Solving the all pairs shortest path problem with Algorithm 1.ipynb

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   "source": [
    "# Algorithm 1 (MMJ distance by recursion) can be revised to solve the all pairs shortest path problem. \n",
    "# By replacing the \"max\" operator with the \"+\" operator."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "c8f75cd9",
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import random\n",
    "import time"
   ]
  },
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   "id": "0d1ffe36",
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    "\n",
    "# create distance matrix of a directed graph\n",
    "\n",
    "def create_distance_matrix(N):\n",
    "\n",
    "    edges = []\n",
    "\n",
    "    for u in range(N):  \n",
    "        for v in range(N): \n",
    "            if u != v:\n",
    "                w = random.randint(1, 1000)\n",
    "                # w = random.uniform(1, 100)\n",
    "                \n",
    "                edges.append((u, v, w))    \n",
    "    \n",
    "    dist_matrix = np.zeros((N, N))\n",
    " \n",
    "    for u, v, w in edges:\n",
    "        dist_matrix[u, v] = w\n",
    "    \n",
    "    return dist_matrix.astype('float64')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "b5de4469",
   "metadata": {},
   "outputs": [],
   "source": [
    "def floyd_warshall(distance_matrix):\n",
    "    N = len(distance_matrix)\n",
    "    shortest_path_matrix = distance_matrix.copy()\n",
    "   \n",
    "    for k in range(N):\n",
    "        for i in range(N):\n",
    "            for j in range(N):\n",
    "                shortest_path_matrix[i][j] = min(shortest_path_matrix[i][j], shortest_path_matrix[i][k] + shortest_path_matrix[k][j])\n",
    "    return shortest_path_matrix\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "5faa263b",
   "metadata": {},
   "outputs": [],
   "source": [
    "def shortest_path_n_to_r(distance_matrix, shortest_path_matrix, n, r):\n",
    "    max_jump_list = []\n",
    "    for t in range(n):\n",
    "        m_jump =  distance_matrix[n,t] + shortest_path_matrix[t,r]\n",
    "        max_jump_list.append(m_jump)\n",
    "    return np.min(max_jump_list)\n",
    "\n",
    "def shortest_path_r_to_n(distance_matrix, shortest_path_matrix, n, r):\n",
    "    max_jump_list = []\n",
    "    for t in range(n):\n",
    "        m_jump =  shortest_path_matrix[r,t] + distance_matrix[t, n] \n",
    "        max_jump_list.append(m_jump)\n",
    "    return np.min(max_jump_list)\n",
    "\n",
    "     \n",
    "def cal_n_shortest_path(distance_matrix, shortest_path_matrix, n):\n",
    "    for i in range(n):\n",
    "        shortest_path_matrix[n,i] = shortest_path_n_to_r(distance_matrix, shortest_path_matrix, n, i)\n",
    "        shortest_path_matrix[i,n] = shortest_path_r_to_n(distance_matrix, shortest_path_matrix, n, i)\n",
    "        \n",
    "    for i in range(n):        \n",
    "        for j in range(n):\n",
    "            if i < j:\n",
    "                shortest_path_matrix[i,j] =  update_shortest_path_ij(distance_matrix, shortest_path_matrix, n, i, j)\n",
    "                shortest_path_matrix[j,i] =  update_shortest_path_ij(distance_matrix, shortest_path_matrix, n, j, i)\n",
    "                \n",
    "def update_shortest_path_ij(distance_matrix, shortest_path_matrix, n, i,j):\n",
    "    m1 = shortest_path_matrix[i,j]\n",
    "    m2 =  shortest_path_matrix[i,n] + shortest_path_matrix[n,j] \n",
    "    return np.min((m1,m2))\n",
    "\n",
    "def cal_shortest_path_matrix_algo_1_python(distance_matrix):\n",
    "    \n",
    "  \n",
    "    N = len(distance_matrix)\n",
    "   \n",
    "    shortest_path_matrix = np.zeros((N,N))\n",
    "\n",
    "    shortest_path_matrix[0,1] = distance_matrix[0,1]\n",
    "    shortest_path_matrix[1,0] = distance_matrix[1,0]\n",
    " \n",
    "    for kk in range(2,N):\n",
    "        cal_n_shortest_path(distance_matrix, shortest_path_matrix, kk)\n",
    "    return shortest_path_matrix"
   ]
  },
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   "cell_type": "code",
   "execution_count": 6,
   "id": "562c9604",
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   "outputs": [],
   "source": [
    "N = 100\n",
    "\n",
    "dist_matrix = create_distance_matrix(N)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "f7fa1eb6",
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   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Time used: 1.032s\n"
     ]
    }
   ],
   "source": [
    "start = time.time()\n",
    "shortest_path_matrix_floyd_warshall = floyd_warshall(dist_matrix)\n",
    "end = time.time()\n",
    "\n",
    " \n",
    "time_used = end - start\n",
    "time_used = np.round(time_used, 3)\n",
    "\n",
    "print(f\"Time used: {time_used}s\" )"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "id": "0a3bfee8",
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   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Time used: 2.239s\n"
     ]
    }
   ],
   "source": [
    "start = time.time()\n",
    "shortest_path_matrix_algo_1 = cal_shortest_path_matrix_algo_1_python(dist_matrix)\n",
    "end = time.time()\n",
    "\n",
    " \n",
    "time_used = end - start\n",
    "time_used = np.round(time_used, 3)\n",
    "\n",
    "print(f\"Time used: {time_used}s\" )"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
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   "execution_count": 9,
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     "text": [
      "True\n",
      "0.0\n"
     ]
    }
   ],
   "source": [
    "print(np.allclose(shortest_path_matrix_algo_1, shortest_path_matrix_floyd_warshall)) \n",
    "print(np.sum(np.abs(shortest_path_matrix_algo_1 - shortest_path_matrix_floyd_warshall))) \n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "id": "d369a7cf",
   "metadata": {},
   "outputs": [],
   "source": [
    "# Although Algorithm 1 (MMJ distance by recursion) is slower than the Floyd-Warshall algorithm,\n",
    "# it has the merit of warm start."
   ]
  },
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