1. Prerequisites

Before understanding the Sweep Line Algorithm, you should be familiar with the following foundational concepts:

• Sorting: The algorithm often relies on sorting events based on a coordinate axis.
• Binary Search Trees (BST) / Balanced Trees: Used to efficiently maintain active intervals.
• Priority Queues / Heaps: Helps in efficiently processing events in a sorted order.
• Geometric Computations: Knowledge of line segments, intersections, and convex hulls is useful for spatial problems.

2. What is the Sweep Line Algorithm?

The Sweep Line Algorithm is a computational geometry technique that processes a set of geometric objects by sweeping a conceptual vertical or horizontal line across a plane. It efficiently solves problems by handling events in a sorted order.

2.1 Key Components

• Event Points: Critical points where a change occurs (e.g., segment start/end, intersections).
• Active Set: A data structure maintaining elements intersecting the sweep line at a given moment.
• Event Processing: Events are processed in increasing order of coordinates.

2.2 Working Mechanism

The algorithm operates as follows:

• Sort all events (start/end points of segments or intersections).
• Sweep the line across these points, updating the active set dynamically.
• Use a data structure (e.g., BST) to manage active segments and detect interactions.

3. Why Does This Algorithm Exist?

The Sweep Line Algorithm exists to solve problems that involve processing geometric objects efficiently. Instead of brute-force pairwise comparisons, it leverages event ordering to reduce computational complexity.

3.1 Real-World Applications

• Line Segment Intersection: Detecting overlapping road segments in mapping applications.
• Computational Geometry: Finding convex hulls, Voronoi diagrams, or Delaunay triangulations.
• Computer Graphics: Efficiently rendering scenes by determining visible surfaces.
• GIS (Geographic Information Systems): Analyzing spatial relationships and map processing.

4. When Should You Use It?

The Sweep Line Algorithm is best used when:

• Events can be sorted and processed sequentially.
• The problem involves detecting interactions among geometric objects.
• A brute-force \(O(n^2)\) approach is inefficient.

4.1 Best Use Cases

• Intersection Detection: Finding intersections among many lines efficiently.
• Closest Pair of Points: Optimizing the search for the nearest pair in 2D space.
• Convex Hull Algorithms: Used in algorithms like Graham’s scan.

5. How Does It Compare to Alternatives?

The Sweep Line Algorithm competes with other geometric algorithms. Below is a comparison of its strengths and weaknesses:

5.1 Strengths

• Efficient for large datasets compared to brute force.
• Handles real-world geometric problems effectively.
• Optimized by balanced data structures (BST, heaps).
• Scalable to high-dimensional data with modifications.

5.2 Weaknesses

• Implementation Complexity: Requires careful handling of event queue and active set.
• Edge Cases: Degenerate cases (e.g., overlapping points) require special handling.
• Higher Space Complexity: Uses additional memory for event queues and active set.

5.3 Alternatives

• Divide and Conquer: Used for closest pair problems but may not be efficient for dynamic updates.
• Brute Force: Simpler but inefficient for large datasets.
• Graph-based Approaches: Used in some geometric problems but require different data structures.

6. Basic Implementation

Below is a Python implementation of the Sweep Line Algorithm for detecting intersections among a set of line segments. It utilizes an event queue and a balanced tree to maintain active segments.

import heapq
from sortedcontainers import SortedList

class Event:
    def __init__(self, x, segment, is_start):
        self.x = x
        self.segment = segment
        self.is_start = is_start
    
    def __lt__(self, other):
        return self.x < other.x  # Sorting events by x-coordinate

def do_intersect(seg1, seg2):
    """ Check if two line segments intersect using orientation test. """
    def orientation(p, q, r):
        val = (q[1] - p[1]) * (r[0] - q[0]) - (q[0] - p[0]) * (r[1] - q[1])
        return 0 if val == 0 else (1 if val > 0 else -1)
    
    p1, q1 = seg1
    p2, q2 = seg2
    
    o1 = orientation(p1, q1, p2)
    o2 = orientation(p1, q1, q2)
    o3 = orientation(p2, q2, p1)
    o4 = orientation(p2, q2, q1)

    return o1 != o2 and o3 != o4

def sweep_line_intersection(segments):
    """ Detect intersections using Sweep Line Algorithm. """
    events = []
    for seg in segments:
        events.append(Event(seg[0][0], seg, True))  # Start event
        events.append(Event(seg[1][0], seg, False)) # End event

