We consider the problem of computing the time convex hull of a set of points in the presence of a straight-line highway in the plane. The traveling speed in the plane is assumed to be much slower than that along the highway. The shortest time path between two arbitrary points is either the straight-line segment connecting these two points or a path that passes through the highway. The time convex hull, CH t (P ), of a set P of n points is the smallest set containing P such that all the shortest time paths between any two points lie in CH t (P ). In this paper we give a Θ(n log n) time algorithm for solving the time convex hull problem for a set of n points in the presence of a highway.

Abstract-The k-means clustering algorithm is a widely used scheme to solve the clustering problem which classifies a given set of n data points in m-dimensional space into k clusters, whose centers are obtained by the centroids of the points in the same cluster. The problem with privacy consideration has been studied, when the data is distributed among different parties and the privacy of the distributed data is to be preserved. In this paper, we apply the concept of parallel computing to solve the privacy-preserving multi-party k-means clustering problem, when the data is vertically partitioned and horizontally partitioned respectively among different parties. We present algorithms for solving the problems for these two data partition models that run in O(nk) time and in O(m(k + log(n/k))) time respectively. The time complexities of the algorithms are much better than others without parallel computing.

No abstract