add problems 81, 82, 83

notebooks/problem0082.ipynb

@@ -0,0 +1,117 @@
+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "id": "bcfe0c52",
+   "metadata": {},
+   "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)"
+   ]
+  }
+ ],
+ "metadata": {
+  "kernelspec": {
+   "display_name": "SageMath 9.5",
+   "language": "sage",
+   "name": "sagemath"
+  },
+  "language_info": {
+   "codemirror_mode": {
+    "name": "ipython",
+    "version": 3
+   },
+   "file_extension": ".py",
+   "mimetype": "text/x-python",
+   "name": "python",
+   "nbconvert_exporter": "python",
+   "pygments_lexer": "ipython3",
+   "version": "3.11.2"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 5
+}