Voronoi Diagrams & Fortune’s Algorithm - CSU083 | Shoolini University

Voronoi Diagrams (Fortune’s Algorithm)

1. Prerequisites

Before understanding Voronoi Diagrams and Fortune’s Algorithm, the following foundational concepts are essential:

1.1 Computational Geometry

The study of algorithms that solve geometric problems, such as triangulation, convex hulls, and spatial partitioning.

1.2 Planar Graphs

A graph that can be drawn on a plane without edges crossing. Voronoi diagrams are a form of planar subdivision.

1.3 Delaunay Triangulation

A dual graph of a Voronoi diagram that connects points to form non-overlapping triangles with an empty circumcircle property.

1.4 Event-driven Sweep Line Algorithm

Fortune’s Algorithm is an event-based algorithm that processes a set of points dynamically using a sweep line approach.

1.5 Priority Queues

Fortune’s Algorithm relies on a priority queue to process site and circle events efficiently.

2. What is a Voronoi Diagram?

A Voronoi Diagram partitions a plane into regions based on distance to a set of given points, called "sites." Each region consists of all points closest to a particular site.

2.1 Definition

For a set of points \( S = \{p_1, p_2, \dots, p_n\} \) in a 2D plane, the Voronoi cell \( V(p_i) \) for a site \( p_i \) is defined as:

$$ V(p_i) = \{ x \in \mathbb{R}^2 \mid d(x, p_i) < d(x, p_j), \forall j \neq i \} $$

2.2 Fortune’s Algorithm

A sweep-line algorithm that efficiently constructs Voronoi diagrams in \( O(n \log n) \) time using two primary data structures:

2.3 Steps of Fortune’s Algorithm

  1. Initialize an event queue with all site events.
  2. Process site events by updating the beach line.
  3. Detect and process circle events when three arcs converge.
  4. Maintain Voronoi edges dynamically as arcs disappear.
  5. Terminate when all events are processed.

3. Why Does This Algorithm Exist?

Voronoi diagrams provide a structured way to divide space based on proximity, enabling numerous real-world applications:

3.1 Nearest Neighbor Queries

Used in computational geometry, GIS, and search problems to find the closest data point efficiently.

3.2 Pathfinding & Robotics

Robot navigation uses Voronoi graphs to define optimal paths avoiding obstacles.

3.3 Cellular Networks

Telecommunication companies use Voronoi diagrams to optimize cell tower coverage areas.

3.4 Meteorology & Climate Modeling

Used in spatial interpolation methods like Thiessen polygons to estimate environmental parameters.

3.5 Medical Imaging & Biology

Voronoi structures appear in biological growth patterns, molecular modeling, and tumor analysis.

4. When Should You Use It?

Voronoi diagrams and Fortune’s Algorithm are most effective when dealing with spatial partitioning problems where locality matters.

4.1 Best Use Cases

5. Comparison with Alternatives

While Voronoi diagrams solve many spatial problems efficiently, alternative methods exist:

5.1 Strengths of Voronoi Diagrams (Fortune’s Algorithm)

5.2 Weaknesses

5.3 Alternatives

6. Basic Implementation

Below is a basic implementation of Fortune’s Algorithm in Python, using a priority queue and beach line representation.

6.1 Python Implementation

The code constructs a Voronoi diagram given a set of points.


import heapq
import math

class Event:
    def __init__(self, x, y, site, is_site=True):
        self.x = x
        self.y = y
        self.site = site
        self.is_site = is_site

    def __lt__(self, other):
        return self.y < other.y if self.y != other.y else self.x < other.x

class Voronoi:
    def __init__(self, points):
        self.points = sorted(points, key=lambda p: p[1])  # Sort by y-coordinate
        self.event_queue = []
        self.beach_line = []
        self.edges = []

    def process(self):
        for x, y in self.points:
            heapq.heappush(self.event_queue, Event(x, y, (x, y)))

        while self.event_queue:
            event = heapq.heappop(self.event_queue)
            if event.is_site:
                self.handle_site_event(event)
            else:
                self.handle_circle_event(event)

    def handle_site_event(self, event):
        # Insert arc into beach line
        self.beach_line.append(event.site)

    def handle_circle_event(self, event):
        # Process circle event (arc disappears)
        pass  # Full implementation involves checking circumcenters

    def get_edges(self):
        return self.edges

points = [(2, 5), (6, 8), (3, 1), (9, 7)]
voronoi = Voronoi(points)
voronoi.process()
print("Voronoi edges:", voronoi.get_edges())

Explanation:

7. Dry Run on a Small Input Set

We run the algorithm on four points: (2,5), (6,8), (3,1), (9,7).

