Incremental algorithm

Definition

An incremental algorithm is a computational method used to construct the convex hull of a set of points by progressively adding points and updating the current hull. This approach allows for efficient handling of dynamic datasets, enabling the hull to be adjusted as new points are introduced. It contrasts with batch algorithms, which require all points upfront before determining the hull.

5 Must Know Facts For Your Next Test

1. Incremental algorithms can efficiently manage large datasets where points are added dynamically, making them suitable for real-time applications.
2. The incremental approach typically involves maintaining a list of the current convex hull vertices and updating it when new points are processed.
3. One common method for incrementally building the convex hull is the 'Gift Wrapping' or 'Jarvis March' algorithm, which adds points one by one.
4. Incremental algorithms usually have a time complexity of O(n log h) where n is the number of input points and h is the number of vertices in the final convex hull.
5. The incremental nature allows these algorithms to be adaptable for scenarios where the dataset is not static and evolves over time.

Review Questions

• How does an incremental algorithm differ from batch algorithms in constructing convex hulls?
  • An incremental algorithm differs from batch algorithms by allowing points to be added one at a time, adjusting the convex hull dynamically with each new point. In contrast, batch algorithms require all data points to be present before calculating the convex hull. This flexibility makes incremental algorithms more suitable for applications where data continuously changes or grows over time.
• Discuss how the incremental algorithm can be applied in real-time applications and what advantages it offers over traditional methods.
  • The incremental algorithm is particularly advantageous in real-time applications such as robotics, computer graphics, or geographic information systems where data may continuously arrive. It enables quick updates to the convex hull without recalculating everything from scratch. This adaptability leads to faster processing times and efficiency gains, especially when working with dynamic datasets.
• Evaluate the performance implications of using an incremental algorithm for constructing convex hulls versus other techniques like Graham's Scan or Divide and Conquer methods.
  • Using an incremental algorithm can be more efficient in scenarios with ongoing data input since it processes each point individually rather than requiring all points at once. While Graham's Scan has a fixed O(n log n) complexity regardless of dynamic data input, incremental algorithms have time complexities that depend on both the number of input points and the vertices in the final hull. This means that while both methods are effective, incremental algorithms provide better adaptability and responsiveness for changing datasets, making them preferable in specific applications.

Related terms

Convex Hull:

The smallest convex shape that can enclose a set of points in a Euclidean space.

Graham's Scan:

A popular algorithm for finding the convex hull of a set of points in two-dimensional space using a sorting approach.

Divide and Conquer: An algorithmic technique that divides a problem into smaller subproblems, solves them independently, and combines their results, often used in computational geometry.