{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Το πρόβλημα της μέγιστης υποακολουθίας"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Τελευταία ενημέρωση: 2/10/2024"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "def brute_force(a_list):\n",
    "    \"\"\"\n",
    "    Find max subarray from a_list using brute force.\n",
    "    \"\"\"\n",
    "\n",
    "    # initialize returns\n",
    "    max_sum, start, end = a_list[0], 0, 0\n",
    "\n",
    "    # brute force\n",
    "    for i in range(len(a_list)):\n",
    "        for j in range(i, len(a_list)):\n",
    "            total = 0\n",
    "            for x in range(i, j + 1):\n",
    "                total += a_list[x]\n",
    "            # replace start and end positions if a larger sum is found\n",
    "            if max_sum < total:\n",
    "                max_sum, start, end = total, i, j\n",
    "\n",
    "    return max_sum, start, end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "max_sum=3623, start=389, end=718\n",
      "max_sum=1826, start=707, end=987\n",
      "max_sum=1084, start=118, end=280\n",
      "max_sum=1905, start=718, end=994\n",
      "max_sum=1996, start=63, end=305\n",
      "max_sum=2379, start=321, end=891\n",
      "max_sum=3150, start=22, end=806\n",
      "max_sum=2932, start=415, end=960\n",
      "5.23 s ± 65.6 ms per loop (mean ± std. dev. of 7 runs, 1 loop each)\n"
     ]
    }
   ],
   "source": [
    "%%timeit\n",
    "import random\n",
    "\n",
    "a_list = [random.randint(-100, 100) for _ in range(1000)]\n",
    "max_sum, start, end = brute_force(a_list)\n",
    "print(f\"{max_sum=}, {start=}, {end=}\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "def prefix_sums(a_list):\n",
    "    \"\"\"\n",
    "    Find max subarray from a_list using prefix sums.\n",
    "    \"\"\"\n",
    "    # initialize returns\n",
    "    max_sum, start, end = a_list[0], 0, 0\n",
    "\n",
    "    # prefix table generation\n",
    "    prefix_table = list()\n",
    "    for index, num in enumerate(a_list):\n",
    "        prefix_table.append(num if index == 0 else num + prefix_table[index - 1])\n",
    "\n",
    "    # use prefix table\n",
    "    for i in range(len(a_list)):\n",
    "        for j in range(i, len(a_list)):\n",
    "            total = prefix_table[j] if i == 0 else prefix_table[j] - prefix_table[i - 1]\n",
    "            # replace starting and ending positions if a larger sum is found\n",
    "            if max_sum < total:\n",
    "                max_sum, start, end = total, i, j\n",
    "\n",
    "    return max_sum, start, end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "max_sum=3446, start=415, end=964\n",
      "max_sum=2905, start=673, end=943\n",
      "max_sum=3566, start=119, end=568\n",
      "max_sum=1206, start=252, end=412\n",
      "max_sum=2941, start=23, end=414\n",
      "max_sum=1544, start=282, end=445\n",
      "max_sum=2676, start=4, end=501\n",
      "max_sum=2936, start=76, end=640\n",
      "max_sum=1973, start=91, end=522\n",
      "max_sum=3105, start=37, end=522\n",
      "max_sum=2506, start=283, end=762\n",
      "max_sum=3677, start=10, end=964\n",
      "max_sum=2374, start=573, end=994\n",
      "max_sum=5604, start=41, end=948\n",
      "max_sum=2250, start=561, end=998\n",
