{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Game Tree Search\n",
    "\n",
    "We start with defining the abstract class `Game`, for turn-taking *n*-player games. We rely on, but do not define yet, the concept of a `state` of the game; we'll see later how individual games define states. For now, all we require is that a state has a `state.to_move` attribute, which gives the name of the player whose turn it is. (\"Name\" will be something like `'X'` or `'O'` for tic-tac-toe.) \n",
    "\n",
    "We also define `play_game`, which takes a game and a dictionary of  `{player_name: strategy_function}` pairs, and plays out the game, on each turn checking `state.to_move` to see whose turn it is, and then getting the strategy function for that player and applying it to the game and the state to get a move."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "from collections import namedtuple, Counter, defaultdict\n",
    "import random\n",
    "import math\n",
    "import functools \n",
    "cache = functools.lru_cache(10**6)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 73,
   "metadata": {},
   "outputs": [],
   "source": [
    "class Game:\n",
    "    \"\"\"A game is similar to a problem, but it has a terminal test instead of \n",
    "    a goal test, and a utility for each terminal state. To create a game, \n",
    "    subclass this class and implement `actions`, `result`, `is_terminal`, \n",
    "    and `utility`. You will also need to set the .initial attribute to the \n",
    "    initial state; this can be done in the constructor.\"\"\"\n",
    "\n",
    "    def actions(self, state):\n",
    "        \"\"\"Return a collection of the allowable moves from this state.\"\"\"\n",
    "        raise NotImplementedError\n",
    "\n",
    "    def result(self, state, move):\n",
    "        \"\"\"Return the state that results from making a move from a state.\"\"\"\n",
    "        raise NotImplementedError\n",
    "\n",
    "    def is_terminal(self, state):\n",
    "        \"\"\"Return True if this is a final state for the game.\"\"\"\n",
    "        return not self.actions(state)\n",
    "    \n",
    "    def utility(self, state, player):\n",
    "        \"\"\"Return the value of this final state to player.\"\"\"\n",
    "        raise NotImplementedError\n",
    "        \n",
    "\n",
    "def play_game(game, strategies: dict, verbose=False):\n",
    "    \"\"\"Play a turn-taking game. `strategies` is a {player_name: function} dict,\n",
    "    where function(state, game) is used to get the player's move.\"\"\"\n",
    "    state = game.initial\n",
    "    while not game.is_terminal(state):\n",
    "        player = state.to_move\n",
    "        move = strategies[player](game, state)\n",
    "        state = game.result(state, move)\n",
    "        if verbose: \n",
    "            print('Player', player, 'move:', move)\n",
    "            print(state)\n",
    "    return state"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Minimax-Based Game Search Algorithms\n",
    "\n",
    "We will define several game search algorithms. Each takes two inputs, the game we are playing and the current state of the game, and returns a a `(value, move)` pair, where `value` is the utility that the algorithm computes for the player whose turn it is to move, and `move` is the move itself.\n",
    "\n",
    "First we define `minimax_search`, which exhaustively searches the game tree to find an optimal move (assuming both players play optimally), and `alphabeta_search`, which does the same computation, but prunes parts of the tree that could not possibly have an affect on the optimnal move.  "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 66,
   "metadata": {},
   "outputs": [],
   "source": [
    "def minimax_search(game, state):\n",
    "    \"\"\"Search game tree to determine best move; return (value, move) pair.\"\"\"\n",
    "\n",
    "    player = state.to_move\n",
    "\n",
    "    def max_value(state):\n",
    "        if game.is_terminal(state):\n",
    "            return game.utility(state, player), None\n",
    "        v, move = -infinity, None\n",
    "        for a in game.actions(state):\n",
    "            v2, _ = min_value(game.result(state, a))\n",
    "            if v2 > v:\n",
    "                v, move = v2, a\n",
    "        return v, move\n",
    "\n",
    "    def min_value(state):\n",
    "        if game.is_terminal(state):\n",
    "            return game.utility(state, player), None\n",
    "        v, move = +infinity, None\n",
    "        for a in game.actions(state):\n",
    "            v2, _ = max_value(game.result(state, a))\n",
    "            if v2 < v:\n",
    "                v, move = v2, a\n",
    "        return v, move\n",
    "\n",
    "    return max_value(state)\n",
    "\n",
    "infinity = math.inf\n",
    "\n",
    "def alphabeta_search(game, state):\n",
    "    \"\"\"Search game to determine best action; use alpha-beta pruning.\n",
    "    As in [Figure 5.7], this version searches all the way to the leaves.\"\"\"\n",
    "\n",
    "    player = state.to_move\n",
    "\n",
    "    def max_value(state, alpha, beta):\n",
    "        if game.is_terminal(state):\n",
    "            return game.utility(state, player), None\n",
    "        v, move = -infinity, None\n",
    "        for a in game.actions(state):\n",
    "            v2, _ = min_value(game.result(state, a), alpha, beta)\n",
    "            if v2 > v:\n",
    "                v, move = v2, a\n",
    "                alpha = max(alpha, v)\n",
