An algebraic approach to complexity of data stream computations

We consider a basic problem in the general data streaming model, namely, to estimate a vector f ∈ Z that is arbitrarily updated (i.e., incremented or decremented) coordinatewise. The estimate f̂ ∈ Z must satisfy ‖f̂ − f‖∞ ≤ ǫ‖f‖1, that is, ∀i (|f̂i − fi| ≤ ǫ‖f‖1). It is known to have Õ(ǫ) randomized space upper bound [4], Ω(ǫ log(ǫn)) space lower bound [2] and deterministic space upper bound of Ω̃(ǫ) bits. We show that any deterministic algorithm for this problem requires space Ω(ǫ−2(log‖f‖1)) bits.