      "max_sum=3347, start=166, end=963\n",
      "max_sum=1944, start=35, end=422\n",
      "max_sum=3762, start=12, end=589\n",
      "max_sum=3074, start=36, end=645\n",
      "max_sum=2048, start=41, end=617\n",
      "max_sum=3202, start=79, end=949\n",
      "max_sum=1149, start=26, end=77\n",
      "max_sum=2207, start=278, end=438\n",
      "max_sum=1587, start=741, end=969\n",
      "max_sum=3600, start=112, end=626\n",
      "max_sum=1752, start=318, end=988\n",
      "max_sum=1772, start=721, end=907\n",
      "max_sum=1995, start=272, end=554\n",
      "max_sum=1597, start=735, end=978\n",
      "max_sum=1809, start=28, end=223\n",
      "max_sum=3462, start=6, end=999\n",
      "max_sum=1958, start=81, end=347\n",
      "max_sum=4071, start=36, end=865\n",
      "max_sum=2720, start=153, end=469\n",
      "max_sum=1099, start=396, end=647\n",
      "max_sum=1832, start=183, end=318\n",
      "max_sum=2810, start=207, end=821\n",
      "max_sum=948, start=846, end=999\n",
      "max_sum=3954, start=30, end=987\n",
      "max_sum=1834, start=52, end=311\n",
      "max_sum=2677, start=236, end=966\n",
      "max_sum=5766, start=5, end=955\n",
      "max_sum=1852, start=245, end=376\n",
      "max_sum=4198, start=632, end=981\n",
      "max_sum=2057, start=572, end=826\n",
      "max_sum=2188, start=600, end=925\n",
      "max_sum=1789, start=255, end=831\n",
      "max_sum=1117, start=25, end=129\n",
      "max_sum=4007, start=322, end=635\n",
      "max_sum=1664, start=787, end=974\n",
      "max_sum=3202, start=179, end=944\n",
      "max_sum=1466, start=16, end=497\n",
      "max_sum=1373, start=437, end=856\n",
      "max_sum=2815, start=25, end=551\n",
      "max_sum=1848, start=3, end=263\n",
      "max_sum=1054, start=377, end=525\n",
      "max_sum=2036, start=301, end=510\n",
      "max_sum=2670, start=40, end=221\n",
      "max_sum=1359, start=562, end=797\n",
      "max_sum=1459, start=447, end=554\n",
      "max_sum=2516, start=400, end=572\n",
      "max_sum=1752, start=267, end=718\n",
      "max_sum=4160, start=43, end=707\n",
      "max_sum=561, start=415, end=495\n",
      "max_sum=2009, start=87, end=526\n",
      "max_sum=3805, start=141, end=978\n",
      "max_sum=3057, start=182, end=990\n",
      "max_sum=2348, start=22, end=225\n",
      "max_sum=3746, start=317, end=999\n",
      "max_sum=1514, start=24, end=187\n",
      "max_sum=2120, start=67, end=402\n",
      "max_sum=2784, start=87, end=653\n",
      "max_sum=2766, start=638, end=953\n",
      "max_sum=1781, start=314, end=983\n",
      "max_sum=3129, start=101, end=676\n",
      "max_sum=3045, start=1, end=539\n",
      "max_sum=1612, start=679, end=878\n",
      "max_sum=1970, start=493, end=628\n",
      "max_sum=1206, start=93, end=230\n",
      "max_sum=967, start=850, end=973\n",
      "max_sum=3217, start=86, end=605\n",
      "34.3 ms ± 236 µs per loop (mean ± std. dev. of 7 runs, 10 loops each)\n"
     ]
    }
   ],
   "source": [
    "%%timeit\n",
    "import random\n",
    "\n",
    "a_list = [random.randint(-100, 100) for _ in range(1000)]\n",
    "max_sum, start, end = prefix_sums(a_list)\n",
    "print(f\"{max_sum=}, {start=}, {end=}\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [],
   "source": [
    "def kadane(a_list):\n",
    "    \"\"\"\n",
