Efficient Parallel Algorithms for Geometric Partitioning Problems through Parallel Range Searching

We present efficient parallel algorithms for some geometric bipartitioning problems. Our algorithms are designed to run in the CREW PRAM model of parallel computation. These bipartition problems are the following. Given a planar point set S (left| S right| = n), a measure mu acting on S and a pair of values fmu_1 and mu_2, does there exist a bipartition S = S_1 cup S_2 such that mu(S_{1}) leqslant mu_i for i = 1,2? We present efficient parallel algorithms for several measures like diameter under L_infty and L_1 metric; area, perimeter or length of diagonal of the smallest enclosing axes-parallel rectangle and the side length of the smallest enclosing axes-parallel square. All our parallel algorithms run in O(logn) time using O{n) processors in the CREW PRAM. The work done (processor-time product) by our algorithms matches the time complexity of the best known sequential algorithms for most of these problems. As a by product of our algorithms, we can perform report mode orthogonal range queries in optimal O(logn) time using 0(1 + k/logn) processors, where k is the number of points inside the query range. The processor-time product for this range reporting algorithm is optimal.