CDQ partition sets slope optimization

CDQ partition sets slope optimization

There are some special dynamic programming problem, seemingly be turned into \ (\ frac {f_j-f_k } {g_j-g_k} <K_i \) slope inequality, but in fact, \ (K_i \) and \ (G_J \) and \ (G_k \) but not monotonic, it can not be monotonic queue / stack implementation binary search monotonous slope optimization. Thus, there will be huge guy came up CDQ partition sets the slope of the optimization.

How CDQ?

According to the sub-state \ (K \) from small to large, partition, before recursion, the sequence index according to the original state is divided into two sub-sub-sequences, sequences before the internal order of sorting remains unchanged;

Recursive sequence of length to \ (1 \) , the optimization of the same construction as an ordinary slope point;

When back, the state of the left insertion sequence monotone queue, the same as ordinary optimization of the slope of the right in a state update sequence.

In this case, when the recursion to the bottom, the current state of the sub-index has been all the original state before their updated.

Example: [SDOI2012] task scheduling

Code:

#include<iostream>
#include<cstring>
#include<cstdio>
#include<algorithm>
#include<vector>
#include<queue>
#define int long long
#define maxn 300000
#define INF 0x3f3f3f3f3f3f3f3f
using namespace std;
void read(int &x){
    int f=1;x=0;char ch=getchar();
    while(!isdigit(ch)){if(ch=='-') f=-1;ch=getchar();}
    while(isdigit(ch)){x=x*10+ch-'0';ch=getchar();}
    x*=f;
}
int A[maxn+5],B[maxn+5],F[maxn+5];
struct vec{
    int i,x,y;vec(){i=x=y=0;}
    vec(int x,int y){i=0,this->x=x,this->y=y;}
    vec(int i,int x,int y){this->i=i,this->x=x,this->y=y;}
    friend vec operator-(vec a,vec b){return vec(a.x-b.x,a.y-b.y);}
    friend bool operator<(vec a,vec b){if(a.x==0) return 0;return 1.0*a.y/a.x<1.0*b.y/b.x;}
} s[maxn+5],q[maxn+5],t[maxn+5];
int head=1,tail=0;
void push(vec x){
    while(tail>head&&(x-q[tail-1])<(q[tail]-q[tail-1])) tail--;
    q[++tail]=x;
}
void pop(int k){
    while(tail>head&&(q[head+1]-q[head])<vec(1,k)) head++;
}
int n,ss;
void CDQ(int left,int right){
    if(left==right){s[left].x=B[left],s[left].y=F[left];return;}
    int mid=(left+right)/2,x1=left-1,x2=mid;
    for(int i=left;i<=right;i++)
        s[i].i<=mid?t[++x1]=s[i]:t[++x2]=s[i];
    for(int i=left;i<=right;i++) s[i]=t[i];
    CDQ(left,mid),head=1,tail=0;
    for(int i=left;i<=mid;i++) push(s[i]);
    for(int i=mid+1;i<=right;i++){
        pop(ss+A[s[i].i]);
        int j=q[head].i;
        F[s[i].i]=min(F[s[i].i],F[j]+ss*(B[n]-B[j])+A[s[i].i]*(B[s[i].i]-B[j]));
    }
    CDQ(mid+1,right),x1=left,x2=mid+1;
    for(int i=left;i<=right;i++)
        if(x1<=mid&&(x2>right||s[x1].x<s[x2].x||(s[x1].x==s[x2].x&&s[x1].y<s[x2].y))) t[i]=s[x1++];
        else t[i]=s[x2++];
    for(int i=left;i<=right;i++) s[i]=t[i];
}
bool cmp(vec a,vec b){return A[a.i]<A[b.i];}
#undef int
int main(){
#define int long long
    read(n),read(ss);
    for(int i=1;i<=n;i++) read(A[i]),read(B[i]),A[i]+=A[i-1],B[i]+=B[i-1],s[i].i=i;
    memset(F,0x3f,sizeof(F));
    F[0]=0,sort(s+1,s+n+1,cmp);
    CDQ(0,n);
    printf("%lld\n",F[n]);
}