    "    Find max subarray from a_list using kadane's algorithm (max suffix sums).\n",
    "    \"\"\"\n",
    "    # initialize returns\n",
    "    max_sum, start, end = a_list[0], 0, 0\n",
    "\n",
    "    # generate kadane table\n",
    "    kadane_table = list()\n",
    "    for index, num in enumerate(a_list):\n",
    "        total = num if index == 0 else num + kadane_table[index - 1]\n",
    "        kadane_table.append(0 if total < 0 else total)\n",
    "\n",
    "    # find max and ending position\n",
    "    for index, total in enumerate(kadane_table):\n",
    "        if max_sum < total:\n",
    "            max_sum, end = total, index\n",
    "\n",
    "    # find starting position\n",
    "    for i in range(end, 0, -1):\n",
    "        if kadane_table[i] == 0:\n",
    "            start = i + 1\n",
    "            break\n",
    "\n",
    "    return max_sum, start, end\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "max_sum=2076, start=429, end=790\n",
      "max_sum=978, start=380, end=430\n",
      "max_sum=1464, start=69, end=214\n",
      "max_sum=1738, start=0, end=209\n",
      "max_sum=2002, start=115, end=575\n",
      "max_sum=2208, start=569, end=977\n",
      "max_sum=2925, start=9, end=841\n",
      "max_sum=1884, start=548, end=963\n",
      "max_sum=2129, start=406, end=998\n",
      "max_sum=1210, start=577, end=691\n",
      "max_sum=1989, start=6, end=942\n",
      "max_sum=1048, start=2, end=123\n",
      "max_sum=1842, start=760, end=938\n",
      "max_sum=1759, start=690, end=804\n",
      "max_sum=1924, start=642, end=873\n",
      "max_sum=2773, start=347, end=767\n",
      "max_sum=3059, start=27, end=949\n",
      "max_sum=1119, start=672, end=755\n",
      "max_sum=2586, start=314, end=994\n",
      "max_sum=2701, start=307, end=998\n",
      "max_sum=1741, start=677, end=990\n",
      "max_sum=2605, start=146, end=792\n",
      "max_sum=1061, start=748, end=792\n",
      "max_sum=1559, start=462, end=775\n",
      "max_sum=2797, start=147, end=999\n",
      "max_sum=2737, start=190, end=915\n",
      "max_sum=2156, start=529, end=951\n",
      "max_sum=3621, start=250, end=999\n",
      "max_sum=3132, start=2, end=668\n",
      "max_sum=2498, start=352, end=593\n",
      "max_sum=1864, start=40, end=574\n",
      "max_sum=1193, start=744, end=997\n",
      "max_sum=2837, start=6, end=960\n",
      "max_sum=2066, start=175, end=975\n",
      "max_sum=1953, start=348, end=521\n",
      "max_sum=1682, start=0, end=249\n",
      "max_sum=1674, start=109, end=300\n",
      "max_sum=3701, start=89, end=987\n",
      "max_sum=1516, start=663, end=847\n",
      "max_sum=1316, start=684, end=894\n",
      "max_sum=2312, start=78, end=970\n",
      "max_sum=4363, start=390, end=999\n",
      "max_sum=2682, start=494, end=925\n",
      "max_sum=1788, start=125, end=337\n",
      "max_sum=772, start=755, end=878\n",
      "max_sum=1646, start=307, end=413\n",
      "max_sum=2977, start=334, end=828\n",
      "max_sum=1751, start=732, end=896\n",
      "max_sum=3309, start=126, end=869\n",
      "max_sum=1635, start=682, end=975\n",
      "max_sum=1240, start=277, end=428\n",
      "max_sum=1450, start=38, end=521\n",
      "max_sum=885, start=32, end=143\n",
      "max_sum=3517, start=409, end=891\n",
      "max_sum=1593, start=785, end=999\n",
