在流模型中识别带括号的表达式的空间复杂度：重新审视索引函数
The space complexity of recognizing well-parenthesized expressions in the streaming model: the Index function revisited
摘要：
我们展示了一个欧米茄（sqrt{n}/T）下界所需的空间，任何单向恒定误差的随机T通流算法，识别是否在两种类型的括号表达式是很好的括号。这证明了Magniez，Mathieu和Nayak（2009）的猜想，并严格建立了双向流与单向流相比在空间使用方面的效率呈指数级提高。我们通过建立最小量的信息，这是必要的，揭示了他们各自的输入在一个两方通信协议的指数函数的变体，即增强指数的球员。信息成本的权衡是通过一个新的应用程序的概念简单和熟悉的想法，如平均编码和剪切和粘贴随机协议的属性。受最近通过流量子算法在空间上指数节省的例子的启发，我们还研究了增强索引的量子协议。为量子协议定义一个合适的信息成本概念涉及到在它的适用性和我们可以分析它的容易性之间进行微妙的平衡。我们定义了一个量子信息成本的概念，它反映了量子信息的一些非直观属性，并给出了这个概念的权衡。虽然这种权衡证明了我们的证明技术的强度，但它并没有导致检查括号的空间下限。我们将量子流算法的这种含义作为一个有趣的悬而未决的问题。
英文摘要：
We show an Omega(sqrt{n}/T) lower bound for the space required by any unidirectional constant-error randomized T-pass streaming algorithm that recognizes whether an expression over two types of parenthesis is well-parenthesized. This proves a conjecture due to Magniez, Mathieu, and Nayak (2009) and rigorously establishes that bidirectional streams are exponentially more efficient in space usage as compared with unidirectional ones. We obtain the lower bound by establishing the minimum amount of information that is necessarily revealed by the players about their respective inputs in a two-party communication protocol for a variant of the Index function, namely Augmented Index. The information cost trade-off is obtained by a novel application of the conceptually simple and familiar ideas such as average encoding and the cut-and-paste property of randomized protocols. Motivated by recent examples of exponential savings in space by streaming quantum algorithms, we also study quantum protocols for Augmented Index. Defining an appropriate notion of information cost for quantum protocols involves a delicate balancing act between its applicability and the ease with which we can analyze it. We define a notion of quantum information cost which reflects some of the non-intuitive properties of quantum information and give a trade-off for this notion. While this trade-off demonstrates the strength of our proof techniques, it does not lead to a space lower bound for checking parentheses. We leave such an implication for quantum streaming algorithms as an intriguing open question.
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