Deterministic simulation in LOGSPACE
STOC '87 Proceedings of the nineteenth annual ACM symposium on Theory of computing
Explicit construction of linear sized tolerant networks
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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
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
Probability and Computing: Randomized Algorithms and Probabilistic Analysis
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The PCP theorem by gap amplification
Proceedings of the thirty-eighth annual ACM symposium on Theory of computing
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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-tradeoff (between number of queries and error probability) and the randomness complexity of the homomorphism test simultaneously. For an abelian group G = Zpm, by using the λ-biased set S of G, we show that, on any given bipartite graph H = (V1, V2;E), the graph test for linearity over G is a test with randomness complexity |V1| log |G| + |V2|O(log |S|), query complexity |V1| + |V2| + |E| and error probability at most p-|E| + (1 - p-|E| ċ δ for any f which is 1 - p-1(1 +√δ2-λ/2 )-far from being affine linear. For a non-abelian group G, we introduce a random walk of some length, l say, on expander graphs to design a probabilistic homomorphism test over G with randomness complexity log |G| + O(log log |G|), query complexity 2l +1 and error probability at most 1 - δ2l2/(δl+δ2l2-δ2l)+ 2ψ(λ, l) for any f which is 2δ/(1 - λ)-far from being affine homomorphism, here ψ(λ, l) = Σ0≤ij≤l-1λj-i-1.