1,159 research outputs found
Lower bounds for approximation schemes for Closest String
In the Closest String problem one is given a family of
equal-length strings over some fixed alphabet, and the task is to find a string
that minimizes the maximum Hamming distance between and a string from
. While polynomial-time approximation schemes (PTASes) for this
problem are known for a long time [Li et al., J. ACM'02], no efficient
polynomial-time approximation scheme (EPTAS) has been proposed so far. In this
paper, we prove that the existence of an EPTAS for Closest String is in fact
unlikely, as it would imply that , a highly
unexpected collapse in the hierarchy of parameterized complexity classes. Our
proof also shows that the existence of a PTAS for Closest String with running
time , for any computable function
, would contradict the Exponential Time Hypothesis
Approximate Hamming distance in a stream
We consider the problem of computing a -approximation of the
Hamming distance between a pattern of length and successive substrings of a
stream. We first look at the one-way randomised communication complexity of
this problem, giving Alice the first half of the stream and Bob the second
half. We show the following: (1) If Alice and Bob both share the pattern then
there is an bit randomised one-way communication
protocol. (2) If only Alice has the pattern then there is an
bit randomised one-way communication protocol.
We then go on to develop small space streaming algorithms for
-approximate Hamming distance which give worst case running time
guarantees per arriving symbol. (1) For binary input alphabets there is an
space and
time streaming -approximate Hamming distance algorithm. (2) For
general input alphabets there is an
space and time streaming
-approximate Hamming distance algorithm.Comment: Submitted to ICALP' 201
Online Pattern Matching for String Edit Distance with Moves
Edit distance with moves (EDM) is a string-to-string distance measure that
includes substring moves in addition to ordinal editing operations to turn one
string to the other. Although optimizing EDM is intractable, it has many
applications especially in error detections. Edit sensitive parsing (ESP) is an
efficient parsing algorithm that guarantees an upper bound of parsing
discrepancies between different appearances of the same substrings in a string.
ESP can be used for computing an approximate EDM as the L1 distance between
characteristic vectors built by node labels in parsing trees. However, ESP is
not applicable to a streaming text data where a whole text is unknown in
advance. We present an online ESP (OESP) that enables an online pattern
matching for EDM. OESP builds a parse tree for a streaming text and computes
the L1 distance between characteristic vectors in an online manner. For the
space-efficient computation of EDM, OESP directly encodes the parse tree into a
succinct representation by leveraging the idea behind recent results of a
dynamic succinct tree. We experimentally test OESP on the ability to compute
EDM in an online manner on benchmark datasets, and we show OESP's efficiency.Comment: This paper has been accepted to the 21st edition of the International
Symposium on String Processing and Information Retrieval (SPIRE2014
On Computing Centroids According to the p-Norms of Hamming Distance Vectors
In this paper we consider the p-Norm Hamming Centroid problem which asks to determine whether some given strings have a centroid with a bound on the p-norm of its Hamming distances to the strings. Specifically, given a set S of strings and a real k, we consider the problem of determining whether there exists a string s^* with (sum_{s in S} d^{p}(s^*,s))^(1/p) <=k, where d(,) denotes the Hamming distance metric. This problem has important applications in data clustering and multi-winner committee elections, and is a generalization of the well-known polynomial-time solvable Consensus String (p=1) problem, as well as the NP-hard Closest String (p=infty) problem.
Our main result shows that the problem is NP-hard for all fixed rational p > 1, closing the gap for all rational values of p between 1 and infty. Under standard complexity assumptions the reduction also implies that the problem has no 2^o(n+m)-time or 2^o(k^(p/(p+1)))-time algorithm, where m denotes the number of input strings and n denotes the length of each string, for any fixed p > 1. The first bound matches a straightforward brute-force algorithm. The second bound is tight in the sense that for each fixed epsilon > 0, we provide a 2^(k^(p/((p+1))+epsilon))-time algorithm. In the last part of the paper, we complement our hardness result by presenting a fixed-parameter algorithm and a factor-2 approximation algorithm for the problem
Approximation and Parameterized Complexity of Minimax Approval Voting
We present three results on the complexity of Minimax Approval Voting. First,
we study Minimax Approval Voting parameterized by the Hamming distance from
the solution to the votes. We show Minimax Approval Voting admits no algorithm
running in time , unless the Exponential
Time Hypothesis (ETH) fails. This means that the
algorithm of Misra et al. [AAMAS 2015] is essentially optimal. Motivated by
this, we then show a parameterized approximation scheme, running in time
, which is essentially
tight assuming ETH. Finally, we get a new polynomial-time randomized
approximation scheme for Minimax Approval Voting, which runs in time
,
almost matching the running time of the fastest known PTAS for Closest String
due to Ma and Sun [SIAM J. Comp. 2009].Comment: 14 pages, 3 figures, 2 pseudocode
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