Eecient Parallel Algorithms for Geometric Clustering and Partitioning Problems

We present eecient parallel algorithms for some geometric clustering and partitioning problems. Our algorithms run in the CREW PRAM model of parallel computation. Given a point set P of n points in two dimensions, the clustering problems are to nd a k-point subset such that some measure for this subset is minimized. We consider the problems of nding a k-point subset with minimum L 1 perimeter and minimum L 1 diameter. For the L 1 perimeter case, our algorithm runs in O(log 2 n) time and O(n log 2 n + nk 2 log 2 k) work. For the L 1 diameter case, our algorithm runs in O(log 2 n+ log 2 k log log k log k) time and O(n log 2 n) work. We consider partitioning problems of the following nature. Given a planar point set S (jSj = n), a measure acting on S and a pair of values 1 and 2 , does there exist a bipartition S = S 1 S 2 such that (S 1) i for i = 1; 2? We consider several measures like diameter under L 1 and L 1 metric; area, perimeter of the smallest enclosing axes-parallel rectangle and the side length of the smallest enclosing axes-parallel square. All our parallel algorithms for partitioning problems run in O(log n) time using O(n) processors in the CREW PRAM. In our algorithms, we use an optimal parallel construction of a range tree. We also show how to perform report mode orthogonal range queries in optimal O(log n) time using O(1 + k= log n) processors, where k is the number of points inside the query range. The processor-time product for this range reporting algorithm is optimal.