The Skyline Problem
Problem Description
A city's skyline is the outer contour of the silhouette formed by all the buildings in that city when viewed from a distance. Given the locations and heights of all the buildings, return the skyline formed by these buildings collectively.
The geometric information of each building is given in the array buildings where buildings[i] = [lefti, righti, heighti]:
- lefti is the x coordinate of the left edge of the ith building.
- righti is the x coordinate of the right edge of the ith building.
- heighti is the height of the ith building.
You may assume all buildings are perfect rectangles grounded on an absolutely flat surface at height 0.
The skyline should be represented as a list of "key points" sorted by their x-coordinate in the form [[x1,y1],[x2,y2],...]. Each key point is the left endpoint of some horizontal segment in the skyline except the last point in the list, which always has a y-coordinate 0 and is used to mark the skyline's termination where the rightmost building ends. Any ground between the leftmost and rightmost buildings should be part of the skyline's contour.
Note: There must be no consecutive horizontal lines of equal height in the output skyline. For instance, [...,[2 3],[4 5],[7 5],[11 5],[12 7],...] is not acceptable; the three lines of height 5 should be merged into one in the final output as such: [...,[2 3],[4 5],[12 7],...]
Example 1:
Input: buildings = [[2,9,10],[3,7,15],[5,12,12],[15,20,10],[19,24,8]]
Output: [[2,10],[3,15],[7,12],[12,0],[15,10],[20,8],[24,0]]
Explanation:
Figure A shows the buildings of the input.
Figure B shows the skyline formed by those buildings. The red points in figure B represent the key points in the output list.
Example 2:
Input: buildings = [[0,2,3],[2,5,3]]
Output: [[0,3],[5,0]]
Constraints:
- 1 <= buildings.length <= 104
- 0 <= lefti < righti <= 231 - 1
- 1 <= heighti <= 231 - 1
- buildings is sorted by lefti in non-decreasing order.
Solve it on LeetCode
Approaches
1. Sorted Edges + Priority Queue
Intuition:
The Skyline problem is essentially about managing the heights of buildings as you move from left to right across the plane. The key idea is to focus on the "critical points" where the height changes. These critical points are either the start or end of a building. By sorting these events and using a priority queue to track current buildings, we can determine the skyline at each step.
Steps:
- Extract critical points from the input buildings. Each building provides two points: its starting event and its end event. The start of a building contributes a positive height and the end contributes a negative height.
- Sort the events:
- By the x-coordinate (starting position).
- In case of ties, prioritize by height. If it's a start point, sort from high to low; if it's an end point, sort from low to high.
- Use a max-heap (priority queue) to keep track of all active building heights. The maximum height in the heap represents the current tallest building that influences the skyline.
- Iterate through the sorted events:
- When adding a start event, add its height to the heap.
- When processing an end event, remove its height from the heap.
- Check if the current max height (top of the heap) changes before and after processing the event. If it does, add the new key point to the result.
- Return the list of key points as the skyline.
Code:
- Time Complexity: O(n log n), where n is the number of critical points. Sorting the events takes O(n log n), and each insertion and deletion from the heap takes O(log n).
- Space Complexity: O(n), as we store up to n events and heights in the heap and list.