    "            if v >= beta:\n",
    "                return v, move\n",
    "        return v, move\n",
    "\n",
    "    def min_value(state, alpha, beta):\n",
    "        if game.is_terminal(state):\n",
    "            return game.utility(state, player), None\n",
    "        v, move = +infinity, None\n",
    "        for a in game.actions(state):\n",
    "            v2, _ = max_value(game.result(state, a), alpha, beta)\n",
    "            if v2 < v:\n",
    "                v, move = v2, a\n",
    "                beta = min(beta, v)\n",
    "            if v <= alpha:\n",
    "                return v, move\n",
    "        return v, move\n",
    "\n",
    "    return max_value(state, -infinity, +infinity)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# A Simple Game: Tic-Tac-Toe\n",
    "\n",
    "We have the notion of an abstract game, we have some search functions; now it is time to define a real game; a simple one, tic-tac-toe. Moves are `(x, y)` pairs denoting squares, where `(0, 0)` is the top left, and `(2, 2)` is the bottom right (on a board of size `height=width=3`)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 67,
   "metadata": {},
   "outputs": [],
   "source": [
    "class TicTacToe(Game):\n",
    "    \"\"\"Play TicTacToe on an `height` by `width` board, needing `k` in a row to win.\n",
    "    'X' plays first against 'O'.\"\"\"\n",
    "\n",
    "    def __init__(self, height=3, width=3, k=3):\n",
    "        self.k = k # k in a row\n",
    "        self.squares = {(x, y) for x in range(width) for y in range(height)}\n",
    "        self.initial = Board(height=height, width=width, to_move='X', utility=0)\n",
    "\n",
    "    def actions(self, board):\n",
    "        \"\"\"Legal moves are any square not yet taken.\"\"\"\n",
    "        return self.squares - set(board)\n",
    "\n",
    "    def result(self, board, square):\n",
    "        \"\"\"Place a marker for current player on square.\"\"\"\n",
    "        player = board.to_move\n",
    "        board = board.new({square: player}, to_move=('O' if player == 'X' else 'X'))\n",
    "        win = k_in_row(board, player, square, self.k)\n",
    "        board.utility = (0 if not win else +1 if player == 'X' else -1)\n",
    "        return board\n",
    "\n",
    "    def utility(self, board, player):\n",
    "        \"\"\"Return the value to player; 1 for win, -1 for loss, 0 otherwise.\"\"\"\n",
    "        return board.utility if player == 'X' else -board.utility\n",
    "\n",
    "    def is_terminal(self, board):\n",
    "        \"\"\"A board is a terminal state if it is won or there are no empty squares.\"\"\"\n",
    "        return board.utility != 0 or len(self.squares) == len(board)\n",
    "\n",
    "    def display(self, board): print(board)     \n",
    "\n",
    "\n",
    "def k_in_row(board, player, square, k):\n",
    "    \"\"\"True if player has k pieces in a line through square.\"\"\"\n",
    "    def in_row(x, y, dx, dy): return 0 if board[x, y] != player else 1 + in_row(x + dx, y + dy, dx, dy)\n",
    "    return any(in_row(*square, dx, dy) + in_row(*square, -dx, -dy) - 1 >= k\n",
    "               for (dx, dy) in ((0, 1), (1, 0), (1, 1), (1, -1)))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "States in tic-tac-toe (and other games) will be represented as a `Board`, which is a subclass of `defaultdict` that in general will consist of `{(x, y): contents}` pairs, for example `{(0, 0): 'X', (1, 1): 'O'}` might be the state of the board after two moves. Besides the contents of squares, a board also has some attributes: \n",
    "- `.to_move` to name the player whose move it is; \n",
    "- `.width` and `.height` to give the size of the board (both 3 in tic-tac-toe, but other numbers in related games);\n",
    "- possibly other attributes, as specified by keywords. \n",
    "\n",
    "As a `defaultdict`, the `Board` class has a `__missing__` method, which returns `empty` for squares that have no been assigned but are within the `width` × `height` boundaries, or `off` otherwise. The class has a `__hash__` method, so instances can be stored in hash tables."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 68,
   "metadata": {},
   "outputs": [],
   "source": [
    "class Board(defaultdict):\n",
    "    \"\"\"A board has the player to move, a cached utility value, \n",
    "    and a dict of {(x, y): player} entries, where player is 'X' or 'O'.\"\"\"\n",
    "    empty = '.'\n",
    "    off = '#'\n",
    "    \n",
    "    def __init__(self, width=8, height=8, to_move=None, **kwds):\n",
    "        self.__dict__.update(width=width, height=height, to_move=to_move, **kwds)\n",
    "        \n",
    "    def new(self, changes: dict, **kwds) -> 'Board':\n",
    "        \"Given a dict of {(x, y): contents} changes, return a new Board with the changes.\"\n",
    "        board = Board(width=self.width, height=self.height, **kwds)\n",
    "        board.update(self)\n",
    "        board.update(changes)\n",
    "        return board\n",
    "\n",
    "    def __missing__(self, loc):\n",
    "        x, y = loc\n",
    "        if 0 <= x < self.width and 0 <= y < self.height:\n",
    "            return self.empty\n",
    "        else:\n",
    "            return self.off\n",
    "            \n",
    "    def __hash__(self): \n",
    "        return hash(tuple(sorted(self.items()))) + hash(self.to_move)\n",
    "    \n",
    "    def __repr__(self):\n",
    "        def row(y): return ' '.join(self[x, y] for x in range(self.width))\n",
