diff --git a/recursion_dynamic/matrix_mult/__init__.py b/recursion_dynamic/matrix_mult/__init__.py new file mode 100644 index 0000000..e69de29 diff --git a/recursion_dynamic/matrix_mult/find_min_cost_challenge.ipynb b/recursion_dynamic/matrix_mult/find_min_cost_challenge.ipynb new file mode 100644 index 0000000..5f4a260 --- /dev/null +++ b/recursion_dynamic/matrix_mult/find_min_cost_challenge.ipynb @@ -0,0 +1,183 @@ +{ + "cells": [ + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "This notebook was prepared by [Donne Martin](https://github.com/donnemartin). Source and license info is on [GitHub](https://github.com/donnemartin/interactive-coding-challenges)." + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Challenge Notebook" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Problem: Given a list of 2x2 matrices, minimize the cost of matrix multiplication.\n", + "\n", + "* [Constraints](#Constraints)\n", + "* [Test Cases](#Test-Cases)\n", + "* [Algorithm](#Algorithm)\n", + "* [Code](#Code)\n", + "* [Unit Test](#Unit-Test)\n", + "* [Solution Notebook](#Solution-Notebook)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Constraints\n", + "\n", + "* Do we just want to calculate the cost and not list the actual order of operations?\n", + " * Yes\n", + "* Can we assume the inputs are valid?\n", + " * No\n", + "* Can we assume this fits memory?\n", + " * Yes" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Test Cases\n", + "\n", + "* None -> Exception\n", + "* [] -> 0\n", + "* [Matrix(2, 3), Matrix(3, 6), Matrix(6, 4), Matrix(4, 5)] -> 124" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Algorithm\n", + "\n", + "Refer to the [Solution Notebook](). If you are stuck and need a hint, the solution notebook's algorithm discussion might be a good place to start." + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Code" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": true + }, + "outputs": [], + "source": [ + "class Matrix(object):\n", + "\n", + " def __init__(self, first, second):\n", + " self.first = first\n", + " self.second = second" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "class MatrixMultiplicationCost(object):\n", + "\n", + " def find_min_cost(self, matrices):\n", + " # TODO: Implement me\n", + " pass" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Unit Test" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "**The following unit test is expected to fail until you solve the challenge.**" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "# %load test_find_min_cost.py\n", + "from nose.tools import assert_equal, assert_raises\n", + "\n", + "\n", + "class TestMatrixMultiplicationCost(object):\n", + "\n", + " def test_find_min_cost(self):\n", + " matrix_mult_cost = MatrixMultiplicationCost()\n", + " assert_raises(TypeError, matrix_mult_cost.find_min_cost, None)\n", + " assert_equal(matrix_mult_cost.find_min_cost([]), 0)\n", + " matrices = [Matrix(2, 3),\n", + " Matrix(3, 6),\n", + " Matrix(6, 4),\n", + " Matrix(4, 5)]\n", + " expected_cost = 124\n", + " assert_equal(matrix_mult_cost.find_min_cost(matrices), expected_cost)\n", + " print('Success: test_find_min_cost')\n", + "\n", + "\n", + "def main():\n", + " test = TestMatrixMultiplicationCost()\n", + " test.test_find_min_cost()\n", + "\n", + "\n", + "if __name__ == '__main__':\n", + " main()" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Solution Notebook\n", + "\n", + "Review the [Solution Notebook]() for a discussion on algorithms and code solutions." + ] + } + ], + "metadata": { + "kernelspec": { + "display_name": "Python 3", + "language": "python", + "name": "python3" + }, + "language_info": { + "codemirror_mode": { + "name": "ipython", + "version": 3 + }, + "file_extension": ".py", + "mimetype": "text/x-python", + "name": "python", + "nbconvert_exporter": "python", + "pygments_lexer": "ipython3", + "version": "3.5.0" + } + }, + "nbformat": 4, + "nbformat_minor": 0 +} diff --git a/recursion_dynamic/matrix_mult/find_min_cost_solution.ipynb