Partial Sums in Multidimensional Arrays

This problem comes in two distinct avors In query mode preprocessing is allowed and q is a query to be an swered on line In o line mode we are given the array A and a set of d rectangles q qm and we must com pute the m sums A qi Partial sum is a special case of the classical orthogonal range searching problem Given n weighted points in d space and a query d rectangle q compute the cumulative weight of the points in q see e g The dynamic version of partial sum in query mode was studied by Fredman who showed that a mixed sequence of n insertions deletions and queries may require n log n time which is optimal Willard and Lueker This re sult was partially extended to groups by Willard in For the case where only insertions and queries are al lowed a lower bound of n log n log log n was proven in the one dimensional case Yao and later extended to n log n log log n d for any xed dimension d Chazelle Regarding static one dimensional partial sum Yao proved that if m units of storage are used then any query can be answered in time O m n which is optimal in the arithmetic model The function m n is the func tional inverse of Ackermann s function de ned by Tarjan See also Alon and Schieber for related upper and lower bounds Our main results are a nonlinear lower bound for one dimensional partial sum in o line mode and a space time tradeo for partial sum in query mode in any xed di mension More precisely we prove that for any n and m there exist m partial sums whose evaluations require n m m n time This is a rare case where the func tion arises in an o line problem Noticeable instances are the complexity of union nd Tarjan and the length of Davenport Schinzel sequences Hart and Sharir Agar wal et al Interestingly the proof technique we use does not involve reductions from these problems Our result im plies that given a sequence of n numbers computing partial sums over a well chosen set of n intervals requires a nonlin ear number of additions This might come as a surprise in light of the fact that there is a trivial linear time algorithm as soon as we allow subtraction The lower bound can be regarded as a generalization of a result of Tarjan con cerning the o line evaluation of functions de ned over the paths of a tree As in our result also leads to an im proved lower bound on the minimum depth of a monotone circuit for computing conjunctions The other contribution of this paper is an algorithm which can answer any partial sum query in time O d m n where m is the amount of storage available This generalizes Yao s one dimensional upper bound to xed arbitrary dimension d Since our algorithm works on a RAM we can use it as the inner loop of standard multidimensional search ing structures For example consider the classical orthogo nal range searching problem on n weighted points in d space Lueker and Willard have described a data structure of size O n log n which can answer any range query in time O log n over a semigroup We improve the time bound to O n log n The remainder of this abstract is devoted to the proofs of the lower and upper bounds Except for a few technical lemmas whose proofs have been omitted our exposition is complete and self contained