    "        return '\\n'.join(map(row, range(self.height))) +  '\\n'"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Players\n",
    "\n",
    "We need an interface for players. I'll represent a player as a `callable` that will be passed two arguments: `(game, state)` and will return a `move`.\n",
    "The function `player` creates a player out of a search algorithm, but you can create your own players as functions, as is done with `random_player` below:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 69,
   "metadata": {},
   "outputs": [],
   "source": [
    "def random_player(game, state): return random.choice(list(game.actions(state)))\n",
    "\n",
    "def player(search_algorithm):\n",
    "    \"\"\"A game player who uses the specified search algorithm\"\"\"\n",
    "    return lambda game, state: search_algorithm(game, state)[1]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Playing a Game\n",
    "\n",
    "We're ready to play a game. I'll set up a match between a `random_player` (who chooses randomly from the legal moves) and a `player(alphabeta_search)` (who makes the optimal alpha-beta move; practical for tic-tac-toe, but not for large games). The `player(alphabeta_search)` will never lose, but if `random_player` is lucky, it will be a tie."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 74,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Player X move: (0, 0)\n",
      "X . .\n",
      ". . .\n",
      ". . .\n",
      "\n",
      "Player O move: (1, 1)\n",
      "X . .\n",
      ". O .\n",
      ". . .\n",
      "\n",
      "Player X move: (1, 2)\n",
      "X . .\n",
      ". O .\n",
      ". X .\n",
      "\n",
      "Player O move: (0, 1)\n",
      "X . .\n",
      "O O .\n",
      ". X .\n",
      "\n",
      "Player X move: (2, 1)\n",
      "X . .\n",
      "O O X\n",
      ". X .\n",
      "\n",
      "Player O move: (2, 0)\n",
      "X . O\n",
      "O O X\n",
      ". X .\n",
      "\n",
      "Player X move: (2, 2)\n",
      "X . O\n",
      "O O X\n",
      ". X X\n",
      "\n",
      "Player O move: (0, 2)\n",
      "X . O\n",
      "O O X\n",
      "O X X\n",
      "\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "-1"
      ]
     },
     "execution_count": 74,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "play_game(TicTacToe(), dict(X=random_player, O=player(alphabeta_search)), verbose=True).utility"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The alpha-beta player will never lose, but sometimes the random player can stumble into a draw. When two optimal (alpha-beta or minimax) players compete, it will always be a draw:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 75,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Player X move: (0, 1)\n",
      ". . .\n",
      "X . .\n",
      ". . .\n",
      "\n",
      "Player O move: (0, 0)\n",
      "O . .\n",
      "X . .\n",
      ". . .\n",
      "\n",
      "Player X move: (2, 0)\n",
      "O . X\n",
      "X . .\n",
      ". . .\n",
      "\n",
      "Player O move: (2, 1)\n",
      "O . X\n",
      "X . O\n",
      ". . .\n",
      "\n",
      "Player X move: (1, 2)\n",
      "O . X\n",
      "X . O\n",
      ". X .\n",
      "\n",
      "Player O move: (0, 2)\n",
      "O . X\n",
      "X . O\n",
      "O X .\n",
      "\n",
      "Player X move: (1, 0)\n",
      "O X X\n",
      "X . O\n",
      "O X .\n",
      "\n",
      "Player O move: (1, 1)\n",
      "O X X\n",
      "X O O\n",
      "O X .\n",
      "\n",
      "Player X move: (2, 2)\n",
      "O X X\n",
      "X O O\n",
      "O X X\n",
      "\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "0"
      ]
     },
     "execution_count": 75,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "play_game(TicTacToe(), dict(X=player(alphabeta_search), O=player(minimax_search)), verbose=True).utility"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Connect Four\n",
    "\n",
    "Connect Four is a variant of tic-tac-toe, played on a larger (7 x 6) board, and with the restriction that in any column you can only play in the lowest empty square in the column."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 76,
   "metadata": {},
   "outputs": [],
   "source": [
    "class ConnectFour(TicTacToe):\n",
    "    \n",
    "    def __init__(self): super().__init__(width=7, height=6, k=4)\n",
    "\n",
    "    def actions(self, board):\n",
    "        \"\"\"In each column you can play only the lowest empty square in the column.\"\"\"\n",
    "        return {(x, y) for (x, y) in self.squares - set(board)\n",
    "                if y == board.height - 1 or (x, y + 1) in board}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 77,
   "metadata": {
    "scrolled": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Player X move: (2, 5)\n",
      ". . . . . . .\n",
      ". . . . . . .\n",
      ". . . . . . .\n",
      ". . . . . . .\n",
      ". . . . . . .\n",
      ". . X . . . .\n",
      "\n",
      "Player O move: (1, 5)\n",
      ". . . . . . .\n",
      ". . . . . . .\n",
      ". . . . . . .\n",
      ". . . . . . .\n",
      ". . . . . . .\n",
      ". O X . . . .\n",
      "\n",
      "Player X move: (5, 5)\n",
      ". . . . . . .\n",
      ". . . . . . .\n",
      ". . . . . . .\n",
      ". . . . . . .\n",
      ". . . . . . .\n",
      ". O X . . X .\n",
      "\n",
      "Player O move: (4, 5)\n",
      ". . . . . . .\n",
      ". . . . . . .\n",
      ". . . . . . .\n",
      ". . . . . . .\n",
      ". . . . . . .\n",
      ". O X . O X .\n",
      "\n",
      "Player X move: (4, 4)\n",
      ". . . . . . .\n",
      ". . . . . . .\n",
