Optimal bi-weighted binary trees and the complexity of maintaining partial sums

Let A be an array. The partial sum problem concerns the design of a data structure for implementing the following operations. The operation update(j,x) has the effect, A[j]←A[j]+x, and the query operation sum(j) returns the partial sum, Σ_i=1^j A[i]. Our interest centers upon the optimal efficiency with which sequences of such operations can be performed, and we derive new upper and lower bounds in the semi-group model of computation. Our analysis relates the optimal complexity of the partial sum problem to optimal binary trees relative to a type of weighting scheme that defines the notion of bi-weighted binary tree