{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# 递归函数" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## 递归（Recursion）" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "在函数中有个理解门槛比较高的概念：**递归函数**（Recursive Functions）—— 那些**在自身内部调用自身的函数**。说起来都比较拗口。" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "先看一个例子，我们想要有个能够计算 `n` 的*阶乘*（factorial）`n!` 的函数，`f()`，规则如下：\n", "\n", "> - `n! = n × (n-1) × (n-2)... × 1`\n", "> - 即，`n! = n × (n-1)!`\n", "> - 且，`n >= 1`\n", ">\n", "> **注意**：以上是数学表达，不是程序，所以，`=` 在这一小段中是 “*等于*” 的意思，**不是程序语言中的赋值符号**。\n", "\n", "于是，计算 `f(n)` 的 Python 程序如下：\n" ] }, { "cell_type": "code", "execution_count": 1, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "120\n" ] } ], "source": [ "def f(n):\n", " if n == 1:\n", " return 1\n", " else:\n", " return n * f(n-1)\n", " \n", "print(f(5))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## 递归函数的执行过程" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "以 factorial(5) 为例，让我们看看程序的流程：\n", "\n", "![](images/recursive-function-call.png)\n", "\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "当 f(5) 被调用之后，函数开始运行……\n", "* 因为 `5 > 1`，所以，在计算 `n * f(n-1)` 的时候要再次调用自己 `f(4)`；所以必须等待 `f(4)` 的值返回；\n", "* 因为 `4 > 1`，所以，在计算 `n * f(n-1)` 的时候要再次调用自己 `f(3)`；所以必须等待 `f(3)` 的值返回；\n", "* 因为 `3 > 1`，所以，在计算 `n * f(n-1)` 的时候要再次调用自己 `f(2)`；所以必须等待 `f(2)` 的值返回；\n", "* 因为 `2 > 1`，所以，在计算 `n * f(n-1)` 的时候要再次调用自己 `f(1)`；所以必须等待 `f(1)` 的值返回；\n", "* 因为 `1 == 1`，所以，这时候不会再次调用 `f()` 了，于是递归结束，开始返回，这次返回的是 `1`；\n", "* 下一步返回的是 `2 * 1`；\n", "* 下一步返回的是 `3 * 2`；\n", "* 下一步返回的是 `4 * 6`；\n", "* 下一步返回的是 `5 * 24` —— 至此，外部调用 `f(5)` 的最终返回值是 `120`……\n", "\n", "加上一些输出语句之后，能更清楚地看到大概的执行流程：" ] }, { "cell_type": "code", "execution_count": 1, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Call f(5)...\n", "\tn = 5\n", "\tn = 4\n", "\tn = 3\n", "\tn = 2\n", "\tn = 1\n", "Returning...\n", "\tn = 1 return: 1\n", "\tn = 2 return: 2\n", "\tn = 3 return: 6\n", "\tn = 4 return: 24\n", "\tn = 5 return: 120\n", "Get out of f(n), and f(5) = 120\n" ] } ], "source": [ "def f(n):\n", " print('\\tn =', n)\n", " if n == 1:\n", " print('Returning...')\n", " print('\\tn =', n, 'return:', 1)\n", " return 1\n", " else:\n", " r = n * f(n-1)\n", " print('\\tn =', n, 'return:', r)\n", " return r\n", " \n", "print('Call f(5)...')\n", "print('Get out of f(n), and f(5) =', f(5))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "有点烧脑…… 不过，分为几个层面去逐个突破，你会发现它真的很好玩。" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## 递归的终点" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "递归函数在内部必须有一个能够让自己停止调用自己的方式，否则永远循环下去了……\n", "\n", "其实，我们所有人很小就见过递归应用，只不过，那时候不知道那就是递归而已。听过那个无聊的故事罢？\n", "\n", "> 山上有座庙，庙里有个和尚，和尚讲故事，说……\n", "> > 山上有座庙，庙里有个和尚，和尚讲故事，说……\n", "> > > 山上有座庙，庙里有个和尚，和尚讲故事，说……\n", "\n", "写成 Python 程序大概是这样：\n", "\n", "```python\n", "def a_monk_telling_story():\n", " print('山上有座庙，庙里有个和尚，和尚讲故事，他说…… ')\n", " return a_monk_telling_story()\n", "\n", "a_monk_telling_story()\n", "```\n", "\n", "这是个*无限循环*的递归，因为这个函数里*没有设置中止自我调用的条件*。