UVA - 1025 A Spy in the Metro[DP DAG]

UVA - 1025

A Spy in the Metro

Secret agent Maria was sent to Algorithms City to carry out an especially dangerous mission. After several thrilling events we find her in the first station of Algorithms City Metro, examining the time table. The Algorithms City Metro consists of a single line with trains running both ways, so its time table is not complicated.

Maria has an appointment with a local spy at the last station of Algorithms City Metro. Maria knows that a powerful organization is after her. She also knows that while waiting at a station, she is at great risk of being caught. To hide in a running train is much safer, so she decides to stay in running trains as much as possible, even if this means traveling backward and forward. Maria needs to know a schedule with minimal waiting time at the stations that gets her to the last station in time for her appointment. You must write a program that finds the total waiting time in a best schedule for Maria.

The Algorithms City Metro system has N stations, consecutively numbered from 1 to N. Trains move in both directions: from the first station to the last station and from the last station back to the first station. The time required for a train to travel between two consecutive stations is fixed since all trains move at the same speed. Trains make a very short stop at each station, which you can ignore for simplicity. Since she is a very fast agent, Maria can always change trains at a station even if the trains involved stop in that station at the same time.

Input

The input file contains several test cases. Each test case consists of seven lines with information as follows.

Line 1. The integer N (2 ≤ N ≤ 50), which is the number of stations.

Line 2. The integer T (0 ≤ T ≤ 200), which is the time of the appointment.

Line 3. N − 1 integers: t1, t2, . . . , tN−1 (1 ≤ ti ≤ 20), representing the travel times for the trains between two consecutive stations: t1 represents the travel time between the first two stations, t2 the time between the second and the third station, and so on.

Line 4. The integer M1 (1 ≤ M1 ≤ 50), representing the number of trains departing from the first station.

Line 5. M1 integers: d1, d2, . . . , dM1 (0 ≤ di ≤ 250 and di < di+1), representing the times at which trains depart from the first station.

Line 6. The integer M2 (1 ≤ M2 ≤ 50), representing the number of trains departing from the N-th station.

Line 7. M2 integers: e1, e2, . . . , eM2 (0 ≤ ei ≤ 250 and ei < ei+1) representing the times at which trains depart from the N-th station.

The last case is followed by a line containing a single zero.

Output

For each test case, print a line containing the case number (starting with 1) and an integer representing the total waiting time in the stations for a best schedule, or the word ‘impossible’ in case Maria is unable to make the appointment. Use the format of the sample output.

Sample Input

4
55
5 10 15
4
0 5 10 20
4
0 5 10 15
4
18
123
5
0 3 6 10 12
6
0 3 5 7 12 15
2
30
20
1
20
7
1 3 5 7 11 13 17 0

Sample Output

Case Number 1: 5
Case Number 2: 0
Case Number 3: impossible

---

白书
f[i][j]表示在i号时间为j 
预处理has[i][j][0/1]实现O(1)转移

//
//  main.cpp
//  uva1025
//
//  Created by Candy on 10/18/16.
//  Copyright © 2016 Candy. All rights reserved.
//

#include <iostream>
#include <cstdio>
#include <cstring>
#include <algorithm>
using namespace std;
typedef long long ll;
const int N=55,M=205,INF=1e9;
inline int read(){
    char c=getchar();int x=0,f=1;
    while(c<'0'||c>'9'){if(c=='-')f=-1;c=getchar();}
    while(c>='0'&&c<='9'){x=x*10+c-'0';c=getchar();}
    return x*f;
}
int n,m,t[N],a;
int f[N][M],has[N][M][2];
void dp(){
    f[1][0]=0;
    for(int i=2;i<=n;i++) f[i][0]=INF;
    for(int j=1;j<=m;j++)
        for(int i=1;i<=n;i++){
            f[i][j]=f[i][j-1]+1;
            if(i!=1&&j-t[i-1]>=0&&has[i-1][j-t[i-1]][0])
                f[i][j]=min(f[i][j],f[i-1][j-t[i-1]]);
            if(i!=n&&j-t[i]>=0&&has[i+1][j-t[i]][1])
                f[i][j]=min(f[i][j],f[i+1][j-t[i]]);
            //printf("f %d %d %d\n",i,j,f[i][j]);
        }
}
int main(int argc, const char * argv[]){
    int cas=0;
    while((n=read())){cas++;
        memset(has,0,sizeof(has));
        m=read();
        for(int i=1;i<=n-1;i++) t[i]=read();
        int tmp=read();
        while(tmp--){
            a=read();
            has[1][a][0]=1;
            for(int i=2;i<=n;i++){
                a+=t[i-1];
                if(a>m) break;
                has[i][a][0]=1;
            }
        }
        tmp=read();
        while(tmp--){
            a=read();
            has[n][a][1]=1;
            for(int i=n-1;i>=1;i--){
                a+=t[i];
                if(a>m) break;
                has[i][a][1]=1;
            }
        }
        dp();
        if(f[n][m]<INF) printf("Case Number %d: %d\n",cas,f[n][m]);
        else printf("Case Number %d: impossible\n",cas);
    }
    return 0;
}