{
"cells": [
{
"cell_type": "markdown",
"metadata": {},
"source": [
"This notebook was prepared by [Donne Martin](https://github.com/donnemartin). Source and license info is on [GitHub](https://github.com/donnemartin/interactive-coding-challenges)."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Solution Notebook"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Problem: Given a list of stock prices on each consecutive day, determine the max profits with k transactions.\n",
"\n",
"* [Constraints](#Constraints)\n",
"* [Test Cases](#Test-Cases)\n",
"* [Algorithm](#Algorithm)\n",
"* [Code](#Code)\n",
"* [Unit Test](#Unit-Test)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Constraints\n",
"\n",
"* Is k the number of sell transactions?\n",
" * Yes\n",
"* Can we assume the prices input is an array of ints?\n",
" * Yes\n",
"* Can we assume the inputs are valid?\n",
" * No\n",
"* If the prices are all decreasing and there is no opportunity to make a profit, do we just return 0?\n",
" * Yes\n",
"* Should the output be the max profit and days to buy and sell?\n",
" * Yes\n",
"* Can we assume this fits memory?\n",
" * Yes"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Test Cases\n",
"\n",
"\n",
"* Prices: None or k: None -> None\n",
"* Prices: [] or k <= 0 -> []\n",
"* Prices: [0, -1, -2, -3, -4, -5]\n",
" * (max profit, list of transactions)\n",
" * (0, [])\n",
"* Prices: [2, 5, 7, 1, 4, 3, 1, 3] k: 3\n",
" * (max profit, list of transactions)\n",
" * (10, [Type.SELL day: 7 price: 3, \n",
" Type.BUY day: 6 price: 1, \n",
" Type.SELL day: 4 price: 4, \n",
" Type.BUY day: 3 price: 1, \n",
" Type.SELL day: 2 price: 7, \n",
" Type.BUY day: 0 price: 2])\n",
""
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Algorithm\n",
"\n",
"We'll use bottom up dynamic programming to build a table.\n",
"\n",
"\n",
"\n",
"The rows (i) represent the prices.\n",
"The columns (j) represent the number of transactions (k).\n",
"\n",
"T[i][j] = max(T[i][j - 1],\n",
" prices[j] - price[m] + T[i - 1][m])\n",
"\n",
"m = 0...j-1\n",
"\n",
" 0 1 2 3 4 5 6 7\n",
"--------------------------------------\n",
"| | 2 | 5 | 7 | 1 | 4 | 3 | 1 | 3 |\n",
"--------------------------------------\n",
"| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |\n",
"| 1 | 0 | 3 | 5 | 5 | 5 | 5 | 5 | 5 |\n",
"| 2 | 0 | 3 | 5 | 5 | 8 | 8 | 8 | 8 |\n",
"| 3 | 0 | 3 | 5 | 5 | 8 | 8 | 8 | 10 |\n",
"--------------------------------------\n",
"\n",
"Optimization:\n",
"\n",
"max_diff = max(max_diff,\n",
" T[i - 1][j - 1] - prices[j - 1])\n",
"\n",
"T[i][j] = max(T[i][j - 1],\n",
" prices[j] + max_diff)\n",
"\n",
"\n",
"\n",
"Complexity:\n",
"* Time: O(n * k)\n",
"* Space: O(n * k)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Code"
]
},
{
"cell_type": "code",
"execution_count": 1,
"metadata": {
"collapsed": false
},
"outputs": [],
"source": [
"from enum import Enum # Python 2 users: Run pip install enum34\n",
"\n",
"\n",
"class Type(Enum):\n",
" SELL = 0\n",
" BUY = 1\n",
"\n",
"\n",
"class Transaction(object):\n",
"\n",
" def __init__(self, type, day, price):\n",
" self.type = type\n",
" self.day = day\n",
" self.price = price\n",
"\n",
" def __eq__(self, other):\n",
" return self.type == other.type and \\\n",
" self.day == other.day and \\\n",
" self.price == other.price\n",
"\n",
" def __repr__(self):\n",
" return str(self.type) + ' day: ' + \\\n",
" str(self.day) + ' price: ' + \\\n",
" str(self.price)"
]
},
{
"cell_type": "code",
"execution_count": 2,
"metadata": {
"collapsed": false
},
"outputs": [],
"source": [
"import sys\n",
"\n",
"\n",
"class StockTrader(object):\n",
"\n",
" def find_max_profit(self, prices, k):\n",
" if prices is None or k is None:\n",
