Range updates and range sum queries on multidimensional points with monoid weights

Abstract

Let P be a set of n points in Rd where each point p∈P carries a weight drawn from a commutative monoid (M,+,0). Given a d-rectangle rupd (i.e., an orthogonal rectangle in Rd) and a value Δ∈M, a range update adds Δ to the weight of every point p∈P∩rupd; given a d-rectangle rqry, a range sum query returns the total weight of the points in P∩rqry. The goal is to store P in a structure to support updates and queries with attractive performance guarantees. We describe a structure of O˜(n) space that handles an update in O˜(Tupd) time and a query in O˜(Tqry) time for arbitrary functions Tupd(n) and Tqry(n) satisfying Tupd⋅Tqry=n. The result holds for any fixed dimensionality d≥2. Our query-update tradeoff is tight up to a polylog factor subject to the OMv-conjecture.