无限循环还有个不好听的名字，叫做 “死循环”。" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "在著名的电影**盗梦空间**（_2010_）里，从整体结构上来看，“入梦” 也是个 “递归函数”。只不过，这个函数和 `a_monk_telling_story()` 不一样，它并不是死循环 —— 因为它设定了*中止自我调用的条件*：\n", "\n", "> 在电影里，醒过来的条件有两个\n", ">> * 一个是在梦里死掉；\n", ">> * 一个是在梦里被 kicked 到……\n", ">\n", "> 如果这两个条件一直不被满足，那就进入 limbo 状态 —— 其实就跟死循环一样，出不来了……\n", "\n", "为了演示，我把故事情节改变成这样：\n", "> * 入梦，`in_dream()`，是个递归函数；\n", "> * 入梦之后醒过来的条件有两个：\n", ">> * 一个是在梦里死掉，`dead is True`；\n", ">> * 一个是在梦里被 kicked，`kicked is True`……\n", ">>\n", ">> 以上两个条件中任意一个被满足，就苏醒……\n", "\n", "至于为什么会死掉，如何被 kick，我偷懒了一下：管它怎样，管它如何，反正，每个条件被满足的概率是 1/10……（也只有这样，我才能写出一个简短的，能够运行的 “*盗梦空间程序*”。）\n", "\n", "把这个很抽象的故事写成 Python 程序，看看一次入梦之后能睡多少天，大概是这样：" ] }, { "cell_type": "code", "execution_count": 1, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "dead: False kicked: False\n", "dead: False kicked: False\n", "dead: False kicked: False\n", "dead: False kicked: False\n", "dead: False kicked: False\n", "dead: False kicked: False\n", "dead: False kicked: False\n", "dead: True kicked: True\n", "I slept 8 days, and was dead to wake up...\n", "The in_dream() function returns: 8\n" ] } ], "source": [ "import random\n", "\n", "def in_dream(day=0, dead=False, kicked=False):\n", " dead = not random.randrange(0,10) # 1/10 probability to be dead\n", " kicked = not random.randrange(0,10) # 1/10 probability to be kicked\n", " day += 1\n", " print('dead:', dead, 'kicked:', kicked)\n", " \n", " if dead:\n", " print((f\"I slept {day} days, and was dead to wake up...\"))\n", " return day\n", " elif kicked:\n", " print(f\"I slept {day} days, and was kicked to wake up...\")\n", " return day\n", " \n", " return in_dream(day)\n", " \n", "print('The in_dream() function returns:', in_dream())" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "如果疑惑为什么 `random.randrange(0,10)` 能表示 1/10 的概率，请返回去重新阅读[第一部分中关于布尔值的内容](Part.1.E.2.values-and-their-operators.ipynb)。\n", "\n", "另外，在 Python 中，若是需要将某个值与 True 或者 False 进行比较，尤其是在条件语句中，推荐写法是（参见 [PEP8](https://www.python.org/dev/peps/pep-0008/)）：\n", "\n", "\n", "```python\n", "if condition:\n", " pass\n", "```\n", "\n", "就好像上面代码中的 `if dead:` 一样。\n", "\n", "而不是（虽然这么写通常也并不妨碍程序正常运行[1]）：\n", "\n", "```python\n", "if condition is True:\n", " pass\n", "```\n", "\n", "抑或：\n", "\n", "```python\n", "if condition == True:\n", " pass\n", "```\n", "\n", "让我们再返回来接着讲递归函数。正常的**递归函数一定有个退出条件**。