Convex hull problems(Divide and Conquer)

1. Convex-Hull Problems
Divide-and-Conquer Technique

2. Convex-Hull Problems
Let S be a set of n>1 points p1(x1, y1), . . . , pn(xn, yn)
in the Cartesian plane.
Assume that the points are sorted in nondecreasing
order of their x coordinates, with ties resolved by
order of their x coordinates, with ties resolved by
increasing order of the y coordinates of the points
involved.
Let the leftmost point p1 and the rightmost point pn
are two distinct extreme points of the set’s convex
hull

3. Convex-Hull Problems
 Let p1pn be the straight line through points p1 and pn
directed from p1 to pn
Line separates the points of S into two sets: S1 is the set of
points to the left of this line, and S2 is the set of points to
the right of this line
Points of S on the line p p other than p and p , cannot be
the right of this line
Points of S on the line p1pn other than p1 and pn, cannot be
extreme points of the convex hull

4. Convex-Hull Problems
 The boundary of the convex hull of S is made up of two polygonal chains:
an “upper” boundary and a “lower” boundary
 The “upper” boundary, called theupper hull, is a sequence of line segments
with vertices at p1, some of the points
in S1 (if S1 is not empty) and pn.
 The “lower” boundary, called the lower hull, is
a sequence of line segments with vertices at p1, some of the points in S2 (if S2 is
a sequence of line segments with vertices at p1, some of the points in S2 (if S2 is
not empty) and pn.
 The fact that the convex hull of the entire set S is composed
of the upper and lower hulls

5. Convex-Hull Problems