      "max_sum=3510, start=203, end=956\n",
      "max_sum=1081, start=295, end=382\n",
      "max_sum=2577, start=178, end=503\n",
      "max_sum=3493, start=113, end=368\n",
      "max_sum=2692, start=632, end=975\n",
      "max_sum=1505, start=629, end=919\n",
      "max_sum=1351, start=303, end=510\n",
      "max_sum=1746, start=17, end=156\n",
      "max_sum=1832, start=748, end=931\n",
      "max_sum=2485, start=418, end=999\n",
      "max_sum=1649, start=626, end=930\n",
      "max_sum=2044, start=243, end=452\n",
      "max_sum=1591, start=817, end=999\n",
      "max_sum=2353, start=9, end=799\n",
      "max_sum=3170, start=84, end=968\n",
      "max_sum=2779, start=73, end=738\n",
      "max_sum=5266, start=198, end=956\n",
      "max_sum=2507, start=455, end=756\n",
      "max_sum=2408, start=86, end=509\n",
      "max_sum=984, start=462, end=627\n",
      "max_sum=2162, start=373, end=884\n",
      "max_sum=5022, start=108, end=986\n",
      "max_sum=3282, start=125, end=995\n",
      "max_sum=1915, start=160, end=417\n",
      "max_sum=4620, start=184, end=913\n",
      "max_sum=1058, start=437, end=492\n",
      "max_sum=3253, start=402, end=995\n",
      "max_sum=2998, start=112, end=979\n",
      "max_sum=2106, start=11, end=650\n",
      "max_sum=1578, start=436, end=592\n",
      "max_sum=2235, start=416, end=954\n",
      "max_sum=2321, start=192, end=562\n",
      "max_sum=2545, start=697, end=994\n",
      "max_sum=1898, start=4, end=336\n",
      "max_sum=3417, start=10, end=793\n",
      "max_sum=2860, start=422, end=843\n",
      "max_sum=1689, start=442, end=833\n",
      "max_sum=1155, start=563, end=712\n",
      "max_sum=2554, start=12, end=327\n",
      "max_sum=3795, start=480, end=802\n",
      "max_sum=2319, start=728, end=993\n",
      "max_sum=2724, start=347, end=919\n",
      "max_sum=2107, start=777, end=984\n",
      "max_sum=1442, start=792, end=984\n",
      "max_sum=1684, start=0, end=160\n",
      "max_sum=2689, start=43, end=998\n",
      "max_sum=1439, start=31, end=211\n",
      "max_sum=3445, start=132, end=990\n",
      "max_sum=2162, start=284, end=840\n",
      "max_sum=1273, start=96, end=292\n",
      "max_sum=4872, start=90, end=968\n",
      "max_sum=2685, start=485, end=813\n",
      "max_sum=3622, start=27, end=627\n",
      "max_sum=1498, start=936, end=997\n",
      "max_sum=3223, start=461, end=820\n",
      "max_sum=3654, start=139, end=940\n",
      "max_sum=3492, start=203, end=795\n",
      "max_sum=2013, start=439, end=963\n",
      "max_sum=1421, start=464, end=576\n",
      "max_sum=1323, start=351, end=414\n",
      "max_sum=3335, start=770, end=983\n",
      "max_sum=2295, start=0, end=708\n",
      "max_sum=3483, start=37, end=983\n",
      "max_sum=1946, start=526, end=742\n",
      "max_sum=2011, start=15, end=426\n",
      "max_sum=1935, start=238, end=483\n",
      "max_sum=2120, start=0, end=907\n",
      "max_sum=1758, start=514, end=925\n",
      "max_sum=1322, start=681, end=826\n",
      "max_sum=1228, start=776, end=926\n",
      "max_sum=1484, start=745, end=869\n",
      "max_sum=2361, start=61, end=869\n",
      "max_sum=2958, start=110, end=854\n",
      "max_sum=2536, start=8, end=566\n",
      "max_sum=2129, start=151, end=999\n",
      "max_sum=1433, start=162, end=317\n",
      "max_sum=5526, start=74, end=964\n",
