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

Let P be a set of n points in $R^d$ where each point $p\in P$ carries a weight drawn from a commutative monoid $(M,+,0)$. Given a d-rectangle $r_{\mathrm{upd}}$ (i.e., an orthogonal rectangle in $R^d$) and a value $\Delta \in M$, a range update adds Δ to the weight of every point $p\in P\cap r_{\mathrm{upd}}$; given a d-rectangle $r_{\mathrm{qry}}$, a range sum query returns the total weight of the points in $P\cap r_{\mathrm{qry}}$. The goal is to store P in a structure to support updates and queries with attractive performance guarantees. We describe a structure of $\tilde{O}\left(n\right)$ space that handles an update in $\tilde{O}\left(T_{\mathrm{upd}}\right)$ time and a query in $\tilde{O}\left(T_{\mathrm{qry}}\right)$ time for arbitrary functions $T_{\mathrm{upd}}\left(n\right)$ and $T_{\mathrm{qry}}\left(n\right)$ satisfying $T_{\mathrm{upd}}\cdot T_{\mathrm{qry}}=n$. The result holds for any fixed dimensionality $d\ge 2$. Our query-update tradeoff is tight up to a polylog factor subject to the OMv-conjecture.