Equivalence of Systematic Linear Data Structures and Matrix Rigidity
Equivalence of Systematic Linear Data Structures and Matrix Rigidity
复制标题
系统线性数据结构与矩阵刚性的等价
DOI：
--
复制
发表时间：
2019
期刊：
影响因子：
--
通讯作者：
Cyrus Rashtchian
中科院分区：
文献类型：
--
作者：
Sivaramakrishnan Natarajan Ramamoorthy;Cyrus Rashtchian
最近，Dvir、Golovnev 和 Weinstein 证明，线性数据结构的足够强的下界将意味着刚性矩阵的新界限。然而，他们的结果使用了需要 $NP$ 预言机的算法，因此刚性矩阵并不显式。在这项工作中，我们推导了数据结构的刚性和系统线性模型之间的等价性。对于 $m$ 查询的 $n$ 维内积问题，我们证明查询时间的下限意味着查询集本身的刚性下限。特别是 $omegaleft(frac{n}{r}log m
由于 Alon、Panigrahy 和 Yekhanin，$r$ 冗余存储位的 ight)$ 将产生比最佳边界更好的刚性参数。我们还证明了相反的结果，表明刚性矩阵直接对应于系统线性模型的硬查询集。作为一个应用，我们证明从一阶二元矩阵获得的向量集是刚性的，其参数与显式集的已知结果相匹配。这意味着向量-矩阵-向量问题需要查询时间 $Omega(n^{3/2}/r)$ 来获得系统线性模型中的冗余 $r geq sqrt{n}$，从而改进了 Chakraborty、Kamma 和 Larsen 的结果。最后，我们证明了高误差范围内向量-矩阵-向量问题的单元探针下界，改进了 Chattopadhyay、Koucký、Loff 和 Mukhopadhyay 的结果。
Recently, Dvir, Golovnev, and Weinstein have shown that sufficiently strong lower bounds for linear data structures would imply new bounds for rigid matrices. However, their result utilizes an algorithm that requires an $NP$ oracle, and hence, the rigid matrices are not explicit. In this work, we derive an equivalence between rigidity and the systematic linear model of data structures. For the $n$-dimensional inner product problem with $m$ queries, we prove that lower bounds on the query time imply rigidity lower bounds for the query set itself. In particular, an explicit lower bound of $omegaleft(frac{n}{r}log m
ight)$ for $r$ redundant storage bits would yield better rigidity parameters than the best bounds due to Alon, Panigrahy, and Yekhanin. We also prove a converse result, showing that rigid matrices directly correspond to hard query sets for the systematic linear model. As an application, we prove that the set of vectors obtained from rank one binary matrices is rigid with parameters matching the known results for explicit sets. This implies that the vector-matrix-vector problem requires query time $Omega(n^{3/2}/r)$ for redundancy $r geq sqrt{n}$ in the systematic linear model, improving a result of Chakraborty, Kamma, and Larsen. Finally, we prove a cell probe lower bound for the vector-matrix-vector problem in the high error regime, improving a result of Chattopadhyay, Koucký, Loff, and Mukhopadhyay.
DOI：
10.1109/focs.2015.71
发表时间：
2015-04
期刊：
2015 IEEE 56th Annual Symposium on Foundations of Computer Science
影响因子：
--
作者：
R. Clifford;A. Jørgensen;Kasper Green Larsen
通讯作者：
R. Clifford;A. Jørgensen;Kasper Green Larsen
空间接近最大隐含电路下界的数据结构的下界
影响因子：
1
作者：
Viola, Emanuele
通讯作者：
Viola, Emanuele