      "max_sum=1208, start=269, end=398\n",
      "max_sum=2157, start=11, end=324\n",
      "max_sum=3094, start=72, end=705\n",
      "max_sum=3057, start=10, end=999\n",
      "max_sum=2016, start=512, end=799\n",
      "max_sum=2609, start=478, end=784\n",
      "max_sum=2003, start=334, end=685\n",
      "max_sum=2330, start=770, end=984\n",
      "max_sum=1960, start=553, end=804\n",
      "max_sum=1966, start=302, end=704\n",
      "max_sum=1590, start=153, end=384\n",
      "max_sum=2084, start=567, end=935\n",
      "max_sum=1199, start=644, end=802\n",
      "max_sum=2381, start=256, end=501\n",
      "max_sum=3020, start=53, end=836\n",
      "max_sum=2689, start=164, end=677\n",
      "max_sum=5646, start=47, end=842\n",
      "max_sum=2571, start=360, end=913\n",
      "max_sum=808, start=799, end=856\n",
      "max_sum=2473, start=237, end=949\n",
      "max_sum=2000, start=121, end=400\n",
      "max_sum=4150, start=296, end=961\n",
      "max_sum=986, start=0, end=80\n",
      "max_sum=3104, start=647, end=923\n",
      "max_sum=2239, start=348, end=989\n",
      "max_sum=1105, start=419, end=504\n",
      "max_sum=5420, start=0, end=982\n",
      "max_sum=1396, start=537, end=706\n",
      "max_sum=2650, start=324, end=843\n",
      "max_sum=3212, start=258, end=514\n",
      "max_sum=3219, start=77, end=902\n",
      "max_sum=2531, start=20, end=467\n",
      "max_sum=2355, start=182, end=484\n",
      "max_sum=1819, start=14, end=150\n",
      "max_sum=1027, start=563, end=663\n",
      "max_sum=1614, start=702, end=961\n",
      "max_sum=3881, start=26, end=738\n",
      "max_sum=2134, start=542, end=998\n",
      "max_sum=1143, start=779, end=964\n",
      "max_sum=3431, start=73, end=999\n",
      "max_sum=4364, start=618, end=996\n",
      "max_sum=3059, start=74, end=997\n",
      "max_sum=2499, start=323, end=956\n",
      "max_sum=2919, start=90, end=984\n",
      "max_sum=1987, start=2, end=308\n",
      "max_sum=2784, start=383, end=855\n",
      "max_sum=3259, start=91, end=636\n",
      "max_sum=1274, start=697, end=785\n",
      "max_sum=4134, start=87, end=699\n",
      "max_sum=1290, start=160, end=391\n",
      "max_sum=3412, start=29, end=682\n",
      "max_sum=2161, start=456, end=864\n",
      "max_sum=2192, start=450, end=909\n",
      "max_sum=3224, start=216, end=929\n",
      "max_sum=1198, start=250, end=444\n",
      "max_sum=1494, start=0, end=133\n",
      "max_sum=2614, start=20, end=886\n",
      "max_sum=2708, start=13, end=393\n",
      "max_sum=2195, start=440, end=839\n",
      "max_sum=791, start=355, end=394\n",
      "max_sum=3184, start=133, end=996\n",
      "max_sum=1498, start=288, end=450\n",
      "max_sum=2610, start=581, end=987\n",
      "max_sum=2891, start=135, end=769\n",
      "max_sum=2143, start=151, end=972\n",
      "max_sum=1614, start=508, end=650\n",
      "max_sum=3078, start=88, end=952\n",
      "max_sum=1612, start=14, end=959\n",
      "max_sum=1750, start=736, end=949\n",
      "max_sum=1973, start=366, end=712\n",
      "max_sum=1454, start=456, end=786\n",
      "max_sum=2710, start=3, end=941\n",
      "max_sum=1920, start=126, end=997\n",
      "max_sum=1911, start=409, end=673\n",
      "max_sum=1692, start=625, end=840\n",
      "max_sum=3289, start=151, end=623\n",
