Linear Sketching over F_2
Linear Sketching over F_2
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DOI：
10.4230/lipics.ccc.2018.8
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发表时间：
2018-06
期刊：
影响因子：
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通讯作者：
Sampath Kannan;Elchanan Mossel;Swagato Sanyal;G. Yaroslavtsev
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文献类型：
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作者：
Sampath Kannan;Elchanan Mossel;Swagato Sanyal;G. Yaroslavtsev
我们对F2上的线性素描进行了系统的研究。 ）概率很高。 - 函数。 ，4）递归多数函数。 F2上的随机更新可以作为均匀分布的F2-Sketches构造，与Li，Nguyen和Woodruff（Stoc'14）的先前作品相比不需要在n中三重指数的流长度，并且可以保留通过均匀随机更新构建的长度O（n）的流。
We initiate a systematic study of linear sketching over F2. For a given Boolean function treated as f: Fn2 → F2 a randomized F2-sketch is a distribution M over d × n matrices with elements over F2 such that Mx suffices for computing f(x) with high probability. Such sketches for d L n can be used to design small-space distributed and streaming algorithms. Motivated by these applications we study a connection between F2-sketching and a two-player one-way communication game for the corresponding XOR-function. We conjecture that F2-sketching is optimal for this communication game. Our results confirm this conjecture for multiple important classes of functions: 1) low-degree F2-polynomials, 2) functions with sparse Fourier spectrum, 3) most symmetric functions, 4) recursive majority function. These results rely on a new structural theorem that shows that F2-sketching is optimal (up to constant factors) for uniformly distributed inputs. Furthermore, we show that (non-uniform) streaming algorithms that have to process random updates over F2 can be constructed as F2-sketches for the uniform distribution. In contrast with the previous work of Li, Nguyen and Woodruff (STOC'14) who show an analogous result for linear sketches over integers in the adversarial setting our result does not require the stream length to be triply exponential in n and holds for streams of length O(n) constructed through uniformly random updates.