On the Complexity of Maintaining Partial Sums

Abstract

Let $F = \{ ({\bf r}_i ,s_i )|0 \leqq i < n\} $ be a file of n records, where ${\bf r}_i $ are d-dimensional vectors and $s_i $ are elements of a commutative semigroup S. We are interested in the partial sum problem, in which queries of the form “$\sum_{r_i \leqq {\bf a}} {s_i = ?} $” are to be answered. A space-time tradeoff $t = \Omega (\log n/\log (m\log n/ n)$ is established for storing a static two-dimensional file. It will also be shown that, for the one-dimensional problem, any dynamic algorithm must have a worst-case time $\Omega (n\log n/\log \log n)$ in processing a sequence $O(n)$ INSERT and QUERY instructions.