      "max_sum=5510, start=0, end=946\n",
      "max_sum=2078, start=510, end=679\n",
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      "max_sum=2328, start=624, end=964\n",
      "max_sum=1179, start=768, end=974\n",
      "max_sum=2378, start=391, end=999\n",
      "max_sum=2901, start=246, end=793\n",
      "max_sum=1402, start=0, end=616\n",
      "max_sum=1343, start=104, end=709\n",
      "max_sum=1984, start=118, end=973\n",
      "max_sum=2773, start=84, end=665\n",
      "max_sum=4082, start=24, end=826\n",
      "max_sum=1234, start=784, end=899\n",
      "max_sum=3541, start=66, end=848\n",
      "max_sum=2185, start=353, end=553\n",
      "max_sum=1681, start=850, end=967\n",
      "max_sum=1884, start=24, end=299\n",
      "max_sum=1646, start=543, end=671\n",
      "max_sum=2926, start=238, end=534\n",
      "max_sum=2262, start=515, end=990\n",
      "max_sum=4957, start=149, end=998\n",
      "max_sum=3237, start=555, end=821\n",
      "max_sum=1385, start=588, end=983\n",
      "max_sum=1843, start=124, end=210\n",
      "max_sum=1849, start=675, end=894\n",
      "max_sum=2568, start=276, end=716\n",
      "max_sum=3276, start=2, end=564\n",
      "max_sum=1690, start=697, end=928\n",
      "max_sum=3265, start=72, end=417\n",
      "max_sum=835, start=600, end=616\n",
      "max_sum=1452, start=873, end=989\n",
      "max_sum=2032, start=35, end=369\n",
      "max_sum=2111, start=209, end=830\n",
      "max_sum=1615, start=280, end=874\n",
      "max_sum=810, start=185, end=235\n",
      "max_sum=1108, start=589, end=692\n",
      "max_sum=1125, start=749, end=821\n",
      "max_sum=946, start=47, end=198\n",
      "max_sum=2952, start=14, end=982\n",
      "max_sum=1476, start=599, end=886\n",
      "max_sum=2489, start=50, end=719\n",
      "max_sum=2651, start=279, end=542\n",
      "max_sum=1809, start=626, end=813\n",
      "max_sum=1092, start=275, end=457\n",
      "max_sum=1546, start=196, end=338\n",
      "max_sum=1170, start=886, end=999\n",
      "max_sum=1907, start=421, end=795\n",
      "max_sum=2122, start=126, end=659\n",
      "max_sum=2319, start=114, end=450\n",
      "max_sum=1873, start=42, end=559\n",
      "max_sum=3385, start=227, end=979\n",
      "max_sum=2179, start=590, end=896\n",
      "max_sum=3020, start=143, end=562\n",
      "max_sum=2904, start=391, end=772\n",
      "max_sum=1279, start=227, end=474\n",
      "max_sum=1388, start=412, end=562\n",
      "max_sum=3467, start=70, end=753\n",
      "max_sum=1022, start=561, end=671\n",
      "max_sum=2423, start=223, end=569\n",
      "max_sum=2375, start=388, end=706\n",
      "max_sum=3621, start=401, end=825\n",
      "max_sum=1666, start=200, end=852\n",
      "max_sum=2161, start=122, end=465\n",
      "max_sum=2051, start=376, end=837\n",
      "max_sum=1549, start=764, end=942\n",
      "max_sum=3441, start=17, end=999\n",
      "max_sum=1821, start=0, end=654\n",
      "max_sum=1733, start=804, end=998\n",
      "max_sum=1811, start=623, end=857\n",
      "max_sum=1533, start=2, end=86\n",
      "max_sum=1653, start=6, end=760\n",
      "max_sum=2016, start=58, end=410\n",
      "max_sum=3026, start=161, end=703\n",
      "max_sum=2555, start=189, end=721\n",
      "max_sum=3489, start=125, end=817\n",
      "max_sum=3094, start=23, end=708\n",
