Problem:
Score : 400 points

Problem Statement

There are N towns in the State of Atcoder, connected by M bidirectional roads.

The i-th road connects Town A_i and B_i and has a length of C_i.

Joisino is visiting R towns in the state, r_1,r_2,..,r_R (not necessarily in this order).

She will fly to the first town she visits, and fly back from the last town she visits, but for the rest of the trip she will have to travel by road.

If she visits the towns in the order that minimizes the distance traveled by road, what will that distance be?

Constraints

2≤N≤200

1≤M≤N×(N-1)/2

2≤R≤min(8,N) (min(8,N) is the smaller of 8 and N.)

r_i≠r_j (i≠j)

1≤A_i,B_i≤N, A_i≠B_i

(A_i,B_i)≠(A_j,B_j),(A_i,B_i)≠(B_j,A_j) (i≠j)

1≤C_i≤100000

Every town can be reached from every town by road.

All input values are integers.

Input

Input is given from Standard Input in the following format:

N M R
r_1 ... r_R
A_1 B_1 C_1
:
A_M B_M C_M

Output

Print the distance traveled by road if Joisino visits the towns in the order that minimizes it.

Sample Input 1

3 3 3
1 2 3
1 2 1
2 3 1
3 1 4

Sample Output 1

2

For example, if she visits the towns in the order of 1, 2, 3, the distance traveled will be 2, which is the minimum possible.

Sample Input 2

3 3 2
1 3
2 3 2
1 3 6
1 2 2

Sample Output 2

4

The shortest distance between Towns 1 and 3 is 4. Thus, whether she visits Town 1 or 3 first, the distance traveled will be 4.

Sample Input 3

4 6 3
2 3 4
1 2 4
2 3 3
4 3 1
1 4 1
4 2 2
3 1 6

Sample Output 3

3

Solution in C (GCC 5.4.1) (judge verdict: Accepted)

#include <stdio.h>
#include <string.h>
#include <stdlib.h>
#include <stdint.h> // uint64_t

#define max(a,b) ((a) > (b) ? (a) : (b))
#define min(a,b) ((a) > (b) ? (b) : (a))
#define TOWNS_MAX 8
#define VERTEX_MAX 200
#define INF 0x1f1f1f1f
#define BUF_SIZE (TOWNS_MAX*4+50)

int get_int3(int *a1, int *a2, int *a3) {
#ifdef BUF_SIZE
  char line[BUF_SIZE];
  if(!fgets(line, BUF_SIZE, stdin)) return -1;
  sscanf(line, "%d %d %d", a1, a2, a3);
#else
#error
#endif
  return 0;
}


// <arr[0]> <arr[1]> .. <arr[size-1]>
int fget_array(int *arr, int size) {
#ifdef BUF_SIZE
    char line[BUF_SIZE];
    char *tmpbuf = line;
    int i;
    if(!fgets(line, BUF_SIZE, stdin)) return -1;
    for(i = 0; i < size; i++) {
        char *tmp = strtok(tmpbuf, " ");
        arr[i] = strtol(tmp, NULL, 10);
        tmpbuf = NULL;
    }
#else
#error
#endif
    return 0;
}

void swap(int *a, int *b) {
    int tmp = *a;
    *a = *b;
    *b = tmp;
}

void reversed(int *arr, int size) {
    int i;
    // no need to reverse operation
    if(size == 0 || size == 1) return;
    static int brr[TOWNS_MAX];
    for(i = 0; i < size; i++) {
        brr[size-i-1] = arr[i];
    }
    for(i = 0; i < size; i++) {
        arr[i] = brr[i];
    }
    return;
}

int next_permutation(int *arr, int size) {
    if(size == 0 || size == 1) return 0;
    int pos;
    int i;
    int cur = arr[size-1];
    for(pos = size-2; pos >= 0; pos--) {
        if(cur > arr[pos]) break;
        cur = arr[pos];
    }
    if(pos==-1) {
        // arr is the last permutation
        for(i = 0; i < size; i++) arr[i] = i;
        return 0;
    }
    for(i = size-1; i > pos; i--) {
        if(arr[pos] < arr[i]) break;
    }

    // swap pos <-> i
    swap(&arr[pos], &arr[i]);
    reversed(&arr[pos+1], size-1-pos);
    return 1;
}

int main(void) {
    int vs, es, ts;
    get_int3(&vs, &es, &ts);
    static int arr[TOWNS_MAX];
    int i, j, k;
    fget_array(arr, ts);
    // 1-indexed -> 0-indexed
    for(i = 0; i < vs; i++) arr[i]--;

    static int graph[VERTEX_MAX][VERTEX_MAX];
    memset(graph, 0x1f, sizeof(int)*VERTEX_MAX*VERTEX_MAX);
    for(i = 0; i < vs; i++) graph[i][i] = 0;

    for(i = 0; i < es; i++) {
        int v1, v2, dist;
        get_int3(&v1, &v2, &dist);
        // 1-indexed -> 0-indexed
        v1--; v2--;
        graph[v1][v2] = graph[v2][v1] = dist;
    }

    // warshall floyd
    for(k = 0; k < vs; k++) {
        for(i = 0; i < vs; i++) {
            for(j = 0; j < vs; j++) {
                int new = graph[i][k] + graph[k][j];
                if(graph[i][j] > new) {
                    graph[i][j] = new;
                }
            }
        }
    }

    // permutation
    static int perm[TOWNS_MAX];
    for(i = 0; i < ts; i++) perm[i] = i;

    int ans = INF; // minimum
    do {
#ifdef DEBUG
        printf("perm: ");
        for(i = 0; i < ts; i++) {
            printf("%d ", arr[perm[i]]);
        }
        putchar('\n');
#endif
        int res = 0;
        int cur = arr[perm[0]];
        for(i = 1; i < ts; i++) {
            int next = arr[perm[i]];
            res += graph[cur][next];
            cur = next;
        }
#ifdef DEBUG
        printf("res: %d\n", res);
#endif
        ans = min(ans, res);
    } while(next_permutation(perm, ts));

    printf("%d\n", ans);
    return 0;
}