Covering Point Patterns

Covering point sets with two convex objects

Let P2n be a point set in the plane with n red and n blue points. Let CR and CB (SR and SB) respectively be red and blue colored and disjoint disks (axisparallel squares). In this paper we prove the following results. Finding the positions for CR and CB that maximizes the number of red points covered by CR plus the number of blue points covered by CB can be done in O(n log n) time. Finding two ...

Covering Paths for Planar Point Sets

Given n points in the plane, a covering path is a polygonal path that visits all the points. If no three points are collinear, every covering path requires at least n/2 segments, and n − 1 straight line segments obviously suffice even if the covering path is required to be noncrossing. We show that every set of n points in the plane admits a (possibly self-crossing) covering path consisting of ...

Covering the Convex Quadrilaterals of Point Sets

For a point set P on the plane, a four element subset S ⊂ P is called a 4-hole of P if the convex hull of S is a quadrilateral and contains no point of P in its interior. Let R be a point set on the plane. We say that a point set B covers all the 4-holes of R if any 4-hole of R contains an element of B in its interior. We show that if |R| ≥ 2|B|+5 then B cannot cover all the 4-holes of R. A sim...

Spatial Point Patterns

In looking at ecological processes, interest in the pattern of occurrences of species, e.g., the pattern of trees in a forest, say junipers and pinions. In spatial epidemiology, we seek to find pattern in disease cases, perhaps different patterns for cases vs. controls. Exple: breast cancer cases; treatment option mastectomy or radiation In syndromic surveillance we seek to identify disease out...

Characterising Linear Point Patterns