Tight bounds for the partial-sums problem
SODA '04 Proceedings of the fifteenth annual ACM-SIAM symposium on Discrete algorithms
Lower bounds for dynamic connectivity
STOC '04 Proceedings of the thirty-sixth annual ACM symposium on Theory of computing
Lower bounds for 2-dimensional range counting
Proceedings of the thirty-ninth annual ACM symposium on Theory of computing
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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 $A[j] \leftarrow A[j]+x \,$, and the query operation $\ssum(j)$ returns the partial sum $\sum_{i=1}^j \, A[i] \,$. 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 semigroup 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 biweighted binary tree.