      ". . . . . . .\n",
      ". . . . . . .\n",
      ". . . . X . .\n",
      ". O X . O X .\n",
      "\n",
      "Player O move: (2, 4)\n",
      ". . . . . . .\n",
      ". . . . . . .\n",
      ". . . . . . .\n",
      ". . . . . . .\n",
      ". . O . X . .\n",
      ". O X . O X .\n",
      "\n",
      "Player X move: (2, 3)\n",
      ". . . . . . .\n",
      ". . . . . . .\n",
      ". . . . . . .\n",
      ". . X . . . .\n",
      ". . O . X . .\n",
      ". O X . O X .\n",
      "\n",
      "Player O move: (1, 4)\n",
      ". . . . . . .\n",
      ". . . . . . .\n",
      ". . . . . . .\n",
      ". . X . . . .\n",
      ". O O . X . .\n",
      ". O X . O X .\n",
      "\n",
      "Player X move: (0, 5)\n",
      ". . . . . . .\n",
      ". . . . . . .\n",
      ". . . . . . .\n",
      ". . X . . . .\n",
      ". O O . X . .\n",
      "X O X . O X .\n",
      "\n",
      "Player O move: (5, 4)\n",
      ". . . . . . .\n",
      ". . . . . . .\n",
      ". . . . . . .\n",
      ". . X . . . .\n",
      ". O O . X O .\n",
      "X O X . O X .\n",
      "\n",
      "Player X move: (5, 3)\n",
      ". . . . . . .\n",
      ". . . . . . .\n",
      ". . . . . . .\n",
      ". . X . . X .\n",
      ". O O . X O .\n",
      "X O X . O X .\n",
      "\n",
      "Player O move: (6, 5)\n",
      ". . . . . . .\n",
      ". . . . . . .\n",
      ". . . . . . .\n",
      ". . X . . X .\n",
      ". O O . X O .\n",
      "X O X . O X O\n",
      "\n",
      "Player X move: (1, 3)\n",
      ". . . . . . .\n",
      ". . . . . . .\n",
      ". . . . . . .\n",
      ". X X . . X .\n",
      ". O O . X O .\n",
      "X O X . O X O\n",
      "\n",
      "Player O move: (6, 4)\n",
      ". . . . . . .\n",
      ". . . . . . .\n",
      ". . . . . . .\n",
      ". X X . . X .\n",
      ". O O . X O O\n",
      "X O X . O X O\n",
      "\n",
      "Player X move: (5, 2)\n",
      ". . . . . . .\n",
      ". . . . . . .\n",
      ". . . . . X .\n",
      ". X X . . X .\n",
      ". O O . X O O\n",
      "X O X . O X O\n",
      "\n",
      "Player O move: (0, 4)\n",
      ". . . . . . .\n",
      ". . . . . . .\n",
      ". . . . . X .\n",
      ". X X . . X .\n",
      "O O O . X O O\n",
      "X O X . O X O\n",
      "\n",
      "Player X move: (0, 3)\n",
      ". . . . . . .\n",
      ". . . . . . .\n",
      ". . . . . X .\n",
      "X X X . . X .\n",
      "O O O . X O O\n",
      "X O X . O X O\n",
      "\n",
      "Player O move: (0, 2)\n",
      ". . . . . . .\n",
      ". . . . . . .\n",
      "O . . . . X .\n",
      "X X X . . X .\n",
      "O O O . X O O\n",
      "X O X . O X O\n",
      "\n",
      "Player X move: (0, 1)\n",
      ". . . . . . .\n",
      "X . . . . . .\n",
      "O . . . . X .\n",
      "X X X . . X .\n",
      "O O O . X O O\n",
      "X O X . O X O\n",
      "\n",
      "Player O move: (0, 0)\n",
      "O . . . . . .\n",
      "X . . . . . .\n",
      "O . . . . X .\n",
      "X X X . . X .\n",
      "O O O . X O O\n",
      "X O X . O X O\n",
      "\n",
      "Player X move: (5, 1)\n",
      "O . . . . . .\n",
      "X . . . . X .\n",
      "O . . . . X .\n",
      "X X X . . X .\n",
      "O O O . X O O\n",
      "X O X . O X O\n",
      "\n",
      "Player O move: (4, 3)\n",
      "O . . . . . .\n",
      "X . . . . X .\n",
      "O . . . . X .\n",
      "X X X . O X .\n",
      "O O O . X O O\n",
      "X O X . O X O\n",
      "\n",
      "Player X move: (6, 3)\n",
      "O . . . . . .\n",
      "X . . . . X .\n",
      "O . . . . X .\n",
      "X X X . O X X\n",
      "O O O . X O O\n",
      "X O X . O X O\n",
      "\n",
      "Player O move: (5, 0)\n",
      "O . . . . O .\n",
      "X . . . . X .\n",
      "O . . . . X .\n",
      "X X X . O X X\n",
      "O O O . X O O\n",
      "X O X . O X O\n",
      "\n",
      "Player X move: (3, 5)\n",
      "O . . . . O .\n",
      "X . . . . X .\n",
      "O . . . . X .\n",
      "X X X . O X X\n",
      "O O O . X O O\n",
      "X O X X O X O\n",
      "\n",
      "Player O move: (1, 2)\n",
      "O . . . . O .\n",
      "X . . . . X .\n",
      "O O . . . X .\n",
      "X X X . O X X\n",
      "O O O . X O O\n",
      "X O X X O X O\n",
      "\n",
      "Player X move: (1, 1)\n",
      "O . . . . O .\n",
      "X X . . . X .\n",
      "O O . . . X .\n",
      "X X X . O X X\n",
      "O O O . X O O\n",
      "X O X X O X O\n",
      "\n",
      "Player O move: (3, 4)\n",
      "O . . . . O .\n",
      "X X . . . X .\n",
      "O O . . . X .\n",
      "X X X . O X X\n",
      "O O O O X O O\n",
      "X O X X O X O\n",
      "\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "-1"
      ]
     },
     "execution_count": 77,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "play_game(ConnectFour(), dict(X=random_player, O=random_player), verbose=True).utility"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Transposition Tables\n",
    "\n",
    "By treating the game tree as a tree, we can arrive at the same state through different paths, and end up duplicating effort. In state-space search, we kept a table of `reached` states to prevent this. For game-tree search, we can achieve the same effect by applying the `@cache` decorator to the `min_value` and `max_value` functions. We'll use the suffix `_tt` to indicate a function that uses these transisiton tables."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [],
   "source": [
    "def minimax_search_tt(game, state):\n",
    "    \"\"\"Search game to determine best move; return (value, move) pair.\"\"\"\n",
    "\n",
    "    player = state.to_move\n",
    "\n",
