Problem:
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15 Puzzle

The goal of the 15 puzzle problem is to complete pieces on $4 \times 4$ cells where one of the cells is empty space.

In this problem, the space is represented by 0 and pieces are represented by integers from 1 to 15 as shown below.

1 2 3 4
6 7 8 0
5 10 11 12
9 13 14 15

You can move a piece toward the empty space at one step. Your goal is to make the pieces the following configuration in the shortest move (fewest steps).

1 2 3 4
5 6 7 8
9 10 11 12
13 14 15 0

Write a program which reads an initial state of the puzzle and prints the fewest steps to solve the puzzle.

Input

The $4 \times 4$ integers denoting the pieces or space are given.

Output

Print the fewest steps in a line.

Constraints

The given puzzle is solvable in at most 45 steps.

Sample Input

1 2 3 4
6 7 8 0
5 10 11 12
9 13 14 15

Sample Output

8

Solution in C++ (judge verdict: Accepted)


#include<iostream>
#include<cmath>
#include<string>
#include<map>
#include<queue>

using namespace std;

#define N 4
#define N2 16
#define LIMIT 100

static const int dx[4] = {0, -1, 0, 1};
static const int dy[4] = {1, 0, -1, 0};
static const char dir[4] = {'r', 'u', 'l', 'd'};
int MDT[N2][N2];

struct Puzzle {
  int f[N2];
  int space;
  int MD;
  int cost;

  bool operator < (const Puzzle& p) const {
    for (int i = 0; i < N2; i++) {
      if (f[i] == p.f[i]) {
        continue;
      }
      return f[i] < p.f[i];
    }
    return false;
  }
};

struct State {
  Puzzle puzzle;
  int estimated;
  bool operator < (const State& s) const {
    return estimated > s.estimated;
  }
};

int getAllMD(Puzzle pz) {
  int sum = 0;
  for (int i = 0; i < N2; i++) {
    if (pz.f[i] == N2) {
      continue;
    }
    sum += MDT[i][pz.f[i] - 1];
  }
  return sum;
}

int astar(Puzzle s) {
  priority_queue<State> PQ;
  s.MD = getAllMD(s);
  s.cost = 0;
  map<Puzzle, bool> V;
  Puzzle u, v;
  State initial;
  initial.puzzle = s;
  initial.estimated = getAllMD(s);
  PQ.push(initial);

  while (!PQ.empty()) {
    State st = PQ.top(); PQ.pop();
    u = st.puzzle;

    if (u.MD == 0) {
      return u.cost;
    }
    V[u] = true;

    int sx = u.space / N;
    int sy = u.space % N;

    for (int r = 0; r < 4; r++) {
      int tx = sx + dx[r];
      int ty = sy + dy[r];
      if (tx < 0 || ty < 0 || tx >= N || ty >= N) {
        continue;
      }
      v = u;

      v.MD -= MDT[tx * N + ty][v.f[tx * N + ty] - 1];
      v.MD += MDT[sx * N + sy][v.f[tx * N + ty] - 1];
      swap(v.f[sx * N + sy], v.f[tx * N + ty]);
      v.space = tx * N + ty;
      if (!V[v]) {
        v.cost++;
        State news;
        news.puzzle = v;
        news.estimated = v.cost + v.MD;
        PQ.push(news);
      }
    }
  }
  return -1;
}

int main(){
  for (int i = 0; i < N2; i++) {
    for (int j = 0; j < N2; j++) {
      MDT[i][j] = abs(i / N - j / N) + abs(i % N - j % N);
    }
  }

  Puzzle in;

  for (int i = 0; i < N2; i++) {
    cin >> in.f[i];
    if (in.f[i] == 0) {
      in.f[i] = N2;
      in.space = i;
    }
  }

  cout << astar(in) << endl;

  return 0;
}