133 research outputs found
Linear time Constructions of some -Restriction Problems
We give new linear time globally explicit constructions for perfect hash
families, cover-free families and separating hash functions
Optimal Query Complexity for Reconstructing Hypergraphs
In this paper we consider the problem of reconstructing a hidden weighted
hypergraph of constant rank using additive queries. We prove the following: Let
be a weighted hidden hypergraph of constant rank with n vertices and
hyperedges. For any there exists a non-adaptive algorithm that finds the
edges of the graph and their weights using
additive queries. This solves the open problem in [S. Choi, J. H. Kim. Optimal
Query Complexity Bounds for Finding Graphs. {\em STOC}, 749--758,~2008].
When the weights of the hypergraph are integers that are less than
where is the rank of the hypergraph (and therefore for
unweighted hypergraphs) there exists a non-adaptive algorithm that finds the
edges of the graph and their weights using additive queries.
Using the information theoretic bound the above query complexities are tight
A Simple Algorithm for Hamiltonicity
We develop a new algebraic technique that solves the following problem: Given
a black box that contains an arithmetic circuit over a field of
characteristic of degree~. Decide whether , expressed as an
equivalent multivariate polynomial, contains a multilinear monomial of degree
.
This problem was solved by Williams \cite{W} and Bj\"orklund et. al.
\cite{BHKK} for a white box (the circuit is given as an input) that contains
arithmetic circuit. We show a simple black box algorithm that solves the
problem with the same time complexity.
This gives a simple randomized algorithm for the simple -path problem for
directed graphs of the same time complexity\footnote{ is
} as in \cite{W} and with reusing the same
ideas from \cite{BHKK} with the above gives another algorithm (probably not
simpler) for undirected graphs of the same time complexity as in
\cite{B10,BHKK}
Almost Optimal Distribution-Free Junta Testing
We consider the problem of testing whether an unknown n-variable Boolean function is a k-junta in the distribution-free property testing model, where the distance between functions is measured with respect to an arbitrary and unknown probability distribution over {0,1}^n. Chen, Liu, Servedio, Sheng and Xie [Zhengyang Liu et al., 2018] showed that the distribution-free k-junta testing can be performed, with one-sided error, by an adaptive algorithm that makes O~(k^2)/epsilon queries. In this paper, we give a simple two-sided error adaptive algorithm that makes O~(k/epsilon) queries
Improved Lower Bound for Estimating the Number of Defective Items
Let be a set of items of size that contains some defective items,
denoted by , where . In group testing, a {\it test} refers to
a subset of items . The outcome of a test is if contains
at least one defective item, i.e., , and otherwise.
We give a novel approach to obtaining lower bounds in non-adaptive randomized
group testing. The technique produced lower bounds that are within a factor of
of the existing upper bounds for any
constant~. Employing this new method, we can prove the following result.
For any fixed constants , any non-adaptive randomized algorithm that, for
any set of defective items , with probability at least , returns an
estimate of the number of defective items to within a constant factor
requires at least tests.
Our result almost matches the upper bound of and solves the open
problem posed by Damaschke and Sheikh Muhammad [COCOA 2010 and Discrete Math.,
Alg. and Appl., 2010]. Additionally, it improves upon the lower bound of
previously established by Bshouty [ISAAC 2019]
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