" raise TypeError('prices or k cannot be None')\n",
" if not prices or k <= 0:\n",
" return []\n",
" num_rows = k + 1 # 0th transaction for dp table\n",
" num_cols = len(prices)\n",
" T = [[None] * num_cols for _ in range(num_rows)]\n",
" for i in range(num_rows):\n",
" for j in range(num_cols):\n",
" if i == 0 or j == 0:\n",
" T[i][j] = 0\n",
" continue\n",
" max_profit = -sys.maxsize\n",
" for m in range(j):\n",
" profit = prices[j] - prices[m] + T[i - 1][m]\n",
" if profit > max_profit:\n",
" max_profit = profit\n",
" T[i][j] = max(T[i][j - 1], max_profit)\n",
" return self._find_max_profit_transactions(T, prices)\n",
"\n",
" def find_max_profit_optimized(self, prices, k):\n",
" if prices is None or k is None:\n",
" raise TypeError('prices or k cannot be None')\n",
" if not prices or k <= 0:\n",
" return []\n",
" num_rows = k + 1\n",
" num_cols = len(prices)\n",
" T = [[None] * num_cols for _ in range(num_rows)]\n",
" for i in range(num_rows):\n",
" max_diff = prices[0] * -1\n",
" for j in range(num_cols):\n",
" if i == 0 or j == 0:\n",
" T[i][j] = 0\n",
" continue\n",
" max_diff = max(\n",
" max_diff,\n",
" T[i - 1][j - 1] - prices[j - 1])\n",
" T[i][j] = max(\n",
" T[i][j - 1],\n",
" prices[j] + max_diff)\n",
" return self._find_max_profit_transactions(T, prices)\n",
"\n",
" def _find_max_profit_transactions(self, T, prices):\n",
" results = []\n",
" i = len(T) - 1\n",
" j = len(T[0]) - 1\n",
" max_profit = T[i][j]\n",
" while i != 0 and j != 0:\n",
" if T[i][j] == T[i][j - 1]:\n",
" j -= 1\n",
" else:\n",
" sell_price = prices[j]\n",
" results.append(Transaction(Type.SELL, j, sell_price))\n",
" profit = T[i][j] - T[i - 1][j - 1]\n",
" i -= 1\n",
" j -= 1\n",
" for m in range(j + 1)[::-1]:\n",
" if sell_price - prices[m] == profit:\n",
" results.append(Transaction(Type.BUY, m, prices[m]))\n",
" break\n",
" return (max_profit, results)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Unit Test"
]
},
{
"cell_type": "code",
"execution_count": 3,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Overwriting test_max_profit.py\n"
]
}
],
"source": [
"%%writefile test_max_profit.py\n",
"from nose.tools import assert_equal\n",
"from nose.tools import assert_raises\n",
"from nose.tools import assert_true\n",
"\n",
"\n",
"class TestMaxProfit(object):\n",
"\n",
" def test_max_profit(self):\n",
" stock_trader = StockTrader()\n",
" assert_raises(TypeError, stock_trader.find_max_profit, None, None)\n",
" assert_equal(stock_trader.find_max_profit(prices=[], k=0), [])\n",
" prices = [5, 4, 3, 2, 1]\n",
" k = 3\n",
" assert_equal(stock_trader.find_max_profit(prices, k), (0, []))\n",
" prices = [2, 5, 7, 1, 4, 3, 1, 3]\n",
" profit, transactions = stock_trader.find_max_profit(prices, k)\n",
" assert_equal(profit, 10)\n",
" assert_true(Transaction(Type.SELL,\n",
" day=7,\n",
" price=3) in transactions)\n",
" assert_true(Transaction(Type.BUY,\n",
" day=6,\n",
" price=1) in transactions)\n",
" assert_true(Transaction(Type.SELL,\n",
" day=4,\n",
" price=4) in transactions)\n",
" assert_true(Transaction(Type.BUY,\n",
" day=3,\n",
" price=1) in transactions)\n",
" assert_true(Transaction(Type.SELL,\n",
" day=2,\n",
" price=7) in transactions)\n",
" assert_true(Transaction(Type.BUY,\n",
" day=0,\n",
" price=2) in transactions)\n",
" print('Success: test_max_profit')\n",
"\n",
"\n",
"def main():\n",
" test = TestMaxProfit()\n",
" test.test_max_profit()\n",
"\n",
"\n",
"if __name__ == '__main__':\n",
" main()"
]
},
{
"cell_type": "code",
"execution_count": 4,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Success: test_max_profit\n"
]
}
],
"source": [
"%run -i test_max_profit.py"
]
}
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