Fast sequential and parallel algorithms for finding the largest rectangle separating two sets - Datta, Amitava; Srikant, R.; Krithivasan, Kamala (International journal of computer mathematics, 1990)
Given a bounding isothetic rectangle BR and two sets of points A and B with cardinalities n and m inside it, we have to find an isothetic rectangle containing maximum number of points from set A and no point from set B. We consider two limiting cases of this problem when the cardinalities of set A (resp. set B) is much greater than that of set B (resp. set ,A). We present efficient sequential and parallel algorithms for these two problems. Our sequential algorithms run in O((n + m)log m + m 2 ) and O((m+ n) log n + n 2 ) time respectively. The parallel algorithms in CREW PRAM run in o(log n) ando(log m 2 ) time using O(max(n,m 2 /logm)) and O(max(m,n 2 /logn)) processors respectively. Our sequential algorithms are faster than a previous algorithm under these constraints on cardinality. No previous parallel algorithm was known for this problem. We also present an optimal systolic algorithm for the original problem.