      "max_sum=3515, start=177, end=995\n",
      "max_sum=1763, start=120, end=340\n",
      "max_sum=2179, start=9, end=495\n",
      "max_sum=1548, start=806, end=924\n",
      "max_sum=1966, start=131, end=420\n",
      "max_sum=1725, start=30, end=654\n",
      "max_sum=2772, start=125, end=938\n",
      "max_sum=1805, start=722, end=998\n",
      "max_sum=1698, start=811, end=969\n",
      "max_sum=4052, start=0, end=918\n",
      "max_sum=1968, start=689, end=816\n",
      "max_sum=1803, start=720, end=979\n",
      "max_sum=1835, start=730, end=893\n",
      "max_sum=4084, start=0, end=962\n",
      "max_sum=1533, start=250, end=995\n",
      "max_sum=1253, start=123, end=222\n",
      "max_sum=1553, start=338, end=610\n",
      "max_sum=2537, start=25, end=988\n",
      "max_sum=1629, start=645, end=815\n",
      "max_sum=4341, start=29, end=900\n",
      "max_sum=2199, start=297, end=992\n",
      "max_sum=1225, start=37, end=560\n",
      "max_sum=1737, start=431, end=535\n",
      "max_sum=2780, start=340, end=880\n",
      "max_sum=3888, start=147, end=999\n",
      "max_sum=3304, start=144, end=909\n",
      "max_sum=1750, start=448, end=692\n",
      "max_sum=1270, start=637, end=827\n",
      "max_sum=3390, start=112, end=386\n",
      "max_sum=1832, start=277, end=486\n",
      "max_sum=2336, start=69, end=905\n",
      "max_sum=2283, start=191, end=412\n",
      "max_sum=2105, start=56, end=486\n",
      "max_sum=1340, start=586, end=980\n",
      "max_sum=2003, start=700, end=922\n",
      "max_sum=2180, start=611, end=969\n",
      "max_sum=3395, start=122, end=612\n",
      "max_sum=4158, start=50, end=826\n",
      "max_sum=2326, start=0, end=173\n",
      "max_sum=1655, start=709, end=841\n",
      "max_sum=1309, start=128, end=344\n",
      "max_sum=2137, start=746, end=980\n",
      "max_sum=3731, start=213, end=660\n",
      "max_sum=2562, start=80, end=978\n",
      "max_sum=1973, start=278, end=481\n",
      "max_sum=3495, start=111, end=992\n",
      "max_sum=1670, start=101, end=242\n",
      "max_sum=1551, start=812, end=977\n",
      "max_sum=2669, start=194, end=896\n",
      "max_sum=2652, start=162, end=559\n",
      "max_sum=1173, start=748, end=982\n",
      "max_sum=1204, start=477, end=803\n",
      "max_sum=1348, start=668, end=866\n",
      "max_sum=3931, start=50, end=911\n",
      "max_sum=1549, start=136, end=640\n",
      "max_sum=2493, start=175, end=453\n",
      "max_sum=1805, start=54, end=220\n",
      "max_sum=1518, start=308, end=448\n",
      "max_sum=1502, start=256, end=558\n",
      "max_sum=2974, start=16, end=715\n",
      "max_sum=779, start=556, end=589\n",
      "max_sum=2792, start=7, end=721\n",
      "max_sum=1455, start=592, end=897\n",
      "max_sum=1821, start=752, end=977\n",
      "max_sum=3275, start=332, end=854\n",
      "max_sum=1211, start=438, end=517\n",
      "max_sum=3042, start=122, end=424\n",
      "max_sum=1135, start=556, end=698\n",
      "max_sum=2044, start=257, end=615\n",
      "max_sum=1507, start=350, end=484\n",
      "max_sum=2974, start=122, end=711\n",
      "max_sum=1489, start=583, end=733\n",
      "max_sum=1628, start=118, end=446\n",
      "max_sum=1680, start=693, end=971\n",
      "max_sum=1191, start=117, end=263\n",
      "max_sum=1911, start=529, end=846\n",
