Index Cartesian Product.ipynb

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 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "I want to produce the value at a specific index in the output of a cartesian product, without having to iterate my way to that value. As long as the arrays going into the cartesian product are the same length, this seems to be just a base conversion problem."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "import torch"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "tensor([[0, 0],\n",
      "        [0, 1],\n",
      "        [0, 2],\n",
      "        [1, 0],\n",
      "        [1, 1],\n",
      "        [1, 2],\n",
      "        [2, 0],\n",
      "        [2, 1],\n",
      "        [2, 2]])\n"
     ]
    }
   ],
   "source": [
    "# concretely\n",
    "x = torch.arange(3)\n",
    "y = torch.arange(3)\n",
    "prod = torch.cartesian_prod(x, y)\n",
    "print(prod)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "tensor([1, 2])"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# I want to produce the value at index `i`, without computing the array `prod`\n",
    "i = 5\n",
    "prod[5]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "5 is 12 in base 3."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "tensor([3, 1])\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "tensor([1, 2])"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "b = 3**(1-torch.arange(2))\n",
    "print(b)\n",
    "(i//b)%3"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Defining an arbitrarily large cartesian product:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Length of cartesian product:  13841287201\n"
     ]
    }
   ],
   "source": [
    "k, n = 7, 12\n",
    "tensors = [torch.arange(k) for i in range(n)]\n",
    "print(\"Length of cartesian product: \", k**n)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "tensor([3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3])"
      ]
     },
     "execution_count": 37,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# index element halfway through this cartesian product\n",
    "i = (k**n)//2\n",
    "b = k**((n-1) - torch.arange(n))\n",
    "basek = (i//b)%k\n",
    "basek"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "tensor(0)"
      ]
     },
     "execution_count": 38,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(basek*b).sum() - i"
   ]
  }
 ],
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