1539C - Stable Groups.cpp

/*
There are n students numerated from 1 to n. The level of the i-th student is ai. You need to split the students into stable groups. A group of students is called stable, if in the sorted array of their levels no two neighboring elements differ by more than x.

For example, if x=4, then the group with levels [1,10,8,4,4] is stable (because 4-1=x, 4-4=x, 8-4=x, 10-8=x), while the group with levels [2,10,10,7] is not stable (7-2=5>x).

Apart from the n given students, teachers can invite at most k additional students with arbitrary levels (at teachers' choice). Find the minimum number of stable groups teachers can form from all students (including the newly invited).

For example, if there are two students with levels 1 and 5; x=2; and k=1, then you can invite a new student with level 3 and put all the students in one stable group.

Input
The first line contains three integers n, k, x (1=n=200000, 0=k=1018, 1=x=1018) — the initial number of students, the number of students you can additionally invite, and the maximum allowed level difference.

The second line contains n integers a1,a2,…,an (1=ai=1018) — the students levels.

Output
In the only line print a single integer: the minimum number of stable groups you can split the students into.

Examples
inputCopy
8 2 3
1 1 5 8 12 13 20 22
outputCopy
2
inputCopy
13 0 37
20 20 80 70 70 70 420 5 1 5 1 60 90
outputCopy
3
Note
In the first example you can invite two students with levels 2 and 11. Then you can split the students into two stable groups:

[1,1,2,5,8,11,12,13],
[20,22].
In the second example you are not allowed to invite new students, so you need 3 groups:

[1,1,5,5,20,20]
[60,70,70,70,80,90]
[420]
*/




There are n students numerated from 1 to n. The level of the i-th student is ai. You need to split the students into stable groups. A group of students is called stable, if in the sorted array of their levels no two neighboring elements differ by more than x.

For example, if x=4, then the group with levels [1,10,8,4,4] is stable (because 4-1=x, 4-4=x, 8-4=x, 10-8=x), while the group with levels [2,10,10,7] is not stable (7-2=5>x).

Apart from the n given students, teachers can invite at most k additional students with arbitrary levels (at teachers' choice). Find the minimum number of stable groups teachers can form from all students (including the newly invited).

For example, if there are two students with levels 1 and 5; x=2; and k=1, then you can invite a new student with level 3 and put all the students in one stable group.

Input
The first line contains three integers n, k, x (1=n=200000, 0=k=1018, 1=x=1018) — the initial number of students, the number of students you can additionally invite, and the maximum allowed level difference.

The second line contains n integers a1,a2,…,an (1=ai=1018) — the students levels.

Output
In the only line print a single integer: the minimum number of stable groups you can split the students into.

Examples
inputCopy
8 2 3
1 1 5 8 12 13 20 22
outputCopy
2
inputCopy
13 0 37
20 20 80 70 70 70 420 5 1 5 1 60 90
outputCopy
3
Note
In the first example you can invite two students with levels 2 and 11. Then you can split the students into two stable groups:

[1,1,2,5,8,11,12,13],
[20,22].
In the second example you are not allowed to invite new students, so you need 3 groups:

[1,1,5,5,20,20]
[60,70,70,70,80,90]
[420]





#include<bits/stdc++.h>
using namespace std;
 
void solve() {
	long long n,  k, x;
	cin >> n >> k >> x;
	
	vector <long long> v(n), b;
	
	for (auto &x: v) cin >> x;
	
	sort(v.begin(), v.end());
	
	int groups = 1; // atleast one
	
	for (int i = 1; i < n; i++) {
		if (v[i] - v[i - 1] <= x) continue;
		else {
			long long diff = v[i] - v[i - 1] - 1;
			b.push_back(diff / x);
			groups++;
		}
	}
	
	sort(b.begin(), b.end());
	
	for (auto &x: b) {
		if (x <= k) {
			k -= x;
			groups--;
		}
		if (k == 0) break;
	}
	
	cout << groups;
}
 
int main(){
    ios_base::sync_with_stdio(false);
	cin.tie(NULL);
	
	int t = 1;
	//cin >> t;
	
	while (t--)  solve();
	
    return 0;
}