Atcoder Grand Contest 014F - Strange Sorting

Problem Statement

Takahashi loves sorting.

He has a permutation (p1,p2,…,pN) of the integers from 1 through N. Now, he will repeat the following operation until the permutation becomes (1,2,…,N):

• First, we will define high and low elements in the permutation, as follows. The i-th element in the permutation is high if the maximum element between the 1-st and i-th elements, inclusive, is the i-th element itself, and otherwise the i-th element is low.
• Then, let a1,a2,…,ak be the values of the high elements, and b1,b2,…,bN−k be the values of the low elements in the current permutation, in the order they appear in it.
• Lastly, rearrange the permutation into (b1,b2,…,bN−k,a1,a2,…,ak).

How many operations are necessary until the permutation is sorted?

Constraints

• 1≤N≤2×105
• (p1,p2,…,pN) is a permutation of the integers from 1 through N.

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Input

Input is given from Standard Input in the following format:

N
p1 p2 ... pN

Output

Print the number of operations that are necessary until the permutation is sorted.

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Sample Input 1

Copy

5
3 5 1 2 4

Sample Output 1

Copy

3

The initial permutation is (3,5,1,2,4), and each operation changes it as follows:

• In the first operation, the 1-st and 2-nd elements are high, and the 3-rd, 4-th and 5-th are low. The permutation becomes: (1,2,4,3,5).
• In the second operation, the 1-st, 2-nd, 3-rd and 5-th elements are high, and the 4-th is low. The permutation becomes: (3,1,2,4,5).
• In the third operation, the 1-st, 4-th and 5-th elements are high, and the 2-nd and 3-rd and 5-th are low. The permutation becomes: (1,2,3,4,5).

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Sample Input 2

Copy

5
5 4 3 2 1

Sample Output 2

Copy

4

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Sample Input 3

Copy

10
2 10 5 7 3 6 4 9 8 1

Sample Output 3

Copy

6

题意：给一个排列，

 每次把max(a[1] to a[i]) = a[i]的放进vector A（记为high）里，其他放进vector B（记为low）里
然后把序列改成B + A
问升序排序需要搞多少次

题解：

我基本就是在翻译题解...

考虑去掉1剩下2到n的序列，剩下的答案是T，1并不影响其他数的操作
那么最后的答案一定是T或者T + 1
如果T = 0，直接判
如果T > 0，
考虑T - 1次操作之后的情况
记f为这个序列的第一项，显然f > 2
(如果f = 2，那么要么这个序列已经被排好序或者下一次f会被移到后面)
如果最后是(f, 1, 2)答案是T否则是T + 1

结论1：
(考虑序列2到n)
在前T - 1次， f不会出现它在不第一个位置且它又是high的情况
证明1：
考虑某个时刻x是high但是不在第一个位置， 显然第一个位置是high， 故vectorA中x前面还有一个数，记为y， y < x
如果x, y都是high或者都是low，它们的相对位置不变
如果x, y是high而y变成low了，这时y仍然在x左边，如果x不在第一个位置，和之前一样了

结论2：
定义循环序列(a, b, c) = (b, c, a) = (c, a, b)
则1, 2, f在前T - 1次组成的(关于它们位置的)循环序列不会变
证明2：
如果1是第一个元素
{
如果2是第二个元素，那么1, 2是high， f是low
如果f是第二个元素，那么1, f是high， 2是low
否则2, f都是low
}
如果2是第一个元素，那么2是high，1和f是low
如果f是第一个元素，那么f是high，1和2是low
否则1, 2, f都是low
然后就证明完了

#include <bits/stdc++.h>
#define xx first
#define yy second
#define mp make_pair
#define pb push_back
#define fill( x, y ) memset( x, y, sizeof x )
#define copy( x, y ) memcpy( x, y, sizeof x )
using namespace std;

typedef long long LL;
typedef pair < int, int > pa;

inline int read()
{
	int sc = 0, f = 1; char ch = getchar();
	while( ch < '0' || ch > '9' ) { if( ch == '-' ) f = -1; ch = getchar(); }
	while( ch >= '0' && ch <= '9' ) sc = sc * 10 + ch - '0', ch = getchar();
	return sc * f;
}

const int MAXN = 200020;

int q[MAXN], p[MAXN], n, T[MAXN], f[MAXN];

int main()
{
#ifdef wxh010910
	freopen( "data.in", "r", stdin );
#endif
	n = read();
	for( int i = 1 ; i <= n ; i++ ) q[ p[ i ] = read() ] = i;
	for( int i = n - 1 ; i ; i-- )
	{
		if( !T[ i + 1 ] )
		{
			if( q[ i ] > q[ i + 1 ] ) T[ i ] = 1, f[ i ] = i + 1;
			else T[ i ] = 0;
		}
		else
		{
			int cnt = 0;
			cnt += q[ f[ i + 1 ] ] < q[ i ];
			cnt += q[ i ] < q[ i + 1 ];
			cnt += q[ i + 1 ] < q[ f[ i + 1 ] ];
			if( cnt == 2 ) T[ i ] = T[ i + 1 ], f[ i ] = f[ i + 1 ];
			else T[ i ] = T[ i + 1 ] + 1, f[ i ] = i + 1;
		}
	}
	return printf( "%d\n", T[ 1 ] ), 0;
}