What is a voronoi diagram?
A Voronoi Diagram splits space based on closeness to points. Imagine you drop some seeds on paper: every spot on the paper belongs to the closest seed. The diagram draws lines (or "walls") to separate these zones—each one is called a cell. Each point owns a region where every location is closer to it than to any other point.
Where Its Used
Geography (e.g. “closest hospital” maps)
Robotics (for navigation and sensor fields)
Cellular networks (signal coverage zones)
Game development (for territory division or procedural generation)
Art & design (generating organic-looking patterns)
Strengths
Gives a natural partitioning of space
Works well for proximity queries (e.g. nearest neighbor)
Visually intuitive and helpful for interactive tools
Fast to compute with small to medium point sets
Weaknesses
Recomputing can be slow if points are moving frequently
Not ideal for very large dynamic datasets
Harder to extend directly into high-dimensional space
Complexities
| Operation |
Average Case |
Worst Case |
| Bulk Insertion (Bowyer-Watson) | O(n log n) | O(n²) |
| Insertion | O(logn) | O(logn) |
| Deletion | O(logn) | O(logn) |
| Range Query | O(√n + m) | O(n) |
| k-NN Query | O(k + log n) | O(kn) |
Bulk Insertion (Bowyer-Watson):
This method incrementally adds points to an existing triangulation. It generally runs in O(n log n), but in degenerate or worst-case scenarios, it may degrade to O(n²).
Insertion:
When a new point is added to a Voronoi diagram, only the nearby regions (cells) affected by that point need to be recalculated. This involves updating boundaries of neighboring cells and inserting a new one. In Delaunay-based methods, this triggers a local re-triangulation. The operation is typically O(log n).
Deletion:
Deleting a point removes its associated region from the diagram. Its neighboring regions merge and form new boundaries. Like insertion, in Delaunay-based approaches, a local reconfiguration is applied. The complexity remains O(log n).
k-NN Search:
For k-nearest neighbors, after locating the cell of the query point, nearby cells are explored. With suitable data structures (like Voronoi graphs or adjacency lists), this can be done in O(k + log n).
Range Query:
To find all points within a region, cells intersecting the query area are identified and scanned. The complexity is O(√n + m), where m is the number of results returned, and √n (or pn) represents the number of intersecting cells.