Abstract :
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.
