Gift Wrapping Algorithm

Definition

The gift wrapping algorithm, also known as the Jarvis march, is a method for finding the convex hull of a set of points in the plane. It works by wrapping a 'gift' around the outermost points of a set, systematically selecting the next point that makes the smallest angle with the line formed by the last two points added to the hull. This algorithm highlights concepts of convexity and geometric properties, which are central to understanding algorithms for constructing convex hulls.

5 Must Know Facts For Your Next Test

1. The gift wrapping algorithm has a time complexity of O(nh), where n is the number of points in the set and h is the number of points in the convex hull.
2. It is intuitive and easy to understand, making it an excellent choice for teaching concepts related to convex hulls.
3. The algorithm begins at the leftmost point of the set and wraps around to find all outer points, effectively outlining the convex boundary.
4. This method may not be efficient for large datasets compared to other algorithms like Graham's scan or QuickHull due to its O(nh) complexity.
5. The gift wrapping algorithm can also be applied in higher dimensions, although it becomes more complex and less practical than in two dimensions.

Review Questions

• How does the gift wrapping algorithm determine which point to add next to the convex hull?
  • The gift wrapping algorithm determines which point to add next by selecting the point that makes the smallest angle with the line formed by the last two points on the hull. Starting from an initial point, usually the leftmost point, it evaluates all other points and chooses the one that maintains the convexity of the hull. This process continues until it wraps around back to the starting point, completing the convex hull.
• Compare and contrast the efficiency of the gift wrapping algorithm with Graham's scan when finding convex hulls.
  • While both algorithms are designed to find convex hulls, their efficiency differs significantly. The gift wrapping algorithm has a time complexity of O(nh), which can be inefficient for large datasets since it depends on the number of points in the hull. In contrast, Graham's scan operates in O(n log n) time due to its initial sorting step, making it more suitable for larger sets of points. Overall, Graham's scan is generally preferred for its efficiency in practical applications.
• Evaluate how understanding the gift wrapping algorithm enhances your comprehension of other convex hull algorithms.
  • Understanding the gift wrapping algorithm lays a foundational grasp on how convex hulls are constructed through geometric relationships between points. It emphasizes key concepts like angle comparison and boundary definition, which are crucial in various other algorithms such as Graham's scan or QuickHull. By analyzing how different approaches tackle similar problems, you develop a broader perspective on computational geometry and can apply this knowledge to optimize or modify existing algorithms based on specific conditions or requirements.

Related terms

Convex Hull: The smallest convex polygon that can contain a set of points in a plane.

Jarvis March:

Another name for the gift wrapping algorithm, named after its developer, R. A. Jarvis.

Graham's Scan:

An efficient algorithm for finding the convex hull of a set of points, which sorts points by angle and uses a stack to build the hull.