Turnstile streaming algorithms might as well be linear sketches

In the turnstile model of data streams, an underlying vector x ∈ {--m,--m+1,..., m--1,m}ⁿ is presented as a long sequence of positive and negative integer updates to its coordinates. A randomized algorithm seeks to approximate a function f(x) with constant probability while only making a single pass over this sequence of updates and using a small amount of space. All known algorithms in this model are linear sketches: they sample a matrix A from a distribution on integer matrices in the preprocessing phase, and maintain the linear sketch A·x while processing the stream. At the end of the stream, they output an arbitrary function of A · x. One cannot help but ask: are linear sketches universal?

In this work we answer this question by showing that any 1-pass constant probability streaming algorithm for approximating an arbitrary function f of x in the turnstile model can also be implemented by sampling a matrix A from the uniform distribution on O(n logm) integer matrices, with entries of magnitude poly(n), and maintaining the linear sketch Ax. Furthermore, the logarithm of the number of possible states of Ax, as x ranges over {--m,--m + 1,..., m}ⁿ, plus the amount of randomness needed to store A, is at most a logarithmic factor larger than the space required of the space-optimal algorithm. Our result shows that to prove space lower bounds for 1-pass streaming algorithms, it suffices to prove lower bounds in the simultaneous model of communication complexity, rather than the stronger 1-way model. Moreover, the fact that we can assume we have a linear sketch with polynomially-bounded entries further simplifies existing lower bounds, e.g., for frequency moments we present a simpler proof of the Ω(n^1-2/k) bit complexity lower bound without using communication complexity.