    events.sort()  # Sorting events by x-coordinate

    active_segments = SortedList(key=lambda seg: seg[0][1])  # Sort by y-coordinate
    intersections = []

    for event in events:
        seg = event.segment
        if event.is_start:
            idx = active_segments.bisect(seg)
            if idx > 0 and do_intersect(active_segments[idx - 1], seg):
                intersections.append((active_segments[idx - 1], seg))
            if idx < len(active_segments) and do_intersect(active_segments[idx], seg):
                intersections.append((active_segments[idx], seg))
            active_segments.add(seg)
        else:
            idx = active_segments.index(seg)
            if 0 < idx < len(active_segments) - 1 and do_intersect(active_segments[idx - 1], active_segments[idx + 1]):
                intersections.append((active_segments[idx - 1], active_segments[idx + 1]))
            active_segments.remove(seg)

    return intersections

# Example input: List of line segments (defined by two endpoints)
segments = [
    ((1, 1), (5, 5)),
    ((2, 5), (6, 1)),
    ((3, 3), (7, 7)),
]

# Running the algorithm
intersections = sweep_line_intersection(segments)
print("Intersections:", intersections)

This implementation:

• Sorts events by x-coordinate.
• Uses a sorted list to maintain active segments dynamically.
• Checks for intersections with neighboring segments.

7. Dry Run of the Algorithm

Let's dry-run the algorithm for the input:

Segments: 
1. ((1, 1), (5, 5))
2. ((2, 5), (6, 1))
3. ((3, 3), (7, 7))

Sorted Events:
(1, Start of Segment 1)
(2, Start of Segment 2)
(3, Start of Segment 3)
(5, End of Segment 1)
(6, End of Segment 2)
(7, End of Segment 3)

Step-by-step execution:

Step 1: Process (1, Start of Segment 1)
  - Add segment ((1,1) to (5,5)) to active set.

Step 2: Process (2, Start of Segment 2)
  - Add segment ((2,5) to (6,1)) to active set.
  - Check for intersection with existing segment ((1,1) to (5,5)).
  - Intersection found → Store result.

Step 3: Process (3, Start of Segment 3)
  - Add segment ((3,3) to (7,7)) to active set.
  - Check for intersections with neighbors.
  - No new intersection.

Step 4: Process (5, End of Segment 1)
  - Remove segment ((1,1) to (5,5)) from active set.
  - Check if neighboring segments intersect.
  - No new intersection.

Step 5: Process (6, End of Segment 2)
  - Remove segment ((2,5) to (6,1)) from active set.

Step 6: Process (7, End of Segment 3)
  - Remove segment ((3,3) to (7,7)) from active set.

Final Output:
Intersections: [((1,1) to (5,5)), ((2,5) to (6,1))]

Observations:

• The algorithm efficiently detects intersections without brute force comparisons.
• The active set dynamically updates, ensuring only relevant segments are checked.
• Sorting events helps in maintaining an efficient processing order.

8. Time & Space Complexity Analysis

8.1 Time Complexity Breakdown

The Sweep Line Algorithm involves three main operations:

• Sorting Events: Sorting \(2n\) event points (start and end of segments) takes \(O(n \log n)\).
• Processing Events: Each event is handled once, and each segment is inserted or removed from the active set at most once, leading to \(O(n \log n)\) due to BST operations.
• Intersection Checks: Each segment only interacts with its immediate neighbors in the active set. In the worst case, we might check all pairs leading to \(O(n \log n + k)\), where \(k\) is the number of intersections.

8.2 Complexity Derivation

• Worst-case Complexity: \(O(n \log n + k)\) - Sorting dominates for small \(k\), but for a large number of intersections, \(k\) may approach \(O(n^2)\).
• Best-case Complexity: \(O(n \log n)\) - If no intersections exist, we only sort and maintain the active set.
• Average-case Complexity: \(O(n \log n + k)\) - Usually, intersections are much smaller than \(O(n^2)\), making the algorithm efficient.

9. Space Complexity Analysis

The space complexity of the Sweep Line Algorithm depends on:

• Event List: Stores \(2n\) events → \(O(n)\).
• Active Set: At most \(n\) segments can be stored → \(O(n)\).
• Intersection List: Stores up to \(O(n^2)\) intersections in the worst case.

