{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Minimizers"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "One way to partition the space of all possible $k$-mers is by minimal $l$-mer, where $l < k$.  For example, the minimal 2-mer in the string `ABC` is `AB` and the minimal 4-mer in the string `abracadabra` is `abra`.  In this context, the minimal $l$-mer is called a *minimizer*, and we'll call such a partitioning scheme a $k, l$-minimizing scheme."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "def string_to_kmers(s, length):\n",
    "    return [s[i:i+length] for i in range(len(s)-length+1)]\n",
    "\n",
    "def minimizer(k, l):\n",
    "    \"\"\" Given k-mer, return its minimal l-mer \"\"\"\n",
    "    assert l <= len(k)\n",
    "    return min(string_to_kmers(k, l))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "'AB'"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "minimizer('ABC', 2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "'abra'"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "minimizer('abracadabra', 4)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "But if our goal is to partition the space of $k$-mers, couldn't we use a hash function instead?  Say $k$ is 10 and $l$ is 4.  A 10,4-minimizing scheme is a way for dividing the space of $4^{10}$ 10-mers (a million or so) into $4^4 = 256$ partitions.  We can accomplish this with a hash function that maps $k$-mers to integers in $[0, 255]$.  Why would we prefer minimizers over hash functions?\n",
    "\n",
    "The answer is that two strings that share long substrings tend to have the same minimizer, but not the same hash value.  For example, the strings `abracadabr` and `bracadabra` have the substring `bracadabr` in common, and they have the same minimal 4-mer:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "('abra', 'abra')"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "minimizer('abracadabr', 4), minimizer('bracadabra', 4)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "But their hash values (modulo 256) are not the same:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(26, 224)"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# you might need to 'pip install mmh3' first\n",
    "import mmh3\n",
    "mmh3.hash('abracadabr') % 256, mmh3.hash('bracadabra') % 256"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Partition size distribution"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "A feature of hash functions is that they divide the 10-mers quite uniformly (evenly) among the 256 buckets.   10,4-minimzers divide them much less uniformly.  This becomes clear when you consider that, given a random 10-mer, the 4-mer `TTTT` is very unlikely to be its minimizer, whereas the 4-mer `AAAA` is much more likely.\n",
    "\n",
    "We can also show this empirically by partitioning a collection of random 10-mers:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [],
   "source": [
    "import random\n",
    "random.seed(629)\n",
    "\n",
    "def random_kmer(k):\n",
    "    return ''.join([random.choice('ACGT') for _ in range(k)])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [],
   "source": [
    "%matplotlib inline\n",
    "import matplotlib.pyplot as plt \n",
    "\n",
    "def plot_counts(counter, title=None):\n",
    "    idx = range(256)\n",
    "    cnts = list(map(lambda x: counter.get(x, 0), idx))\n",
    "    plt.bar(idx, cnts, ec='none')\n",
    "    plt.xlim(0, 256)\n",
    "    plt.ylim(0, 35)\n",
    "    if title is not None:\n",
    "        plt.title(title)\n",
    "    plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "from collections import Counter\n",
    "# hash 1000 random 10-mers\n",
    "cnt = Counter([mmh3.hash(s) % 256 for s in [random_kmer(10) for _ in range(1000)]])\n",
    "plot_counts(cnt, 'Frequency of partitions using hash mod 256')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [],
   "source": [
    "def lmer_to_int(mer):\n",
    "    \"\"\" Maps AAAA to 0, AAAC to 1, etc.  Works for any length argument. \"\"\"\n",
    "    cum = 0\n",
    "    charmap = {'A':0, 'C':1, 'G':2, 'T':3}\n",
    "    for c in mer:\n",
    "        cum *= 4\n",
    "        cum += charmap[c]\n",
    "    return cum"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# get minimal 4-mers from 1000 random 10-mers\n",
    "cnt = Counter([lmer_to_int(minimizer(s, 4)) for s in [random_kmer(10) for _ in range(1000)]])\n",
    "plot_counts(cnt, 'Frequency of partitions using minimal 4-mer; AAAA at left, TTTT at right')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### TODO: Salting minimizers"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### TODO: Making minimizer partitionings more uniform"
   ]
  }
 ],
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   "name": "python3"
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