Reducing Randomness via Irrational Numbers
Zhi-Zhong Chen,Ming-Yang Kao +1 more
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A general methodology for testing whether a given polynomial with integer coefficients is identically zero is proposed, which can decrease the error probability by increasing the precision of the approximations instead of using more random bits and improve randomized algorithms that use the classical technique.Abstract:
We propose a general methodology for testing whether a given polynomial with integer coefficients is identically zero. The methodology evaluates the polynomial at efficiently computable approximations of suitable irrational points. In contrast to the classical technique of DeMillo, Lipton, Schwartz, and Zippel, this methodology can decrease the error probability by increasing the precision of the approximations instead of using more random bits. Consequently, randomized algorithms that use the classical technique can generally be improved using the new methodology. To demonstrate the methodology, we discuss two nontrivial applications. The first is to decide whether a graph has a perfect matching in parallel. Our new NC algorithm uses fewer random bits while doing less work than the previously best NC algorithm by Chari, Rohatgi, and Srinivasan. The second application is to test the equality of two multisets of integers. Our new algorithm improves upon the previously best algorithms by Blum and Kannan and can speed up their checking algorithm for sorting programs on a large range of inputs.read more
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Arithmetic Circuits: A Survey of Recent Results and Open Questions
Amir Shpilka,Amir Yehudayoff +1 more
TL;DR: The goal of this monograph is to survey the field of arithmetic circuit complexity, focusing mainly on what it finds to be the most interesting and accessible research directions, with an emphasis on works from the last two decades.
Journal ArticleDOI
Derandomizing polynomial identity tests means proving circuit lower bounds
TL;DR: If Permanent requires superpolynomial-size arithmetic circuits, then one can test in subexponential time whether a given arithmetic circuit of polynomially bounded degree computes an identically zero polynomial.
Journal ArticleDOI
Locally Decodable Codes with Two Queries and Polynomial Identity Testing for Depth 3 Circuits
Zeev Dvir,Amir Shpilka +1 more
TL;DR: This work gives new PIT algorithms for $\Sigma\Pi\Sigma$ circuits with a bounded top fan-in: a deterministic algorithm that runs in quasipolynomial time, and a randomized algorithm that run in polynomial time and uses only a polylogarithmic number of random bits.
Journal ArticleDOI
Polynomial Identity Testing for Depth 3 Circuits
Neeraj Kayal,Nitin Saxena +1 more
TL;DR: The first deterministic polynomial time identity test for depth 3 arithmetic circuits with bounded top fanin is given, showing that the rank of a minimal and simple circuit with boundedTopFanin, computing zero, can be unbounded.
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Diagonal Circuit Identity Testing and Lower Bounds
TL;DR: This paper gives the first deterministic polynomial time algorithm for testing whether a diagonaldepth-3 circuit C (i.e. a sum of powers of linear functions) is zero and proves an exponential lower bound showing that such a circuit will compute determinant or permanent only if there are exponentially many linear functions.