    "    @cache\n",
    "    def max_value(state):\n",
    "        if game.is_terminal(state):\n",
    "            return game.utility(state, player), None\n",
    "        v, move = -infinity, None\n",
    "        for a in game.actions(state):\n",
    "            v2, _ = min_value(game.result(state, a))\n",
    "            if v2 > v:\n",
    "                v, move = v2, a\n",
    "        return v, move\n",
    "\n",
    "    @cache\n",
    "    def min_value(state):\n",
    "        if game.is_terminal(state):\n",
    "            return game.utility(state, player), None\n",
    "        v, move = +infinity, None\n",
    "        for a in game.actions(state):\n",
    "            v2, _ = max_value(game.result(state, a))\n",
    "            if v2 < v:\n",
    "                v, move = v2, a\n",
    "        return v, move\n",
    "\n",
    "    return max_value(state)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "For alpha-beta search, we can still use a cache, but it should be based just on the state, not on whatever values alpha and beta have."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 79,
   "metadata": {},
   "outputs": [],
   "source": [
    "def cache1(function):\n",
    "    \"Like lru_cache(None), but only considers the first argument of function.\"\n",
    "    cache = {}\n",
    "    def wrapped(x, *args):\n",
    "        if x not in cache:\n",
    "            cache[x] = function(x, *args)\n",
    "        return cache[x]\n",
    "    return wrapped\n",
    "\n",
    "def alphabeta_search_tt(game, state):\n",
    "    \"\"\"Search game to determine best action; use alpha-beta pruning.\n",
    "    As in [Figure 5.7], this version searches all the way to the leaves.\"\"\"\n",
    "\n",
    "    player = state.to_move\n",
    "\n",
    "    @cache1\n",
    "    def max_value(state, alpha, beta):\n",
    "        if game.is_terminal(state):\n",
    "            return game.utility(state, player), None\n",
    "        v, move = -infinity, None\n",
    "        for a in game.actions(state):\n",
    "            v2, _ = min_value(game.result(state, a), alpha, beta)\n",
    "            if v2 > v:\n",
    "                v, move = v2, a\n",
    "                alpha = max(alpha, v)\n",
    "            if v >= beta:\n",
    "                return v, move\n",
    "        return v, move\n",
    "\n",
    "    @cache1\n",
    "    def min_value(state, alpha, beta):\n",
    "        if game.is_terminal(state):\n",
    "            return game.utility(state, player), None\n",
    "        v, move = +infinity, None\n",
    "        for a in game.actions(state):\n",
    "            v2, _ = max_value(game.result(state, a), alpha, beta)\n",
    "            if v2 < v:\n",
    "                v, move = v2, a\n",
    "                beta = min(beta, v)\n",
    "            if v <= alpha:\n",
    "                return v, move\n",
    "        return v, move\n",
    "\n",
    "    return max_value(state, -infinity, +infinity)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 81,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "CPU times: user 593 ms, sys: 52 ms, total: 645 ms\n",
      "Wall time: 655 ms\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "O X X\n",
       "X O O\n",
       "O X X"
      ]
     },
     "execution_count": 81,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "%time play_game(TicTacToe(), {'X':player(alphabeta_search_tt), 'O':player(minimax_search_tt)})"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 82,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "CPU times: user 3.07 s, sys: 30.7 ms, total: 3.1 s\n",
      "Wall time: 3.15 s\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "O X X\n",
       "X O O\n",
       "O X X"
      ]
     },
     "execution_count": 82,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "%time play_game(TicTacToe(), {'X':player(alphabeta_search), 'O':player(minimax_search)})"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Heuristic Cutoffs"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 63,
   "metadata": {},
   "outputs": [],
   "source": [
    "def cutoff_depth(d):\n",
    "    \"\"\"A cutoff function that searches to depth d.\"\"\"\n",
    "    return lambda game, state, depth: depth > d\n",
    "\n",
    "def h_alphabeta_search(game, state, cutoff=cutoff_depth(6), h=lambda s, p: 0):\n",
    "    \"\"\"Search game to determine best action; use alpha-beta pruning.\n",
    "    As in [Figure 5.7], this version searches all the way to the leaves.\"\"\"\n",
    "\n",
    "    player = state.to_move\n",
    "\n",
    "    @cache1\n",
    "    def max_value(state, alpha, beta, depth):\n",
    "        if game.is_terminal(state):\n",
    "            return game.utility(state, player), None\n",
    "        if cutoff(game, state, depth):\n",
    "            return h(state, player), None\n",
    "        v, move = -infinity, None\n",
    "        for a in game.actions(state):\n",
    "            v2, _ = min_value(game.result(state, a), alpha, beta, depth+1)\n",
    "            if v2 > v:\n",
    "                v, move = v2, a\n",
    "                alpha = max(alpha, v)\n",
    "            if v >= beta:\n",
    "                return v, move\n",
    "        return v, move\n",
    "\n",
    "    @cache1\n",
    "    def min_value(state, alpha, beta, depth):\n",
    "        if game.is_terminal(state):\n",
    "            return game.utility(state, player), None\n",
    "        if cutoff(game, state, depth):\n",
    "            return h(state, player), None\n",
