Binary_search_for_DNA.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Searching for DNA string matches\n",
    "\n",
    "Jacob L. Fine\n",
    "\n",
    "June 3rd, 2024\n",
    "\n",
    "We are often interested in searching through a list of strings to return if a query string matches any of the strings present in the list. \n",
    "\n",
    "Here we will implement two approaches: the binary search algorithm and the brute-force approach. The binary search algorithm works by partitioning a sorted list into parts based on whether the string being searched is alphabetically greater/less than the lower/upper index bound, so that only portions of the list are searched, rather than the entire list. The brute-force approach does just this, i.e., searching every item in the list until a match is found.\n",
    "\n",
    "Here, we will implement both methods and show that the binary search requires less time to compute then the brute-force approach. We will demonstrate this in the context of finding the index of a specific DNA sequence in a list of many DNA sequences, with lists of up to 1000000 sequences."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 157,
   "metadata": {},
   "outputs": [],
   "source": [
    "import random\n",
    "\n",
    "random.seed(101010)\n",
    "def generate_random_dna_sequence(length: int) -> str:\n",
    "    # generates DNA sequence from the alphabet 'ATCG'\n",
    "    return ''.join(random.choice('ATCG') for _ in range(length))\n",
    "\n",
    "def generate_dna_list(num_sequences: int, sequence_length: int) -> list:\n",
    "    # generates a list of DNA sequences using the generator for a given sequence\n",
    "    return [generate_random_dna_sequence(sequence_length) for _ in range(num_sequences)]\n",
    "\n",
    "def binary_search(dna_sequences: list, query: str) -> int:\n",
    "    '''\n",
    "    A binary search on a sorted list of DNA sequences.\n",
    "\n",
    "    Args:\n",
    "    dna_sequences (list of str): A sorted list of DNA sequences\n",
    "    query (str): The DNA sequence to find\n",
    "\n",
    "    Returns:\n",
    "    int: The index of the query sequence in the list, or -1 if not found.\n",
    "    '''\n",
    "    # the initial lower and upper bounds of the list to search\n",
    "    low = 0\n",
    "    high = len(dna_sequences) - 1\n",
    "\n",
    "    while low <= high: \n",
    "        # sets teh middle index based on the upper and lower indices\n",
    "        mid = (low + high) // 2\n",
    "        # gets the item associated with the middle index\n",
    "        mid_val = dna_sequences[mid]\n",
    "\n",
    "        # finally, once the item with middle index is equal to the query, return the middle index\n",
    "        if mid_val == query:\n",
    "            return mid\n",
    "        # otherwise, (since the list is sorted alphabetically), if the query occurs after the middle value, we now just need to consider\n",
    "        # strings after the middle value, so we update the lower bound to the middle index.\n",
    "        elif query > mid_val:\n",
    "            low = mid + 1\n",
    "        # otherwise, do the opposite\n",
    "        else:\n",
    "            high = mid - 1\n",
    "\n",
    "    return -1  # we return -1 if nothing was found\n",
    "\n",
    "\n",
    "def brute_force_search(dna_sequences, query):\n",
    "    '''\n",
    "    A brute force seach on a sorted list of DNA sequences.\n",
    "\n",
    "    Args:\n",
    "    dna_sequences (list of str): A sorted list of DNA sequences\n",
    "    query (str): The DNA sequence to find\n",
    "\n",
    "    Returns:\n",
    "    int: The index of the query sequence in the list, or -1 if not found.\n",
    "    '''\n",
    "    # simply scans every element in the list until a match is found\n",
    "    for index, sequence in enumerate(dna_sequences):\n",
    "        if sequence == query:\n",
    "            return index\n",
    "    return -1  # we return -1 if nothing was found"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 158,
   "metadata": {},
   "outputs": [],
   "source": [
    "import timeit\n",
    "import math\n",
    "\n",
    "random.seed(101010)\n",
    "\n",
    "# max value of input size\n",
    "max_value = 1000000\n",
    "\n",
    "# generates different input sizes\n",
    "input_sizes = [10**i for i in range(int(math.log10(1)), int(math.log10(max_value)) + 1)]\n",
    "\n",
    "# to store the list of running times for binary search\n",
    "time_list_binary = []\n",
    "time_list_brute = []\n",
    "# iterates through each running time and implements binary search\n",
    "for size in input_sizes:\n",
    "    num_sequences = size  # number of DNA sequences to generate\n",
    "    sequence_length = 10  # length of each DNA sequence\n",
    "\n",
    "    # generates the list of DNA sequences\n",
    "    dna_sequences = generate_dna_list(num_sequences, sequence_length)\n",
    "\n",
    "    # sorts the list, necessary for binary search\n",
    "    sorted_dna_sequences = sorted(dna_sequences)\n",
    "\n",
    "    # generates a search to search for\n",
    "    query = random.choice(dna_sequences)\n",
    "    # performs the search, and times it (for binary search)\n",
    "    time_taken_binary = timeit.timeit(lambda: binary_search(sorted_dna_sequences, query), number=1)\n",
    "    # appends to the list of times (for binary search)\n",
    "    time_list_binary.append(time_taken_binary)\n",
    "\n",
    "    # performs the search, and times it (for brute force approach)\n",
    "    time_taken_brute = timeit.timeit(lambda: brute_force_search(sorted_dna_sequences, query), number=1)\n",
    "    # appends to the list of times (for brute force approach)\n",
    "    time_list_brute.append(time_taken_brute)\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 161,
   "metadata": {},
   "outputs": [
    {
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",
      "text/plain": [
       "<Figure size 1000x600 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "import seaborn as sns\n",
    "import matplotlib.pyplot as plt\n",
    "import pandas as pd\n",
    "\n",
    "\n",
    "# makes a dataframe out of the running time lists for different inputs.\n",
    "data = {\n",
    "    'input size': input_sizes,\n",
    "    'binary search': time_list_binary,\n",
    "    'brute force': time_list_brute\n",
    "}\n",
    "\n",
    "df = pd.DataFrame(data)\n",
    "\n",
    "# plots the data\n",
    "\n",
    "plt.figure(figsize=(10, 6))\n",
    "sns.lineplot(x='input size', y='binary search', data=df, marker='o', label='binary search')\n",
    "sns.lineplot(x='input size', y='brute force', data=df, marker='o', label='brute-force')\n",
    "plt.xlabel('Input Size')\n",
    "plt.ylabel('running time (s)')\n",
    "plt.title('Binary search vs. brute-force approach')\n",
    "plt.legend()\n",
    "plt.show()\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can observe that the binary search algorithm has consistently lower running time for different input sizes, than the brute force approach. \n",
    "\n",
    "The actual time complexity of the binary search can be shown to be O(log n), where n is the number of items in the list being searched, while the brute force approach has a time complexity of O(n)."
   ]
  }
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