消息传递模型下线性代数问题的通信复杂性
On The Communication Complexity of Linear Algebraic Problems in the Message Passing Model
摘要：
研究了多人消息传递模型中有限域上线性代数问题的通信复杂性，证明了一些紧下界。具体来说，对于分布在多个参与者中的矩阵，我们考虑确定其秩、计算其逆元素以及求解线性方程组的问题。我们还考虑相关的问题，如计算广义内积的向量在不同的服务器上举行。我们给出了一个一般的框架，减少这些多玩家的问题，他们的两个球员的同行，表明这些问题的随机$s$球员的通信复杂性至少是$s$倍的随机两个球员的通信复杂性。提供的问题有一定数量的代数对称性，我们正式定义，我们可以显示最难的输入分布是一个对称分布，因此适用于最近的多玩家下限技术的菲利普斯等人。此外，我们给新的两个玩家的下界为这些问题的数量。特别是，我们的最佳下界的两个玩家版本的矩阵秩问题解决了一个公开的问题孙和王。我们的下界的一个共同特征是，它们甚至适用于这些问题的特殊“阈值承诺”版本，其中潜在的数量，例如，rank被保证是两个值之一，一个在某个临界阈值的两侧。这些类型的承诺问题是常见的文献中的数据流的来源，减少给空间下限的难度。
英文摘要：
We study the communication complexity of linear algebraic problems over finite fields in the multi-player message passing model, proving a number of tight lower bounds. Specifically, for a matrix which is distributed among a number of players, we consider the problem of determining its rank, of computing entries in its inverse, and of solving linear equations. We also consider related problems such as computing the generalized inner product of vectors held on different servers. We give a general framework for reducing these multi-player problems to their two-player counterparts, showing that the randomized $s$-player communication complexity of these problems is at least $s$ times the randomized two-player communication complexity. Provided the problem has a certain amount of algebraic symmetry, which we formally define, we can show the hardest input distribution is a symmetric distribution, and therefore apply a recent multi-player lower bound technique of Phillips et al. Further, we give new two-player lower bounds for a number of these problems. In particular, our optimal lower bound for the two-player version of the matrix rank problem resolves an open question of Sun and Wang. A common feature of our lower bounds is that they apply even to the special "threshold promise" versions of these problems, wherein the underlying quantity, e.g., rank, is promised to be one of just two values, one on each side of some critical threshold. These kinds of promise problems are commonplace in the literature on data streaming as sources of hardness for reductions giving space lower bounds.
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