Query Time Versus Redundancy Trade-Offs for Range Queries

Let A(l),A(2),...,A(n) be an array which stores values in a commutative semi-group S. We are concerned with the design of data structures for representing the array A(i) which facilitate implementation of the range queries Retrieve(j, k) defined as follows. Retrieve(j, k) (l<j<k&n) Return i A(i). i=j An obvious data structure would be the array A(j) itself. This approach has the disadvantage that the Retrieve task can take time n in the worst case. Another natural data structure would consist of a minimal height binary tree with n leaves. In the jth leaf we store the value A(j). In each internal node we store the sum of the values in the leaves of the subtree beneath that node. It is easily seen that the quantity returned by Retrieve(j, k) can be computed by summing the values stored in no more than 2 log, n nodes of the tree. On the other hand, the number of nodes in the tree whose values depend on A(j) is log, n for each j. We refer to this latter quantity as the degree of redundancy of A(j) in the data structure. Observe that the data structure consisting only of array A(j) has a redundancy of 1. The degree of redundancy is significant if we wish to change the value of A(j). Another reason to be interested in redundancy will be mentioned below. The purpose of this paper is to explore the extent of possible trade-offs between query time and redundancy relative to a large class of possible data structures. We begin by providing a formal statement of this problem. This is followed by a presentation of our main results. Lastly, we conclude with a brief description of related work.