Cybersecurity and Cyberforensics Conference最新文献_第7页

Nearly optimal separations between communication (or query) complexity and partitions 通信(或查询)复杂性与分区之间几乎最佳的分离

Cybersecurity and Cyberforensics Conference Pub Date : 2015-12-03 DOI: 10.4230/LIPIcs.CCC.2016.4

Robin Kothari

引用次数: 27

Polynomials, Quantum Query Complexity, and Grothendieck's Inequality 多项式，量子查询复杂度，和Grothendieck不等式

Cybersecurity and Cyberforensics Conference Pub Date : 2015-11-27 DOI: 10.4230/LIPIcs.CCC.2016.25

S. Aaronson, A. Ambainis, Janis Iraids, M. Kokainis, Juris Smotrovs

{"title":"Polynomials, Quantum Query Complexity, and Grothendieck's Inequality","authors":"S. Aaronson, A. Ambainis, Janis Iraids, M. Kokainis, Juris Smotrovs","doi":"10.4230/LIPIcs.CCC.2016.25","DOIUrl":"https://doi.org/10.4230/LIPIcs.CCC.2016.25","url":null,"abstract":"We show an equivalence between 1-query quantum algorithms and representations by degree-2 polynomials. Namely, a partial Boolean function $f$ is computable by a 1-query quantum algorithm with error bounded by $epsilon<1/2$ iff $f$ can be approximated by a degree-2 polynomial with error bounded by $epsilon'<1/2$. This result holds for two different notions of approximation by a polynomial: the standard definition of Nisan and Szegedy and the approximation by block-multilinear polynomials recently introduced by Aaronson and Ambainis (STOC'2015, arXiv:1411.5729). \u0000We also show two results for polynomials of higher degree. First, there is a total Boolean function which requires $tilde{Omega}(n)$ quantum queries but can be represented by a block-multilinear polynomial of degree $tilde{O}(sqrt{n})$. Thus, in the general case (for an arbitrary number of queries), block-multilinear polynomials are not equivalent to quantum algorithms. \u0000Second, for any constant degree $k$, the two notions of approximation by a polynomial (the standard and the block-multilinear) are equivalent. As a consequence, we solve an open problem of Aaronson and Ambainis, showing that one can estimate the value of any bounded degree-$k$ polynomial $p:{0, 1}^n rightarrow [-1, 1]$ with $O(n^{1-frac{1}{2k}})$ queries.","PeriodicalId":246506,"journal":{"name":"Cybersecurity and Cyberforensics Conference","volume":"33 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2015-11-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127890844","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 21

A Linear Time Algorithm for Quantum 2-SAT 量子2-SAT的线性时间算法

Cybersecurity and Cyberforensics Conference Pub Date : 2015-08-28 DOI: 10.4230/LIPIcs.CCC.2016.27

N. D. Beaudrap, Sevag Gharibian

引用次数: 12

How to Compress Asymmetric Communication 如何压缩非对称通信

Cybersecurity and Cyberforensics Conference Pub Date : 2015-06-17 DOI: 10.4230/LIPIcs.CCC.2015.102

Sivaramakrishnan Natarajan Ramamoorthy, Anup Rao

引用次数: 12

Correlation Bounds Against Monotone NC^1 单调NC^1的相关界

Cybersecurity and Cyberforensics Conference Pub Date : 2015-06-17 DOI: 10.4230/LIPIcs.CCC.2015.392

Benjamin Rossman

{"title":"Correlation Bounds Against Monotone NC^1","authors":"Benjamin Rossman","doi":"10.4230/LIPIcs.CCC.2015.392","DOIUrl":"https://doi.org/10.4230/LIPIcs.CCC.2015.392","url":null,"abstract":"This paper gives the first correlation bounds under product distributions, including the uniform distribution, against the class mNC1 of polynomial-size O(log n)-depth monotone circuits. Our main theorem, proved using the pathset complexity framework introduced in [56], shows that the average-case k-CYCLE problem (on Erdos-Renyi random graphs with an appropriate edge density) is [EQUATION] hard for mNC1. Combining this result with O'Donnell's hardness amplification theorem [43], we obtain an explicit monotone function of n variables (in the class mSAC1) which is [EQUATION] hard for mNC1 under the uniform distribution for any desired constant e > 0. This bound is nearly best possible, since every monotone function has agreement [EQUATION] with some function in mNC1 [44]. \u0000 \u0000Our correlation bounds against mNC1 extend smoothly to non-monotone NC1 circuits with a bounded number of negation gates. Using Holley's monotone coupling theorem [30], we prove the following lemma: with respect to any product distribution, if a balanced monotone function f is [EQUATION] hard for monotone circuits of a given size and depth, then f is [EQUATION] hard for (non-monotone) circuits of the same size and depth with at most t negation gates. We thus achieve a lower bound against NC1 circuits with [EQUATION] log n negation gates, improving the previous record of 1/6 log log n [7]. Our bound on negations is \"half\" optimal, since ⌈log(n + 1)⌉ negation gates are known to be fully powerful for NC1 [3, 21].","PeriodicalId":246506,"journal":{"name":"Cybersecurity and Cyberforensics Conference","volume":"13 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2015-06-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114968163","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 14

