{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "ab5cd7d1-1a57-46fc-8282-dae0a6cc2944",
   "metadata": {},
   "outputs": [],
   "source": [
    "import matplotlib.pyplot as plt\n",
    "import numpy as np\n",
    "import random"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "3d1ad0b9-f6a8-4e98-84aa-6e02e4279954",
   "metadata": {},
   "outputs": [],
   "source": [
    "DATABASE_OBJECT_COUNT = 100\n",
    "CACHE_SIZE = DATABASE_OBJECT_COUNT/2\n",
    "ZIPF_CONSTANT = 2\n",
    "\n",
    "CACHE_MISS_COST = 1\n",
    "CACHE_REFRESH_COST = 1\n",
    "\n",
    "SEED = 42\n",
    "np.random.seed(SEED)\n",
    "random.seed(SEED)\n",
    "\n",
    "LAMBDA_VALUES = np.array([np.random.zipf(ZIPF_CONSTANT) for i in np.arange(1, DATABASE_OBJECT_COUNT + 1,1)])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "9cc83cf6-5c78-4f0d-b7cb-08cdb80c362e",
   "metadata": {},
   "outputs": [],
   "source": [
    "# LAMBDA_VALUES = np.array([0.03, 0.04,0.05,0.06,0.07,1,1.1,1.2,1.3,1.4,1.5])\n",
    "# DATABASE_OBJECT_COUNT = len(LAMBDA_VALUES)\n",
    "# CACHE_SIZE = 4.4\n",
    "# CACHE_MISS_COST = 7\n",
    "# CACHE_REFRESH_COST = 1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "3dc07233-0b56-4fee-a93b-212836c18b42",
   "metadata": {},
   "outputs": [],
   "source": [
    "db_object_count = DATABASE_OBJECT_COUNT\n",
    "cache_sz = CACHE_SIZE\n",
    "\n",
    "lambda_vals = LAMBDA_VALUES\n",
    "c_f = CACHE_MISS_COST\n",
    "c_delta = CACHE_REFRESH_COST"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "5a27d416-8f98-4814-af9e-6c6bef95f4ef",
   "metadata": {},
   "outputs": [],
   "source": [
    "def eta_star(db_object_count, c_f, cache_sz, c_delta, lambda_vals):\n",
    "    num = (db_object_count * c_f - cache_sz * c_delta)\n",
    "    denom = np.sum(1.0/lambda_vals)\n",
    "    return max(0, num/denom)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "6276a9ce-f839-4fe6-90f2-2195cf065fc8",
   "metadata": {},
   "outputs": [],
   "source": [
    "def h_i_star(c_f, eta, lambda_vals, c_delta):\n",
    "    optimized_hitrate = (c_f - (eta/lambda_vals)) / c_delta\n",
    "    return optimized_hitrate"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "dcd31a8c-6864-4b9a-8bb3-998f0c32baf6",
   "metadata": {},
   "outputs": [],
   "source": [
    "def get_index_of_furthest_hitrate_from_boundary(hitrates):\n",
    "    local_hitrates = hitrates[(hitrates < 0) | (hitrates > 1)]\n",
    "    smallest_delta = np.abs(np.min(local_hitrates))\n",
    "    biggest_delta = np.max(local_hitrates) - 1\n",
    "    if smallest_delta > biggest_delta:\n",
    "        index = np.where(hitrates == np.min(local_hitrates))[0][0]\n",
    "        return index\n",
    "    else:\n",
    "        index = np.where(hitrates == np.max(local_hitrates))[0][0]\n",
    "        return index"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "id": "ccd4b95d-1cdd-4c99-a22e-4b31338993cf",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([0.30256805, 0.76752268, 0.30256805, 0.30256805, 0.65128403,\n",
       "       0.30256805, 0.86051361, 0.30256805, 0.30256805, 0.30256805,\n",
       "       0.65128403, 0.30256805, 0.30256805, 0.30256805, 0.65128403,\n",
       "       0.65128403, 0.30256805, 0.30256805, 0.76752268, 0.30256805,\n",
       "       0.30256805, 0.30256805, 0.30256805, 0.65128403, 0.30256805,\n",
       "       0.30256805, 0.30256805, 0.86051361, 0.30256805, 0.30256805,\n",
       "       0.30256805, 0.82564201, 0.30256805, 0.82564201, 0.30256805,\n",
       "       0.30256805, 0.30256805, 0.76752268, 0.91282101, 0.30256805,\n",
       "       0.82564201, 0.82564201, 0.65128403, 0.30256805, 0.30256805,\n",
       "       0.30256805, 0.93025681, 0.30256805, 0.30256805, 0.30256805,\n",
       "       0.86051361, 0.92250756, 0.30256805, 0.30256805, 0.30256805,\n",
       "       0.30256805, 0.30256805, 0.95897459, 0.65128403, 0.30256805,\n",
