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 Go (1.14.1) (judge verdict: Wrong Answer)

package main

import (
	"bufio"
	"fmt"
	"math"
	"os"
)

var mod = 1000000007

func main() {
	r := bufio.NewReader(os.Stdin)
	var n, m, R int
	fmt.Fscan(r, &n)
	fmt.Fscan(r, &m)
	fmt.Fscan(r, &R)

	rs := make([]int, R)
	for i := 0; i < R; i++ {
		fmt.Fscan(r, &rs[i])
		rs[i]--
	}

	costs := make([][]int, n)
	for i := 0; i < n; i++ {
		costs[i] = make([]int, n)
		for j := 0; j < n; j++ {
			costs[i][j] = math.MaxInt64
		}
		costs[i][i] = 0
	}
	var a, b, c int
	for i := 0; i < m; i++ {
		fmt.Fscan(r, &a)
		fmt.Fscan(r, &b)
		fmt.Fscan(r, &c)
		a--
		b--
		costs[a][b] = c
		costs[b][a] = c
	}

	// ワーシャルフロイド法で最短経路を算出
	for k := 0; k < n; k++ {
		for i := 0; i < n; i++ {
			for j := 0; j < n; j++ {
				costs[i][j] = min(costs[i][j], costs[i][k]+costs[k][j])
			}
		}
	}

	// 順列で訪問先の順序をすべて試す
	// これでは重複して訪問する町がでてしまうか。。
	perms := permutations(rs)
	ans := math.MaxInt64
	for _, perm := range perms {
		tmp := 0
		for j := 1; j < len(perm); j++ {
			from := perm[j-1]
			to := perm[j]
			tmp += costs[from][to]
		}
		ans = min(ans, tmp)
	}
	fmt.Println(ans)
}

// permutations
func permutations(arr []int) [][]int {
	var helper func([]int, int)
	res := [][]int{}

	helper = func(arr []int, n int) {
		if n == 1 {
			tmp := make([]int, len(arr))
			copy(tmp, arr)
			res = append(res, tmp)
		} else {
			for i := 0; i < n; i++ {
				helper(arr, n-1)
				if n%2 == 1 {
					tmp := arr[i]
					arr[i] = arr[n-1]
					arr[n-1] = tmp
				} else {
					tmp := arr[0]
					arr[0] = arr[n-1]
					arr[n-1] = tmp
				}
			}
		}
	}
	helper(arr, len(arr))
	return res
}

// Union-Find
type unionFind struct {
	d []int
}

func newUnionFind(n int) *unionFind {
	uf := new(unionFind)
	uf.d = make([]int, n)
	for i := 0; i < n; i++ {
		uf.d[i] = -1
	}
	return uf
}

func (uf *unionFind) find(x int) int {
	if uf.d[x] < 0 {
		return x
	}
	uf.d[x] = uf.find(uf.d[x])
	return uf.d[x]
}

func (uf *unionFind) unite(x, y int) bool {
	rootX := uf.find(x)
	rootY := uf.find(y)
	if rootX == rootY {
		return false
	}

	if uf.d[rootX] < uf.d[rootY] {
		uf.d[rootX] += uf.d[rootY]
		uf.d[rootY] = rootX
	} else {
		uf.d[rootY] += uf.d[rootX]
		uf.d[rootX] = rootY
	}

	return true
}

func (uf *unionFind) same(x, y int) bool {
	return uf.find(x) == uf.find(y)
}

func (uf *unionFind) size(x int) int {
	return -uf.d[uf.find(x)]
}

// mod
func modpow(a, n, mod int) int {
	res := 1
	for n > 0 {
		if n%2 != 0 {
			res = res * a % mod
		}
		a = a * a % mod
		n = n / 2
	}
	return res
}

func modinv(n, mod int) int {
	return modpow(n, mod-2, mod)
}

func modcomb(n, a, mod int) int {
	x := 1
	y := 1
	for i := 0; i < a; i++ {
		x = x * (n - i)
		x %= mod
		y = y * (i + 1)
		y %= mod
	}
	return x * modpow(y, mod-2, mod) % mod
}

func modfactorial(n, mod int) int {
	result := 1
	for i := 1; i <= n; i++ {
		result = (result * i) % mod
	}
	return result
}

