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 (1.15.1) (judge verdict: Accepted)

fn main() {
  let (n, m, _): (usize, usize, usize) = get::read_3tuple();
  
  let rs: Vec<usize> = get::read_vec();
  let rs = rs.iter().map(|r|r-1).collect::<Vec<_>>();
  
  let edges: Vec<(usize, usize, isize)> = get::read_3tuples(m);
  let edges = edges.iter().map(|&(a,b,c)|(a-1,b-1,c)).collect::<Vec<_>>();
  
  let g = Graph::new_labeled(n, &edges);
  let dist = g.warshall_floyd();
  
  let ans =
    permutations(&rs).
    iter().
    map(|r| r.windows(2).fold(0, |a,w| a + dist[w[0]][w[1]])).
    min().
    unwrap();
  
  println!("{}", ans);

}

fn permutations<T: Clone + std::fmt::Debug>(vec: &Vec<T>) -> Vec<Vec<T>> {
  fn go<T: Clone + std::fmt::Debug>(vec: &Vec<T>, n: usize, m: usize, k: usize, buf: &mut Vec<usize>, acc: &mut Vec<Vec<T>>) {
    if k == 0 {
      let r: Vec<T> = buf.iter().map(|&i| vec[i-1].clone()).collect();
      acc.push(r);
    } else {
      for x in n .. m + 1 {
        if !buf.contains(&x) {
          buf.push(x);
          go(vec, n, m, k-1, buf, acc);
          buf.pop();
        }
      }
    }
  }

  let mut acc = vec![];
  go(vec, 1, vec.len(), vec.len(), &mut vec![], &mut acc);
  acc
}

struct Graph {
  size: usize,
  adj: Vec<Vec<(usize,isize)>>
}

impl Clone for Graph {
  fn clone(&self) -> Graph {
    Graph {
      size: self.size,
      adj: self.adj.clone()
    }
  }
}

#[allow(dead_code)]
impl Graph {
  fn new_nonlabeled(n: usize, edges: &[(usize,usize)]) -> Graph {
    let mut g: Vec<Vec<(usize,isize)>> = vec![vec![]; n];
    for &(a, b) in edges {
      g[a].push((b,0));
      g[b].push((a,0));  // delete for digraph
    }
    Graph {size: n, adj: g}
  }

  fn new_labeled(n: usize, edges: &[(usize,usize,isize)]) -> Graph {
    let mut g: Vec<Vec<(usize,isize)>> = vec![vec![]; n];
    for &(a, b, c) in edges {
      g[a].push((b,c));
      g[b].push((a,c));  // delete for digraph
    }
    Graph {size: n, adj: g}
  }

  fn is_connected(&self) -> bool {
    self.dfs(0).len() == self.size
  }

  fn size(&self) -> usize {
    self.size
  }

  fn warshall_floyd(&self) -> Vec<Vec<isize>> {
    let inf = isize::max_value() >> 1;
    let mut wf: Vec<Vec<isize>> = vec![vec![inf; self.size]; self.size];
    for i in 0 .. self.size {wf[i][i] = 0}
    for (i, next) in self.adj.iter().enumerate() {
      for &(j, x) in next {
        wf[i][j] = x
      }
    }
    for k in 0 .. self.size {
      for i in 0 .. self.size {
        for j in 0 .. self.size {
          wf[i][j] = std::cmp::min(wf[i][j], wf[i][k] + wf[k][j]);
        }
      }
    }
    wf
  }

  fn coloring2(&self) -> Option<(Vec<usize>, Vec<usize>)> {
    fn paint(v: usize, p: bool, g: &Graph, cv: &mut Vec<Option<bool>>) -> bool {
      match cv[v] {
        None => {
          let next = &g.adj[v];
          cv[v] = Some(p);
          next.iter().all(|&(w,_)|paint(w,!p,g,cv))},
        Some(q) => {q == p}
      }
    }

