add problems 81, 82, 83

notebooks/problem0081.ipynb

@@ -0,0 +1,114 @@
+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "id": "0aba780c",
+   "metadata": {},
+   "source": [
+    "# [Path Sum: Two Ways](https://projecteuler.net/problem=81)\n",
+    "\n",
+    "First things first, we'll read in the matrix."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 1,
+   "id": "ef846cb1",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "with open(\"txt/0081_matrix.txt\") as f:\n",
+    "    mat = matrix((int(n) for n in line.split(',')) for line in f)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "af2036f4",
+   "metadata": {},
+   "source": [
+    "This problem is fundamentally a [shortest path problem](https://en.wikipedia.org/wiki/Shortest_path_problem), a well-studied problem with lots of algorithms to choose from. We'll employ a variant of [Dijkstra's algorithm](https://en.wikipedia.org/wiki/Dijkstra%27s_algorithm) - we would need a different algorithm if the matrix had negative entries.\n",
+    "\n",
+    "In short, this algorithm starts with the top-left entry of the matrix and adds its neighbors below and right of it to a search queue. The queue emits entries in the order of smallest path sum from the top-left entry, so the first time we visit an entry, we know the path taken to it is its minimal path. This means when we visit the bottom-right entry, we'll have computed its minimal path and can exit. We also keep track of nodes we've already visited so we don't waste time re-visiting them.\n",
+    "\n",
+    "Note that we could improve this even further by implementing a [Fibonacci heap](https://en.wikipedia.org/wiki/Fibonacci_heap) for our priority queue, but using a binary heap - [built-in to Python!](https://docs.python.org/3/library/heapq.html) - is plenty fast for this problem."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 2,
+   "id": "100b1cf6",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "import heapq\n",
+    "\n",
+    "def minimal_path_sum(mat):\n",
+    "    m, n = mat.dimensions()\n",
+    "    \n",
+    "    visited = set()\n",
+    "    queue = [(0, (0, 0))]\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) == (m - 1, n - 1):\n",
+    "            break\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": "9fa7d67d",
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "427337"
+      ]
+     },
+     "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
+}