Algorithm
You are given n segments on a number line; each endpoint of every segment has integer coordinates. Some segments can degenerate to points. Segments can intersect with each other, be nested in each other or even coincide.
The intersection of a sequence of segments is such a maximal set of points (not necesserily having integer coordinates) that each point lies within every segment from the sequence. If the resulting set isn't empty, then it always forms some continuous segment. The length of the intersection is the length of the resulting segment or 0 in case the intersection is an empty set.
For example, the intersection of segments [1;5] and [3;10] is [3;5] (length 2), the intersection of segments [1;5] and [5;7] is [5;5] (length 0) and the intersection of segments [1;5] and [6;6] is an empty set (length 0).
Your task is to remove exactly one segment from the given sequence in such a way that the intersection of the remaining (n−1) segments has the maximal possible length.
The first line contains a single integer n (2≤n≤3⋅105) — the number of segments in the sequence.
Each of the next n lines contains two integers li and ri (0≤li≤ri≤109) — the description of the i-th segment.
Print a single integer — the maximal possible length of the intersection of (n−1) remaining segments after you remove exactly one segment from the sequence.
4
1 3
2 6
0 4
3 3
1
5
2 6
1 3
0 4
1 20
0 4
2
3
4 5
1 2
9 20
0
2
3 10
1 5
7
In the first example you should remove the segment [3;3], the intersection will become [2;3] (length 1). Removing any other segment will result in the intersection [3;3] (length 0).
In the second example you should remove the segment [1;3] or segment [2;6], the intersection will become [2;4] (length 2) or [1;3] (length 2), respectively. Removing any other segment will result in the intersection [2;3] (length 1).
In the third example the intersection will become an empty set no matter the segment you remove.
In the fourth example you will get the intersection [3;10] (length 7) if you remove the segment [1;5] or the intersection [1;5] (length 4) if you remove the segment [3;10].
Code Examples
#1 Code Example with C++ Programming
Code -
C++ Programming
#include <bits/stdc++.h>
using namespace std;
int const N = 3e5 + 1;
int n, a[N], b[N], aa[N], bb[N], cs[2 * N + 100];
vector<int> all;
int main() {
scanf("%d", &n);
for(int i = 0; i < n; ++i) {
scanf("%d %d", a + i, b + i);
all.push_back(a[i]);
all.push_back(b[i]);
}
if(n == 2) {
printf("%d\n", max(b[0] - a[0], b[1] - a[1]));
return 0;
}
sort(all.begin(), all.end());
all.resize(unique(all.begin(), all.end()) - all.begin());
for(int i = 0; i < n; ++i) {
aa[i] = lower_bound(all.begin(), all.end(), a[i]) - all.begin();
bb[i] = lower_bound(all.begin(), all.end(), b[i]) - all.begin();
}
for(int i = 0; i < n; ++i) {
cs[aa[i]] += 1;
cs[bb[i] + 1] += -1;
}
int cur = 0;
for(int i = 0; i < 2 * N + 100; ++i) {
cur += cs[i];
cs[i] = cur;
}
int mx = -1;
for(int i = 0; i < 2 * N + 100; ++i)
mx = max(mx, cs[i]);
if(mx < n - 1) {
puts("0");
return 0;
}
int l = a[0], r = b[0], ex[2] = {0, 0};
for(int i = 1; i < n; ++i) {
if(a[i] >= l && a[i] <= r)
l = a[i], ex[0] = i;
if(b[i] >= l && b[i] <= r)
r = b[i], ex[1] = i;
}
int res = -1;
for(int j = 0, l, r; j < 2; ++j) {
if(ex[j] == 0)
l = a[1], r = b[1];
else
l = a[0], r = b[0];
for(int i = 0; i < n; ++i) {
if(i == ex[j])
continue;
if(a[i] >= l && a[i] <= r)
l = a[i];
if(b[i] >= l && b[i] <= r)
r = b[i];
}
res = max(res, r - l);
}
printf("%d\n", res);
return 0;
}Input
1 3
2 6
0 4
3 3
Output
Demonstration
Codeforcess Solution-C. Maximal Intersection-Solution in C, C++, Java, Python,Codeforcess Solution,Maximal Intersection