Region-restricted clustering for geographic data mining

Clustering problems in a complex geographical setting are often required to incorporate the type and extent of land cover within a region. Given a set P of n points in a geographical setting, with the constraint that the points of P can only occur in one type of land cover, an interesting problem is the detection of clusters. First, we extend the definition of clusters and define the concept of a region-restricted cluster that satisfies the following properties: (i) the cluster has sufficient number of points, (ii) the cluster points are confined to a small geographical area, and (iii) the amount of land cover of the specific type in which the points lie is also small. Next, we give efficient exact and approximation algorithms for computing such clusters. The exact algorithm determines all axis-parallel squares with exactly m out of n points inside, size at most some prespecified value, and area of a given land cover type at most another prespecified value, and runs in O(nmlog2n+(nm+nnf)log2nf) time, where nf is the number of edges that bound the regions with the given land cover type. The approximation algorithm allows the square to be a factor 1+ε too large, and runs in O(nlogn+n/ε2+nflog2nf+(nlog2nf)/(mε2)) time. We also show how to compute largest clusters and outliers.