7.1 Initial Conditions

Sorted input: [(3,1), (2,5), (9,7), (6,8)]

Step Event Queue Beach Line Voronoi Edges
1. Process (3,1) [(2,5), (9,7), (6,8)] [(3,1)] []
2. Process (2,5) [(9,7), (6,8)] [(3,1), (2,5)] [Initial edge starts forming]
3. Process (9,7) [(6,8)] [(3,1), (2,5), (9,7)] [Edges between (2,5) and (9,7)]
4. Process (6,8) [] [(3,1), (2,5), (9,7), (6,8)] [More edges complete]

7.2 Observations

Final Result: The Voronoi diagram is formed, but a full implementation would compute exact edge equations and intersections.

8. Time & Space Complexity Analysis

8.1 Worst-Case Time Complexity

Fortune’s Algorithm runs in \( O(n \log n) \) in the worst case, derived as follows:

Final Worst-Case Complexity: \( O(n \log n) \).

8.2 Best-Case Time Complexity

The best case occurs when the input points are structured such that no circle events occur (e.g., points are collinear).

8.3 Average-Case Complexity

In practical cases, we assume a random distribution of points. The average number of events remains \( O(n) \), and each is processed in \( O(\log n) \), leading to:

Average-Case Complexity: \( O(n \log n) \).

9. Space Complexity Analysis

9.1 Breakdown of Space Consumption

9.2 Final Space Complexity

All components contribute at most \( O(n) \), so the overall space complexity is:

Space Complexity: \( O(n) \).

9.3 Growth with Input Size

As the number of input points increases:

10. Trade-offs in Using Fortune’s Algorithm

10.1 Strengths

10.2 Weaknesses

10.3 Alternative Approaches

Fortune’s Algorithm remains a preferred choice when an efficient, deterministic method for Voronoi construction is required.

11. Optimizations & Variants

Fortune’s Algorithm is efficient but can be further optimized in practical applications.

11.1 Common Optimizations

11.1.1 Optimized Data Structures
11.1.2 Numerical Precision Handling
11.1.3 Parallelization

11.2 Variants of Fortune’s Algorithm

11.2.1 Incremental Voronoi Construction

Instead of sweeping the plane, insert points one at a time and update the Voronoi structure dynamically.

11.2.2 Approximate Voronoi Diagrams

For large datasets, exact Voronoi diagrams are expensive. Approximate methods include:

11.2.3 Higher-Dimensional Voronoi Diagrams

Extends Voronoi diagrams to 3D and beyond.

12. Iterative vs. Recursive Implementations

12.1 Iterative Implementation

Most Fortune’s Algorithm implementations are iterative, using a priority queue for event processing.

12.2 Recursive Implementation

A recursive implementation mimics divide-and-conquer approaches.

12.3 Final Comparison

Criteria Iterative Recursive
Time Complexity \( O(n \log n) \) \( O(n \log n) \)
Space Complexity \( O(n) \) \( O(n) \) (plus recursion stack)
Ease of Implementation More complex (explicit data structures) Easier (mirrors mathematical intuition)
Scalability Handles large inputs well Risk of stack overflow

Conclusion: Iterative implementations are more practical for large-scale applications, whereas recursive approaches are useful for educational and conceptual purposes.

13. Edge Cases & Failure Handling

While implementing Fortune’s Algorithm, several edge cases and pitfalls must be handled carefully.

13.1 Common Edge Cases

13.1.1 Collinear Points
13.1.2 Duplicate Points
13.1.3 Floating-Point Precision Issues
13.1.4 Very Close Points
13.1.5 Large Input Sizes

13.2 Failure Handling Mechanisms

14. Test Cases to Verify Correctness

14.1 Basic Test Cases

14.1.1 Single Point

Expected Output: A single cell covering the entire space.


test_points = [(5, 5)]
expected_edges = []  # No edges as only one point exists
14.1.2 Two Points

Expected Output: A single straight boundary dividing space equally.


test_points = [(2, 5), (8, 5)]
expected_edges = [((5, 0), (5, ∞))]  # Vertical line at x = 5

14.2 Edge Case Test Cases

14.2.1 Collinear Points

Expected Output: Parallel Voronoi edges.


test_points = [(1, 5), (3, 5), (7, 5)]
expected_edges = [((2, 0), (2, ∞)), ((5, 0), (5, ∞))]
14.2.2 Randomly Distributed Points

Expected Output: A valid Voronoi diagram with correct edge relationships.


