## Introduction

The Sweep Line Algorithm is a powerful technique used to solve problems that involve geometric objects like points, lines, and rectangles. It helps optimize the solution by avoiding unnecessary comparisons and reducing the time complexity.

In this article, we will discuss the concept of the sweep line algorithm and its applications in different problems like finding the closest pair of points, detecting line segment intersections, calculating the union of rectangles, computing the convex hull, and building the Manhattan minimum spanning tree.

## What is Sweep Line Technique?

The sweep line technique is an algorithmic approach used to solve geometric problems efficiently. Imagine a vertical line (the sweep line) moving from left to right across the plane, sweeping over the geometric objects in the problem. As the line moves, it encounters the objects in a specific order, allowing us to process them sequentially.

The key idea behind the sweep line technique is to maintain a balanced binary search tree (BST) or a similar data structure that keeps track of the objects currently intersecting the sweep line. This data structure is updated as the sweep line progresses, allowing us to efficiently query and update the relevant information.

The sweep line algorithm follows these steps:

1. Sort the objects based on their x-coordinates.

2. Initialize an empty BST to store the objects intersecting the sweep line.

3. Process the objects in the sorted order:

- If the object is a left endpoint, insert it into the BST.

- If the object is a right endpoint, remove it from the BST.

- If the object is a point or a query, perform the necessary operations using the BST.

4. Update the solution based on the processed objects and the information stored in the BST.

By processing the objects in a sorted order and maintaining a BST, the sweep line algorithm reduces the number of comparisons needed and achieves a more efficient solution compared to brute-force approaches.

For example :

Suppose we have a set of points in a 2D plane, and we want to find the closest pair of points. We can use the sweep line technique as follows:

1. Sort the points based on their x-coordinates.

2. Initialize an empty BST to store the points currently intersecting the sweep line.

3. Process the points in the sorted order:

- Insert each point into the BST.

- Check the points in the BST that are within a certain vertical distance from the current point.

- Update the closest pair if a closer pair is found.

4. Return the closest pair of points.