{
 "cells": [
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   "cell_type": "markdown",
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   "source": [
    "## Problem statement\n",
    "\n",
    "Q: *Lobb number* — Calculate the Lobb number for particular values of $m$ and $n$:\n",
    "\n",
    "$$L_{m,n} = \\frac{2m + 1}{m + n + 1}{2n \\choose m + n}$$\n",
    "\n",
    "The left term is a constant time operation, whilst the right term is a factorial. Similar to Fibbonacci, we could solve the right term using a brute-force recursive algorithm. But it's much more efficient to reuse computations from one factorial chunk for the other chunks.\n",
    "\n",
    "The factorial form of the combination calculation is:\n",
    "\n",
    "$$\\frac{2n!}{(m + n)! \\times (n - m)!}$$\n",
    "\n",
    "The key insight: if $m \\geq n$, $(m + n)!$ has the most factorial terms. Otherwise $2n!$ does. We only need to calculate one of these factorials, the rest of the factorial values will follow."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Tests"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "{ [Function: ok]\n",
       "  fail: [Function: fail],\n",
       "  AssertionError: [Function: AssertionError],\n",
       "  ok: [Circular],\n",
       "  equal: [Function: equal],\n",
       "  notEqual: [Function: notEqual],\n",
       "  deepEqual: [Function: deepEqual],\n",
       "  deepStrictEqual: [Function: deepStrictEqual],\n",
       "  notDeepEqual: [Function: notDeepEqual],\n",
       "  notDeepStrictEqual: [Function: notDeepStrictEqual],\n",
       "  strictEqual: [Function: strictEqual],\n",
       "  notStrictEqual: [Function: notStrictEqual],\n",
       "  throws: [Function: throws],\n",
       "  doesNotThrow: [Function: doesNotThrow],\n",
       "  ifError: [Function: ifError] }"
      ]
     },
     "execution_count": 1,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "assert = require('assert');\n",
    "\n",
    "// TODO"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Optimized pseudocode\n",
    "\n",
    "```\n",
    "function lobb(m, n):\n",
    "    mult_part = (2*m + 1) / (m + n + 1)\n",
    "\n",
    "    if (m >= n):\n",
    "        maxfact = m + n\n",
    "    else:\n",
    "        maxfact = 2 * n\n",
    "\n",
    "    memo = fact(maxfact)\n",
    "    comb_part = memo[2* n] / (memo[m + n] * memo[2n - m])\n",
    "\n",
    "    return mult_part * comb_part\n",
    "    \n",
    "    \n",
    "function fact(n):\n",
    "    memo = [1, 1]\n",
    "    if n <= 1: return memo[n]\n",
    "    \n",
    "    for (nv in range(2, n + 1)):\n",
    "        memo[nv] = memo[nv - 1] * nv\n",
    "    \n",
    "    return memo[n]\n",
    "```"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Optimized implementation"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "function fact(n) {\n",
    "    let memo = [1, 1];\n",
    "    if (n <= 1) return memo[n];\n",
    "    \n",
    "    for (nv of [...Array(n).keys()].map(v => v + 2)) memo.push(memo[nv - 1] * nv);\n",
    "    \n",
    "    return memo;\n",
    "}\n",
    "\n",
    "function lobb(m, n) {\n",
    "    mult_part = (2*m + 1) / (m + n + 1);\n",
    "    \n",
    "    let maxfact = (m >= n) ? m + n : 2*n;\n",
    "    let memo = fact(maxfact);\n",
    "    comb_part = memo[2* n] / (memo[m + n] * memo[n - m]);\n",
    "    \n",
    "    return mult_part * comb_part;\n",
    "}"
   ]
  }
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