    "        v, move = +infinity, None\n",
    "        for a in game.actions(state):\n",
    "            v2, _ = max_value(game.result(state, a), alpha, beta, depth + 1)\n",
    "            if v2 < v:\n",
    "                v, move = v2, a\n",
    "                beta = min(beta, v)\n",
    "            if v <= alpha:\n",
    "                return v, move\n",
    "        return v, move\n",
    "\n",
    "    return max_value(state, -infinity, +infinity, 0)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 54,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "CPU times: user 367 ms, sys: 7.9 ms, total: 375 ms\n",
      "Wall time: 375 ms\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "O X X\n",
       "X O O\n",
       "O X X"
      ]
     },
     "execution_count": 54,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "%time play_game(TicTacToe(), {'X':player(h_alphabeta_search), 'O':player(h_alphabeta_search)})"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 60,
   "metadata": {
    "scrolled": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      ". . . . . . .\n",
      ". . . . . . .\n",
      ". . . . . . .\n",
      ". . . . . . .\n",
      ". . . . . . .\n",
      ". . . . . X .\n",
      "\n",
      ". . . . . . .\n",
      ". . . . . . .\n",
      ". . . . . . .\n",
      ". . . . . . .\n",
      ". . . . . . .\n",
      ". . . . . X O\n",
      "\n",
      ". . . . . . .\n",
      ". . . . . . .\n",
      ". . . . . . .\n",
      ". . . . . . .\n",
      ". . . . . . X\n",
      ". . . . . X O\n",
      "\n",
      ". . . . . . .\n",
      ". . . . . . .\n",
      ". . . . . . .\n",
      ". . . . . . .\n",
      ". . . . . O X\n",
      ". . . . . X O\n",
      "\n",
      ". . . . . . .\n",
      ". . . . . . .\n",
      ". . . . . . .\n",
      ". . . . . . .\n",
      ". . . . . O X\n",
      ". . . . X X O\n",
      "\n",
      ". . . . . . .\n",
      ". . . . . . .\n",
      ". . . . . . .\n",
      ". . . . . O .\n",
      ". . . . . O X\n",
      ". . . . X X O\n",
      "\n",
      ". . . . . . .\n",
      ". . . . . . .\n",
      ". . . . . . .\n",
      ". . . . . O .\n",
      ". . . . X O X\n",
      ". . . . X X O\n",
      "\n",
      ". . . . . . .\n",
      ". . . . . . .\n",
      ". . . . . . .\n",
      ". . . . . O .\n",
      ". . . . X O X\n",
      ". O . . X X O\n",
      "\n",
      ". . . . . . .\n",
      ". . . . . . .\n",
      ". . . . . . .\n",
      ". . . . . O .\n",
      ". X . . X O X\n",
      ". O . . X X O\n",
      "\n",
      ". . . . . . .\n",
      ". . . . . . .\n",
      ". . . . . . .\n",
      ". . . . . O O\n",
      ". X . . X O X\n",
      ". O . . X X O\n",
      "\n",
      ". . . . . . .\n",
      ". . . . . . .\n",
      ". . . . . . .\n",
      ". . . . X O O\n",
      ". X . . X O X\n",
      ". O . . X X O\n",
      "\n",
      ". . . . . . .\n",
      ". . . . . . .\n",
      ". . . . . . .\n",
      ". . . . X O O\n",
      ". X . . X O X\n",
      ". O . O X X O\n",
      "\n",
      ". . . . . . .\n",
      ". . . . . . .\n",
      ". . . . . X .\n",
      ". . . . X O O\n",
      ". X . . X O X\n",
      ". O . O X X O\n",
      "\n",
      ". . . . . . .\n",
      ". . . . . O .\n",
      ". . . . . X .\n",
      ". . . . X O O\n",
      ". X . . X O X\n",
      ". O . O X X O\n",
      "\n",
      ". . . . . . .\n",
      ". . . . . O .\n",
      ". . . . . X .\n",
      ". X . . X O O\n",
      ". X . . X O X\n",
      ". O . O X X O\n",
      "\n",
      ". . . . . . .\n",
      ". . . . . O .\n",
      ". . . . . X O\n",
      ". X . . X O O\n",
      ". X . . X O X\n",
      ". O . O X X O\n",
      "\n",
      ". . . . . . .\n",
      ". . . . . O .\n",
      ". X . . . X O\n",
      ". X . . X O O\n",
      ". X . . X O X\n",
      ". O . O X X O\n",
      "\n",
      ". . . . . . .\n",
      ". . . . . O .\n",
      ". X . . . X O\n",
      ". X . . X O O\n",
      ". X . . X O X\n",
      ". O O O X X O\n",
      "\n",
      ". . . . . . .\n",
      ". . . . . O .\n",
      ". X . . . X O\n",
      ". X . . X O O\n",
      ". X . . X O X\n",
      "X O O O X X O\n",
      "\n",
      ". . . . . . .\n",
      ". . . . . O O\n",
      ". X . . . X O\n",
      ". X . . X O O\n",
      ". X . . X O X\n",
      "X O O O X X O\n",
      "\n",
      ". . . . . . X\n",
      ". . . . . O O\n",
      ". X . . . X O\n",
      ". X . . X O O\n",
      ". X . . X O X\n",
      "X O O O X X O\n",
      "\n",
      ". . . . . . X\n",
      ". . . . . O O\n",
      ". X . . . X O\n",
      ". X . . X O O\n",
      "O X . . X O X\n",
      "X O O O X X O\n",
      "\n",
      ". . . . . X X\n",
      ". . . . . O O\n",
      ". X . . . X O\n",
      ". X . . X O O\n",
      "O X . . X O X\n",
      "X O O O X X O\n",
      "\n",
      ". . . . . X X\n",
      ". . . . . O O\n",
      ". X . . . X O\n",
      ". X . . X O O\n",
      "O X . O X O X\n",
      "X O O O X X O\n",
      "\n",
      ". . . . . X X\n",
      ". . . . . O O\n",
      ". X . . . X O\n",
      ". X . X X O O\n",
      "O X . O X O X\n",
      "X O O O X X O\n",
      "\n",
      ". . . . . X X\n",
      ". . . . . O O\n",
      ". X . . O X O\n",
      ". X . X X O O\n",
      "O X . O X O X\n",
      "X O O O X X O\n",
      "\n",
      ". . . . . X X\n",
      ". . . . . O O\n",
      ". X . X O X O\n",
      ". X . X X O O\n",
      "O X . O X O X\n",
      "X O O O X X O\n",
      "\n",
      ". . . . . X X\n",
      ". . . . O O O\n",
      ". X . X O X O\n",
      ". X . X X O O\n",
      "O X . O X O X\n",
      "X O O O X X O\n",
