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 "cells": [
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   "metadata": {},
   "source": [
    "# Homework 3"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Write your name: [Your Name Here]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Recursive Insertion Sort"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "You can also think of insertion sort as a recursive algorithm (`recInsertionSort`). In order to sort $A[0:n-1]$:\n",
    "- recursively sort the subarray $A[0:n-2]$ and then \n",
    "- insert $A[n-1]$ into the sorted subarray $A[0:n-2]$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Exercise 1 (5 Points)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Write a Python code for this recursive version (`recInsertionSort`) of insertion sort.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "def recInsertionSort(A):\n",
    "    # implement me\n",
    "    pass"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Exercise 2 (3 Points)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Give a recurrence equation for its worst-case running time $T(n)$. \n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Exercise 3 (2 Points)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Compare and contrast $T(n)$ with the best and worst-case running times of incremental `insertionSort`.\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Divide-and-Conquer"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Exercise 4 (3 Points)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let $A[0:n-1]$ be an array of $n$ numbers. If $i<j$ and $A[i]>A[j]$, then the \n",
    "pair $(i,j)$ is called an **inversion** of $A$.\n",
    "\n",
    "Design an $\\mathcal O(n\\lg{n})$ algorithm that determines the number of inversions in any permutation of $n$ elements. Provide a Python implementation.\n",
    "\n",
    "**Hint:** Modify `MERGE-SORT`.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "def countInversions(A):\n",
    "    # implement me\n",
    "    pass"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Exercise 5 (2 Points)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Analyze the time complexity of your algorithm in Exercise $4$.\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Exercise 6 (3 Points)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Design an $\\mathcal{O}(n\\lg{n})$ algorithm that, given a set $S$ of $n$ integers and another integer $x$, determines whether $S$ contains two elements that sum to exactly $x$. Provide a Python implementation.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "def findPairWithSum(S, x):\n",
    "    # implement me\n",
    "    pass"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Exercise 7 (2 Points)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Analyze the time complexity of your algorithm in Exercise $6$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Time-complexity of Recursive Algorithms"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Exercise 8 (1 Point)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Use the Master Theorem to get an upper bound (in big-$\\mathcal O$ notation) for the following recurrence:\n",
    "$$T(n)=T(n/2) + n^3.$$\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Exercise 9 (1 Point)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Use the Master Theorem to get an upper bound (in big-$\\mathcal O$ notation) for the following recurrence:\n",
    "$$T(n)=4T(n/2) + n.$$\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Exercise 10 (3 Points)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Given the recurrence relation $T(n)=2T(n/2)+\\mathcal O(n)$, what is the time complexity of the corresponding recursive algorithm?\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Exercise 11 (🏅 Bonus)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "How does recursion impact space complexity, and what factors should be considered when analyzing it?"
   ]
  }
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