Improved Update/Query Algorithms for the Interval Valuation Problem

Abstract Let I be the set of intervals with endpoints in the integers 1, 2, …, 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(log n). For the semigroup problem, the use of range trees enables both uptade and query algorithms of time complexity O(log 2 n). Data structures are presented for the semigroup problem with (update, query) algorithms of complexities (log n, log 2 n) and (log n log log n, log n).