215 research outputs found
Why is it hard to beat for Longest Common Weakly Increasing Subsequence?
The Longest Common Weakly Increasing Subsequence problem (LCWIS) is a variant
of the classic Longest Common Subsequence problem (LCS). Both problems can be
solved with simple quadratic time algorithms. A recent line of research led to
a number of matching conditional lower bounds for LCS and other related
problems. However, the status of LCWIS remained open.
In this paper we show that LCWIS cannot be solved in strongly subquadratic
time unless the Strong Exponential Time Hypothesis (SETH) is false.
The ideas which we developed can also be used to obtain a lower bound based
on a safer assumption of NC-SETH, i.e. a version of SETH which talks about NC
circuits instead of less expressive CNF formulas
Distributed PCP Theorems for Hardness of Approximation in P
We present a new distributed model of probabilistically checkable proofs
(PCP). A satisfying assignment to a CNF formula is
shared between two parties, where Alice knows , Bob knows
, and both parties know . The goal is to have
Alice and Bob jointly write a PCP that satisfies , while
exchanging little or no information. Unfortunately, this model as-is does not
allow for nontrivial query complexity. Instead, we focus on a non-deterministic
variant, where the players are helped by Merlin, a third party who knows all of
.
Using our framework, we obtain, for the first time, PCP-like reductions from
the Strong Exponential Time Hypothesis (SETH) to approximation problems in P.
In particular, under SETH we show that there are no truly-subquadratic
approximation algorithms for Bichromatic Maximum Inner Product over
{0,1}-vectors, Bichromatic LCS Closest Pair over permutations, Approximate
Regular Expression Matching, and Diameter in Product Metric. All our
inapproximability factors are nearly-tight. In particular, for the first two
problems we obtain nearly-polynomial factors of ; only
-factor lower bounds (under SETH) were known before
Approximating Dynamic Time Warping and Edit Distance for a Pair of Point Sequences
We give the first subquadratic-time approximation schemes for dynamic time
warping (DTW) and edit distance (ED) of several natural families of point
sequences in , for any fixed . In particular, our
algorithms compute -approximations of DTW and ED in time
near-linear for point sequences drawn from k-packed or k-bounded curves, and
subquadratic for backbone sequences. Roughly speaking, a curve is
-packed if the length of its intersection with any ball of radius
is at most , and a curve is -bounded if the sub-curve
between two curve points does not go too far from the two points compared to
the distance between the two points. In backbone sequences, consecutive points
are spaced at approximately equal distances apart, and no two points lie very
close together. Recent results suggest that a subquadratic algorithm for DTW or
ED is unlikely for an arbitrary pair of point sequences even for . Our
algorithms work by constructing a small set of rectangular regions that cover
the entries of the dynamic programming table commonly used for these distance
measures. The weights of entries inside each rectangle are roughly the same, so
we are able to use efficient procedures to approximately compute the cheapest
paths through these rectangles
String Sanitization Under Edit Distance
Let W be a string of length n over an alphabet Σ, k be a positive integer, and be a set of length-k substrings of W. The ETFS problem asks us to construct a string X_{ED} such that: (i) no string of occurs in X_{ED}; (ii) the order of all other length-k substrings over Σ is the same in W and in X_{ED}; and (iii) X_{ED} has minimal edit distance to W. When W represents an individual’s data and represents a set of confidential substrings, algorithms solving ETFS can be applied for utility-preserving string sanitization [Bernardini et al., ECML PKDD 2019]. Our first result here is an algorithm to solve ETFS in (kn²) time, which improves on the state of the art [Bernardini et al., arXiv 2019] by a factor of |Σ|. Our algorithm is based on a non-trivial modification of the classic dynamic programming algorithm for computing the edit distance between two strings. Notably, we also show that ETFS cannot be solved in (n^{2-δ}) time, for any δ>0, unless the strong exponential time hypothesis is false. To achieve this, we reduce the edit distance problem, which is known to admit the same conditional lower bound [Bringmann and Künnemann, FOCS 2015], to ETFS
Why walking the dog takes time: Frechet distance has no strongly subquadratic algorithms unless SETH fails
The Frechet distance is a well-studied and very popular measure of similarity
of two curves. Many variants and extensions have been studied since Alt and
Godau introduced this measure to computational geometry in 1991. Their original
algorithm to compute the Frechet distance of two polygonal curves with n
vertices has a runtime of O(n^2 log n). More than 20 years later, the state of
the art algorithms for most variants still take time more than O(n^2 / log n),
but no matching lower bounds are known, not even under reasonable complexity
theoretic assumptions.
To obtain a conditional lower bound, in this paper we assume the Strong
Exponential Time Hypothesis or, more precisely, that there is no
O*((2-delta)^N) algorithm for CNF-SAT for any delta > 0. Under this assumption
we show that the Frechet distance cannot be computed in strongly subquadratic
time, i.e., in time O(n^{2-delta}) for any delta > 0. This means that finding
faster algorithms for the Frechet distance is as hard as finding faster CNF-SAT
algorithms, and the existence of a strongly subquadratic algorithm can be
considered unlikely.
Our result holds for both the continuous and the discrete Frechet distance.
We extend the main result in various directions. Based on the same assumption
we (1) show non-existence of a strongly subquadratic 1.001-approximation, (2)
present tight lower bounds in case the numbers of vertices of the two curves
are imbalanced, and (3) examine realistic input assumptions (c-packed curves)
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