wqs二分+斜率优化：1019T4 / P9338

本文搬运自本人高中时期CSDN博客，若图片加载不出来，可到原文查看：https://blog.csdn.net/zhangtingxiqwq/article/details/133955885

https://www.luogu.com.cn/problem/P9338

考虑暴力前 ii 个分 jj 段 fi,k=fj−1,k−1+gj,if_{i,k}=f_{j-1,k-1}+g_{j,i} ， O(n3)O(n^3)

然后划分段数，段数显然越多越优，那么就上wqs二分， O(n2log⁡n)O(n^2\log n)

然后我们发现 gg 可以拆，拆完之后拿斜率dp优化即可。 O(nlog⁡n)O(n\log n)

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
#include<bits/stdc++.h>
using namespace std;
#define int long long
inline int read(){int x=0,f=1;char ch=getchar(); while(ch<'0'||
ch>'9'){if(ch=='-')f=-1;ch=getchar();}while(ch>='0'&&ch<='9'){
x=(x<<1)+(x<<3)+(ch^48);ch=getchar();}return x*f;}
#define Z(x) (x)*(x)
#define pb push_back
//mt19937 rand(time(0));
//mt19937_64 rand(time(0));
//srand(time(0));
#define N 2000010
//#define M
//#define mo
void Mn(int &a, int b) {
	a = min(a, b); 
}
struct F {
	int a, t; 
	F operator + (const F &A) const {
		F B; B.a = a + A.a; 
		B.t = t + A.t; 
		return B; 
	}
	F operator + (const int &x) const {
		F B; B.a=a+x; B.t=t; 
		return B; 
	}
	bool operator < (const F &A) const {
		if(a == A.a) return t < A.t; 
		return a < A.a; 
	}
}f[N];
int n, m, i, j, k, T;
int a[N], b[N]; 
int sa[N], sb[N], sum[N]; 
int ida[N], idb[N], h[N]; 
int X[N], Y[N]; 
int l, r, pos, ans, ja, jb; 
int X1, X2, X3, Y1, Y2, Y3; 
char s[N]; 
int q[N]; 

int calc(int x) { //一段的代价为x 
	memset(f, 0x3f, sizeof(f)); 
	auto suan = [&] (int j) -> int {
		return Y[j] - X[j] * i; 
	}; 
	int l, r;  
	l = r = 0; 
	f[0]={0, 0}; q[0]=1;
	X[0]=1; Y[0]=h[1]; 
	for(i=1; i<=n; ++i) {
		while(r-l+1 >= 2 && suan(l) > suan(l+1)) ++l; 
		f[i].a = suan(l); f[i].t = f[q[l]-1].t; 
		f[i] = f[i] + sum[i] + i + x; f[i].t++; 
		while(r-l+1 >= 2) {
			X1 = X[r-1]; Y1 = Y[r-1]; 
			X2 = X[r]; Y2 = Y[r]; 
			X3 = (i+1); Y3 = f[i].a + h[i+1]; 
			if((Y2-Y1)*(X3-X2) > (Y3-Y2)*(X2-X1)) --r; 
			else break; 
		}
		q[++r] = i+1; X[r] = (i+1); Y[r] = f[i].a + h[i+1]; 
	}
	return f[n].t; 
}

signed main()
{
	freopen("easy.in", "r", stdin);
	freopen("easy.out", "w", stdout);
	n=read(); m=read(); ans=1e18; 
	scanf("%s", s+1); 
	for(i=1; i<=2*n; ++i) if(s[i]=='A') a[++ja]=i, sa[i]=1; 
	else b[++jb]=i, sb[i]=1; 
	partial_sum(sa+1, sa+2*n+1, sa+1); 
	partial_sum(sb+1, sb+2*n+1, sb+1); 
	for(i=1; i<=n; ++i) ida[i] = sb[a[i]]; //第i个a前面有多少个b
	for(i=1; i<=n; ++i) idb[i] = sa[b[i]]; 
	for(i=1; i<=n; ++i) idb[i] = max(idb[i], i - 1); 
	partial_sum(ida+1, ida+n+1, sum+1); 
	for(i=1; i<=n; ++i) h[i] = idb[i]*(i-1)-sum[idb[i]]; 
	l=0; r=1e12; //一段的代价
	while(l<r)  {
		int mid = (l+r)>>1; 
		if(calc(mid)<=m) r=mid; 
		else l=mid+1; 
	}
	calc(l); 
	printf("%lld", f[n].a - m * l); 
	return 0;
}