Interval finding and its application to data mining

In this paper, we investigate inverse problems of the interval query problem in application to data mining. Let I be the set of all intervals on U = {1, 2, . . . , n}. Consider an objective function f(I), conditional functions ui(I) on I, and define an optimization problem of finding the interval I maximizing f(I) subject to ui(I) > Ki for given real numbers Ki (i = 1, 2, . . . , h). We propose efficient algorithms to solve the above optimization problem if the objective function is either additive or quotient, and the conditional functions are additive, where a function f is additive if f(I) = ∑i∈I f̂(i) extending a function f̂ on U, and quotient if it is represented as a quotient of two additive functions. We use computational-geometric methods such as convex hull, range searching, and multidimensional divide-and-conquer.