9.3 Line Segment Intersection

Line segment intersection is a crucial problem in computational geometry. The Bentley-Ottmann algorithm uses a sweep line technique to efficiently detect intersections among multiple line segments, employing an event queue and status structure to process events.

The algorithm's time complexity is O((n + k) log n), where n is the number of input segments and k is the number of intersections. This makes it more efficient than naive approaches, especially for sparse arrangements with few intersections.

Sweep Line Algorithm

Sweep Line Technique and Event Queue

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• Bentley-Ottmann algorithm employs sweep line technique to efficiently detect intersections among multiple line segments
• Sweep line consists of an imaginary vertical line moving from left to right across the plane
• Event queue maintains ordered list of upcoming events (segment endpoints and intersections)
• Events in queue sorted by x-coordinate, then y-coordinate for ties
• Two main event types processed during sweep:
  • Segment endpoint events (left and right endpoints)
  • Intersection events (points where two segments cross)
• Algorithm processes events in order, updating data structures and reporting intersections
• Sweep line status tracks currently active line segments intersecting the sweep line
• Active segments sorted by y-coordinate of their intersection with sweep line

Status Structure and Intersection Detection

• Status structure typically implemented as balanced binary search tree (red-black tree)
• Red-black tree allows efficient insertion, deletion, and neighbor finding operations
• Key operations performed on status structure:
  • Insert new segment when left endpoint encountered
  • Remove segment when right endpoint reached
  • Find neighbors of newly inserted or intersecting segments
  • Swap positions of intersecting segments in the structure
• Intersection detection occurs by checking adjacent segments in status structure
• When intersection found, new event added to event queue for future processing
• Algorithm continues until all events in queue processed

Intersection Detection

Intersection Point Calculation and Reporting

• Intersection point calculated using line equations of two segments
• General form of line equation: $ax + by + c = 0$
• Intersection point $(x, y)$ found by solving system of two line equations
• Coordinates of intersection point computed as:
  • $x = \frac{b_1c_2 - b_2c_1}{a_1b_2 - a_2b_1}$
  • $y = \frac{a_2c_1 - a_1c_2}{a_1b_2 - a_2b_1}$
• Algorithm reports intersection point when detected during sweep
• Intersection points stored or output depending on specific implementation requirements

Red-Black Tree for Efficient Status Management

• Red-black tree self-balancing binary search tree used for status structure
• Key properties of red-black tree:
  • Every node colored either red or black
  • Root and leaves (NIL) always black
  • Red node cannot have red child
  • Every path from root to leaf contains same number of black nodes
• These properties ensure tree remains balanced, guaranteeing $O(log n)$ time for basic operations
• Red-black tree operations used in sweep line algorithm:
  • Insertion: add new segment to status structure
  • Deletion: remove segment when right endpoint reached
  • Search: find neighbors of newly inserted or intersecting segments
  • Predecessor/Successor: identify adjacent segments for intersection checks

Algorithm Analysis

Time and Space Complexity

• Time complexity of Bentley-Ottmann algorithm: $O((n + k) log n)$
  • $n$: number of input line segments
  • $k$: number of intersection points
• Breakdown of time complexity:
  • Sorting initial endpoints: $O(n log n)$
  • Processing $2n + k$ events: $O((n + k) log n)$
  • Each event involves $O(log n)$ operations on red-black tree
• Space complexity: $O(n)$ for storing segments and maintaining data structures
• Comparison with naive approach:
  • Naive algorithm checks all pairs of segments: $O(n^2)$ time
  • Bentley-Ottmann more efficient for sparse arrangements with few intersections

Performance Considerations and Optimizations

• Algorithm performs best when number of intersections $(k)$ small compared to $n^2$
• Degeneracies require special handling:
  • Multiple segments intersecting at single point
  • Overlapping collinear segments
• Numerical precision issues may arise in floating-point calculations
• Potential optimizations:
  • Use of more efficient data structures for specific input distributions
  • Parallel processing of non-overlapping regions of input
  • Approximate algorithms for scenarios where exact results not required