    let mut canvas: Vec<Option<bool>> = vec![None; self.size];
    let ans = paint(0, false, self, &mut canvas);
    let bs = canvas.iter().enumerate().filter(|&(_,&v)|v==Some(false)).map(|(i,_)|i).collect::<Vec<_>>();
    let ws = canvas.iter().enumerate().filter(|&(_,&v)|v==Some(true)).map(|(i,_)|i).collect::<Vec<_>>();
    if ans {Some((bs,ws))} else {None}
  }

  fn dfs(&self, s: usize) -> Vec<usize> {
    fn go(g: &Graph, current: usize, mut path: &mut Vec<usize>, mut visited: &mut Vec<bool>) {
      for &(next, _) in &g.adj[current] {
        if visited[next] {
          continue
        } else {
          visited[next] = true;
          path.push(next);
          go(&g, next, &mut path, &mut visited)
        }
      }
    }
    let mut path = vec![s];
    let mut visited = vec![false; self.adj.len()];
    visited[s] = true;
    go(&self, s, &mut path, &mut visited);
    path
  }
  
  fn cut(&mut self, v: usize, w: usize) {
    self.adj[v].retain(|&(t,_)| t != w);
    self.adj[w].retain(|&(t,_)| t != v);
  }
}

#[allow(dead_code)]
mod get {
  use std::io::*;
  use std::str::*;

  pub fn read<T: FromStr>() -> T {
    let mut buf = String::new();
    stdin().read_line(&mut buf).ok();
    buf.trim().parse::<T>().ok().unwrap()
  }

  pub fn reads<T: FromStr>(n: usize) -> Vec<T> {
    let mut vec: Vec<T> = vec![];
    for _ in 0 .. n {
      vec.push(read());
    }
    vec
  }

  pub fn read_tuple<T1: FromStr, T2: FromStr>() -> (T1, T2) {
    let mut buf = String::new();
    stdin().read_line(&mut buf).ok();
    let mut it = buf.trim().split_whitespace();
    let x = it.next().unwrap().parse::<T1>().ok().unwrap();
    let y = it.next().unwrap().parse::<T2>().ok().unwrap();
    (x, y)
  }

  pub fn read_tuples<T1: FromStr, T2: FromStr>(n: usize) -> Vec<(T1, T2)> {
    let mut vec: Vec<(T1, T2)> = vec![];
    for _ in 0 .. n {
      vec.push(read_tuple());
    }
    vec
  }

  pub fn read_3tuple<T1: FromStr, T2: FromStr, T3: FromStr>() -> (T1, T2, T3) {
    let mut buf = String::new();
    stdin().read_line(&mut buf).ok();
    let mut it = buf.trim().split_whitespace();
    let x = it.next().unwrap().parse::<T1>().ok().unwrap();
    let y = it.next().unwrap().parse::<T2>().ok().unwrap();
    let z = it.next().unwrap().parse::<T3>().ok().unwrap();
    (x, y, z)
  }

  pub fn read_3tuples<T1: FromStr, T2: FromStr, T3: FromStr>(n: usize) -> Vec<(T1, T2, T3)> {
    let mut vec: Vec<(T1, T2, T3)> = vec![];
    for _ in 0 .. n {
      vec.push(read_3tuple());
    }
    vec
  }

  pub fn read_vec<T: FromStr>() -> Vec<T> {
    let mut buf = String::new();
    stdin().read_line(&mut buf).ok();
    buf.trim().split_whitespace().map(|t| t.parse::<T>().ok().unwrap()).collect()
  }

  pub fn read_mat<T: FromStr>(h: usize) -> Vec<Vec<T>> {
    let mut mat: Vec<Vec<T>> = vec![];
    for _ in 0 .. h {
      mat.push(read_vec());
    }
    mat
  }
}

#[allow(dead_code)]
mod put {
  use std::string::*;

  pub fn print_vec<T: ToString>(vec: &Vec<T>, sep: &str) {
    let out = vec.iter().map(|e| e.to_string()).collect::<Vec<_>>().as_slice().join(sep);
    println!("{}", out);
  }

  pub fn print_mat<T: ToString>(mat: &Vec<Vec<T>>, sep: &str) {
    for v in mat {
      print_vec(v, sep);
    }
  }
}