Deterministic simulation in LOGSPACE
STOC '87 Proceedings of the nineteenth annual ACM symposium on Theory of computing
Explicit construction of linear sized tolerant networks
Discrete Mathematics - First Japan Conference on Graph Theory and Applications
Approximations of general independent distributions
STOC '92 Proceedings of the twenty-fourth annual ACM symposium on Theory of computing
Small-bias probability spaces: efficient constructions and applications
SIAM Journal on Computing
Self-testing/correcting with applications to numerical problems
Journal of Computer and System Sciences - Special issue: papers from the 22nd ACM symposium on the theory of computing, May 14–16, 1990
&egr;-discrepancy sets and their application for interpolation of sparse polynomials
Information Processing Letters
A PCP characterization of NP with optimal amortized query complexity
STOC '00 Proceedings of the thirty-second annual ACM symposium on Theory of computing
Simple analysis of graph tests for linearity and PCP
Random Structures & Algorithms
Randomness-efficient low degree tests and short PCPs via epsilon-biased sets
Proceedings of the thirty-fifth annual ACM symposium on Theory of computing
Linearity testing in characteristic two
FOCS '95 Proceedings of the 36th Annual Symposium on Foundations of Computer Science
Derandomizing homomorphism testing in general groups
STOC '04 Proceedings of the thirty-sixth annual ACM symposium on Theory of computing
The PCP theorem by gap amplification
Proceedings of the thirty-eighth annual ACM symposium on Theory of computing
| Hi-index | 5.23 |
In this article, we study the randomness-efficient graph tests for homomorphism over arbitrary groups (which can be used in locally testing the Hadamard code and PCP construction). We try to optimize both the amortized query complexity and the randomness complexity of the homomorphism test simultaneously. For abelian groups G=Z"p^m, @C=Z"p and function f:G-@C, by using a @l-biased set S of size poly(log|G|), we show that, on any given bipartite graph H=(V"1,V"2;E), there exists a graph test for linearity over G with randomness complexity |V"1|log|G|+|V"2|O(loglog|G|), query complexity |V"1|+|V"2|+|E| and if the test accepts f:G-@C with probability at least p^-^|^E^|+(1-p^-^|^E^|)@d, then f has agreement =p^-^1(1+@d^2-@l2) with some affine linear function. It is a derandomized version of the graph test for linearity of Samorodnitsky and Trevisan (2000) [13]. For general groups G, @C and function f:G-@C, we introduce k random walks of some length, @? say, on expander graphs to design a probabilistic homomorphism test, which could be thought as a graph test on a graph which is the union of k paths. This gives a homomorphism test over general groups with randomness complexity klog|G|+@?O(loglog|G|), query complexity k+@?+k@? and if the test accepts f with probability at least 1-k@m@?^2k@?(1+@m@?-@m)+2@j(@l,@?), then f is 2@m/(1-@l)-far from being affine homomorphism, here @j(@l,@?)=@?"t"="1^@?^-^1t@l^@?^-^1^-^t. It is a graph test version of the derandomized test for homomorphism of Shpilka and Wigderson (2004) [14].