// Combination ...
type Combination struct {
	facts, invs []int
	mod         int
}

// NewCombination ...
func NewCombination(n, mod int) Combination {
	return Combination{
		facts: make([]int, n+1),
		invs:  make([]int, n+1),
		mod:   mod,
	}
}

func (cmb *Combination) calc(n, k int) int {
	ret := cmb.facts[n] * cmb.invs[k]
	ret %= cmb.mod
	ret = ret * cmb.invs[n-k]
	ret %= cmb.mod
	return ret
}

func (cmb *Combination) init(n int) {
	cmb.facts[0] = 1
	// 階乗を算出
	for i := 1; i <= n; i++ {
		cmb.facts[i] = cmb.facts[i-1] * i % cmb.mod
	}
	// 逆元を算出
	cmb.invs[n] = modinv(cmb.facts[n], cmb.mod)
	for i := n - 1; i >= 0; i-- {
		cmb.invs[i] = cmb.invs[i+1] * (i + 1) % cmb.mod
	}
}

// Utility
func min(x, y int) int {
	if x < y {
		return x
	}
	return y
}

func max(x, y int) int {
	if x > y {
		return x
	}
	return y
}

func abs(x int) int {
	if x < 0 {
		return -x
	}
	return x
}

func primeFactor(x int) map[int]int {
	res := make(map[int]int)
	for i := 2; i*i <= x; i++ {
		for x%i == 0 {
			res[i]++
			x = x / i
		}
	}
	if x != 1 {
		res[x] = 1
	}
	return res
}

func divisor(x int) []int {
	res := make([]int, 0)
	for i := 1; i*i <= x; i++ {
		if x%i == 0 {
			res = append(res, i)
			if i != x/i {
				res = append(res, x/i)
			}
		}
	}
	return res
}

func gcd(x, y int) int {
	if y == 0 {
		return x
	}
	return gcd(y, x%y)
}

func lcm(x, y int) int {
	return x / gcd(x, y) * y
}

func reverseString(s string) string {
	r := []rune(s)
	for i, j := 0, len(r)-1; i < len(r)/2; i, j = i+1, j-1 {
		r[i], r[j] = r[j], r[i]
	}
	return string(r)
}

func pow(a, n int) int {
	ret := 1
	for i := 1; i <= n; i++ {
		ret *= a
	}
	return ret
}

// Queue ...
type Queue []int

// pop ...
func (q *Queue) empty() bool {
	return len(*q) == 0
}

// push ...
func (q *Queue) push(i int) {
	*q = append(*q, i)
}

// pop ...
func (q *Queue) pop() (int, bool) {
	if q.empty() {
		return 0, false
	} else {
		res := (*q)[0]
		*q = (*q)[1:]
		return res, true
	}
}

// Stack ...
type Stack []int

// pop ...
func (s *Stack) empty() bool {
	return len(*s) == 0
}

// push ...
func (s *Stack) push(i int) {
	*s = append(*s, i)
}

// pop ...
func (s *Stack) pop() (int, bool) {
	if s.empty() {
		return 0, false
	} else {
		index := len(*s) - 1
		res := (*s)[index]
		*s = (*s)[:index]
		return res, true
	}
}

// priority_queue
type intHeap []int

func (h intHeap) Len() int           { return len(h) }
func (h intHeap) Less(i, j int) bool { return h[i] > h[j] }
func (h intHeap) Swap(i, j int)      { h[i], h[j] = h[j], h[i] }
func (h *intHeap) Push(x interface{}) {
	*h = append(*h, x.(int))
}
func (h *intHeap) Pop() interface{} {
	old := *h
	n := len(old)
	x := old[n-1]
	*h = old[0 : n-1]
	return x
}

// sortable points
type point struct {
	x int
	y int
}

type points []point

func (p points) Len() int {
	return len(p)
}

func (p points) Less(i, j int) bool {
	return p[i].x < p[j].x
}

func (p points) Swap(i, j int) {
	p[i], p[j] = p[j], p[i]
}