Matter, Table of Contents, Preface
No full textFront Matter, Table of Contents, Preface, List of Author
Proceedings of the 27th Annual Symposium on Combinatorial Pattern Matching.
No full textThe objective of the Annual Symposium on Combinatorial Pattern Matching is to provide an international forum for research in combinatorial pattern matching and related applications. It addresses issues of searching and matching strings and more complicated patterns such as trees, regular expressions, graphs, point sets, and arrays. The goal is to derive combinatorial properties of such structures and to exploit these properties in order to achieve a superior performance for the corresponding computational problems. The meeting also deals with problems in bioinformatics and computational biology, coding and data compression, combinatorics on words, data mining, information retrieval, natural language processing, pattern discovery, string algorithms, string processing in databases, symbolic computing, and text searching.
This volume contains the papers presented at the 27th Annual Symposium on Combinatorial Pattern Matching (CPM 2016) held during June 27-29, 2016, in Tel Aviv, Israe
Parameterized matching with mismatches
No full textThe problem of approximate parameterized string searching consists of ¯nding, for a
given text t = t1t2:::tn and pattern p = p1p2:::pm over respective alphabets §t and §p,
the injection ¼i from §p to §t maximizing the number of matches between ¼i(p) and
titi+1:::ti+m¡1 (i = 1; 2; :::n¡m+1). We examine the special case where both strings are
run-length encoded, and further restrict to the case where one of the alphabets is binary.
For this case, we give a construction working in time O(n + (rp £ rt) ®(rt) log(rt)),
where rp and rt denote the number of runs in the corresponding encodings for y and x,
respectively, and ® is the inverse of the Ackermann's function
How Hard is it to Find (Honest) Witnesses?
No full textIn recent years much effort has been put into developing polynomial-time conditional lower bounds for algorithms and data structures in both static and dynamic settings. Along these lines we introduce a framework for proving conditional lower bounds based on the well-known 3SUM conjecture. Our framework creates a compact representation of an instance of the 3SUM problem using hashing and domain specific encoding. This compact representation admits false solutions to the original 3SUM problem instance which we reveal and eliminate until we find a true solution. In other words, from all witnesses (candidate solutions) we figure out if an honest one (a true solution) exists. This enumeration of witnesses is used to prove conditional lower bounds on reporting problems that generate all witnesses. In turn, these reporting problems are then reduced to various decision problems using special search data structures which are able to enumerate the witnesses while only using solutions to decision variants. Hence, 3SUM-hardness of the decision problems is deduced.
We utilize this framework to show conditional lower bounds for several variants of convolutions, matrix multiplication and string problems. Our framework uses a strong connection between all of these problems and the ability to find witnesses.
Specifically, we prove conditional lower bounds for computing partial outputs of convolutions and matrix multiplication for sparse inputs. These problems are inspired by the open question raised by Muthukrishnan 20 years ago. The lower bounds we show rule out the possibility (unless the 3SUM conjecture is false) that almost linear time solutions to sparse input-output convolutions or matrix multiplications exist. This is in contrast to standard convolutions and matrix multiplications that have, or assumed to have, almost linear solutions.
Moreover, we improve upon the conditional lower bounds of Amir et al. for histogram indexing, a problem that has been of much interest recently. The conditional lower bounds we show apply for both reporting and decision variants. For the well-studied decision variant, we show a full tradeoff between preprocessing and query time for every alphabet size > 2. At an extreme, this implies that no solution to this problem exists with subquadratic preprocessing time and ~O(1) query time for every alphabet size > 2, unless the 3SUM conjecture is false. This is in contrast to a recent result by Chan and Lewenstein for a binary alphabet.
While these specific applications are used to demonstrate the techniques of our framework, we believe that this novel framework is useful for many other problems as well
On the Hardness of Set Disjointness and Set Intersection with Bounded Universe
No full textIn the SetDisjointness problem, a collection of m sets S_1,S_2,...,S_m from some universe U is preprocessed in order to answer queries on the emptiness of the intersection of some two query sets from the collection. In the SetIntersection variant, all the elements in the intersection of the query sets are required to be reported. These are two fundamental problems that were considered in several papers from both the upper bound and lower bound perspective.
