Python 3.11+
The examples use the standard Python syntax and containers supported by the browser Judge; the Sweep Line invariant is independent of a minor Python release.
Verify in Python docs(opens in a new tab)Pattern1 Practice1 Interview
Sweep Line
Sort boundary events and process them in order to track active intervals or geometric state. Learn its decision rule, maintained state, Python cost model, and transfer from a focused drill to a full Interview Problem.
Recognize it when
Consider Sweep Line when the prompt's constraints and required operations match this shape: Sort boundary events and process them in order to track active intervals or geometric state.
On this page · Turn Intervals into Ordered Events
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Sweep Line Code Labs
Turn Intervals into Ordered Events
A sweep line replaces continuous intervals with discrete start and end events. Between consecutive coordinates, the active state is constant.
Define Tie Order at Equal Coordinates
Tie order encodes endpoint semantics. For half-open intervals, process ends before starts so intervals touching at one coordinate do not overlap; closed intervals may require the reverse.
Trace Sweep Events
Reference
def active_trace(intervals):
events=[]
for start,end in intervals:events.append((start,1));events.append((end,-1))
active=0;trace=[]
for coordinate,delta in sorted(events):active+=delta;trace.append([coordinate,delta,active])
return tracePractice
Implement active_trace(intervals). Intervals are half-open; process end before start at equal coordinates and return [coordinate,delta,active].
Public tests
- Apply half-open tie order
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Maintain Active State Between Events
Apply each event delta to the active count or data structure, then measure the quantity required by the contract. Group equal coordinates when intermediate same-coordinate states should not be observable.
Find Maximum Overlap
Reference
def maximum_overlap(intervals):
events=[]
for start,end in intervals:events.append((start,1));events.append((end,-1))
active=best=0
for _,delta in sorted(events):active+=delta;best=max(best,active)
return bestPractice
Implement maximum_overlap(intervals). Return the maximum number of active half-open intervals.
Public tests
- Measure active state after each event
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Choose Sweep Line or Interval Merging
Choose merging to compute the union of intervals. Choose a sweep line for concurrency, coverage counts, weighted activity, or geometry events. Sorting events costs O(n log n); the scan is linear unless active state uses a tree or heap.
Explain It in an Interview
Say: “Intervals become ordered boundary events, and active state describes the open span until the next coordinate. This tie order implements half-open semantics.” Explain when the answer is measured and how equal coordinates are grouped.
Python Version Note
Complexity & Invariants
Interview bounds depend on the stated representation, input model, and real Python operations.
| Operation | Average | Worst | Interview note |
|---|---|---|---|
| Sweep Line decision loop | O(n log n) | O(n log n) | Sweep sorted signed boundaries, relying on end-before-start tie order for half-open sessions. |
Space
O(n) for the focused Find Peak Active Sessions implementation.
Assumptions
- The running delta sum equals active sessions after each ordered boundary. Processing -1 before +1 at equal times removes ended sessions before adding new ones, matching the half-open contract.
- The bound includes real Python operations; no repeated slicing, linear list membership, or hidden sorting is treated as free.
Invariants Worth Saying Aloud
- State the precise Sweep Line invariant before coding and preserve it after every update, traversal step, or recursive return.
- Active state equals all events processed at the current sweep position.
- Tie ordering matches whether touching boundaries overlap.
- Convert intervals into signed events. Preserve this property after every transition.
- Process end events before starts at equal timestamps. Preserve this property after every transition.
When To Use Or Avoid Sweep Line
Use It When
- Use Sweep Line when the prompt exposes the ordering, monotonicity, recurrence, or constraint that makes this decision safe.
- Use it when active state equals all events processed at the current sweep position.
Choose Another Tool When
- Avoid Sweep Line when its decision rule cannot eliminate work or preserve the required state.
- Avoid forcing the label onto a prompt whose simpler direct scan already meets the constraints.
Common Pitfalls
These are the mistakes most likely to survive a happy-path example and fail a boundary case.
Moving state without a proof
Applying the pattern without proving its ordering, monotonicity, or window invariant can silently miss valid candidates.
Prevent it: Write the decision rule beside the loop and verify it against Find Peak Active Sessions before optimizing.
Avoid
def peak_active_sessions(sessions):
passUse instead
def peak_active_sessions(sessions):
events = [(time, delta) for start, end in sessions for time, delta in ((start, 1), (end, -1))]
active = peak = 0
for _, delta in sorted(events):
active += delta
peak = max(peak, active)
return peakWhere you will hit this: Find Peak Active Sessions(opens in a new tab)
Breaking the maintained state
Processes starts before ends at equal timestamps and counts touching sessions as overlapping.
Prevent it: Use the public tests and preserve this state: Active state equals all events processed at the current sweep position.
Avoid
def peak_active_sessions(sessions):
events=[]
for s,e in sessions: events += [(s,1),(e,-1)]
active=peak=0
for _,d in sorted(events,key=lambda event:(event[0],-event[1])):
active+=d; peak=max(peak,active)
return peakUse instead
def peak_active_sessions(sessions):
events = [(time, delta) for start, end in sessions for time, delta in ((start, 1), (end, -1))]
active = peak = 0
for _, delta in sorted(events):
active += delta
peak = max(peak, active)
return peakWhere you will hit this: Find Peak Active Sessions(opens in a new tab)
Hiding Python work in the hot path
Python slicing, copying, membership in lists, or repeated sorting inside the loop can invalidate the intended complexity.
Prevent it: Count every slice, copy, sort, membership check, and container update before claiming O(n log n).
Avoid
def peak_active_sessions(sessions):
events=[]
for s,e in sessions: events += [(s,1),(e,-1)]
active=peak=0
for _,d in sorted(events,key=lambda event:(event[0],-event[1])):
active+=d; peak=max(peak,active)
return peakUse instead
def peak_active_sessions(sessions):
events = [(time, delta) for start, end in sessions for time, delta in ((start, 1), (end, -1))]
active = peak = 0
for _, delta in sorted(events):
active += delta
peak = max(peak, active)
return peakWhere you will hit this: First and Last Target Position(opens in a new tab)
Reviewed References
Official Python documentation supports language behavior. Canonical problem pages provide additional practice context.
- Python More on Lists(opens in a new tab)
python-docs · checked 2026-07-27
- LeetCode 56 — Merge Intervals(opens in a new tab)
leetcode · checked 2026-07-26