Bichromatic Separating Circles

Problem Given two sets of points, R and B ∈ R ² where | R | = n and | B | = m: – Find the smallest circle containing all of R and as few points as possible from B

Example Smallest enclosing circle Smallest separating circle

Possible Applications Communications jamming Minimizing civilian casualties Data set separation

Previous Work Fisk - Pure separability in O(nm) time O’Rourke et al. - Pure separability in O(n+m) time using linear programming – Linear Programming methods taken from Meggido

Previous Work Hurtado et al. – Color grouping within axis parallel rectangles and strips Aronov et al. – Separating red and blue sets using a line

Possible Separating Circles If it contains 3 points from R the answer is the smallest enclosing circle for R Must contain at least two points from R

No Less Than Two Red Points Single red point on boundary - contradiction

Two Red Points Two red points lying on the boundary of a circle – Farthest two red points since all red points must be contained in the circle

Farthest Neighbor Voronoi Diagram An edge is the locus of all points that have the same two farthest points from R Can be thought of as a tree with the root being the center of the minimum enclosing circle of R

Finding the Boundary Center circle on farthest Neighbor Voronoi edge Find event point for each point in B Event point is defined as when a blue point enters or exits the circle

Finding the Boundary Two red points and the edge of FVD that they define

Finding the Boundary Finding the event points

Finding the Boundary

The Algorithm Compute the FVD for R For each of the O(n) edges of the diagram, we perform an event point sweep – Find event points for all blue points – Sort in order of increasing distance from the red points – Calculate # of blue points within the circle Only need to increment/decrement after first count – Constant time update

Running Time/Space Analysis (Smallest Circle) Computing the FVD for R takes O(n logn) time and uses O(n) space For each of the O(n) edges of the diagram, we perform the sweep – This uses O(m) space and O(mlogm) time Thus, overall we use O(n+m) space and O(nmlogm + nlogn) time

Event Points Sweep algorithm analyzes all event points Each edge may have O(m) event points As the circle grows, it will eventually enclose all blue points in the halfspace

Event Points Each edge may have O(m) event points

Event Points Each edge may have O(m) event points

Event Points Intersection of halfspaces Place O(m) points within this region – All of those points will cause event points for each of the edges

Exit Event Points Size of enclosing circle increases along path from root to leaf Not necessary to analyze all event points – Only analyze exit events

Exit Event Points Each edge of FVD( R) defines an exit region Exit regions are disjoint – <= m exit events

Counting Blue Points Sweeping allowed count to be found in constant time Use circular range counting

Running Time/Space Analysis (Smallest Circle) Second Algorithm Find edge of FVD( R) defining the exit region for each point in B – O(nm) time Perform O(m) circular range counting queries – O*(m 1.5 ) time and space * notation hides a polylogarithmic factor Faster than the sweeping algorithm if n=ω(√m)

Bound on the Number of Smallest Circles There are n separating circles in the worst case Lower bound

Bound on the Number of Smallest Circles There are n separating circles in the worst case Lower bound

Bound on the Number of Smallest Circles There are n separating circles in the worst case Upper bound Center of smallest enclosing circle Root to leaf path increases circle size, so smallest separating circles can only exist on disjoint root to leaf paths

Largest Separating Circle Use largest separating circle algorithm for purely separable sets as a subroutine Remove all points from B that are inside circles with the minimum number of blue points encountered during the sweep

Largest Separating Circle Subroutine takes O((n+m)log(n+m)) time Overall running time O(m(n+m)log(n+m))