Doubly-connected edge list

Definition

A doubly-connected edge list (DCEL) is a data structure used in computational geometry to represent a planar subdivision. It provides a way to efficiently store and access information about the edges, vertices, and faces of a polygonal representation. Each edge in a DCEL has pointers to both its incident vertices and the adjacent edges, allowing for easy traversal of the structure while maintaining the relationships among faces and edges.

5 Must Know Facts For Your Next Test

1. In a DCEL, each edge is represented by two directed half-edges, allowing for bidirectional traversal along the edge.
2. DCEL provides efficient storage for planar subdivisions, enabling fast access to adjacent edges and incident vertices.
3. The structure supports dynamic operations such as insertion and deletion of edges, making it suitable for algorithms that modify geometric structures.
4. Each face in the DCEL is represented by a pointer to one of its bounding edges, which makes it easy to retrieve all edges that enclose a face.
5. DCEL is particularly useful in algorithms related to Voronoi diagrams and monotone polygons, as it facilitates efficient computations involving polygonal regions.

Review Questions

• How does the doubly-connected edge list enhance the representation of planar subdivisions compared to simpler data structures?
  • The doubly-connected edge list improves the representation of planar subdivisions by allowing easy access to not only the vertices and edges but also the faces. Unlike simpler structures that might only track vertices or edges individually, DCEL maintains pointers that connect these elements, facilitating efficient traversal and manipulation. This interconnectedness is particularly beneficial in applications where understanding the relationships among these components is crucial.
• Discuss how the DCEL structure can be applied in constructing Voronoi diagrams and the advantages it offers in this context.
  • When constructing Voronoi diagrams, the DCEL structure allows for efficient management of edges and faces formed by the diagram's cells. The ability to quickly access adjacent edges and incident vertices helps in dynamically updating the structure as new points are added or removed. This interconnectedness aids in maintaining the integrity of the diagram during algorithm execution, ensuring fast performance when querying or modifying elements within the Voronoi diagram.
• Evaluate the impact of using a doubly-connected edge list on algorithms that work with monotone polygons, considering efficiency and complexity.
  • Using a doubly-connected edge list significantly enhances algorithms dealing with monotone polygons by optimizing both efficiency and complexity. Since DCEL allows for rapid access to adjacent edges and vertices, algorithms can quickly perform necessary operations like merging or splitting polygons without excessive overhead. This leads to improved runtime performance compared to other data structures that may not support such efficient relational queries, ultimately making it easier to implement complex geometric algorithms while minimizing computational costs.

Related terms

Half-edge data structure: A data structure similar to DCEL that represents a subdivision of space, focusing on edges and their relationships with faces and vertices.

Planar graph: A graph that can be drawn on a plane without any edges crossing, often represented using data structures like DCEL.

Face:

A bounded region in a planar subdivision, defined by edges and vertices, which can be easily accessed and manipulated using DCEL.