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 LuaJIT (2.0.4) (judge verdict: Accepted)

local mce, mfl, msq, mmi, mma = math.ceil, math.floor, math.sqrt, math.min, math.max

local n, m, rnum = io.read("*n", "*n", "*n")
local r = {}
for i = 1, rnum do r[i] = io.read("*n") end
local edge = {}
for i = 1, n do edge[i] = {} end
for i = 1, m do
  local u, v, l = io.read("*n", "*n", "*n")
  edge[u][v] = l
  edge[v][u] = l
end

local inf = 30000001

local function getlength(start_idx, dst_idx)
  local taskstate = {}
  for i = 1, n do taskstate[i] = false end
  local tasks = {}
  local tasknum = 0
  local done = 0

  local len = {}
  for i = 1, n do len[i] = inf end
  len[start_idx] = 0

  local function addtask(idx)
    if(not taskstate[idx]) then
      taskstate[idx] = true
      tasknum = tasknum + 1
      tasks[tasknum] = idx
    end
  end
  addtask(start_idx)

  while(done < tasknum) do
    done = done + 1
    local src = tasks[done]
    taskstate[src] = false
    for dst, cost in pairs(edge[src]) do
      if len[src] + cost < len[dst] then
        len[dst] = len[src] + cost
        addtask(dst)
      end
    end
  end
  return len[dst_idx]
end

local function heldkarp_prepare(n)
  local tot = 1
  local alltask = {}
  local stagetask = {}
  for i = 1, n do
    tot = tot * 2
    alltask[i] = {}
    stagetask[i] = {}
  end
  for i = 1, n do
    for j = 1, tot - 1 do
      alltask[i][j] = 160000001
    end
  end
  for i = 1, tot - 1 do
    local ti = i
    local cnt = 0
    for j = 1, n do
      if ti % 2 == 1 then
        cnt = cnt + 1
        ti = ti - 1
      end
      ti = ti / 2
    end
    table.insert(stagetask[cnt], i)
  end
  -- set first state
  local tmp = 1
  for i = 1, n do
    alltask[i][tmp] = 0
    tmp = tmp * 2
  end
  return tot, alltask, stagetask
end

local function heldkarp_doall(n, alltask, stagetask, len)
  for stage = 1, n - 1 do
    local stlen = #stagetask[stage]
    for i_stagetask = 1, stlen do
      local used = stagetask[stage][i_stagetask]
      local t_used = used
      for i = 1, n do
        if t_used % 2 == 1 then
          local val = alltask[i][used]
          local mul = 1
          local tmp = used
          for j = 1, n do
            if tmp % 2 == 0 then
              alltask[j][used + mul] = mmi(alltask[j][used + mul], val + len[(i - 1) * n + j])
            else
              tmp = tmp - 1
            end
            tmp = tmp / 2
            mul = mul * 2
          end
          t_used = t_used - 1
        end
        t_used = t_used / 2
      end
    end
  end
end

local function heldkarp_getresult(n, tot, alltask)
  local ret = alltask[1][tot - 1]
  for i = 2, n do
    ret = mmi(ret, alltask[i][tot - 1])
  end
  return ret
end


local len = {}
for i = 1, rnum * rnum do
  len[i] = 0
end
for i = 1, rnum - 1 do
  for j = i + 1, rnum do
    local l = getlength(r[i], r[j])
    len[(i - 1) * rnum + j] = l
    len[(j - 1) * rnum + i] = l
  end
end
local t1, t2, t3 = heldkarp_prepare(rnum)
heldkarp_doall(rnum, t2, t3, len)
print(heldkarp_getresult(rnum, t1, t2))