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Pseudorandomness for multilinear readonce algebraic branching programs
 in any order. Electronic Colloquium on Computational Complexity (ECCC
"... We give deterministic blackbox polynomial identity testing algorithms for multilinear readonce oblivious algebraic branching programs (ROABPs), in nO(lg 2 n) time.1 Further, our algorithm is oblivious to the order of the variables. This is the first subexponential time algorithm for this model. F ..."
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We give deterministic blackbox polynomial identity testing algorithms for multilinear readonce oblivious algebraic branching programs (ROABPs), in nO(lg 2 n) time.1 Further, our algorithm is oblivious to the order of the variables. This is the first subexponential time algorithm for this model. Furthermore, our result has no known analogue in the model of readonce oblivious boolean branching programs with unknown order, as despite recent work (eg. [BPW11, IMZ12, RSV13]) there is no known pseudorandom generator for this model with subpolynomial seedlength (for unboundedwidth branching programs). This result extends and generalizes the result of Forbes and Shpilka [FS12b] that obtained a nO(lgn)time algorithm when given the order. We also extend and strengthen the work of Agrawal, Saha and Saxena [ASS12] that gave a blackbox algorithm running in time exp((lg n)Ω(d)) for setmultilinear formulas of depth d. We note that the model of multilinear ROABPs contains the model of setmultilinear algebraic branching programs, which itself contains the model of setmultilinear formulas of arbitrary depth. We obtain our results by recasting,
Deterministic identity testing for sum of readonce oblivious arithmetic branching programs
 In 30th Conference on Computational Complexity, CCC 2015
"... A readonce oblivious arithmetic branching program (ROABP) is an arithmetic branching program (ABP) where each variable occurs in at most one layer. We give the first polynomial time whitebox identity test for a polynomial computed by a sum of constantly many ROABPs. We also give a corresponding bl ..."
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A readonce oblivious arithmetic branching program (ROABP) is an arithmetic branching program (ABP) where each variable occurs in at most one layer. We give the first polynomial time whitebox identity test for a polynomial computed by a sum of constantly many ROABPs. We also give a corresponding blackbox algorithm with quasipolynomial time complexity nO(logn). In both the cases, our time complexity is double exponential in the number of ROABPs. ROABPs are a generalization of setmultilinear depth3 circuits. The prior results for the sum of constantly many setmultilinear depth3 circuits were only slightly better than bruteforce, i.e. exponentialtime. Our techniques are a new interplay of three concepts for ROABP: low evaluation dimension, basis isolating weight assignment and lowsupport rank concentration. We relate basis isolation to rank concentration and extend it to a sum of two ROABPs using evaluation dimension (or partial derivatives).
Polynomial Identity Testing of ReadOnce Oblivious Algebraic Branching Programs
, 2014
"... We study the problem of obtaining efficient, deterministic, blackbox polynomial identity testing algorithms (PIT) for algebraic branching programs (ABPs) that are readonce and oblivious. This class has an efficient, deterministic, whitebox polynomial identity testing algorithm (due to Raz and Shp ..."
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We study the problem of obtaining efficient, deterministic, blackbox polynomial identity testing algorithms (PIT) for algebraic branching programs (ABPs) that are readonce and oblivious. This class has an efficient, deterministic, whitebox polynomial identity testing algorithm (due to Raz and Shpilka [RS05]), but prior to this work there was no known such blackbox algorithm. The main result of this work gives the first quasipolynomial sized hitting set for size S circuits from this class, when the order of the variables is known. As our hitting set is of size exp(lg2 S), this is analogous (in the terminology of boolean pseudorandomness) to a seedlength of lg2 S, which is the seed length of the pseudorandom generators of Nisan [Nis92] and ImpagliazzoNisanWigderson [INW94] for readonce oblivious
CERTIFICATE
, 2007
"... It is certified that the work contained in the thesis entitled “KMoteDesign and Implementation of a low cost, low power hardware platform for wireless sensor networks ” by Naveen Madabhushi has been carried out under our supervision and that this work has not been submitted elsewhere for a degree. ..."
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It is certified that the work contained in the thesis entitled “KMoteDesign and Implementation of a low cost, low power hardware platform for wireless sensor networks ” by Naveen Madabhushi has been carried out under our supervision and that this work has not been submitted elsewhere for a degree.
cb Licensed under a Creative Commons Attribution License (CCBY) DOI: 10.4086/toc.2014.v010a018
, 2012
"... Abstract: An arithmetic readonce formula (ROF for short) is a formula (a circuit whose underlying graph is a tree) in which the operations are {+,×} and each input variable labels at most one leaf. A preprocessed ROF (PROF for short) is a ROF in which we are allowed to replace each variable xi with ..."
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Abstract: An arithmetic readonce formula (ROF for short) is a formula (a circuit whose underlying graph is a tree) in which the operations are {+,×} and each input variable labels at most one leaf. A preprocessed ROF (PROF for short) is a ROF in which we are allowed to replace each variable xi with a univariate polynomial Ti(xi). We obtain a deterministic nonadaptive reconstruction algorithm for PROFs, that is, an algorithm that, given blackbox access to a PROF, constructs a PROF computing the same polynomial. The running time of the algorithm is (nd)O(logn) for PROFs of individual degrees at most d. To the best of our knowledge our results give the first subexponentialtime deterministic reconstruction algorithms for ROFs. Another question that we study is the following generalization of the polynomial identity testing (PIT) problem. Given an arithmetic circuit computing a polynomial P(x̄), decide whether there is a PROF computing P(x̄), and find one if one exists. We call this question the readonce testing problem (ROT for short). Previous (randomized) algorithms for reconstruction of ROFs imply that there exists a randomized algorithm for the ROT problem.
Technion
, 2010
"... We present a polynomialtime deterministic algorithm for testing whether constantread multilinear arithmetic formulae are identically zero. In such a formula each variable occurs only a constant number of times and each subformula computes a multilinear polynomial. Our algorithm runs in time sO(1) ..."
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We present a polynomialtime deterministic algorithm for testing whether constantread multilinear arithmetic formulae are identically zero. In such a formula each variable occurs only a constant number of times and each subformula computes a multilinear polynomial. Our algorithm runs in time sO(1) · nk O(k), where s denotes the size of the formula, n denotes the number of variables, and k bounds the number of occurrences of each variable. Before our work no subexponentialtime deterministic algorithm was known for this class of formulae. We also present a deterministic algorithm that works in a blackbox fashion and runs in time nk O(k)+O(k logn) in general, and time nk O(k2)+O(kd) for depth d. Finally, we extend our results and allow the inputs to be replaced with sparse polynomials. Our results encompass recent deterministic identity tests for sums of a constant number of readonce formulae, and for multilinear depthfour formulae. ∗Partially supported by NSF grants 0728809 and 1017597. †Partially supported by NSF grants 0728809 and 1017597.