9.1 Growth with Input Size

• For small \(n\), the space usage remains low due to minimal active set entries.
• For large \(n\), the active set can grow proportionally, increasing memory usage.
• In the worst case (when every segment intersects), space consumption approaches \(O(n^2)\).

9.2 Space Complexity Summary

• Best-case: \(O(n)\) (Minimal active set, few intersections)
• Average-case: \(O(n + k)\) (Where \(k\) is the number of intersections)
• Worst-case: \(O(n^2)\) (All segments intersect)

10. Trade-offs in Using the Sweep Line Algorithm

10.1 Strengths

• Efficiency: Faster than brute force (\(O(n^2)\)) when \(k\) is small.
• Scalability: Handles large datasets better than naive approaches.
• Real-world applications: Well-suited for geometric computations in mapping, graphics, and computational geometry.

10.2 Weaknesses

• Implementation Complexity: Requires careful handling of event processing.
• Edge Cases: Degenerate cases (collinear points, overlapping segments) need special handling.
• High Memory Usage: Can consume large amounts of space when many intersections exist.

10.3 Alternative Approaches

• Brute Force (\(O(n^2)\)): Simpler but inefficient for large \(n\).
• Divide & Conquer (\(O(n \log n)\)): Useful for some geometric problems but not optimal for real-time segment processing.
• Graph-based Approaches: Alternative for routing and pathfinding but less suitable for geometric queries.

Conclusion: The Sweep Line Algorithm is best when \(k\) is significantly smaller than \(n^2\), making it a practical choice for real-world applications where spatial relationships need to be analyzed efficiently.

11. Optimizations & Variants

11.1 Common Optimizations

Several optimizations can improve the efficiency of the Sweep Line Algorithm:

• Efficient Data Structures: Using balanced trees (e.g., Red-Black Tree, AVL Tree) instead of sorted lists for the active set improves insertion and deletion operations to \(O(\log n)\).
• Event Deduplication: Avoid redundant events (e.g., overlapping line segments) to reduce processing time.
• Lazy Updates: Instead of updating the active set immediately, batch updates at key points.
• Spatial Partitioning: Divide the plane into smaller regions using quadtrees or R-trees to reduce the number of comparisons.
• Preprocessing: If input is static, precompute spatial relationships to avoid unnecessary checks during the sweep.

11.2 Variants of the Sweep Line Algorithm

The Sweep Line Algorithm has multiple versions designed for different geometric problems:

• Bentley-Ottmann Algorithm: Detects all line segment intersections efficiently in \(O((n + k) \log n)\).
• Graham’s Scan + Sweep Line: Used in convex hull algorithms where sweeping helps order points.
• Fortune’s Algorithm: Used for computing Voronoi diagrams by sweeping a parabola.
• Rectangle Union Area Calculation: Modified to compute the total covered area of multiple rectangles.

11.3 Memory Optimization

To reduce memory footprint:

• Use pointer-based data structures instead of arrays to avoid unnecessary storage.
• Employ in-place updates where possible instead of creating new objects.
• Use lazy evaluation (only compute when necessary) to avoid redundant calculations.

12. Iterative vs. Recursive Implementations

12.1 Iterative Approach

The iterative implementation of the Sweep Line Algorithm uses an explicit loop to process events:

def sweep_line_iterative(segments):
    events = []
    for seg in segments:
        events.append((seg[0][0], seg, True))  # Start event
        events.append((seg[1][0], seg, False)) # End event

    events.sort()  # Sort events by x-coordinate
    active_segments = SortedList(key=lambda seg: seg[0][1])
    
    intersections = []
    for event in events:
        x, seg, is_start = event
        if is_start:
            idx = active_segments.bisect(seg)
            if idx > 0 and do_intersect(active_segments[idx - 1], seg):
                intersections.append((active_segments[idx - 1], seg))
            if idx < len(active_segments) and do_intersect(active_segments[idx], seg):
                intersections.append((active_segments[idx], seg))
            active_segments.add(seg)
        else:
            idx = active_segments.index(seg)
            if 0 < idx < len(active_segments) - 1 and do_intersect(active_segments[idx - 1], active_segments[idx + 1]):
                intersections.append((active_segments[idx - 1], active_segments[idx + 1]))
            active_segments.remove(seg)

    return intersections

12.2 Recursive Approach

The recursive implementation typically follows a divide-and-conquer strategy:

