Cell-Probe Lower Bounds for Prefix Sums

We prove that to store n bits x ∈ {0, 1}n so that each prefix sum (a.k.a. rank) query Sum(i) := ∑ k≤i xk can be answered by non-adaptively probing q cells of lg n bits, one needs memory n + n/ log n. This matches a recent upper bound of n+ n/ log n by Pǎtraşcu (FOCS 2008), also non-adaptive. We also obtain a n + n/ log O(q) n lower bound for storing a string of balanced brackets so that each Match(i) query can be answered by non-adaptively probing q cells. To obtain these bounds we show that a too efficient data structure allows us to break the correlations between query answers. ∗Supported by NSF grant CCF-0845003. Email: [email protected]