b/recursion_dynamic/matrix_mult/find_min_cost_solution.ipynb new file mode 100644 index 0000000..7eb3c9a --- /dev/null +++ b/recursion_dynamic/matrix_mult/find_min_cost_solution.ipynb @@ -0,0 +1,316 @@ +{ + "cells": [ + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "This notebook was prepared by [Donne Martin](https://github.com/donnemartin). Source and license info is on [GitHub](https://github.com/donnemartin/interactive-coding-challenges)." + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Solution Notebook" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Problem: Given a list of 2x2 matrices, minimize the cost of matrix multiplication.\n", + "\n", + "* [Constraints](#Constraints)\n", + "* [Test Cases](#Test-Cases)\n", + "* [Algorithm](#Algorithm)\n", + "* [Code](#Code)\n", + "* [Unit Test](#Unit-Test)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Constraints\n", + "\n", + "* Do we just want to calculate the cost and not list the actual order of operations?\n", + " * Yes\n", + "* Can we assume the inputs are valid?\n", + " * No\n", + "* Can we assume this fits memory?\n", + " * Yes" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Test Cases\n", + "\n", + "* None -> Exception\n", + "* [] -> 0\n", + "* [Matrix(2, 3), Matrix(3, 6), Matrix(6, 4), Matrix(4, 5)] -> 124" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Algorithm\n", + "\n", + "We'll use bottom up dynamic programming to build a table.\n", + "\n", + "
\n",
+    "\n",
+    "  0    1    2    3\n",
+    "[2,3][3,6][6,4][4,5]\n",
+    "\n",
+    "Case: 0 * 1\n",
+    "2 * 3 * 6 = 36\n",
+    "\n",
+    "Case: 1 * 2\n",
+    "3 * 6 * 4 = 72\n",
+    "\n",
+    "Case: 2 * 3\n",
+    "6 * 4 * 5 = 120\n",
+    "\n",
+    "Case: 0 * 1 * 2\n",
+    "0 * (1 * 2) = 2 * 3 * 4 + 72 = 96\n",
+    "(0 * 1) * 2 = 36 + 2 * 6 * 4 = 84\n",
+    "min: 84\n",
+    "\n",
+    "Case: 1 * 2 * 3\n",
+    "1 * (2 * 3) = 3 * 6 * 5 + 120 = 210\n",
+    "(1 * 2) * 3 = 72 + 3 * 4 * 5 = 132\n",
+    "min: 132\n",
+    "\n",
+    "Case: 0 * 1 * 2 * 3\n",
+    "0 * (1 * 2 * 3) = 2 * 3 * 5 + 132 = 162\n",
+    "(0 * 1) * (2 * 3) = 36 + 120 + 2 * 6 * 5 = 216\n",
+    "(0 * 1 * 2) * 3 = 84 + 2 * 4 * 5 = 124\n",
+    "min: 124\n",
+    "\n",
+    "  ---------------------\n",
+    "  | 0 |  1 |  2 |   3 |\n",
+    "  ---------------------\n",
+    "0 | 0 | 36 | 84 | 124 |\n",
+    "1 | x |  0 | 72 | 132 |\n",
+    "2 | x |  x |  0 | 120 |\n",
+    "3 | x |  x |  x |   0 |\n",
+    "  ---------------------\n",
+    "\n",
+    "min cost = T[0][cols-1] = 124\n",
+    "\n",
+    "for k in range(i, j):\n",
+    "    T[i][j] = minimum of (T[i][k] + T[k+1][j] +\n",
+    "                          m[i].first * m[k].second * m[j].second) for all k\n",
+    "
" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "### Explanation of k" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "
\n",
+    "  0    1    2    3\n",
+    "[2,3][3,6][6,4][4,5]\n",
+    "\n",
+    "Fill in the missing cell, where i = 0, j = 3\n",
+    "\n",
+    "  ---------------------\n",
+    "  | 0 |  1 |  2 |   3 |\n",
+    "  ---------------------\n",
+    "0 | 0 | 36 | 84 | ??? |\n",
+    "1 | x |  0 | 72 | 132 |\n",
+    "2 | x |  x |  0 | 120 |\n",
+    "3 | x |  x |  x |   0 |\n",
+    "  ---------------------\n",
+    "\n",
+    "Case: 0 * (1 * 2 * 3), k = 0\n",
+    "i = 0, j = 3\n",
+    "\n",
+    "0 * (1 * 2 * 3) = 2 * 3 * 5 + 132 = 162\n",
+    "T[i][k] + T[k+1][j] + m[i].first * m[k].second * m[j].second\n",
+    "T[0][0] + T[1][3] + 2 * 3 * 5\n",
+    "0 + 132 + 30 = 162\n",
+    "\n",
+    "Case: (0 * 1) * (2 * 3), k = 1\n",
+    "i = 0, j = 3\n",
+    "\n",
+    "(0 * 1) * (2 * 3) = 36 + 120 + 2 * 6 * 5 = 216\n",
+    "T[i][k] + T[k+1][j] + m[i].first * m[k].second * m[j].second\n",
+    "T[0][1] + T[2][3] + 2 * 6 * 5\n",
+    "36 + 120 + 60 = 216\n",
+    "\n",
+    "Case: (0 * 1 * 2) * 3, k = 2\n",
+    "i = 0, j = 3\n",
+    "\n",
+    "(0 * 1 * 2) * 3 = 84 + 2 * 4 * 5 = 124\n",