Simplified Lower Bounds on the Multiparty Communication Complexity of Disjointness 不连接下多方通信复杂度的简化下界

Cybersecurity and Cyberforensics Conference Pub Date : 2015-06-17 DOI: 10.4230/LIPIcs.CCC.2015.88

Anup Rao, A. Yehudayoff

引用次数: 30

A Depth-Five Lower Bound for Iterated Matrix Multiplication 迭代矩阵乘法的深度五下界

Cybersecurity and Cyberforensics Conference Pub Date : 2015-06-17 DOI: 10.4230/LIPIcs.CCC.2015.183

S. Bera, Amit Chakrabarti

引用次数: 9

Adaptivity Helps for Testing Juntas 适应性有助于测试Juntas

Cybersecurity and Cyberforensics Conference Pub Date : 2015-06-17 DOI: 10.4230/LIPIcs.CCC.2015.264

R. Servedio, Li-Yang Tan, John Wright

引用次数: 25

Circuits with Medium Fan-In 具有中等风扇输入的电路

Cybersecurity and Cyberforensics Conference Pub Date : 2015-06-17 DOI: 10.4230/LIPIcs.CCC.2015.381

P. Hrubes, Anup Rao

引用次数: 21

Sums of products of polynomials in few variables : lower bounds and polynomial identity testing 少变量多项式乘积和:下界与多项式恒等检验

Cybersecurity and Cyberforensics Conference Pub Date : 2015-04-23 DOI: 10.4230/LIPIcs.CCC.2016.35

Mrinal Kumar, Shubhangi Saraf

{"title":"Sums of products of polynomials in few variables : lower bounds and polynomial identity testing","authors":"Mrinal Kumar, Shubhangi Saraf","doi":"10.4230/LIPIcs.CCC.2016.35","DOIUrl":"https://doi.org/10.4230/LIPIcs.CCC.2016.35","url":null,"abstract":"We study the complexity of representing polynomials as a sum of products of polynomials in few variables. More precisely, we study representations of the form $$P = sum_{i = 1}^T prod_{j = 1}^d Q_{ij}$$ such that each $Q_{ij}$ is an arbitrary polynomial that depends on at most $s$ variables. We prove the following results. \u00001. Over fields of characteristic zero, for every constant $mu$ such that $0 leq mu < 1$, we give an explicit family of polynomials ${P_{N}}$, where $P_{N}$ is of degree $n$ in $N = n^{O(1)}$ variables, such that any representation of the above type for $P_{N}$ with $s = N^{mu}$ requires $Td geq n^{Omega(sqrt{n})}$. This strengthens a recent result of Kayal and Saha [KS14a] which showed similar lower bounds for the model of sums of products of linear forms in few variables. It is known that any asymptotic improvement in the exponent of the lower bounds (even for $s = sqrt{n}$) would separate VP and VNP[KS14a]. \u00002. We obtain a deterministic subexponential time blackbox polynomial identity testing (PIT) algorithm for circuits computed by the above model when $T$ and the individual degree of each variable in $P$ are at most $log^{O(1)} N$ and $s leq N^{mu}$ for any constant $mu < 1/2$. We get quasipolynomial running time when $s < log^{O(1)} N$. The PIT algorithm is obtained by combining our lower bounds with the hardness-randomness tradeoffs developed in [DSY09, KI04]. To the best of our knowledge, this is the first nontrivial PIT algorithm for this model (even for the case $s=2$), and the first nontrivial PIT algorithm obtained from lower bounds for small depth circuits.","PeriodicalId":246506,"journal":{"name":"Cybersecurity and Cyberforensics Conference","volume":"119 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2015-04-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133182711","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 22