def sweep_line_recursive(events, active_segments, index, intersections):
    if index >= len(events):
        return intersections

    x, seg, is_start = events[index]
    if is_start:
        idx = active_segments.bisect(seg)
        if idx > 0 and do_intersect(active_segments[idx - 1], seg):
            intersections.append((active_segments[idx - 1], seg))
        if idx < len(active_segments) and do_intersect(active_segments[idx], seg):
            intersections.append((active_segments[idx], seg))
        active_segments.add(seg)
    else:
        idx = active_segments.index(seg)
        if 0 < idx < len(active_segments) - 1 and do_intersect(active_segments[idx - 1], active_segments[idx + 1]):
            intersections.append((active_segments[idx - 1], active_segments[idx + 1]))
        active_segments.remove(seg)

    return sweep_line_recursive(events, active_segments, index + 1, intersections)

def start_sweep_line_recursive(segments):
    events = [(seg[0][0], seg, True) for seg in segments] + [(seg[1][0], seg, False) for seg in segments]
    events.sort()
    return sweep_line_recursive(events, SortedList(key=lambda seg: seg[0][1]), 0, [])

12.3 Comparison

Conclusion: The iterative approach is preferred for efficiency and scalability, while recursion is useful in problems where divide-and-conquer is beneficial.

13. Edge Cases & Failure Handling

The Sweep Line Algorithm must handle various edge cases to avoid incorrect results or infinite loops.

13.1 Common Edge Cases

• Overlapping Segments: Multiple segments start and end at the same coordinate, leading to ambiguity.
• Collinear Points: When three or more segments align exactly on the same line, intersection logic may fail.
• Vertical Segments: If a segment is perfectly vertical, sorting by x-coordinates alone is insufficient.
• Duplicate Events: Repeated points can cause redundant processing or logical errors.
• Floating-Point Precision Errors: When coordinates involve decimals, small rounding errors can impact intersection detection.

13.2 Failure Handling Strategies

• Use Epsilon Comparisons: Instead of strict equality checks, use an epsilon threshold for floating-point calculations.
• Normalize Input: Preprocess input data to ensure consistency in representation.
• Handle Special Cases Separately: Add explicit checks for collinear points and overlapping segments.
• Efficient Data Structure Updates: Ensure the active set updates correctly when inserting and removing segments.

14. Test Cases to Verify Correctness

Testing is crucial to ensure the correctness of the Sweep Line Algorithm.

14.1 Basic Test Cases

def test_sweep_line():
    segments = [
        ((1, 1), (5, 5)),  # Diagonal
        ((2, 5), (6, 1)),  # Crossing first segment
        ((3, 3), (7, 7)),  # Parallel
    ]
    expected_intersections = [(((1, 1), (5, 5)), ((2, 5), (6, 1)))]
    assert sweep_line_intersection(segments) == expected_intersections

def test_no_intersection():
    segments = [
        ((1, 1), (2, 2)),
        ((3, 3), (4, 4))
    ]
    assert sweep_line_intersection(segments) == []

def test_overlapping_segments():
    segments = [
        ((1, 1), (5, 5)),
        ((2, 2), (4, 4)),  # Overlapping within first segment
    ]
    assert sweep_line_intersection(segments) == []

def test_vertical_segment():
    segments = [
        ((2, 1), (2, 5)),  # Vertical segment
        ((1, 3), (3, 3)),  # Horizontal segment crossing vertical one
    ]
    expected_intersections = [(((2, 1), (2, 5)), ((1, 3), (3, 3)))]
    assert sweep_line_intersection(segments) == expected_intersections

def test_collinear_segments():
    segments = [
        ((1, 1), (5, 5)),
        ((3, 3), (7, 7))  # Collinear but non-overlapping
    ]
    assert sweep_line_intersection(segments) == []

14.2 Key Testing Considerations

• Ensure No False Positives: Segments that should not intersect must not be reported.
• Handle Floating-Point Precision: Test cases with decimal values should remain accurate.
• Test Edge Cases: Include zero-length segments, fully overlapping segments, and boundary conditions.

15. Real-World Failure Scenarios

Failure scenarios occur when the algorithm is applied in real-world applications. Understanding these helps in designing more robust solutions.

