606 research outputs found
Binary Jumbled String Matching for Highly Run-Length Compressible Texts
The Binary Jumbled String Matching problem is defined as: Given a string
over of length and a query , with non-negative
integers, decide whether has a substring with exactly 's and
's. Previous solutions created an index of size O(n) in a pre-processing
step, which was then used to answer queries in constant time. The fastest
algorithms for construction of this index have running time
[Burcsi et al., FUN 2010; Moosa and Rahman, IPL 2010], or in
the word-RAM model [Moosa and Rahman, JDA 2012]. We propose an index
constructed directly from the run-length encoding of . The construction time
of our index is , where O(n) is the time for computing
the run-length encoding of and is the length of this encoding---this
is no worse than previous solutions if and better if . Our index can be queried in time. While
in the worst case, preliminary investigations have
indicated that may often be close to . Furthermore, the algorithm
for constructing the index is conceptually simple and easy to implement. In an
attempt to shed light on the structure and size of our index, we characterize
it in terms of the prefix normal forms of introduced in [Fici and Lipt\'ak,
DLT 2011].Comment: v2: only small cosmetic changes; v3: new title, weakened conjectures
on size of Corner Index (we no longer conjecture it to be always linear in
size of RLE); removed experimental part on random strings (these are valid
but limited in their predictive power w.r.t. general strings); v3 published
in IP
Conditional Lower Bounds for Space/Time Tradeoffs
In recent years much effort has been concentrated towards achieving
polynomial time lower bounds on algorithms for solving various well-known
problems. A useful technique for showing such lower bounds is to prove them
conditionally based on well-studied hardness assumptions such as 3SUM, APSP,
SETH, etc. This line of research helps to obtain a better understanding of the
complexity inside P.
A related question asks to prove conditional space lower bounds on data
structures that are constructed to solve certain algorithmic tasks after an
initial preprocessing stage. This question received little attention in
previous research even though it has potential strong impact.
In this paper we address this question and show that surprisingly many of the
well-studied hard problems that are known to have conditional polynomial time
lower bounds are also hard when concerning space. This hardness is shown as a
tradeoff between the space consumed by the data structure and the time needed
to answer queries. The tradeoff may be either smooth or admit one or more
singularity points.
We reveal interesting connections between different space hardness
conjectures and present matching upper bounds. We also apply these hardness
conjectures to both static and dynamic problems and prove their conditional
space hardness.
We believe that this novel framework of polynomial space conjectures can play
an important role in expressing polynomial space lower bounds of many important
algorithmic problems. Moreover, it seems that it can also help in achieving a
better understanding of the hardness of their corresponding problems in terms
of time
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
On the Parikh-de-Bruijn grid
We introduce the Parikh-de-Bruijn grid, a graph whose vertices are
fixed-order Parikh vectors, and whose edges are given by a simple shift
operation. This graph gives structural insight into the nature of sets of
Parikh vectors as well as that of the Parikh set of a given string. We show its
utility by proving some results on Parikh-de-Bruijn strings, the abelian analog
of de-Bruijn sequences.Comment: 18 pages, 3 figures, 1 tabl
Normal, Abby Normal, Prefix Normal
A prefix normal word is a binary word with the property that no substring has
more 1s than the prefix of the same length. This class of words is important in
the context of binary jumbled pattern matching. In this paper we present
results about the number of prefix normal words of length , showing
that for some and
. We introduce efficient
algorithms for testing the prefix normal property and a "mechanical algorithm"
for computing prefix normal forms. We also include games which can be played
with prefix normal words. In these games Alice wishes to stay normal but Bob
wants to drive her "abnormal" -- we discuss which parameter settings allow
Alice to succeed.Comment: Accepted at FUN '1
- …