Interval finding and its application to data mining

Abstract

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={0, 1, 2,., n}. Consider an objective function f(I), conditional functions u _i(I) on I, and define an optimization problem of finding the interval I maximizing f(I) subject to u _i(I) > K _i for given real numbers K _i (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 $f\left( I \right) = \sum _{i \in I} \hat f\left( i \right)$ extending a function $\hat f$ 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.