# sweep line algorithm

> class of algorithms in computational geometry that uses a conceptual sweep line/surface to solve various problems in Euclidean space

**Wikidata**: [Q2372426](https://www.wikidata.org/wiki/Q2372426)  
**Wikipedia**: [English](https://en.wikipedia.org/wiki/Sweep_line_algorithm)  
**Source**: https://4ort.xyz/entity/sweep-line-algorithm

## Summary
A sweep line algorithm is a class of algorithms in computational geometry that uses a conceptual sweep line or surface to systematically solve geometric problems in Euclidean space. This approach provides an efficient method for processing geometric objects by sweeping across the plane while maintaining data structures to manage active elements.

## Key Facts
- It is a class of algorithms specifically designed for computational geometry problems
- It utilizes a conceptual sweep line/surface that moves through the Euclidean space
- It solves various geometric problems by systematically processing objects as the sweep line advances
- It has an alternative name: plane sweep algorithm
- It is classified both as an algorithm and a subclass of computational geometry
- It has Wikipedia coverage in 8 languages: German (de), English (en), Persian (fa), French (fr), Hebrew (he), Romanian (ro), Russian (ru), and Ukrainian (uk)
- It has 8 sitelinks across different language editions of Wikipedia
- It had a Microsoft Academic ID: 52676573 (now discontinued)
- Its Freebase identifier is /m/027fgjv

## FAQs
### Q: What is a sweep line algorithm?
A: A sweep line algorithm is a method in computational geometry that processes geometric problems by moving a conceptual line (or surface) across the Euclidean space, maintaining and updating data structures as it encounters different geometric objects.

### Q: What problems can sweep line algorithms solve?
A: Sweep line algorithms can solve various geometric problems including line segment intersections, finding closest pairs of points, constructing Voronoi diagrams, polygon triangulation, and solving visibility problems in computational geometry.

### Q: How does the sweep line algorithm work?
A: The algorithm works by simulating a line that sweeps across the plane from left to right (or in another consistent direction), processing events (like line segment endpoints or intersections) as they occur along the sweep path, and maintaining an active set of relevant objects.

### Q: What makes sweep line algorithms efficient?
A: Their efficiency comes from processing each event only once and maintaining data structures that allow for O(log n) time operations for updates and queries, leading to O(n log n) time complexity for many geometric problems.

## Why It Matters
Sweep line algorithms represent a fundamental paradigm in computational geometry, enabling efficient solutions to problems that would be computationally expensive with brute-force methods. By systematically processing geometric objects through a sweeping approach, these algorithms reduce the complexity of many spatial problems from O(n²) to O(n log n), making them practical for real-world applications in computer graphics, geographic information systems, robotics, and computer-aided design. The sweep line approach has influenced numerous subsequent geometric algorithms and remains a cornerstone technique taught in computational geometry courses worldwide.

## Notable For
- Its unique paradigm of using a conceptual moving line to solve geometric problems
- Its ability to solve numerous geometric problems with optimal O(n log n) time complexity
- Its influence on the field of computational geometry as a fundamental algorithmic approach
- Its broad recognition across international language editions of Wikipedia
- Its practical application in solving real-world geometric problems in computer graphics and GIS

## Body

### Core Concept
The sweep line algorithm operates on a simple yet powerful principle: a conceptual line (or surface) moves through Euclidean space, processing geometric events as they are encountered. This line typically moves from left to right across a 2D plane, though other directions may be used depending on the specific problem.

### Algorithm Components
- **Sweep Line**: A conceptual line that moves in a predetermined direction through the space
- **Event Points**: Geometric features (such as line segment endpoints) that trigger processing as the sweep line encounters them
- **Status Structure**: A data structure that maintains the "active" elements intersected by the sweep line
- **Output Structure**: Collects the results of the computation as events are processed

### Implementation Approach
1. Sort all relevant event points along the sweep direction
2. Initialize empty status and output structures
3. Process events in sorted order:
   - Update the status structure to reflect current active elements
   - Perform geometric calculations and store results in output structure
4. Continue until all events are processed

### Applications
- Line segment intersection detection
- Finding the closest pair of points
- Polygon triangulation
- Constructing Voronoi diagrams
- Visibility and ray shooting problems

### Time Complexity
Most sweep line algorithms achieve O(n log n) time complexity, which is optimal for many geometric problems. The sorting step typically dominates the complexity, while the sweep phase processes each event once, with O(log n) time operations on the status structure.