Convex Hull and Searching in Design and Analysis of Algorithms
Convex Hull:
In computational geometry, the convex hull of a set of points in the Euclidean plane is the smallest convex polygon that encloses all the points in the set. It's a fundamental concept with various applications in fields like computer graphics, pattern recognition, and geographic information systems.
The design and analysis of algorithms related to convex hulls involve techniques to efficiently compute the convex hull of a given set of points. One of the most well-known algorithms for this purpose is the Graham Scan algorithm. It sorts the points based on their polar angles with respect to the lowest point and then iteratively constructs the convex hull.
Here's a simple example:
Searching:
Searching is a fundamental operation in computer science and is extensively used in various algorithms and data structures. It involves finding a particular element within a collection of items.
There are numerous searching algorithms designed and analyzed to optimize the search process based on factors such as time complexity, space complexity, and the nature of the data structure. Some of the commonly used searching algorithms include Linear Search, Binary Search, and Hashing-based techniques.
Linear Search is a simple searching algorithm that sequentially checks each element of the collection until it finds the desired one or reaches the end of the collection. While simple, it may not be efficient for large datasets.
Binary Search, on the other hand, is more efficient for sorted arrays. It works by repeatedly dividing the search interval in half until the desired element is found or the interval becomes empty.
Hashing-based techniques involve mapping keys to their corresponding values using a hash function. This allows for constant-time retrieval of elements in an ideal scenario, making it highly efficient for search operations.
Here's a comparison of Linear Search and Binary Search: