{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Homework 2"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Your Name: "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Big-$O$ Notation"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Question 1 (3 points)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Is $2^{n+1}=O(2^n)$? Justify your answer using [Definition 3.1](https://6001-majhi.datasci.land/notes/big-o.html#def-o)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Question 2 (3 points)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Is $2^{2n}=O(2^n)$? Justify your answer using [Definition 3.1](https://6001-majhi.datasci.land/notes/big-o.html#def-o)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Question 3 (3 points)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Using [Definition 3.1](https://6001-majhi.datasci.land/notes/big-o.html#def-o), show that the function $100n\\lg{n}$ is $O(n^2-1)$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Question 4 (1+1 points)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Describe the difference between an upper bound and a tight upper bound? Explain using an example.\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Question 5 (1+1 points)\n",
    "Provide an example of an $O(1)$ and $O(n)$ algorithm.\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Question 6 (🏅 Bonus)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "If $f(n)=O(g(n))$, then $2^{f(n)} = O(2^{g(n)})$. True or false? Justify your answer.\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Powerset Generation"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Question 7 (5 points)\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Write a Python code to generate all the subsets of a given list.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "def powerset(A):\n",
    "    # input: a list\n",
    "    # output: a list of lists\n",
    "    # implement me\n",
    "    pass\n",
    "\n",
    "powerset(['a', 'c', 'b'])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Question 8 (2 points)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "If the input list has $n$ elements, describe the time complexity of your code. A thorough explanation is required.\n"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "base",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "name": "python",
   "version": "3.12.2"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}