      "\n",
      ". . . . . X X\n",
      ". . . X O O O\n",
      ". X . X O X O\n",
      ". X . X X O O\n",
      "O X . O X O X\n",
      "X O O O X X O\n",
      "\n",
      ". . . . O X X\n",
      ". . . X O O O\n",
      ". X . X O X O\n",
      ". X . X X O O\n",
      "O X . O X O X\n",
      "X O O O X X O\n",
      "\n",
      ". . . X O X X\n",
      ". . . X O O O\n",
      ". X . X O X O\n",
      ". X . X X O O\n",
      "O X . O X O X\n",
      "X O O O X X O\n",
      "\n",
      "CPU times: user 8.82 s, sys: 146 ms, total: 8.96 s\n",
      "Wall time: 9.19 s\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "1"
      ]
     },
     "execution_count": 60,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "%time play_game(ConnectFour(), {'X':player(h_alphabeta_search), 'O':random_player}, verbose=True).utility"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 61,
   "metadata": {
    "scrolled": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      ". . . . . . .\n",
      ". . . . . . .\n",
      ". . . . . . .\n",
      ". . . . . . .\n",
      ". . . . . . .\n",
      ". . . . . X .\n",
      "\n",
      ". . . . . . .\n",
      ". . . . . . .\n",
      ". . . . . . .\n",
      ". . . . . . .\n",
      ". . . . . O .\n",
      ". . . . . X .\n",
      "\n",
      ". . . . . . .\n",
      ". . . . . . .\n",
      ". . . . . . .\n",
      ". . . . . . .\n",
      ". . . . . O .\n",
      ". . . . X X .\n",
      "\n",
      ". . . . . . .\n",
      ". . . . . . .\n",
      ". . . . . . .\n",
      ". . . . . . .\n",
      ". . . . . O .\n",
      ". . O . X X .\n",
      "\n",
      ". . . . . . .\n",
      ". . . . . . .\n",
      ". . . . . . .\n",
      ". . . . . . .\n",
      ". . . . X O .\n",
      ". . O . X X .\n",
      "\n",
      ". . . . . . .\n",
      ". . . . . . .\n",
      ". . . . . . .\n",
      ". . . . . . .\n",
      ". . . . X O .\n",
      ". . O O X X .\n",
      "\n",
      ". . . . . . .\n",
      ". . . . . . .\n",
      ". . . . . . .\n",
      ". . . . . . .\n",
      ". . . . X O .\n",
      ". X O O X X .\n",
      "\n",
      ". . . . . . .\n",
      ". . . . . . .\n",
      ". . . . . . .\n",
      ". . . . . . .\n",
      ". O . . X O .\n",
      ". X O O X X .\n",
      "\n",
      ". . . . . . .\n",
      ". . . . . . .\n",
      ". . . . . . .\n",
      ". X . . . . .\n",
      ". O . . X O .\n",
      ". X O O X X .\n",
      "\n",
      ". . . . . . .\n",
      ". . . . . . .\n",
      ". O . . . . .\n",
      ". X . . . . .\n",
      ". O . . X O .\n",
      ". X O O X X .\n",
      "\n",
      ". . . . . . .\n",
      ". . . . . . .\n",
      ". O . . . . .\n",
      ". X . . . . .\n",
      ". O . . X O .\n",
      "X X O O X X .\n",
      "\n",
      ". . . . . . .\n",
      ". . . . . . .\n",
      ". O . . . . .\n",
      ". X . . . . .\n",
      "O O . . X O .\n",
      "X X O O X X .\n",
      "\n",
      ". . . . . . .\n",
      ". . . . . . .\n",
      ". O . . . . .\n",
      ". X . . X . .\n",
      "O O . . X O .\n",
      "X X O O X X .\n",
      "\n",
      ". . . . . . .\n",
      ". . . . . . .\n",
      ". O . . O . .\n",
      ". X . . X . .\n",
      "O O . . X O .\n",
      "X X O O X X .\n",
      "\n",
      ". . . . . . .\n",
      ". . . . X . .\n",
      ". O . . O . .\n",
      ". X . . X . .\n",
      "O O . . X O .\n",
      "X X O O X X .\n",
      "\n",
      ". . . . . . .\n",
      ". . . . X . .\n",
      ". O . . O . .\n",
      ". X . . X O .\n",
      "O O . . X O .\n",
      "X X O O X X .\n",
      "\n",
      ". . . . . . .\n",
      ". . . . X . .\n",
      ". O . . O X .\n",
      ". X . . X O .\n",
      "O O . . X O .\n",
      "X X O O X X .\n",
      "\n",
      ". . . . . . .\n",
      ". . . . X . .\n",
      ". O . . O X .\n",
      ". X . . X O .\n",
      "O O O . X O .\n",
      "X X O O X X .\n",
      "\n",
      ". . . . . . .\n",
      ". . . . X . .\n",
      ". O . . O X .\n",
      ". X . . X O .\n",
      "O O O X X O .\n",
      "X X O O X X .\n",
      "\n",
      ". . . . . . .\n",
      ". . . . X O .\n",
      ". O . . O X .\n",
      ". X . . X O .\n",
      "O O O X X O .\n",
      "X X O O X X .\n",
      "\n",
      ". . . . . . .\n",
      ". . . . X O .\n",
      ". O . . O X .\n",
      ". X . X X O .\n",
      "O O O X X O .\n",
      "X X O O X X .\n",
      "\n",
      ". . . . . . .\n",
      ". . . . X O .\n",
      ". O . O O X .\n",
      ". X . X X O .\n",
      "O O O X X O .\n",
      "X X O O X X .\n",
      "\n",
      ". . . . . . .\n",
      ". . . X X O .\n",
      ". O . O O X .\n",
      ". X . X X O .\n",
      "O O O X X O .\n",
      "X X O O X X .\n",
      "\n",
      ". . . O . . .\n",
      ". . . X X O .\n",
      ". O . O O X .\n",
      ". X . X X O .\n",
      "O O O X X O .\n",
      "X X O O X X .\n",
      "\n",
      ". . . O . . .\n",
      ". . . X X O .\n",
      ". O . O O X .\n",
      ". X X X X O .\n",
      "O O O X X O .\n",
      "X X O O X X .\n",
      "\n",
      "CPU times: user 12.1 s, sys: 237 ms, total: 12.4 s\n",
      "Wall time: 12.9 s\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "1"
      ]
     },
     "execution_count": 61,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "%time play_game(ConnectFour(), {'X':player(h_alphabeta_search), 'O':player(h_alphabeta_search)}, verbose=True).utility"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 83,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Result states:   6,589; Terminal tests:   3,653; for alphabeta_search_tt\n",