       "       0.97317569, 0.30256805, 0.30256805, 0.65128403, 0.30256805,\n",
       "       0.93025681, 0.30256805, 0.98989229, 0.30256805, 0.30256805,\n",
       "       0.65128403, 0.30256805, 0.30256805, 0.30256805, 0.76752268,\n",
       "       0.65128403, 0.65128403, 0.76752268, 0.95350454, 0.30256805,\n",
       "       0.30256805, 0.86051361, 0.65128403, 0.30256805, 0.30256805,\n",
       "       0.65128403, 0.30256805, 0.65128403, 0.30256805, 0.30256805,\n",
       "       0.65128403, 0.65128403, 0.76752268, 0.30256805, 0.65128403,\n",
       "       0.30256805, 0.30256805, 0.98115049, 0.82564201, 0.65128403])"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "eta = eta_star(db_object_count, c_f, cache_sz, c_delta, lambda_vals)\n",
    "optimized_hitrates = (c_f - eta / lambda_vals) / c_delta\n",
    "optimized_hitrates"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "id": "0e21c26f-058a-4e56-a5ad-1c47bf28656c",
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "All values optimized.\n"
     ]
    }
   ],
   "source": [
    "\"\"\"\n",
    "Perform theoretical optimization to compute optimal hit probabilities.\n",
    "\n",
    "Parameters:\n",
    "- lambda_vals (numpy array): Request rates for each item.\n",
    "- B (float): Total cache size.\n",
    "- c_f (float): Fetching linear cost (cache miss cost).\n",
    "- c_delta (float): Age linear cost.\n",
    "\n",
    "Returns:\n",
    "- h_opt (numpy array): Optimal hit probabilities for each item.\n",
    "\"\"\"\n",
    "optimized_hitrates = np.zeros(DATABASE_OBJECT_COUNT)\n",
    "current_db_object_count = DATABASE_OBJECT_COUNT\n",
    "current_cache_size = CACHE_SIZE\n",
    "\n",
    "differenc_set = np.arange(DATABASE_OBJECT_COUNT)\n",
    "fix_i = []\n",
    "\n",
    "while True:\n",
    "    if current_db_object_count == 0:\n",
    "        print(\"No objects left to optimize.\")\n",
    "        if current_cache_size > 0:\n",
    "            # Redistribute unused cache size among items with zero hit probability\n",
    "            differenc_set = np.where(optimized_hitrates == 0)[0]\n",
    "            fix_i = np.setdiff1d(np.arange(DATABASE_OBJECT_COUNT), differenc_set)\n",
    "            current_db_object_count = len(differenc_set)\n",
    "            continue\n",
    "        else:\n",
    "            optimized_hitrates[differenc_set] = 0\n",
    "            break\n",
    "    # Compute Lagrangian multiplier and optimal hit probabilities\n",
    "    eta = eta_star(current_db_object_count, c_f, current_cache_size, c_delta, lambda_vals[differenc_set])\n",
    "    optimized_hitrates[differenc_set] = (c_f - eta / lambda_vals[differenc_set]) / c_delta\n",
    "\n",
    "    if eta < 0:\n",
    "        print(\"eta was negative.\")\n",
    "        current_cache_size = current_db_object_count * c_f / c_delta  # Adjust cache size for next iteration\n",
    "        continue\n",
    "    \n",
    "    if len((optimized_hitrates[differenc_set])[((optimized_hitrates[differenc_set]) < 0) | ((optimized_hitrates[differenc_set])> 1)]) == 0:\n",
    "        print(\"All values optimized.\")\n",
    "        break\n",
    "    \n",
    "    max_outbound_index = get_index_of_furthest_hitrate_from_boundary(optimized_hitrates)\n",
    "    optimized_hitrates[max_outbound_index] = (1 if optimized_hitrates[max_outbound_index] > 1 else 0)\n",
    "\n",
    "    current_cache_size =- optimized_hitrates[max_outbound_index]\n",
    "    fix_i.append(max_outbound_index)\n",
    "    differenc_set = np.setdiff1d(np.arange(DATABASE_OBJECT_COUNT), fix_i)\n",
    "    current_db_object_count -= 1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "11682b36-e705-4bd9-9d75-79012791d1ee",
   "metadata": {},
   "outputs": [],
   "source": []
  }
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