Jarvis March

Definition

Jarvis March, also known as the Gift Wrapping algorithm, is a geometric algorithm used to compute the convex hull of a set of points in the plane. This algorithm works by wrapping a 'string' around the outermost points, effectively identifying the boundary of the convex shape formed by those points. It connects to various concepts like the Minkowski sum and applications of convex hulls, highlighting its significance in computational geometry.

5 Must Know Facts For Your Next Test

1. Jarvis March operates in O(nh) time complexity, where n is the number of input points and h is the number of points on the convex hull.
2. The algorithm starts with the leftmost point and repeatedly selects the next point that makes the smallest angle with respect to the current point.
3. It is particularly useful when the number of hull points (h) is much smaller than the total number of input points (n).
4. While it is easy to understand and implement, Jarvis March is not as efficient as other algorithms like Graham's Scan for large datasets.
5. The algorithm can handle collinear points gracefully, as it can include them along the boundary of the convex hull.

Review Questions

• How does Jarvis March compare to other algorithms for computing convex hulls, such as Graham's Scan?
  • Jarvis March and Graham's Scan are both popular algorithms for computing convex hulls but differ significantly in their approach and efficiency. While Jarvis March has a time complexity of O(nh), making it suitable for situations where h is small relative to n, Graham's Scan operates in O(n log n) time due to its sorting step. This means that for larger datasets or when many points are on the hull, Graham's Scan is generally more efficient than Jarvis March, despite both algorithms being conceptually straightforward.
• Discuss how Jarvis March could be applied in real-world scenarios involving convex hulls.
  • Jarvis March has practical applications in fields such as computer graphics, robotics, and geographic information systems (GIS). For instance, in robotics, it can be used to determine collision-free paths by identifying the outer boundaries of obstacles within an environment. Additionally, GIS can utilize Jarvis March to define areas or regions by calculating the outer limits of geographical data points collected from surveys. The simplicity of its implementation makes it attractive for smaller datasets where speed may not be as critical.
• Evaluate the implications of using Jarvis March for calculating Minkowski sums in computational geometry.
  • Using Jarvis March to calculate Minkowski sums involves understanding both the properties of convex hulls and how they interact geometrically when combining two sets of points. The efficiency of this approach can be affected by the number of output points produced by the Minkowski sum, which could lead to an increased computation time if many points fall on or within the resultant convex shape. However, leveraging Jarvis March allows for a straightforward way to visualize and compute these sums, which is particularly useful in applications like motion planning where understanding spatial relationships between objects is crucial.

Related terms

Convex Hull: The smallest convex set that contains all points in a given set, forming the 'shape' that can be wrapped around the points.

Graham's Scan: An efficient algorithm for computing the convex hull of a set of points, which sorts points and then constructs the hull using a stack.

Minkowski Sum: The result of adding two sets of points in space, often used in robotics and motion planning, which can be analyzed using convex hull algorithms.