Calculating the cardinality of some classes of binary matrices using bitwise operations — A polynomial algorithm

Let [Formula: see text] be the set of all [Formula: see text] binary matrices with exactly [Formula: see text] units in each row and each column, [Formula: see text]. A matrix [Formula: see text] will be called primitive, if there is no [Formula: see text] submatrix of [Formula: see text] that belongs to the set [Formula: see text], [Formula: see text]. The article describes a polynomial algorithm, which works in time [Formula: see text] for verifying whether a [Formula: see text]-matrix is primitive. The work applies this algorithm for finding all primitive [Formula: see text]-matrices which rows and columns are in lexicographically nondecreasing order (semi-canonical binary matrices) for some integers [Formula: see text] and [Formula: see text].

Single Machine Group Scheduling with Position Dependent Processing Times and Ready Times

We investigate a single machine group scheduling problem with position dependent processing times and ready times. The actual processing time of a job is a function of positive group-dependent job-independent positional factors. The actual setup time of the group is a linear function of the total completion time of the former group. Each job has a release time. The decision should be taken regarding possible sequences of jobs in each group and group sequence to minimize the makespan. We show that jobs in each group are scheduled in nondecreasing order of its release time and the groups are arranged in nondecreasing order of some certain conditions. We also present a polynomial time solution procedure for the special case of the proposed problem.

SIMPLE ALGORITHMS FOR ENUMERATING INTERPOINT DISTANCES AND FINDING k NEAREST NEIGHBORS

We present an O(n log n+k log k) time and O(n+k) space algorithm which takes as input a set of n points in the plane and enumerates the k smallest distances between pairs of points in nondecreasing order. We also present an O(n log n+kn log k) solution to the problem of finding the k nearest neighbors for each of n points. Both algorithms are conceptually very simple, are easy to implement, and are based on a common data structure: the Delaunay triangulation. Variants of the algorithms work for any convex distance function metric.