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   "source": [
    "# [Path Sum: Three Ways](https://projecteuler.net/problem=82)\n",
    "\n",
    "This is the same matrix as [problem 81](https://projecteuler.net/problem=81)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "098b70b9",
   "metadata": {},
   "outputs": [],
   "source": [
    "with open(\"txt/0082_matrix.txt\") as f:\n",
    "    mat = matrix((int(n) for n in line.split(',')) for line in f)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "aaf2f803",
   "metadata": {},
   "source": [
    "We can reuse our solution from problem 81 as well, with some slight modifications.\n",
    "1. Instead of our queue initially containing only the top-left entry of the matrix, it will initially hold *all* the entries from the first column.\n",
    "2. Additionally, instead of checking if we've reached the bottom-right entry, we'll stop on reaching *any entry* in the last column.\n",
    "3. Finally, when adding new entries to the queue, we'll add the entry *above* our current entry, along with the entries to the right and below, as before."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "3b7872ef",
   "metadata": {},
   "outputs": [],
   "source": [
    "import heapq\n",
    "\n",
    "def minimal_path_sum(mat):\n",
    "    m, n = mat.dimensions()\n",
    "    \n",
    "    destinations = {(i, n - 1) for i in range(0, m)}\n",
    "    visited = set()\n",
    "    queue = [(0, (i, 0)) for i in range(0, m)]\n",
    "    while queue != []:\n",
    "        cost, (i, j) = heapq.heappop(queue)\n",
    "        \n",
    "        if (i, j) in visited:\n",
    "            continue\n",
    "        visited.add((i, j))\n",
    "        \n",
    "        cost += mat[i, j]\n",
    "        \n",
    "        if (i, j) in destinations:\n",
    "            break\n",
    "        \n",
    "        if i - 1 >= 0:\n",
    "            heapq.heappush(queue, (cost, (i - 1, j)))\n",
    "            \n",
    "        if i + 1 < m:\n",
    "            heapq.heappush(queue, (cost, (i + 1, j)))\n",
    "            \n",
    "        if j + 1 < n:\n",
    "            heapq.heappush(queue, (cost, (i, j + 1)))\n",
    "            \n",
    "    return cost"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "62aba318",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "260324"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "minimal_path_sum(mat)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "db7b11ca",
   "metadata": {},
   "source": [
    "#### Copyright (C) 2025 filifa\n",
    "\n",
    "This work is licensed under the [Creative Commons Attribution-ShareAlike 4.0 International license](https://creativecommons.org/licenses/by-sa/4.0/) and the [BSD Zero Clause license](https://spdx.org/licenses/0BSD.html)."
   ]
  }
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