CodeForces - 311B:Cats Transport (DP+斜率优化)

Posted 2020-11-20 hua-dong

Zxr960115 is owner of a large farm. He feeds m cute cats and employs p feeders. There‘s a straight road across the farm and n hills along the road, numbered from 1 to n from left to right. The distance between hill i and (i - 1) is d_i meters. The feeders live in hill 1.

One day, the cats went out to play. Cat i went on a trip to hill h_i, finished its trip at time t_i, and then waited at hill h_i for a feeder. The feeders must take all the cats. Each feeder goes straightly from hill 1 to n without waiting at a hill and takes all the waiting cats at each hill away. Feeders walk at a speed of 1 meter per unit time and are strong enough to take as many cats as they want.

For example, suppose we have two hills (d₂ = 1) and one cat that finished its trip at time 3 at hill 2 (h₁ = 2). Then if the feeder leaves hill 1 at time 2 or at time 3, he can take this cat, but if he leaves hill 1 at time 1 he can‘t take it. If the feeder leaves hill 1 at time 2, the cat waits him for 0 time units, if the feeder leaves hill 1 at time 3, the cat waits him for 1 time units.

Your task is to schedule the time leaving from hill 1 for each feeder so that the sum of the waiting time of all cats is minimized.

Input

The first line of the input contains three integers n, m, p (2 ≤ n ≤ 10⁵, 1 ≤ m ≤ 10⁵, 1 ≤ p ≤ 100).

The second line contains n - 1 positive integers d₂, d₃, ..., d_n (1 ≤ d_i < 10⁴).

Each of the next m lines contains two integers h_i and t_i (1 ≤ h_i ≤ n, 0 ≤ t_i ≤ 10⁹).

Output

Output an integer, the minimum sum of waiting time of all cats.

Please, do not write the %lld specifier to read or write 64-bit integers in С++. It is preferred to use the cin, cout streams or the %I64d specifier.

Examples

Input

4 6 2
1 3 5
1 0
2 1
4 9
1 10
2 10
3 12

Output

3

题意：有一些猫，放在一些位置，人走到每个猫的时间已知，给个猫出现的时间已知，假设派出一个人，可以自由安排其出发时间，沿途已经出现的猫pick掉，猫等待的时间是被pick的时间减去出现的时间t，t>=0。现在有P个人，问总时间T最小是多少。

思路：对猫: 人time+猫dis-猫time。把c[i]-t[i]排序，那么就成为了把M个数划分位P个区间，每个区间的值=所有数与最大数的差值。

      DP[i][j]=min DP[k][j-1]+c[i]*(i-k)-(sum[i]-sum[k]); 

      转化：B=-c[i]*k+(dp[k][j-1]+sum[k])+c[i]*i-sum[i];

      方程的斜率为k=c[i]；y= (dp[k][j-1]+sum[k]) ；截距B=DP[i][j]；常数C=c[i]*i-sum[i];

#include<bits/stdc++.h>
#define ll long long
using namespace std;
const int maxn=100010;
ll d[maxn],c[maxn],sum[maxn],dp[maxn][101],t;
int q[maxn],head,tail;
ll getans(int i,int j,int k){ return dp[k][j-1]+c[i]*(i-k)-(sum[i]-sum[k]); }
ll Y(int k,int j){ return dp[k][j-1]+sum[k]; }
int main()
{
    int N,M,P,i,j,h;
    scanf("%d%d%d",&N,&M,&P);
    for(i=2;i<=N;i++) scanf("%I64d",&d[i]),d[i]+=d[i-1];
    for(i=1;i<=M;i++){
        scanf("%d%I64d",&h,&t);
        c[i]=t-d[h];
    }
    sort(c+1,c+M+1);
    for(i=1;i<=M;i++) sum[i]=sum[i-1]+c[i];
    for(i=1;i<=M;i++) dp[i][1]=c[i]*(i-1)-sum[i-1];
    for(j=2;j<=P;j++){
        head=tail=0;
        for(i=1;i<=M;i++){
            while(tail>head&&Y(q[head+1],j)-Y(q[head],j)<c[i]*(q[head+1]-q[head])) head++;
            dp[i][j]=getans(i,j,q[head]);
            while(tail>head&&(Y(i,j)-Y(q[tail],j))*(q[tail]-q[tail-1])<(Y(q[tail],j)-Y(q[tail-1],j))*(i-q[tail])) tail--;
            q[++tail]=i;
        }
    }
    printf("%I64d
",dp[M][P]);
    return 0;
} 

 经验：弹出队首时，可以直接通过比较结果获得。

#include<bits/stdc++.h>
#define ll long long
using namespace std;
const int maxn=100011;
ll d[maxn],c[maxn],sum[maxn],dp[maxn][101],t;
int q[maxn],head,tail;
ll getans(int i,int j,int k){ return dp[k][j-1]+c[i]*(i-k)-(sum[i]-sum[k]); }
ll Y(int k,int j){ return dp[k][j-1]+sum[k]; }
int main()
{
    int N,M,P,i,j,h;
    scanf("%d%d%d",&N,&M,&P);
    for(i=2;i<=N;i++) scanf("%I64d",&d[i]),d[i]+=d[i-1];
    for(i=1;i<=M;i++){
        scanf("%d%I64d",&h,&t);
        c[i]=t-d[h];
    }
    sort(c+1,c+M+1);
    for(i=1;i<=M;i++) sum[i]=sum[i-1]+c[i];
    for(i=1;i<=M;i++) dp[i][1]=c[i]*(i-1)-sum[i-1];
    for(j=2;j<=P;j++){
        head=tail=0;
        for(i=1;i<=M;i++){
            while(tail>head&&getans(i,j,q[head])>getans(i,j,q[head+1])) head++;
            dp[i][j]=getans(i,j,q[head]);
            while(tail>head&&(Y(i,j)-Y(q[tail],j))*(q[tail]-q[tail-1])<(Y(q[tail],j)-Y(q[tail-1],j))*(i-q[tail])) tail--;
            q[++tail]=i; //队首可以getans维护，队尾不行，必须维护斜率！ 
        }
    }
    printf("%I64d
",dp[M][P]);
    return 0;
} 

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