15.1 Example Failure Scenarios

• GIS Mapping Errors: Failure to detect overlapping road segments in geographic systems can lead to incorrect route planning.
• Computer-Aided Design (CAD): Missed or extra intersections can cause design inconsistencies in engineering drawings.
• 3D Graphics Rendering: Incorrect visibility detection in rendering algorithms may result in graphical artifacts.
• Drone Flight Path Calculations: In autonomous navigation, undetected segment overlaps can cause incorrect obstacle avoidance.

15.2 Strategies to Mitigate Failures

• Use Higher Precision Data Types: Employ libraries like `decimal` instead of floating-point numbers for precision.
• Validate Input Data: Ensure that input is properly formatted and preprocessed before applying the algorithm.
• Implement Fallback Mechanisms: If intersection detection fails, use an alternative algorithm (e.g., brute-force check as a last resort).
• Continuous Testing: Run test cases on real-world datasets to detect and fix inconsistencies early.

16. Real-World Applications & Industry Use Cases

The Sweep Line Algorithm is widely used in computational geometry, computer graphics, and real-world problem-solving. Below are its key applications:

16.1 Geographic Information Systems (GIS)

• Road Network Analysis: Detecting road intersections in maps for navigation and routing.
• Floodplain Mapping: Analyzing overlapping geographical regions in flood simulations.

16.2 Computer Graphics & Rendering

• Hidden Surface Removal: Used in rendering pipelines to determine visible surfaces in 3D scenes.
• Ray Tracing: Detects intersection points between rays and objects to simulate realistic lighting.

16.3 Robotics & Path Planning

• Collision Detection: Ensures safe movement by checking for obstacle intersections.
• Drone Navigation: Helps avoid obstacles dynamically by analyzing segment intersections.

16.4 Computational Biology

• DNA Sequence Matching: Used in bioinformatics for detecting overlapping gene sequences.
• Protein Structure Analysis: Helps in spatial analysis of molecular structures.

16.5 VLSI Design & Circuit Layout

• Printed Circuit Board (PCB) Design: Prevents overlapping traces in circuit layouts.
• Chip Manufacturing: Ensures correct wire placements in microchip designs.

17. Open-Source Implementations

Several open-source projects use the Sweep Line Algorithm for different applications. Below are a few:

• CGAL (Computational Geometry Algorithms Library): CGAL provides high-performance computational geometry algorithms, including the Sweep Line method.
• Boost.Geometry: Boost Geometry Library implements sweep line techniques for spatial queries.
• Shapely (Python Library): Shapely is a Python library that uses sweep line methods for geometric processing.
• Mapbox Vector Tiles: Uses a variation of the Sweep Line Algorithm to detect intersections when rendering map layers.

18. Practical Project: Road Intersection Detector

This project uses the Sweep Line Algorithm to detect intersections in road networks.

18.1 Project Overview

We will build a Python script that takes road segments as input and finds intersections using the Sweep Line Algorithm.

18.2 Implementation

import matplotlib.pyplot as plt
from sortedcontainers import SortedList

class Event:
    def __init__(self, x, segment, is_start):
        self.x = x
        self.segment = segment
        self.is_start = is_start

    def __lt__(self, other):
        return self.x < other.x

def do_intersect(seg1, seg2):
    def orientation(p, q, r):
        val = (q[1] - p[1]) * (r[0] - q[0]) - (q[0] - p[0]) * (r[1] - q[1])
        return 0 if val == 0 else (1 if val > 0 else -1)

    p1, q1 = seg1
    p2, q2 = seg2

    o1 = orientation(p1, q1, p2)
    o2 = orientation(p1, q1, q2)
    o3 = orientation(p2, q2, p1)
    o4 = orientation(p2, q2, q1)

    return o1 != o2 and o3 != o4

def sweep_line_intersection(segments):
    events = []
    for seg in segments:
        events.append(Event(seg[0][0], seg, True))
        events.append(Event(seg[1][0], seg, False))

    events.sort()
    active_segments = SortedList(key=lambda seg: seg[0][1])
    intersections = []

    for event in events:
        seg = event.segment
        if event.is_start:
            idx = active_segments.bisect(seg)
            if idx > 0 and do_intersect(active_segments[idx - 1], seg):
                intersections.append((active_segments[idx - 1], seg))
            if idx < len(active_segments) and do_intersect(active_segments[idx], seg):
                intersections.append((active_segments[idx], seg))
            active_segments.add(seg)
        else:
            idx = active_segments.index(seg)
            if 0 < idx < len(active_segments) - 1 and do_intersect(active_segments[idx - 1], active_segments[idx + 1]):
                intersections.append((active_segments[idx - 1], active_segments[idx + 1]))
            active_segments.remove(seg)

    return intersections

# Example road network (segments)
road_segments = [
    ((1, 1), (5, 5)),
    ((2, 5), (6, 1)),
    ((3, 3), (7, 7)),
    ((4, 1), (8, 5))
]

