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The least weight subsequence problem
The least weight subsequence (LWS) problem is introduced, and is shown to be equivalent to the classic minimum path problem for directed graphs. A special case of the LWS problem is shown to be solvable in O(n log n) time generally and, for certain weight functions, in linear time. A number of applications are given, including an optimum paragraph formation problem and the problem of finding a minimum height B-tree, whose solutions realize improvement in asymptotic time complexity
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A Linear-Time Algorithm for Concave One-Dimensional Dynamic Programming
The least weight subsequence problem is a special case of the one-dimensional dynamic programming problem where D[i] = E[i]. The modified edit distance problem, which arises in molecular biology. geology, and speech recognition, can be decomposed into 2n copies of the problem
Sketching, Streaming, and Fine-Grained Complexity of (Weighted) LCS
We study sketching and streaming algorithms for the Longest Common Subsequence problem (LCS) on strings of small alphabet size |Sigma|. For the problem of deciding whether the LCS of strings x,y has length at least L, we obtain a sketch size and streaming space usage of O(L^{|Sigma| - 1} log L). We also prove matching unconditional lower bounds.
As an application, we study a variant of LCS where each alphabet symbol is equipped with a weight that is given as input, and the task is to compute a common subsequence of maximum total weight. Using our sketching algorithm, we obtain an O(min{nm, n + m^{|Sigma|}})-time algorithm for this problem, on strings x,y of length n,m, with n >= m. We prove optimality of this running time up to lower order factors, assuming the Strong Exponential Time Hypothesis
Davenport constant with weights
For the cyclic group and any non-empty
. We define the Davenport constant of with weight ,
denoted by , to be the least natural number such that for any
sequence with , there exists a non-empty
subsequence and such that
. Similarly, we define the constant to be
the least such that for all sequences with
, there exist indices , and with . In the present paper, we show that
. This solve the problem raised by Adhikari and Rath
\cite{ar06}, Adhikari and Chen \cite{ac08}, Thangadurai \cite{th07} and
Griffiths \cite{gr08}.Comment: 6page
Faster optimal univariate microgaggregation
Microaggregation is a method to coarsen a dataset, by optimally clustering
data points in groups of at least points, thereby providing a -anonymity
type disclosure guarantee for each point in the dataset. Previous algorithms
for univariate microaggregation had a time complexity. By rephrasing
microaggregation as an instance of the concave least weight subsequence
problem, in this work we provide improved algorithms that provide an optimal
univariate microaggregation on sorted data in time and space. We further
show that our algorithms work not only for sum of squares cost functions, as
typically considered, but seamlessly extend to many other cost functions used
for univariate microaggregation tasks. In experiments we show that the
presented algorithms lead to real world performance improvements
On the Fine-Grained Complexity of One-Dimensional Dynamic Programming
In this paper, we investigate the complexity of one-dimensional dynamic programming, or more specifically, of the Least-Weight Subsequence (LWS) problem: Given a sequence of n data items together with weights for every pair of the items, the task is to determine a subsequence S minimizing the total weight of the pairs adjacent in S. A large number of natural problems can be formulated as LWS problems, yielding obvious O(n^2)-time solutions.
In many interesting instances, the O(n^2)-many weights can be succinctly represented. Yet except for near-linear time algorithms for some specific special cases, little is known about when an LWS instantiation admits a subquadratic-time algorithm and when it does not. In particular, no lower bounds for LWS instantiations have been known before. In an attempt to remedy this situation, we provide a general approach to study the fine-grained complexity of succinct instantiations of the LWS problem: Given an LWS instantiation we identify a highly parallel core problem that is subquadratically equivalent. This provides either an explanation for the apparent hardness of the problem or an avenue to find improved algorithms as the case may be.
More specifically, we prove subquadratic equivalences between the following pairs (an LWS instantiation and the corresponding core problem) of problems: a low-rank version of LWS and minimum inner product, finding the longest chain of nested boxes and vector domination, and a coin change problem which is closely related to the knapsack problem and (min,+)-convolution. Using these equivalences and known SETH-hardness results for some of the core problems, we deduce tight conditional lower bounds for the corresponding LWS instantiations. We also establish the (min,+)-convolution-hardness of the knapsack problem. Furthermore, we revisit some of the LWS instantiations which are known to be solvable in near-linear time and explain their easiness in terms of the easiness of the corresponding core problems
The zero exemplar distance problem
Given two genomes with duplicate genes, \textsc{Zero Exemplar Distance} is
the problem of deciding whether the two genomes can be reduced to the same
genome without duplicate genes by deleting all but one copy of each gene in
each genome. Blin, Fertin, Sikora, and Vialette recently proved that
\textsc{Zero Exemplar Distance} for monochromosomal genomes is NP-hard even if
each gene appears at most two times in each genome, thereby settling an
important open question on genome rearrangement in the exemplar model. In this
paper, we give a very simple alternative proof of this result. We also study
the problem \textsc{Zero Exemplar Distance} for multichromosomal genomes
without gene order, and prove the analogous result that it is also NP-hard even
if each gene appears at most two times in each genome. For the positive
direction, we show that both variants of \textsc{Zero Exemplar Distance} admit
polynomial-time algorithms if each gene appears exactly once in one genome and
at least once in the other genome. In addition, we present a polynomial-time
algorithm for the related problem \textsc{Exemplar Longest Common Subsequence}
in the special case that each mandatory symbol appears exactly once in one
input sequence and at least once in the other input sequence. This answers an
open question of Bonizzoni et al. We also show that \textsc{Zero Exemplar
Distance} for multichromosomal genomes without gene order is fixed-parameter
tractable if the parameter is the maximum number of chromosomes in each genome.Comment: Strengthened and reorganize
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
Viterbi Sequences and Polytopes
A Viterbi path of length n of a discrete Markov chain is a sequence of n+1
states that has the greatest probability of ocurring in the Markov chain. We
divide the space of all Markov chains into Viterbi regions in which two Markov
chains are in the same region if they have the same set of Viterbi paths. The
Viterbi paths of regions of positive measure are called Viterbi sequences. Our
main results are (1) each Viterbi sequence can be divided into a prefix,
periodic interior, and suffix, and (2) as n increases to infinity (and the
number of states remains fixed), the number of Viterbi regions remains bounded.
The Viterbi regions correspond to the vertices of a Newton polytope of a
polynomial whose terms are the probabilities of sequences of length n. We
characterize Viterbi sequences and polytopes for two- and three-state Markov
chains.Comment: 15 pages, 2 figures, to appear in Journal of Symbolic Computatio
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