Communication Complexity and the Log-rank Conjecture
نویسنده
چکیده
Def: A deterministic protocol computing a function f(x, y) is a binary tree T whose internal nodes specify which party speaks and the value of the bit they communicate, as a function of their input. The leaves of the tree are labelled with 0 or 1, in such a way that if Alice and Bob’s path through the tree given inputs (x, y) ends up in that leaf, the label on the leaf is f(x, y). The cost of such a protocol is the length of the longest root-leaf path in T . We denote by D(f) the minimum cost among all deterministic protocols computing f .
منابع مشابه
An additive combinatorics approach to the log-rank conjecture in communication complexity
For a {0, 1}-valued matrixM let CC(M) denote the deterministic communication complexity of the boolean function associated with M . The log-rank conjecture of Lovász and Saks [FOCS 1988] states that CC(M) ≤ logc(rank(M)) for some absolute constant c where rank(M) denotes the rank of M over the field of real numbers. We show that CC(M) ≤ c · rank(M)/ log rank(M) for some absolute constant c, ass...
متن کاملRecent advances on the log-rank conjecture in communication complexity
The log-rank conjecture is one of the fundamental open problems in communication complexity. It speculates that the deterministic communication complexity of any two-party function is equal to the log of the rank of its associated matrix, up to polynomial factors. Despite much research, we still know very little about this conjecture. Recently, there has been renewed interest in this conjecture...
متن کاملThe Computational Complexity Column
The log-rank conjecture is one of the fundamental open problems in communication complexity. It speculates that the deterministic communication complexity of any two-party function is equal to the log of the rank of its associated matrix, up to polynomial factors. Despite much research, we still know very little about this conjecture. Recently, there has been renewed interest in this conjecture...
متن کاملThe Log-Rank Conjecture for Read- k XOR Functions
The log-rank conjecture states that the deterministic communication complexity of a Boolean function g (denoted by D(g)) is polynomially related to the logarithm of the rank of the communication matrixMg whereMg is the communication matrix defined byMg(x, y) = g(x, y). An XOR function F : {0, 1} × {0, 1} → {0, 1} with respect to f : {0, 1} → {0, 1} is a function defined by F (x, y) = f(x⊕ y). I...
متن کاملEn Route to the Log-Rank Conjecture: New Reductions and Equivalent Formulations
We prove that several measures in communication complexity are equivalent, up to polynomial factors in the logarithm of the rank of the associated matrix: deterministic communication complexity, randomized communication complexity, information cost and zero-communication cost. This shows that in order to prove the log-rank conjecture, it suffices to show that low-rank matrices have efficient pr...
متن کاملA direct proof for Lovett's bound on the communication complexity of low rank matrices
The log-rank conjecture in communication complexity suggests that the deterministic communication complexity of any Boolean rank-r function is bounded by polylog(r). Recently, major progress was made by Lovett who proved that the communication complexity is bounded by O( √ r · log r). Lovett’s proof is based on known estimates on the discrepancy of low-rank matrices. We give a simple, direct pr...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2017