【BZOJ】【1911】【APIO2010】特别行动队commando

DP/斜率优化

　　嗯……第三道斜率优化的题目了。

　　定义 $s[i]=sum_{k=1}^{i} x[k] $

　　方程：$f[i]=max{ f[j]+a*(s[i]-s[j])^2+b*(s[i]-s[j])+c } $

　　对于 $ j > k $

　　若决策 j 比 k 更优：[ egin{aligned} {f[j] + a*(s[i]-s[j])^2+b*(s[i]-s[j])+c} &> {f[k]+a*(s[i]-s[k])^2+b*(s[i]-s[k])+c} \ {f[j]-f[k]+a*(s[j]^2-s[k]^2)-b*(s[j]-s[k])} &> {2a*s[i]*(s[j]-s[k])} \ frac{ f[j]-f[k]+a*(s[j]^2-s[k]^2)-b*(s[j]-s[k]) } { 2a*(s[k]-s[j]) } &> {s[i]} end{aligned} ]

 1 /**************************************************************
 2     Problem: 1911
 3     User: Tunix
 4     Language: C++
 5     Result: Accepted
 6     Time:1260 ms
 7     Memory:28616 kb
 8 ****************************************************************/
 9  
10 //BZOJ 1911
11 #include<cmath>
12 #include<vector>
13 #include<cstdio>
14 #include<cstring>
15 #include<cstdlib>
16 #include<iostream>
17 #include<algorithm>
18 #define rep(i,n) for(int i=0;i<n;++i)
19 #define F(i,j,n) for(int i=j;i<=n;++i)
20 #define D(i,j,n) for(int i=j;i>=n;--i)
21 #define pb push_back
22 using namespace std;
23 int getint(){
24     int v=0,sign=1; char ch=getchar();
25     while(ch<'0'||ch>'9'){ if (ch=='-') sign=-1; ch=getchar();}
26     while(ch>='0'&&ch<='9'){ v=v*10+ch-'0'; ch=getchar();}
27     return v*=sign;
28 }
29 const int N=1e6+10;
30 typedef long long LL;
31 /******************tamplate*********************/
32 LL f[N],x[N],s[N];
33 int q[N],a,b,c;
34 double slop(int k,int j){
35     return double(f[j]-f[k]+a*(s[j]*s[j]-s[k]*s[k])+b*(s[k]-s[j]))/
36         double(2*a*(s[j]-s[k]));
37 }
38 int main(){
39 #ifndef ONLINE_JUDGE
40     freopen("1911.in","r",stdin);
41     freopen("1911.out","w",stdout);
42 #endif
43     int n=getint();
44     a=getint(),b=getint(),c=getint();
45     F(i,1,n){ x[i]=getint(); s[i]=s[i-1]+x[i];}
46     int l=0,r=0;
47     F(i,1,n){
48         while(l<r && slop(q[l],q[l+1])<s[i])l++;
49         int t=q[l];
50         f[i]=f[t]+a*((s[i]-s[t])*(s[i]-s[t]))+b*(s[i]-s[t])+c;
51         while(l<r && slop(q[r-1],q[r])>slop(q[r],i))r--;
52         q[++r]=i;
53     }
54     printf("%lld
",f[n]);
55     return 0;
56 }

View Code

1911: [Apio2010]特别行动队

Time Limit: 4 Sec Memory Limit: 64 MB
Submit: 3036 Solved: 1367
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Description

Input

Output

Sample Input

4
-1 10 -20
2 2 3 4

Sample Output

HINT

Source

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