Several conditional lower bounds for these problems were proven for the tradeoff between preprocessing and query time or the tradeoff between space and query time. Moreover, there are several unconditional hardness results for these problems in some specific computational models. The fundamental nature of the SetDisjointness and SetIntersection problems makes them useful for proving the conditional hardness of other problems from various areas. However, the universe of the elements in the sets may be very large, which may cause the reduction to some other problems to be inefficient and therefore it is not useful for proving their conditional hardness.
In this paper, we prove the conditional hardness of SetDisjointness and SetIntersection with bounded universe. This conditional hardness is shown for both the interplay between preprocessing and query time and the interplay between space and query time. Moreover, we present several applications of these new conditional lower bounds. These applications demonstrates the strength of our new conditional lower bounds as they exploit the limited universe size. We believe that this new framework of conditional lower bounds with bounded universe can be useful for further significant applications
Structure and Hardness in P (Dagstuhl Seminar 16451)
No full textThis document contains description of the talks at the Dagstuhl seminar 16451 "Structure and Hardness in P". The main goal of the seminar was to bring together researchers from several disciplines and connect those who work on proving conditional lower bounds with those who or may benefit from it. This resulted in an extensive list of open problems which is also provided
Orthogonal Vectors Indexing
No full textIn the recent years, intensive research work has been dedicated to prove conditional lower bounds in order to reveal the inner structure of the class P. These conditional lower bounds are based on many popular conjectures on well-studied problems. One of the most heavily used conjectures is the celebrated Strong Exponential Time Hypothesis (SETH). It turns out that conditional hardness proved based on SETH goes, in many cases, through an intermediate problem - the Orthogonal Vectors (OV) problem.
Almost all research work regarding conditional lower bound was concentrated on time complexity. Very little attention was directed toward space complexity. In a recent work, Goldstein et al.[WADS '17] set the stage for proving conditional lower bounds regarding space and its interplay with time. In this spirit, it is tempting to investigate the space complexity of a data structure variant of OV which is called OV indexing. In this problem n boolean vectors of size clogn are given for preprocessing. As a query, a vector v is given and we are required to verify if there is an input vector that is orthogonal to it or not.
This OV indexing problem is interesting in its own, but it also likely to have strong implications on problems known to be conditionally hard, in terms of time complexity, based on OV. Having this in mind, we study OV indexing in this paper from many aspects. We give some space-efficient algorithms for the problem, show a tradeoff between space and query time, describe how to solve its reporting variant, shed light on an interesting connection between this problem and the well-studied SetDisjointness problem and demonstrate how it can be solved more efficiently on random input
Improved Space-Time Tradeoffs for kSUM
No full textIn the kSUM problem we are given an array of numbers a_1,a_2,...,a_n and we are required to determine if there are k different elements in this array such that their sum is 0. This problem is a parameterized version of the well-studied SUBSET-SUM problem, and a special case is the 3SUM problem that is extensively used for proving conditional hardness. Several works investigated the interplay between time and space in the context of SUBSET-SUM. Recently, improved time-space tradeoffs were proven for kSUM using both randomized and deterministic algorithms.
In this paper we obtain an improvement over the best known results for the time-space tradeoff for kSUM. A major ingredient in achieving these results is a general self-reduction from kSUM to mSUM where m1. (iv) An algorithm for 6SUM running in O(n^4) time using just O(n^{2/3}) space. (v) A solution to 3SUM on random input using O(n^2) time and O(n^{1/3}) space, under the assumption of a random read-only access to random bits
Space-Efficient String Indexing for Wildcard Pattern Matching
No full textIn this paper we describe compressed indexes that support pattern matching queries for strings with wildcards. For a constant size alphabet our data structure uses O(n.log^e(n)) bits for any e>0 and reports all occ occurrences of a wildcard string in O(m+s^g.M(n)+occ) time, where M(n)=o(log(log(log(n)))), s is the alphabet size, m is the number of alphabet symbols and g is the number of wildcard symbols in the query string. We also present an O(n)-bit index with O((m+s^g+occ).log^e(n)) query time and an O(n{log(log(n))}^2)-bit index with O((m+s^g+occ).log(log(n))) query time. These are the first non-trivial data structures for this problem that need o(n.log(n)) bits of space
- …
