Staying in the Middle: Exact and Approximate Medians in R and R for Moving Points

We study the problem of “staying in the middle”: we have a set of points moving in a geometric space and wish to maintain another point (possibly one of the given points, but not necessarily) that stays continuously “in the middle” of the moving set. More precisely, in R we wish to maintain the median or, more generally, a point of rank k. In R we wish to maintain suitable analogs of the median and point of rank k, defined as follows. Let P be a point set in R, and define the depth ∆(p) of a point p ∈ R as the minimum number of points of P on either side of any hyperplane passing through p. A point with depth at least δn is called a δ-center point. A 1/(d + 1)-center point is called just a center point, and it generalizes the concept of median. It has been proven using Helly’s Theorem that any point set has a center point [10]. For R, given some k ≤ n/2, the set of points with depth at least k is called k-th depth contour, denoted by Dk. We study both exact and approximate algorithms for kinetic medians (in R) and kinetic center points and depth contours (in R). As we will see, in both cases the approximate algorithms offer far greater stability and maintenance efficiency, for a very modest loss in the quality of the partitions they can generate. Another