Range Updates and Range Sum Queries on Multidimensional Points with Monoid Weights

Abstract

Let P be a set of n points in R d where each point p ∈ P carries a weight drawn from a commutative monoid ( M , + , 0). Given a d -rectangle r upd (i.e., an orthogonal rectangle in R d ) and a value ∆ ∈ M , a range update adds ∆ to the weight of every point p ∈ P ∩ r upd ; given a d -rectangle r qry , a range sum query returns the total weight of the points in P ∩ r qry . 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 ( T upd ) time and a query in ˜ O ( T qry ) time for arbitrary functions T upd ( n ) and T qry ( n ) satisfying T upd · T qry = 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. 2012 ACM Subject Classification Theory of computation → Data structures design and analysis