Lower bounds on the complexity of multidimensional searching

We establish new lower bounds on the complexity of several searching problems. We show that the time for solving the partial sum problem on n points in d dimensions is at least proportional to (log n/log 2m/n)d-1 in both the worst and average cases; m denotes the amount of storage used. This bound is provably tight for m = Ω(nlogcn) and any c ≫ d- 1. We also prove a lower bound of Ω(n(log n/log log n)d) on the time required for executing n inserts and queries. Other results include a lower bound on the complexity of orthogonal range searching in d dimensions (in report-mode). We show that on a pointer machine a query time of O(s+polylog(n)) time can only be achieved at the expense of Ω(n(log n/log log n)d-1) space, which is optimal; n and s denote respectively the input and output sizes.