scholarly journals Property Testing on Graphs and Games

2021 ◽  
pp. 13-29
Author(s):  
Hiro Ito
Keyword(s):  
Property Testing ◽  
Complete Characterization ◽  
Constant Time ◽  
Complete Problems ◽  
Discrete Algorithms ◽  
First Results ◽  
Hereditary Properties ◽  
Testing Algorithm ◽  
Necessary And Sufficient ◽  
Constant Time Algorithms

AbstractConstant-time algorithms are powerful tools, since they run by reading only a constant-sized part of each input. Property testing is the most popular research framework for constant-time algorithms. In property testing, an algorithm determines whether a given instance satisfies some predetermined property or is far from satisfying the property with high probability by reading a constant-sized part of the input. A property is said to be testable if there is a constant-time testing algorithm for the property. This chapter covers property testing on graphs and games. The fields of graph algorithms and property testing are two of the main streams of research on discrete algorithms and computational complexity. In the section on graphs in this chapter, we present some important results, particularly on the characterization of testable graph properties. At the end of the section, we show results that we published in 2020 on a complete characterization (necessary and sufficient condition) of testable monotone or hereditary properties in the bounded-degree digraphs. In the section on games, we present results that we published in 2019 showing that the generalized chess, Shogi (Japanese chess), and Xiangqi (Chinese chess) are all testable. We believe that this is the first results for testable EXPTIME-complete problems.

scholarly journals Multigenerator Gabor Frames on Local Fields

2018 ◽  
Vol 33 (2) ◽  
pp. 307
Author(s):  
Owais Ahmad ◽  
Neyaz Ahmad Sheikh
Keyword(s):  
Sufficient Conditions ◽  
Complete Characterization ◽  
Necessary And Sufficient Conditions ◽  
Local Fields ◽  
Gabor Frames ◽  
Gabor System ◽  
Necessary And Sufficient

The main objective of this paper is to provide complete characterization of multigenerator Gabor frames on a periodic set $\Omega$ in $K$. In particular, we provide some necessary and sufficient conditions for the multigenerator Gabor system to be a frame for $L^2(\Omega)$. Furthermore, we establish the complete characterizations of multigenerator Parseval Gabor frames.

scholarly journals Helly-Type Theorems in Property Testing

2018 ◽  
Vol 28 (04) ◽  
pp. 365-379
Author(s):  
Sourav Chakraborty ◽  
Rameshwar Pratap ◽  
Sasanka Roy ◽  
Shubhangi Saraf
Keyword(s):  
High Probability ◽  
Convex Sets ◽  
Property Testing ◽  
Geometric Object ◽  
Fundamental Result ◽  
Symmetric Convex ◽  
Helly’S Theorem ◽  
Convex Object ◽  
Constant Size ◽  
Testing Algorithm

Helly’s theorem is a fundamental result in discrete geometry, describing the ways in which convex sets intersect with each other. If [Formula: see text] is a set of [Formula: see text] points in [Formula: see text], we say that [Formula: see text] is [Formula: see text]-clusterable if it can be partitioned into [Formula: see text] clusters (subsets) such that each cluster can be contained in a translated copy of a geometric object [Formula: see text]. In this paper, as an application of Helly’s theorem, by taking a constant size sample from [Formula: see text], we present a testing algorithm for [Formula: see text]-clustering, i.e., to distinguish between the following two cases: when [Formula: see text] is [Formula: see text]-clusterable, and when it is [Formula: see text]-far from being [Formula: see text]-clusterable. A set [Formula: see text] is [Formula: see text]-far [Formula: see text] from being [Formula: see text]-clusterable if at least [Formula: see text] points need to be removed from [Formula: see text] in order to make it [Formula: see text]-clusterable. We solve this problem when [Formula: see text], and [Formula: see text] is a symmetric convex object. For [Formula: see text], we solve a weaker version of this problem. Finally, as an application of our testing result, in the case of clustering with outliers, we show that with high probability one can find the approximate clusters by querying only a constant size sample.

On *-clean group rings

2014 ◽  
Vol 14 (01) ◽  
pp. 1550004 ◽  
Author(s):  
Yuanlin Li ◽  
M. M. Parmenter ◽  
Pingzhi Yuan
Keyword(s):  
Local Ring ◽  
Prime Order ◽  
Sufficient Conditions ◽  
Complete Characterization ◽  
Group Rings ◽  
Clean Ring ◽  
Commutative Local Ring ◽  
Irreducible Factorization ◽  
Necessary And Sufficient

A ring with involution * is called *-clean if each of its elements is the sum of a unit and a projection. Clearly a *-clean ring is clean. Vaš asked whether there exists a clean ring with involution * that is not *-clean. In a recent paper, Gao, Chen and the first author investigated when a group ring RG with classical involution * is *-clean and obtained necessary and sufficient conditions for RG to be *-clean, where R is a commutative local ring and G is one of C3, C4, S3 and Q8. As a consequence, the authors provided many examples of group rings which are clean, but not *-clean. In this paper, we continue this investigation and we give a complete characterization of when the group algebra 𝔽Cp is *-clean, where 𝔽 is a field and Cp is the cyclic group of prime order p. Our main result is related closely to the irreducible factorization of a pth cyclotomic polynomial over the field 𝔽. Among other results we also obtain a complete characterization of when RCn (3 ≤ n ≤ 6) is *-clean where R is a commutative local ring.

scholarly journals Multiplication of Matrices With Different Sparseness Properties on Dynamically Reconfigurable Meshes

VLSI Design ◽  
1999 ◽  
Vol 9 (1) ◽  
pp. 69-81 ◽  
Author(s):  
Martin Middendorf ◽  
Hartmut Schmeck ◽  
Heiko Schröder ◽  
Gavin Turner
Keyword(s):  
Lower Bound ◽  
Sparse Matrix ◽  
Sparse Matrices ◽  
The Other ◽  
Constant Time ◽  
Dynamically Reconfigurable ◽  
Reconfigurable Meshes ◽  
Constant Time Algorithms

Algorithms for multiplying several types of sparse n x n-matrices on dynamically reconfigurable n x n-arrays are presented. For some classes of sparse matrices constant time algorithms are given, e.g., when the first matrix has at most kn elements in each column or in each row and the second matrix has at most kn nonzero elements in each row, where k is a constant. Moreover, O(kn ) algorithms are obtained for the case that one matrix is a general sparse matrix with at most kn nonzero elements and the other matrix has at most k nonzero elements in every row or in every column. Also a lower bound of Ω(Kn ) is proved for this and other cases which shows that the algorithms are close to the optimum.

scholarly journals Constant Time Algorithms for Computing the Contour of Maximal Elements on a Reconfigurable Mesh

1998 ◽  
Vol 08 (03) ◽  
pp. 351-361 ◽  
Author(s):  
M. Manzur Murshed ◽  
Richard P. Brent
Keyword(s):  
Parallel Architectures ◽  
Constant Time ◽  
Maximal Elements ◽  
Reconfigurable Mesh ◽  
Constant Time Algorithms

There has recently been an interest in the introduction of reconfigurable buses to existing parallel architectures. Among them the Reconfigurable Mesh (RM) draws much attention because of its simplicity. This paper presents three constant time algorithms to compute the contour of the maximal elements of N planar points on the RM. The first algorithm employs an RM of size N × N while the second one uses a 3-D RM of size [Formula: see text]. We further extend the result to k-D RM of size N1/(k - 1) × N1/(k - 1) × … × N1/(k - 1).

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