      "Result states:  25,703; Terminal tests:  25,704; for alphabeta_search\n",
      "Result states:   4,687; Terminal tests:   2,805; for h_alphabeta_search\n",
      "Result states:  16,167; Terminal tests:   5,478; for minimax_search_tt\n"
     ]
    }
   ],
   "source": [
    "class CountCalls:\n",
    "    \"\"\"Delegate all attribute gets to the object, and count them in ._counts\"\"\"\n",
    "    def __init__(self, obj):\n",
    "        self._object = obj\n",
    "        self._counts = Counter()\n",
    "        \n",
    "    def __getattr__(self, attr):\n",
    "        \"Delegate to the original object, after incrementing a counter.\"\n",
    "        self._counts[attr] += 1\n",
    "        return getattr(self._object, attr)\n",
    "    \n",
    "def report(game, searchers):\n",
    "    for searcher in searchers:\n",
    "        game = CountCalls(game)\n",
    "        searcher(game, game.initial)\n",
    "        print('Result states: {:7,d}; Terminal tests: {:7,d}; for {}'.format(\n",
    "            game._counts['result'], game._counts['is_terminal'], searcher.__name__))\n",
    "    \n",
    "report(TicTacToe(), (alphabeta_search_tt,  alphabeta_search, h_alphabeta_search, minimax_search_tt))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Monte Carlo Tree Search"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "class Node:\n",
    "    def __init__(self, parent, )\n",
    "def mcts(state, game, N=1000):"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Heuristic Search Algorithms"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "t = CountCalls(TicTacToe())\n",
    "    \n",
    "play_game(t, dict(X=minimax_player, O=minimax_player), verbose=True)\n",
    "t._counts"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "for tactic in (three, fork, center, opposite_corner, corner, any):\n",
    "    for s in squares:\n",
    "        if tactic(board, s,player): return s\n",
    "    for s ins quares:\n",
    "        if tactic(board, s, opponent): return s"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "\n",
    "\n",
    "def ucb(U, N, C=2**0.5, parentN=100):\n",
    "    return round(U/N + C * math.sqrt(math.log(parentN)/N), 2)\n",
    "\n",
    "{C: (ucb(60, 79, C), ucb(1, 10, C), ucb(2, 11, C)) \n",
    "    for C in (1.4, 1.5)}\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "def ucb(U, N, parentN=100, C=2):\n",
    "    return U/N + C * math.sqrt(math.log(parentN)/N)\n",
    "\n",
    "\n",
    "C = 1.4 \n",
    "\n",
    "class Node:\n",
    "    def __init__(self, name, children=(), U=0, N=0, parent=None, p=0.5):\n",
    "        self.__dict__.update(name=name, U=U, N=N, parent=parent, children=children, p=p)\n",
    "        for c in children:\n",
    "            c.parent = self\n",
    "            \n",
    "    def __repr__(self):\n",
    "        return '{}:{}/{}={:.0%}{}'.format(self.name, self.U, self.N, self.U/self.N, self.children)\n",
    "    \n",
    "def select(n):\n",
    "    if n.children:\n",
    "        return select(max(n.children, key=ucb))\n",
    "    else:\n",
    "        return n\n",
    "    \n",
    "def back(n, amount):\n",
    "    if n:\n",
    "        n.N += 1\n",
    "        n.U += amount\n",
    "        back(n.parent, 1 - amount)\n",
    "    \n",
    "    \n",
    "def one(root): \n",
    "    n = select(root)\n",
    "    amount = int(random.uniform(0, 1) < n.p)\n",
    "    back(n, amount)\n",
    "        \n",
    "def ucb(n): \n",
    "    return (float('inf') if n.N == 0 else\n",
    "            n.U / n.N + C * math.sqrt(math.log(n.parent.N)/n.N))\n",
    "\n",
    "\n",
    "tree = Node('root', [Node('a', p=.8, children=[Node('a1', p=.05), \n",
    "                                               Node('a2', p=.25,\n",
    "                                                    children=[Node('a2a', p=.7), Node('a2b')])]),\n",
    "                     Node('b', p=.5, children=[Node('b1', p=.6,\n",
    "                                                   children=[Node('b1a', p=.3), Node('b1b')]), \n",
    "                                               Node('b2', p=.4)]),\n",
    "                     Node('c', p=.1)])\n",
    "\n",
    "for i in range(100):\n",
    "    one(tree); \n",
    "for c in tree.children: print(c)\n",
    "'select', select(tree), 'tree', tree\n",
    "\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "us = (100, 50, 25, 10, 5, 1)\n",
    "infinity = float('inf')\n",
    "\n",
    "@lru_cache(None)\n",
    "def f1(n, denom):\n",
    "    return (0 if n == 0 else\n",
    "            infinity if n < 0 or not denom else\n",
    "            min(1 + f1(n - denom[0], denom),\n",
    "                    f1(n, denom[1:])))\n",
    "    \n",
    "@lru_cache(None)\n",
    "def f2(n, denom):\n",
    "    @lru_cache(None)\n",
    "    def f(n):\n",
    "        return (0 if n == 0 else\n",
    "                infinity if n < 0 else\n",
    "                1 + min(f(n - d) for d in denom))\n",
    "    return f(n)\n",
    "\n",
    "@lru_cache(None)\n",
    "def f3(n, denom):\n",
    "    return (0 if n == 0 else\n",
    "            infinity if n < 0 or not denom else\n",
    "            min(k + f2(n - k * denom[0], denom[1:]) \n",
    "                for k in range(1 + n // denom[0])))\n",
    "    \n",
    "\n",
    "def g(n, d=us): return f1(n, d), f2(n, d), f3(n, d)\n",
    "    \n",
    "n = 12345\n",
    "%time f1(n, us)\n",
    "%time f2(n, us)\n",
    "%time f3(n, us)\n",
    "        "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
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