      "max_sum=2435, start=445, end=799\n",
      "max_sum=3288, start=148, end=997\n",
      "max_sum=2146, start=318, end=608\n",
      "max_sum=5110, start=111, end=905\n",
      "max_sum=3526, start=15, end=998\n",
      "max_sum=1977, start=9, end=411\n",
      "max_sum=1334, start=21, end=165\n",
      "max_sum=927, start=306, end=427\n",
      "max_sum=2793, start=114, end=485\n",
      "max_sum=1079, start=752, end=868\n",
      "max_sum=2258, start=11, end=326\n",
      "max_sum=2566, start=127, end=764\n",
      "max_sum=2108, start=297, end=998\n",
      "max_sum=2191, start=243, end=971\n",
      "max_sum=1282, start=672, end=799\n",
      "max_sum=3486, start=138, end=664\n",
      "max_sum=1030, start=274, end=409\n",
      "max_sum=1738, start=0, end=146\n",
      "max_sum=1454, start=112, end=797\n",
      "max_sum=2210, start=125, end=674\n",
      "max_sum=2596, start=69, end=291\n",
      "max_sum=1820, start=693, end=837\n",
      "max_sum=2303, start=183, end=893\n",
      "max_sum=1990, start=3, end=286\n",
      "max_sum=1121, start=853, end=988\n",
      "max_sum=2162, start=602, end=988\n",
      "max_sum=2514, start=768, end=998\n",
      "max_sum=3666, start=16, end=977\n",
      "max_sum=1677, start=518, end=715\n",
      "max_sum=1215, start=518, end=569\n",
      "max_sum=4730, start=80, end=876\n",
      "max_sum=2559, start=757, end=976\n",
      "max_sum=1374, start=785, end=947\n",
      "max_sum=2782, start=75, end=680\n",
      "max_sum=2476, start=0, end=310\n",
      "max_sum=1431, start=0, end=534\n",
      "max_sum=1521, start=174, end=543\n",
      "max_sum=3824, start=421, end=930\n",
      "max_sum=1803, start=837, end=998\n",
      "max_sum=3064, start=365, end=896\n",
      "max_sum=3107, start=264, end=788\n",
      "max_sum=3328, start=17, end=732\n",
      "max_sum=1519, start=53, end=379\n",
      "max_sum=5024, start=31, end=951\n",
      "max_sum=1707, start=220, end=999\n",
      "max_sum=2598, start=18, end=451\n",
      "max_sum=2526, start=401, end=998\n",
      "max_sum=1854, start=133, end=399\n",
      "max_sum=2959, start=25, end=793\n",
      "max_sum=1362, start=358, end=555\n",
      "max_sum=4188, start=79, end=900\n",
      "max_sum=2303, start=411, end=998\n",
      "max_sum=2344, start=389, end=721\n",
      "max_sum=1633, start=295, end=408\n",
      "max_sum=1029, start=364, end=515\n",
      "max_sum=1848, start=747, end=951\n",
      "max_sum=4317, start=26, end=837\n",
      "max_sum=1103, start=859, end=906\n",
      "max_sum=2937, start=68, end=984\n",
      "max_sum=1382, start=943, end=999\n",
      "max_sum=3568, start=32, end=718\n",
      "max_sum=1061, start=732, end=973\n",
      "max_sum=1234, start=131, end=253\n",
      "max_sum=3462, start=277, end=954\n",
      "max_sum=2082, start=579, end=870\n",
      "max_sum=1201, start=368, end=710\n",
      "max_sum=809, start=510, end=590\n",
      "479 µs ± 8.49 µs per loop (mean ± std. dev. of 7 runs, 1,000 loops each)\n"
     ]
    }
   ],
   "source": [
    "%%timeit\n",
    "import random\n",
    "\n",
    "a_list = [random.randint(-100, 100) for _ in range(1000)]\n",
    "max_sum, start, end = kadane(a_list)\n",
    "print(f\"{max_sum=}, {start=}, {end=}\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
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