SLIDE 1
Problem Dicusssion
Part 2: Some more problems
Lucca Siaudzionis and Jack Spalding-Jamieson 2020/03/19
University of British Columbia
Announcements
- A4 is now due Wednesday March 25th.
1
Problem Dicusssion Part 2: Some more problems Lucca Siaudzionis and - - PowerPoint PPT Presentation
Problem Dicusssion Part 2: Some more problems Lucca Siaudzionis and - - PowerPoint PPT Presentation
Problem Dicusssion Part 2: Some more problems Lucca Siaudzionis and Jack Spalding-Jamieson 2020/03/19 University of British Columbia Announcements A4 is now due Wednesday March 25th. 1 Maximum Area Rectangle Find the maximum area of a
SLIDE 2
SLIDE 3
Maximum Area Rectangle
Find the maximum area of a rectangle that can fit in a histogram.
1 3 6 2 3 5 7 4 1 3
The histogram has 1 ≤ n ≤ 5 · 106 different entries of non-negative integers. Try it yourself: https://leetcode.com/problems/largest-rectangle-in-histogram/
2
SLIDE 4
Maximum Area Rectangle - General Ideas
This problem can be solved with a segment tree in Θ(n log n) time (a good exercise), but we can do better. We will find all the maximal rectangles, i.e. the rectangles that cannot be expanded any more within the bounds of the histogram. 1 3 6 2 3 5 7 4 1 3 1 3 6 2 3 5 7 4 1 3
Figure 1: Two different rectangles. One is maximal (left), the other is not (right). The right rectangle could be expanded to the left.
3
SLIDE 5
Maximum Area Rectangle - Line Sweep
To find all these maximal rectangles, we’re going to do a line sweep. As we move our line, we’re going to keep track of all the maximal rectangles that touch the sweep line, sorted by their height. Note that since the sweep line constrains the rectangles, these rectangles may not be maximal in the entire problem. Let’s look at an example.
4
SLIDE 6
Maximum Area Rectangle - Line Sweep Demo (1)
1 3 6 2 3 5 7 4 1 3
5
SLIDE 7
Maximum Area Rectangle - Line Sweep Demo (2)
1 3 6 2 3 5 7 4 1 3
6
SLIDE 8
Maximum Area Rectangle - Line Sweep Demo (3)
1 3 6 2 3 5 7 4 1 3
7
SLIDE 9
Maximum Area Rectangle - Line Sweep Demo (4)
1 3 6 2 3 5 7 4 1 3
8
SLIDE 10
Maximum Area Rectangle - Line Sweep Demo (5)
1 3 6 2 3 5 7 4 1 3
9
SLIDE 11
Maximum Area Rectangle - Line Sweep Demo (6)
1 3 6 2 3 5 7 4 1 3
10
SLIDE 12
Maximum Area Rectangle - Line Sweep Demo (7)
1 3 6 2 3 5 7 4 1 3
11
SLIDE 13
Maximum Area Rectangle - Line Sweep Demo (8)
1 3 6 2 3 5 7 4 1 3
12
SLIDE 14
Maximum Area Rectangle - Line Sweep Demo (9)
1 3 6 2 3 5 7 4 1 3
13
SLIDE 15
Maximum Area Rectangle - Line Sweep Demo (10)
1 3 6 2 3 5 7 4 1 3
14
SLIDE 16
Maximum Area Rectangle - Line Sweep Demo (11)
1 3 6 2 3 5 7 4 1 3
15
SLIDE 17
Maximum Area Rectangle - Line Sweep Demo (12)
1 3 6 2 3 5 7 4 1 3
16
SLIDE 18
Maximum Area Rectangle - Line Sweep Demo (13)
1 3 6 2 3 5 7 4 1 3
17
SLIDE 19
Maximum Area Rectangle - Line Sweep Demo (14)
1 3 6 2 3 5 7 4 1 3
18
SLIDE 20
Maximum Area Rectangle - Line Sweep Demo (15)
1 3 6 2 3 5 7 4 1 3
19
SLIDE 21
Maximum Area Rectangle - Line Sweep Demo (16)
1 3 6 2 3 5 7 4 1 3
20
SLIDE 22
Maximum Area Rectangle - Line Sweep Demo (17)
1 3 6 2 3 5 7 4 1 3
21
SLIDE 23
Maximum Area Rectangle - Line Sweep Demo (18)
1 3 6 2 3 5 7 4 1 3
22
SLIDE 24
Maximum Area Rectangle - Line Sweep Demo (19)
1 3 6 2 3 5 7 4 1 3
23
SLIDE 25
Maximum Area Rectangle - Line Sweep Demo (20)
1 3 6 2 3 5 7 4 1 3
24
SLIDE 26
Maximum Area Rectangle - Line Sweep Demo (21)
1 3 6 2 3 5 7 4 1 3
25
SLIDE 27
Maximum Area Rectangle - Line Sweep Demo (22)
1 3 6 2 3 5 7 4 1 3
26
SLIDE 28
Maximum Area Rectangle - Line Sweep Demo (23)
1 3 6 2 3 5 7 4 1 3
27
SLIDE 29
Maximum Area Rectangle - Line Sweep Demo (24)
1 3 6 2 3 5 7 4 1 3
28
SLIDE 30
Maximum Area Rectangle - Line Sweep
When we advance the line, we need to destroy some of the rectangles that can no longer touch the line. Observe that the set of rectangles we destroy corresponds to exactly those with a height greater than the next entry in the histogram. For implementation, we can store our current set of rectangles with a stack!. Some quick observations about this method: The current set of rectangles stored in the stack at any given time is decreasing in both leftmost x coordinate, and height (as we delete things with larger height, they also get wider).
29
SLIDE 31
Maximum Area Rectangle - Implementation (C++)
1
struct bar { int x, y; bar(int x, int y) : x(x), y(y) {} };
2
int largestRectangleArea(vector<int>& heights) {
3
stack<bar> mc; mc.emplace(-1,-1); int best = 0;
4
// sweep line x coordinate
5
int x; for (x = 0; x < heights.size(); ++x) {
6
int& h = heights[x]; // height of next item
7
// look at and delete all the rectangles too tall to touch the new sweep
8
while (mc.size() > 1 && mc.top().y > h) {
9
bar cur = mc.top(); mc.pop();
10
best = max(best, (x-mc.top().x-1)*cur.y); }
11
mc.emplace(x,h); }
12
while (mc.size() > 1) { // look at the rectangles that touch the very end
13
bar cur = mc.top(); mc.pop();
14
best = max(best, (x-mc.top().x-1)*cur.y); }
15
return best;
16
}
30