Space lower bounds for distance approximation in the data stream model

(MATH) We consider the problem of approximating the distance of two <i>d</i>-dimensional vectors <b>x</b> and <b>y</b> in the data stream model. In this model, the 2<i>d</i> coordinates are presented as a "stream" of data in some arbitrary order, where each data item includes the index and value of some coordinate and a bit that identifies the vector (<b>x</b> or <b>y</b>) to which it belongs. The goal is to minimize the amount of memory needed to approximate the distance. For the case of <i>L^p</i>-distance with <i>p</i> ε [1,2], there are good approximation algorithms that run in polylogarithmic space in <i>d</i> (here we assume that each coordinate is an integer with <i>O</i>(log <i>d</i>) bits). Here we prove that they do not exist for <i>p</i>ρ2. In particular, we prove an optimal approximation-space tradeoff of approximating <i>L</i>^&infty; distance of two vectors. We show that any randomized algorithm that approximates <i>L</i>^&infty; distance of two length <i>d</i> vectors within factor of <i>d</i>^δ requires ω(<i>d</i>^1—4δ) space. As a consequence we show that for <i>p</i>ρ2/(1—4δ), any randomized algorithm that approximate <i>L</i>^<i>p</i> distance of two length <i>d</i> vectors within a factor <i>d</i>^δ requires ω(<i>d</i> ^{1— ²< \over _p—4δ}) space.The lower bound follows from a lower bound on the two-party one-round communication complexity of this problem. This lower bound is proved using a combination of information theory and Fourier analysis.