test_points = [(2, 5), (6, 8), (3, 1), (9, 7)]
expected_edges = [...]  # Computed based on Voronoi relations
14.2.3 Floating-Point Precision Test

Expected Output: No incorrect or missing edges due to precision errors.


test_points = [(2.00001, 5), (2.00002, 5)]
expected_edges = [((2.000015, 0), (2.000015, ∞))]

15. Real-World Failure Scenarios

15.1 GIS and Mapping Failures

In geographic mapping, small floating-point errors can cause incorrect region assignments, leading to:

15.2 Robotics and Navigation

Voronoi diagrams are used for robot path planning. Failure scenarios include:

15.3 Wireless Network Coverage

Telecommunication networks use Voronoi diagrams for cell tower placement.

15.4 Computational Biology

Used in modeling cellular growth and molecular structures.

Conclusion: Proper validation, precision handling, and optimization techniques are necessary to prevent failures in real-world applications.

16. Real-World Applications & Industry Use Cases

Voronoi diagrams, constructed using Fortune’s Algorithm, have extensive applications across multiple industries.

16.1 Geographic Information Systems (GIS)

16.2 Telecommunications & Wireless Networks

16.3 Robotics & AI

16.4 Computer Graphics & Game Development

16.5 Medical Imaging & Computational Biology

16.6 Supply Chain & Logistics

17. Open-Source Implementations

Several open-source libraries provide implementations of Fortune’s Algorithm and Voronoi diagram generation:

17.1 SciPy (Python)

17.2 OpenCV (C++)

17.3 CGAL (C++)

17.4 Qhull (C/C++)

18. Practical Project: Voronoi-based City Zone Planner

In this project, we will use Fortune’s Algorithm to generate city zones based on landmark locations.

18.1 Project Overview

18.2 Implementation (Python)


import numpy as np
import matplotlib.pyplot as plt
from scipy.spatial import Voronoi, voronoi_plot_2d

# Define city landmarks (hospitals, fire stations, schools)
landmarks = np.array([[10, 20], [30, 40], [50, 60], [70, 80], [90, 20]])

# Compute Voronoi Diagram
vor = Voronoi(landmarks)

# Plot Voronoi Diagram
fig, ax = plt.subplots(figsize=(8, 6))
voronoi_plot_2d(vor, ax=ax)

# Plot landmarks
ax.scatter(landmarks[:, 0], landmarks[:, 1], color='red', marker='o', label='Facilities')

# Customize plot
plt.title("City Zone Planner using Voronoi Diagram")
plt.legend()
plt.show()

18.3 Explanation

18.4 Extensions

Conclusion: This project demonstrates how Voronoi diagrams help in real-world spatial planning.

19. Competitive Programming & System Design Integration

19.1 Fortune’s Algorithm in Competitive Programming

Voronoi diagrams and Fortune’s Algorithm are rarely used in direct problem-solving but have applications in geometry-based problems:

19.1.1 Typical Problem Types
19.1.2 Tips for Competitive Programming

19.2 Voronoi Diagrams in System Design

System design problems often require spatial partitioning for efficient resource allocation.

19.2.1 Example Use Cases
19.2.2 System Design Considerations

20. Assignments

20.1 Solve 10 Problems Using Voronoi Diagrams

Solve the following problems to strengthen your understanding of Fortune’s Algorithm:

  1. Nearest Restaurant Finder: Given coordinates of restaurants, find the nearest one for each user query.
  2. Cell Tower Optimization: Assign optimal coverage regions for telecom towers.
  3. Convex Hull and Voronoi Edges: Compute both the convex hull and Voronoi edges for a set of points.
  4. Autonomous Vehicle Navigation: Use Voronoi diagrams to compute safe paths for self-driving cars.
  5. Game AI Pathfinding: Divide a game map into AI-controlled territories using Voronoi partitions.
  6. Distributed Databases: Design a sharding strategy where each server is assigned a Voronoi region.
  7. Urban Planning: Given city landmarks, partition the city into zones for efficient infrastructure allocation.
  8. Disaster Relief Zones: Compute emergency response zones based on Voronoi diagrams.
  9. Air Traffic Control: Optimize flight path allocation using Voronoi-based airspace partitioning.
  10. Biological Cell Growth Simulation: Model cell growth using Voronoi structures in a biological simulation.

20.2 System Design Problem

Design a scalable system that assigns users to the nearest available server using Voronoi diagrams.

Requirements:

20.3 Implement Under Time Constraints

To test your speed and accuracy, implement the following within a strict time limit:

Challenge 1 (30 minutes):
Challenge 2 (60 minutes):
Challenge 3 (90 minutes):