Lower Bounds for Dynamic Partial Sums

Let G be a group. The partial sums problem asks to maintain an array A[1 . . n] of group elements, initialized to zeroes (a.k.a. the identity), under the following operations: update(k,∆): modify A[k]← ∆, where ∆ ∈ G. query(k): returns the partial sum ∑k i=1A[i]. For concreteness, let us work on a machine with w-bits words (w ≥ lg n), and take G to be Z/2wZ, i.e. integer arithmetic on machine words (modulo 2w). Then, the partial sums problem can be solved trivially in O(lg n) time per operation, using augmented binary trees. In this note, we describe a simple lower bound, originating in [1], showing that the problem requires Ω(lg n) time per operation.