Convex hull problems(Divide and Conquer)
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This document discusses the divide-and-conquer technique for solving convex hull problems. It describes how to divide a set of points S into two subsets S1 and S2 based on a separating line between the leftmost and rightmost points. The boundary of the convex hull of S is made up of an "upper" and "lower" polygonal chain, with the upper chain composed of segments connecting points in S1 and the lower chain composed of segments connecting points in S2. The convex hull of the overall set S is the combination of the upper and lower hulls of the divided subsets.
Convex hull problems(Divide and Conquer)
- 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