Graham Scan

Definition

Graham Scan is an efficient algorithm used to compute the convex hull of a set of points in the plane. This algorithm sorts the points based on their polar angles relative to a reference point, and then constructs the convex hull by iterating through the sorted points and maintaining a stack of vertices. It connects well with output-sensitive convex hull algorithms because it has a time complexity of O(n log n), mainly due to the sorting step, making it effective when handling larger datasets with varying point distributions.

5 Must Know Facts For Your Next Test

1. Graham Scan operates in two main phases: sorting the points and constructing the convex hull using a stack.
2. The algorithm selects the point with the lowest y-coordinate as the reference point for sorting others by polar angle.
3. During construction, Graham Scan checks for counterclockwise turns to determine whether to add or remove points from the hull.
4. It efficiently handles collinear points by only keeping those that contribute to the outer boundary of the convex shape.
5. While Graham Scan has a time complexity of O(n log n), its space complexity is O(n) due to storing the points and stack.

Review Questions

• How does the sorting step in Graham Scan contribute to its efficiency in finding the convex hull?
  • The sorting step is crucial for Graham Scan as it arranges the points based on their polar angles relative to a chosen reference point. By performing this sorting in O(n log n) time, it sets up the algorithm to efficiently iterate through points in order while maintaining geometric relationships. This structure helps avoid unnecessary comparisons, making it faster to identify which points form the outer boundary of the convex hull.
• Discuss how Graham Scan manages collinear points during its execution.
  • Graham Scan addresses collinear points by only keeping those that are necessary for forming the convex hull. When constructing the hull, if three consecutive points make a straight line, only the outermost points are retained, ensuring that no unnecessary vertices are included in the final convex shape. This management enhances both the accuracy and efficiency of the algorithm, allowing it to focus solely on relevant boundary points.
• Evaluate how Graham Scan compares to other output-sensitive convex hull algorithms in terms of performance and application scenarios.
  • Graham Scan, with its O(n log n) time complexity, offers competitive performance compared to other output-sensitive convex hull algorithms like QuickHull. While QuickHull can be faster for certain distributions of points (especially if they are randomly scattered), Graham Scan is more predictable and stable across different data sets. Its systematic approach ensures a consistent runtime, making it suitable for applications requiring reliable performance regardless of point distribution. Understanding these differences allows developers to choose the best algorithm based on their specific requirements.

Related terms

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

Polar Angle: The angle formed between a reference direction (usually the positive x-axis) and a line connecting the reference point to another point.

Stack Data Structure: A collection of elements that follows the Last In First Out (LIFO) principle, commonly used in algorithms for maintaining order.