Base notebooks for the coursework

DP.ipynb

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+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "# Subset-Sum Problem (SSP)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "The **Subset Sum problem (SSP)** is defined as follows:\n",
+    "\n",
+    ">Given a set $S = \\{x_1, x_2, \\ldots, x_n\\}$ of positive integers,\n",
+    ">and a positive integers $t$, is there a subset of $S$ whose sum is equal to $t$?\n",
+    "\n",
+    "This is the **decision version** of SSP as it only asks for a true/false answer."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "#### Example\n",
+    "\n",
+    "$S=\\{1,2,3,10\\}$ and $t=13$.\n",
+    "\n",
+    "A solution is given by $\\{1,2,10\\}$. There is also another solution: $\\{3,10\\}$."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## Exhaustive search"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "An **exhaustive search algorithm** for this problem can written *iteratively* as follows:\n",
+    "\n",
+    "**INPUT:** A set $S = \\{x_1, x_2, \\ldots, x_n\\}$ of positive integers, and a positive integer $t$.\n",
+    "\n",
+    "**OUTPUT:** $(c_1,\\ldots,c_n)$ such that $\\sum_{i=1}^n c_i x_i = t$\n",
+    "\n",
+    "1. **for all** $(c_1,\\ldots,c_n)\\in\\{0,1\\}^n$ **do**\n",
+    "2. $\\quad$ **if** $\\sum_{i=1}^n c_i x_i = t$ **then**\n",
+    "3. $\\qquad$ **return** $(c_1,\\ldots,c_n)$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "Now think of the exhaustive search as a binary tree, where at each level we decide whether to include the $i^\\text{th}$ number or not. For example, if $S = \\{a, b, c, d\\}$ then we get:\n",
+    "\n",
+    "![](img/ssp-binary-tree.jpg)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "Write a recursive version of the exhaustive search algorithm."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "......................................................................................\n",
+    "......................................................................................\n",
+    "......................................................................................\n",
+    "......................................................................................\n",
+    "......................................................................................\n",
+    "......................................................................................\n",
+    "......................................................................................\n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## Dynamic Programming."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "The above iterative pseudocode considers the subsets of $S$ as the search space.\n",
+    "Another way to search for solutions is to iteratively build the answer for smaller target values $t' = 0, 1, 2, 3, \\ldots$ until we reach $t$.\n",
+    "\n",
+    "If we have built all the possible sums from the subset $\\{x_1,\\ldots,x_k\\}$, what other sums become possible if we add $x_{k+1}$ to the set?"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "............................................................................\n",
+    "............................................................................\n",
+    "............................................................................\n",
+    "............................................................................\n",
+    "............................................................................\n",
+    "............................................................................\n",
+    "............................................................................\n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "Now explain how we can use this for a bottom-up approach to decide SSP."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "# Implementation"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 1,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "from random import randint, sample"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 2,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "def get_S_t(n, MAX_X = 100):\n",
+    "    S = [randint(1,MAX_X) for i in range(n)]\n",
+    "    t = sum(sample(S,randint(1,n)))\n",
+    "    return S,t"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 3,
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "([42, 26, 59, 27, 18, 74, 92, 54, 85, 31], 96)"
+      ]
+     },
+     "execution_count": 3,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "get_S_t(10)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "### 1) Memoization (Top-down approach)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 4,
+   "metadata": {
+    "ExecuteTime": {
+     "end_time": "2022-10-25T11:38:11.266563Z",
+     "start_time": "2022-10-25T11:38:11.256547Z"
+    }
+   },
+   "outputs": [],
+   "source": [
+    "# Implementation"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 5,
+   "metadata": {
+    "ExecuteTime": {
+     "end_time": "2022-10-25T11:38:11.281559Z",
+     "start_time": "2022-10-25T11:38:11.271567Z"
+    }
+   },
+   "outputs": [],
+   "source": [
+    "# Test examples"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "### 2) Bottom-up approach"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 6,
+   "metadata": {
+    "ExecuteTime": {
+     "end_time": "2022-10-25T11:38:11.295714Z",
+     "start_time": "2022-10-25T11:38:11.286560Z"
+    }
+   },
+   "outputs": [],
+   "source": [
+    "# Implementation"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 7,
+   "metadata": {
+    "ExecuteTime": {
+     "end_time": "2022-10-25T11:38:11.311861Z",
+     "start_time": "2022-10-25T11:38:11.295714Z"
+    }
+   },
+   "outputs": [],
+   "source": [
+    "# Test examples"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "# Conclusion\n",
+    "\n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "# List of references\n"
+   ]
+  }
+ ],
+ "metadata": {
+  "kernelspec": {
+   "display_name": "Python 3 (ipykernel)",
+   "language": "python",
+   "name": "python3"
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+   "mimetype": "text/x-python",
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+   "nbconvert_exporter": "python",
+   "pygments_lexer": "ipython3",
+   "version": "3.11.1"
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+  "toc": {
+   "base_numbering": 1,
+   "nav_menu": {},
+   "number_sections": false,
+   "sideBar": true,
+   "skip_h1_title": false,
+   "title_cell": "Table of Contents",
+   "title_sidebar": "Contents",
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+   "toc_window_display": false
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+}