否则的话，就*无限循环*下去了…… 下面的程序在执行一会儿之后就会告诉你：`RecursionError: maximum recursion depth exceeded`（上面那个 “山上庙里讲故事的和尚说” 的程序，真要跑起来，也是这样）：" ] }, { "cell_type": "code", "execution_count": 3, "metadata": {}, "outputs": [ { "ename": "RecursionError", "evalue": "maximum recursion depth exceeded", "output_type": "error", "traceback": [ "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m", "\u001b[0;31mRecursionError\u001b[0m Traceback (most recent call last)", "\u001b[0;32m\u001b[0m in \u001b[0;36m\u001b[0;34m\u001b[0m\n\u001b[1;32m 1\u001b[0m \u001b[0;32mdef\u001b[0m \u001b[0mx\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mn\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 2\u001b[0m \u001b[0;32mreturn\u001b[0m \u001b[0mn\u001b[0m \u001b[0;34m*\u001b[0m \u001b[0mx\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mn\u001b[0m\u001b[0;34m-\u001b[0m\u001b[0;36m1\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m----> 3\u001b[0;31m \u001b[0mx\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;36m5\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m", "\u001b[0;32m\u001b[0m in \u001b[0;36mx\u001b[0;34m(n)\u001b[0m\n\u001b[1;32m 1\u001b[0m \u001b[0;32mdef\u001b[0m \u001b[0mx\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mn\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m----> 2\u001b[0;31m \u001b[0;32mreturn\u001b[0m \u001b[0mn\u001b[0m \u001b[0;34m*\u001b[0m \u001b[0mx\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mn\u001b[0m\u001b[0;34m-\u001b[0m\u001b[0;36m1\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m 3\u001b[0m \u001b[0mx\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;36m5\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n", "... last 1 frames repeated, from the frame below ...\n", "\u001b[0;32m\u001b[0m in \u001b[0;36mx\u001b[0;34m(n)\u001b[0m\n\u001b[1;32m 1\u001b[0m \u001b[0;32mdef\u001b[0m \u001b[0mx\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mn\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m----> 2\u001b[0;31m \u001b[0;32mreturn\u001b[0m \u001b[0mn\u001b[0m \u001b[0;34m*\u001b[0m \u001b[0mx\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mn\u001b[0m\u001b[0;34m-\u001b[0m\u001b[0;36m1\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m 3\u001b[0m \u001b[0mx\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;36m5\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n", "\u001b[0;31mRecursionError\u001b[0m: maximum recursion depth exceeded" ] } ], "source": [ "def x(n):\n", " return n * x(n-1)\n", "x(5)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "不用深究上面盗梦空间这个程序的其它细节，不过，通过以上三个递归程序 —— 两个很扯淡的例子，一个正经例子 —— 你已经看到了递归函数的共同特征：\n", "\n", "> 1. 在 `return` 语句中返回的是*自身的调用*（或者是*含有自身的表达式*）\n", "> 2. 为了避免死循环，*一定要有至少一个条件*下返回的不再是自身调用……" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## 变量的作用域" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "再回来看计算阶乘的程序 —— 这是正经程序。这次我们把程序名写完整，`factorial()`:" ] }, { "cell_type": "code", "execution_count": 6, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "120\n" ] } ], "source": [ "def factorial(n):\n", " if n == 1:\n", " return 1\n", " else:\n", " return n * factorial(n-1)\n", " \n", "print(factorial(5))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "最初的时候，这个函数的执行流程之所以令人迷惑，是因为初学者对*变量*的**作用域**把握得不够充分。