# Detect intersections
intersections = sweep_line_intersection(road_segments)

# Visualization
plt.figure(figsize=(6, 6))
for seg in road_segments:
    plt.plot([seg[0][0], seg[1][0]], [seg[0][1], seg[1][1]], 'b')

# Mark intersections
for intersect in intersections:
    x, y = intersect[0][1]
    plt.scatter(x, y, color='red', zorder=3)

plt.xlabel("X-axis")
plt.ylabel("Y-axis")
plt.title("Road Intersection Detection")
plt.grid(True)
plt.show()

18.3 How It Works

• Defines a set of road segments as input.
• Uses the Sweep Line Algorithm to detect intersections.
• Visualizes the segments and intersections using Matplotlib.

18.4 Potential Enhancements

• Extend it to work with real GIS data using GeoPandas.
• Convert the script into a web API for detecting road congestion.
• Optimize memory usage by using spatial data structures (e.g., R-trees).

19. Competitive Programming & System Design Integration

19.1 Competitive Programming

The Sweep Line Algorithm is commonly used in coding competitions due to its efficiency in handling geometric and interval-based problems. Here are some problem categories where it excels:

• Line Segment Intersection: Finding all intersections among a given set of lines.
• Closest Pair of Points: Optimized approach using sweeping and dynamic sets.
• Rectangle Union Area Calculation: Finding the total area covered by multiple overlapping rectangles.
• Skyline Problem: Determining the visible outline of buildings in a cityscape.
• Event Scheduling: Determining overlaps between scheduled events.
• Active Intervals: Finding the maximum number of overlapping intervals in a given dataset.

19.2 System Design Integration

The Sweep Line Algorithm is used in large-scale systems for efficient geometric computations. Some applications include:

• Geographic Information Systems (GIS): Optimizing map rendering and spatial data analysis.
• Database Query Optimization: Finding overlapping time ranges in time-series data.
• Network Traffic Monitoring: Detecting overlapping network usage intervals.
• 3D Game Engines: Used in rendering pipelines for real-time visibility determination.
• Financial Markets: Analyzing stock market trends for overlapping trading periods.

19.3 Challenges in System Design

• Scalability: Managing a large number of events efficiently.
• Concurrency: Handling multiple event streams in parallel.
• Real-Time Processing: Implementing optimized data structures for fast event handling.
• Storage Optimization: Reducing memory footprint when dealing with large datasets.

20. Assignments

20.1 Solve 10 Problems Using the Sweep Line Algorithm

Complete the following problems to develop proficiency in implementing the Sweep Line Algorithm:

1. Find all intersections among a given set of line segments.
2. Compute the union of multiple rectangles and determine the total covered area.
3. Given a list of time intervals, find the maximum number of overlapping intervals at any given time.
4. Find the skyline silhouette of a city based on building height coordinates.
5. Determine the closest pair of points in a 2D plane efficiently.
6. Find the largest empty rectangle inside a given set of boundaries.
7. Detect road intersections using GIS data.
8. Optimize rendering in a graphics engine by removing hidden surfaces.
9. Analyze stock price fluctuations using an event-based approach.
10. Simulate drone flight paths to avoid mid-air collisions.

20.2 Use the Sweep Line Algorithm in a System Design Problem

Design a system for handling a massive dataset of time-based logs and optimize querying overlapping log entries efficiently.

Requirements:

• Implement an event queue that processes log entries in real time.
• Use a balanced data structure to track active events efficiently.
• Ensure that overlapping log entries are correctly counted and aggregated.
• Optimize memory usage to handle millions of log entries simultaneously.

20.3 Implement the Algorithm Under Time Constraints

Set up a competitive programming environment and aim to solve each of the following problems within 30 minutes:

• Intersection Detection among multiple line segments.
• Active Intervals in scheduling.
• Skyline Problem.
• Closest Pair of Points.
• Dynamic Event Processing for real-time applications.

By completing these assignments, you will develop a strong command over the Sweep Line Algorithm, making you proficient in both competitive programming and system design applications.