Improved Update/Query Algorithms for the Interval Valuation Problem

Let I be the set of intervals with end points in the integers 1 : : :n. Associated with each element in I is a value from a commutative semigroup S. Two operations are to be implemented: update of the value associated with an interval and query of the sum of the values associated with the intervals which include an integer. If the values are from a commutative group (i.e., inverses exist) then there is a data structure which enables both update and query algorithms of time complexity O(logn). For the semigroup problem, the use of range trees enables both update and query algorithms of time complexity O(log 2 n). Data structures are presented for the semigroup problem with (update,query) algorithms of complexities (logn, log 2 n), (logn log logn, logn). Introduction Let I be the set of intervals with end points in the integers 1 : : :n. Associated with each element in I is a value from a commutative semigroup S. Let sum refer to the semigroup operator. Examples of such operators over the integers are addition, multiplication and minimum. Two operations are to be implemented, update and query. An update of an interval changes the value associated with that interval. A query of integer k returns the sum of values associated with the intervals which include k. We consider solutions of the class that involves variables storing values in S. Each variable stores the sum of values associated with some subset of I. A query is answered by summing a subset of these variables. An update is accomplished by recomputing the values of the appropriate variables. This class has been used by Fredman and others [F80, BFK81, F81, F81b] for analysis of query problems. If the values associated with the interval are from a commutative group (i.e., inverses exist) then there is a data structure which enables both update and query algorithms of time complexity O(logn) [F79]. For the semigroup problem, the use of range trees enables both update and query algorithms of time complexity O(log 2 n) [W78, L78, LW82, W85]. 1 We present data structures with update and query algorithms of the following complexities. Update Time Query Time O(logn) O(log 2 n) O(logn log logn) O(logn) Table 1 Data Structure and Algorithm We associate each interval [i; j] with a point in a two dimensional plane whose horizontal (x) and vertical (y) coordinates are i and j, the values of the endpoints of the interval, respectively. These points lie within the upper left triangular region of an n by n square. A query of k needs to retrieve the sum of the values associated with all points whose x coordinate is k and whose y coordinate is k. That is, the query region consists of all points which lie in a rectangle whose upper left corner is at (1; n) and whose lower right corner is at (k; k), as illustrated in Figure 1.