17.9 Sweep Line Algorithms
Many geometric problems involve detecting interactions among large collections of geometric objects.
17.9 Sweep Line Algorithms
Problem
Many geometric problems involve detecting interactions among large collections of geometric objects.
Examples include:
- Finding all segment intersections
- Computing rectangle union area
- Detecting overlapping intervals
- Processing geographic boundaries
- Building visibility graphs
- Event scheduling and timeline analysis
A straightforward approach often compares every object with every other object.
For example, checking whether any two of n line segments intersect requires:
O(n²)
pairwise comparisons.
This becomes prohibitively expensive as datasets grow.
Sweep line algorithms reduce these problems by exploiting geometric order.
Solution
Imagine a vertical line moving from left to right across the plane.
Instead of examining all geometric objects simultaneously, process them in the order they are encountered by the moving line.
|
|
|
------|-------> sweep direction
|
|
|
At any moment, the algorithm only needs to reason about objects currently intersecting the sweep line.
This transforms a two-dimensional problem into a sequence of one-dimensional updates.
The technique typically consists of:
- An event queue
- A status structure
- Event-processing rules
The sweep line processes events in sorted order while maintaining an active set of objects.
A Simple Example: Overlapping Intervals
Consider intervals:
[1,5]
[3,8]
[7,10]
Represent each endpoint as an event:
(1, start)
(5, end)
(3, start)
(8, end)
(7, start)
(10, end)
Sort all events:
1 start
3 start
5 end
7 start
8 end
10 end
Now scan from left to right.
Maintain:
active = 0
Processing:
1 start -> active = 1
3 start -> active = 2
5 end -> active = 1
7 start -> active = 2
8 end -> active = 1
10 end -> active = 0
Whenever:
active > 1
an overlap exists.
The sweep line converts a potentially quadratic comparison problem into a sorting problem.
Components of a Sweep Line
Most sweep-line algorithms share the same structure.
Event Queue
Stores future events.
Typical events:
segment start
segment end
intersection
rectangle boundary
Events are usually sorted by:
x-coordinate
A priority queue is often used:
import heapq
events = []
heapq.heappush(events, event)
The next event is always processed first.
Status Structure
Stores currently active objects.
Examples:
active intervals
active segments
active rectangles
Balanced trees are commonly used because they support:
insert
delete
neighbor lookup
in:
O(log n)
time.
Event Processing
Each event updates the status structure.
For example:
segment start
adds a segment.
segment end
removes a segment.
The algorithm checks only nearby objects rather than the entire dataset.
Segment Intersection Problem
Suppose we have many segments:
\ /
\ /
\ /
\/
The naive solution checks every pair.
For:
n segments
that requires:
O(n²)
comparisons.
The sweep-line approach is dramatically different.
As the sweep line moves:
- insert segments when their left endpoint is encountered
- remove segments when their right endpoint is encountered
- check only neighboring segments in the active set
A crucial observation is that only neighboring active segments can generate new intersections.
This reduces the search space enormously.
The Bentley-Ottmann Algorithm
The classic sweep-line solution for segment intersections is the Bentley-Ottmann algorithm.
Named after entity["people","Jon Bentley","computer scientist"] and entity["people","Thomas Ottmann","computer scientist"], it reports all intersections among line segments.
The algorithm maintains:
event queue
active segment tree
Events include:
segment start
segment end
segment intersection
Whenever neighboring segments exchange order, a new intersection event is generated.
The running time is:
O((n + k) log n)
where:
n = number of segments
k = number of intersections
This is significantly better than:
O(n²)
when the number of actual intersections is small.
Rectangle Union Area
Consider axis-aligned rectangles:
+-----+
| |
| |
+-----+
The goal is to compute total covered area.
A naive approach might discretize the plane or examine every pair of rectangles.
The sweep-line method processes:
left boundary
right boundary
events.
At each vertical position:
covered_y_length
×
horizontal_distance
contributes to the total area.
The active set stores intervals on the y-axis.
This transforms a difficult two-dimensional area problem into a sequence of interval-union computations.
Closest Event Scheduling
Sweep-line ideas appear outside geometry.
Suppose you have:
meeting intervals
resource reservations
network allocations
Sorting start and end events produces a timeline sweep.
The active set contains currently active intervals.
Questions such as:
maximum overlap
resource utilization
peak concurrency
can be answered efficiently.
This is effectively a one-dimensional sweep-line algorithm.
Event Ordering Matters
Consider:
segment end
segment start
occurring at the same coordinate.
Should the end be processed first?
Or the start?
Different choices produce different results.
Many bugs arise from inconsistent tie-breaking.
A typical ordering might be:
intersection
start
end
or:
start
intersection
end
depending on the problem.
The ordering must be defined explicitly.
Example: Maximum Overlap
Given intervals:
[1,6]
[3,7]
[5,10]
Events:
1 start
3 start
5 start
6 end
7 end
10 end
Processing:
active = 1
active = 2
active = 3
active = 2
active = 1
active = 0
Maximum overlap:
3
No interval pair comparisons were required.
Correctness Argument
The sweep-line method works because geometric relationships change only at discrete event locations.
Between events:
nothing important changes
The status structure therefore contains sufficient information to describe the current state of the problem.
By processing all events in sorted order and maintaining the correct invariant, the algorithm never misses a relevant interaction.
This is the central proof strategy for sweep-line algorithms.
Complexity Analysis
Let:
n = number of objects
Sorting events:
O(n log n)
Each event typically performs:
insert
delete
query
operations costing:
O(log n)
Thus many sweep-line algorithms run in:
O(n log n)
or:
O((n + k) log n)
where:
k
represents discovered intersections or reported outputs.
Common Pitfalls
Incorrect Event Ordering
Events occurring at the same coordinate require a deterministic tie-breaking rule.
Forgetting Active Structure Updates
Failing to remove objects when they leave the sweep line causes incorrect results.
Checking Too Many Neighbors
Many sweep-line optimizations rely on checking only adjacent active objects.
Examining all active objects destroys performance.
Numerical Precision
Segment intersection sweeps often depend on orientation tests and geometric comparisons.
Floating-point arithmetic can cause incorrect ordering.
Ignoring Degenerate Cases
Examples include:
overlapping segments
shared endpoints
duplicate events
These frequently require special handling.
Discussion
Sweep-line algorithms are among the most powerful design techniques in computational geometry. They replace global pairwise comparisons with a local view centered on a moving frontier. By maintaining only active objects and processing changes as discrete events, many seemingly quadratic problems become nearly linear after sorting.
The pattern appears repeatedly across geometry, databases, scheduling systems, graphics engines, and geographic information systems. Once you recognize that a problem evolves continuously but changes only at a finite set of event locations, a sweep-line solution often emerges naturally. The next section applies this philosophy to a practical geometric problem: computing the union area of rectangles efficiently.