文章基本信息

标题：Lifting Theorems for Equality
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作者：Bruno Loff ; Sagnik Mukhopadhyay
期刊名称：LIPIcs : Leibniz International Proceedings in Informatics
电子版ISSN：1868-8969
出版年度：2019
卷号：126
页码：1-19
DOI：10.4230/LIPIcs.STACS.2019.50
出版社：Schloss Dagstuhl -- Leibniz-Zentrum fuer Informatik
摘要：We show a deterministic simulation (or lifting) theorem for composed problems f o Eq_n where the inner function (the gadget) is Equality on n bits. When f is a total function on p bits, it is easy to show via a rank argument that the communication complexity of f o Eq_n is Omega(deg(f) * n). However, there is a surprising counter-example of a partial function f on p bits, such that any completion f' of f has deg(f') = Omega(p), and yet f o Eq_n has communication complexity O(n). Nonetheless, we are able to show that the communication complexity of f o Eq_n is at least D(f) * n for a complexity measure D(f) which is closely related to the AND-query complexity of f and is lower-bounded by the logarithm of the leaf complexity of f. As a corollary, we also obtain lifting theorems for the set-disjointness gadget, and a lifting theorem in the context of parity decision-trees, for the NOR gadget. As an application, we prove a tight lower-bound for the deterministic communication complexity of the communication problem, where Alice and Bob are each given p-many n-bit strings, with the promise that either all of the strings are distinct, or all-but-one of the strings are distinct, and they wish to know which is the case. We show that the complexity of this problem is Theta(p * n).
关键词：Communication complexity; Query complexity; Simulation theorem; Equality function