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 Rust (1.42.0) (judge verdict: Accepted)

use proconio::{fastout, input, marker::Usize1};
use std::cmp::min;

/// Permute a slice into its next or previous permutation (in lexical order).
///
/// ```
/// use permutohedron::LexicalPermutation;
///
/// let mut data = [1, 2, 3];
/// let mut permutations = Vec::new();
///
/// loop {
///     permutations.push(data.to_vec());
///     if !data.next_permutation() {
///         break;
///     }
/// }
///
/// assert_eq!(permutations, &[&[1, 2, 3], &[1, 3, 2],
///                            &[2, 1, 3], &[2, 3, 1],
///                            &[3, 1, 2], &[3, 2, 1]]);
///
/// // `data` has been mutated in-place:
/// assert_eq!(data, [3, 2, 1]);
/// ```
pub trait LexicalPermutation {
    /// Return `true` if the slice was permuted, `false` if it is already
    /// at the last ordered permutation.
    fn next_permutation(&mut self) -> bool;
    /// Return `true` if the slice was permuted, `false` if it is already
    /// at the first ordered permutation.
    fn prev_permutation(&mut self) -> bool;
}

impl<T> LexicalPermutation for [T]
where
    T: Ord,
{
    /// Original author in Rust: Thomas Backman <serenity@exscape.org>
    fn next_permutation(&mut self) -> bool {
        // These cases only have 1 permutation each, so we can't do anything.
        if self.len() < 2 {
            return false;
        }

        // Step 1: Identify the longest, rightmost weakly decreasing part of the vector
        let mut i = self.len() - 1;
        while i > 0 && self[i - 1] >= self[i] {
            i -= 1;
        }

        // If that is the entire vector, this is the last-ordered permutation.
        if i == 0 {
            return false;
        }

        // Step 2: Find the rightmost element larger than the pivot (i-1)
        let mut j = self.len() - 1;
        while j >= i && self[j] <= self[i - 1] {
            j -= 1;
        }

        // Step 3: Swap that element with the pivot
        self.swap(j, i - 1);

        // Step 4: Reverse the (previously) weakly decreasing part
        self[i..].reverse();

        true
    }

    fn prev_permutation(&mut self) -> bool {
        // These cases only have 1 permutation each, so we can't do anything.
        if self.len() < 2 {
            return false;
        }

        // Step 1: Identify the longest, rightmost weakly increasing part of the vector
        let mut i = self.len() - 1;
        while i > 0 && self[i - 1] <= self[i] {
            i -= 1;
        }

        // If that is the entire vector, this is the first-ordered permutation.
        if i == 0 {
            return false;
        }

        // Step 2: Reverse the weakly increasing part
        self[i..].reverse();

        // Step 3: Find the rightmost element equal to or bigger than the pivot (i-1)
        let mut j = self.len() - 1;
        while j >= i && self[j - 1] < self[i - 1] {
            j -= 1;
        }

        // Step 4: Swap that element with the pivot
        self.swap(i - 1, j);

        true
    }
}

#[fastout]
fn main() {
    input! {
        n: usize,
        m: usize,
        r: usize,
        mut togo: [Usize1; r],
        abc: [(Usize1, Usize1, usize); m],
    }

    let INF = 1_000_000_000_000_000usize;
    let mut dist = vec![vec![INF; n]; n];
    for i in 0..n {
        dist[i][i] = 0;
    }
    for i in 0..abc.len() {
        dist[abc[i].0][abc[i].1] = min(abc[i].2, dist[abc[i].0][abc[i].1]);
        dist[abc[i].1][abc[i].0] = min(abc[i].2, dist[abc[i].1][abc[i].0]);
    }

    // Warshall Floyd
    for k in 0..n {
        for i in 0..n {
            for j in 0..n {
                dist[i][j] = min(dist[i][j], dist[i][k] + dist[k][j]);
            }
        }
    }

    // println!("{:?}", dist);

    togo.sort();
    let mut res = INF;
    let mut temp = 0;
    for i in 1..togo.len() {
        temp += dist[togo[i]][togo[i - 1]];
    }
    res = min(res, temp);
    while togo.next_permutation() {
        temp = 0;
        for i in 1..togo.len() {
            temp += dist[togo[i]][togo[i - 1]];
        }
        res = min(res, temp);
        // println!("Ord: {:?}, t: {}", togo, temp);
    }

    println!("{}", res);
}