+    "T[i][k] + T[k+1][j] + m[i].first * m[k].second * m[j].second\n",
+    "T[0][2] + T[3][3] + 2 * 4 * 5\n",
+    "84 + 0 + 40 = 124\n",
+    "\n",
+    "
\n", + "\n", + "Complexity:\n", + "* Time: O(n^3)\n", + "* Space: O(n^2)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Code" + ] + }, + { + "cell_type": "code", + "execution_count": 1, + "metadata": { + "collapsed": true + }, + "outputs": [], + "source": [ + "class Matrix(object):\n", + "\n", + " def __init__(self, first, second):\n", + " self.first = first\n", + " self.second = second" + ] + }, + { + "cell_type": "code", + "execution_count": 2, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "import sys\n", + "\n", + "\n", + "class MatrixMultiplicationCost(object):\n", + "\n", + " def find_min_cost(self, matrices):\n", + " if matrices is None:\n", + " raise TypeError('matrices cannot be None')\n", + " if not matrices:\n", + " return 0\n", + " size = len(matrices)\n", + " T = [[0] * size for _ in range(size)]\n", + " for offset in range(1, size):\n", + " for i in range(size-offset):\n", + " j = i + offset\n", + " min_cost = sys.maxsize\n", + " for k in range(i, j):\n", + " cost = (T[i][k] + T[k+1][j] +\n", + " matrices[i].first *\n", + " matrices[k].second *\n", + " matrices[j].second)\n", + " if cost < min_cost:\n", + " min_cost = cost\n", + " T[i][j] = min_cost\n", + " return T[0][size-1]" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Unit Test" + ] + }, + { + "cell_type": "code", + "execution_count": 3, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Overwriting test_find_min_cost.py\n" + ] + } + ], + "source": [ + "%%writefile test_find_min_cost.py\n", + "from nose.tools import assert_equal, assert_raises\n", + "\n", + "\n", + "class TestMatrixMultiplicationCost(object):\n", + "\n", + " def test_find_min_cost(self):\n", + " matrix_mult_cost = MatrixMultiplicationCost()\n", + " assert_raises(TypeError, matrix_mult_cost.find_min_cost, None)\n", + " assert_equal(matrix_mult_cost.find_min_cost([]), 0)\n", + " matrices = [Matrix(2, 3),\n", + " Matrix(3, 6),\n", + " Matrix(6, 4),\n", + " Matrix(4, 5)]\n", + " expected_cost = 124\n", + " assert_equal(matrix_mult_cost.find_min_cost(matrices), expected_cost)\n", + " print('Success: test_find_min_cost')\n", + "\n", + "\n", + "def main():\n", + " test = TestMatrixMultiplicationCost()\n", + " test.test_find_min_cost()\n", + "\n", + "\n", + "if __name__ == '__main__':\n", + " main()" + ] + }, + { + "cell_type": "code", + "execution_count": 4, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Success: test_find_min_cost\n" + ] + } + ], + "source": [ + "%run -i test_find_min_cost.py" + ] + } + ], + "metadata": { + "kernelspec": { + "display_name": "Python 3", + "language": "python", + "name": "python3" + }, + "language_info": { + "codemirror_mode": { + "name": "ipython", + "version": 3 + }, + "file_extension": ".py", + "mimetype": "text/x-python", + "name": "python", + "nbconvert_exporter": "python", + "pygments_lexer": "ipython3", + "version": "3.5.0" + } + }, + "nbformat": 4, + "nbformat_minor": 0 +} diff --git a/recursion_dynamic/matrix_mult/test_find_min_cost.py b/recursion_dynamic/matrix_mult/test_find_min_cost.py new file mode 100644 index 0000000..20c4cc8 --- /dev/null +++ b/recursion_dynamic/matrix_mult/test_find_min_cost.py @@ -0,0 +1,25 @@ +from nose.tools import assert_equal, assert_raises + + +class TestMatrixMultiplicationCost(object): + + def test_find_min_cost(self): + matrix_mult_cost = MatrixMultiplicationCost() + assert_raises(TypeError, matrix_mult_cost.find_min_cost, None) + assert_equal(matrix_mult_cost.find_min_cost([]), 0) + matrices = [Matrix(2, 3), + Matrix(3, 6), + Matrix(6, 4), + Matrix(4, 5)] + expected_cost = 124 + assert_equal(matrix_mult_cost.find_min_cost(matrices), expected_cost) + print('Success: test_find_min_cost') + + +def main(): + test = TestMatrixMultiplicationCost() + test.test_find_min_cost() + + +if __name__ == '__main__': + main() \ No newline at end of file