Burton Rosenberg

Computing 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

THE COMPLEXITY OF COMPUTING PARTIAL SUMS OFF-LINE

Given an array A with n entries in an additive semigroup, and m intervals of the form Ii=[i,j], where 0<i<j≤n, we show that the computation of A[i]+⋯+A[j] for all Ii, requires Ω(n+mα(m,n)) semigroup additions. Here, α is the functional inverse of Ackermanns function. A matching upper bound has already been demonstrated.

Simplex range reporting on a pointer machine

Abstract We give a lower bound on the following problem, known as simplex range reporting: Given a collection P of n points in d-space and an arbitrary simplex q, find all the points in P ∩ q. It is understood that P is fixed and can be preprocessed ahead of time, while q is a query that must be answered on-line. We consider data structures for this problem that can be modeled on a pointer machine and whose query time is bounded by O(nδ + r), where r is the number of points to be reported and δ is an arbitrary fixed real. We prove that any such data structure of that form must occupy storage Ω(n d(1 − δ)− e ) , for any fixed e > 0. This lower bound is tight within a factor of ne.

Constraint programming and graph algorithms

Aconstraint system includes a set of variables and a set of relations among these variables, calledconstraints. The solution of a constraint system is an assignment of values to variables so that all, or many, of the relations are made true. A simple and efficient method for constraint resolution has been proposed in the work of B.N. Freeman-Benson, J. Maloney, and A. Borning. We show how their method is related to the classical problem of graph matching, and from this connection we derive new resolution algorithms.

A SECRETARY PROBLEM WITH TWO DECISION MAKERS

n applicants of similar qualification are on an interview list and their salary demands are from a known and continuous distribution. Two managers, I and II, will interview them one at a time. Right after each interview, manager I always has the first opportunity to decide to hire the applicant or not unless he has hired one already. If manager I decides not to hire the current applicant, then manager II can decide to hire the applicant or not unless he has hired one already. If both managers fail to hire the current applicant, they interview the next applicant, but both lose the chance of hiring the current applicant. If one of the managers does hire the current one, then they proceed with interviews until the other manager also hires an applicant. The interview process continues until both managers hire an applicant each. However, at the end of the process, each manager must have hired an applicant. In this paper, we first derive the optimal strategies for them so that the probability that the one he hired demands less salary than the one hired by the other does is maximized. Then we derive an algorithm for computing manager IIs winning probability when both managers play optimally. Finally, we show that manager IIs winning probability is strictly increasing in n, is always less than c, and converges to c as n -+ co, where c--0.3275624139--- is a solution of the equation ln(2) + x In(x) = x. SECRETARY PROBLEM; OPTIMAL STRATEGY

Fast nondeterministic recognition of context-free languages using two queues

We show how to accept a context-free language nondeterministically in O(n log n) time on a two-queue machine.