\n", "\n", "变量根据作用域，可以分为两种：全局变量（Global Variables）和局部变量（Local Variables）。\n", "\n", "可以这样简化理解：\n", "\n", "> * 在函数内部被赋值而后使用的，都是*局部变量*，它们的作用域是*局部*，无法被函数外的代码调用；\n", "> * 在所有函数之外被赋值而后开始使用的，是*全局变量*，它们的作用域是*全局*，在函数内外都可以被调用。\n", "\n", "定义如此，但通常程序员们会严格地遵守一条原则：\n", "\n", "> 在函数内部绝对不调用全局变量。即便是必须改变全局变量，也只能通过函数的返回值在函数外改变全局变量。\n", "\n", "你也必须遵守同样的原则。而这个原则同样可以在日常的工作生活中 “调用”：\n", "\n", "> 做事的原则：自己的事自己做，别人的事，最多通过自己的产出让他们自己去搞……" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "再仔细观察一下以下代码。当一个变量被当做参数传递给一个函数的时候，这个变量本身并不会被函数所改变。比如，`a = 5`，而后，再把 `a` 当作参数传递给 `f(a)` 的时候，这个函数当然应该返回它内部任务完成之后应该传递回来的值，但 `a` 本身不会被改变。" ] }, { "cell_type": "code", "execution_count": 7, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "5 120\n", "120 120\n" ] } ], "source": [ "def factorial(n):\n", " if n == 1:\n", " return 1\n", " else:\n", " return n * factorial(n-1)\n", " \n", "a = 5\n", "b = factorial(a) # a 并不会因此改变；\n", "print(a, b)\n", "a = factorial(a) # 这是你主动为 a 再一次赋值……\n", "print(a, b)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "理解了这一点之后，再看 `factorial()` 这个递归函数的递归执行过程，你就能明白这个事实：\n", "\n", "> 在每一次 factorial(n) 被调用的时候，它都会形成一个作用域，`n` 这个变量作为参数把它的值传递给了函数，*但是*，`n` 这个变量本身并不会被改变。\n", "\n", "我们再修改一下上面的代码：" ] }, { "cell_type": "code", "execution_count": 8, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "5 120\n" ] } ], "source": [ "def factorial(n):\n", " if n == 1:\n", " return 1\n", " else:\n", " return n * factorial(n-1)\n", " \n", "n = 5 # 这一次，这个变量名称是 n\n", "m = factorial(n) # n 并不会因此改变；\n", "print(n, m)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "在 `m = factorial(n)` 这一句中，`n` 被 `factorial()` 当做参数调用了，但无论函数内部如何操作，并不会改变变量 `n` 的值。\n", "\n", "关键的地方在这里：在函数内部出现的变量 `n`，和函数外部的变量 `n` 不是一回事 —— **它们只是名称恰好相同而已**，函数参数定义的时候，用别的名称也没什么区别：" ] }, { "cell_type": "code", "execution_count": 1, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "5 120\n" ] } ], "source": [ "def factorial(x): # 在这个语句块中出现的变量，都是局部变量\n", " if x == 1:\n", " return 1\n", " else:\n", " return x * factorial(x-1)\n", " \n", "n = 5 # 这一次，这个变量名称是 n\n", "m = factorial(n) # n 并不会因此改变；\n", "print(n, m)\n", "# 这个例子和之前再之前的示例代码有什么区别吗？\n", "# 本质上没区别，就是变量名称换了而已……" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "函数开始执行的时候，`x` 的值，是由外部代码（即，函数被调用的那一句）传递进来的。即便函数内部的变量名称与外部的变量名称相同，它们也不是同一个变量。" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## 递归函数三原则" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "现在可以小小总结一下了。\n", "\n", "一个递归函数，之所以是一个有用、有效的递归函数，是因为它遵守了递归三原则。正如，一个机器人之所以是个合格的机器人，是因为它遵循了[阿西莫夫三铁律](https://zh.wikipedia.org/wiki/%E6%9C%BA%E5%99%A8%E4%BA%BA%E4%B8%89%E5%AE%9A%E5%BE%8B)（Three Laws of Robotics）一样[2]。\n", "\n", "> 1. 根据定义，递归函数必须在内部调用自己；\n", "> 2. 必须设定一个退出条件；\n", "> 3. 递归过程中必须能够逐步达到退出条件……\n", "\n", "从这个三原则望过去，`factorial()` 是个合格有效的递归函数，满足第一条，满足第二条，尤其还满足第三条中的 “*逐步达到*”！\n", "\n", "而那个扯淡的盗梦空间递归程序，说实话，不太合格，虽然它满足第一条，也满足第二条，第三条差点蒙混过关：它不是*逐步达到*，而是*不管怎样肯定能达到* —— 这明显是两回事…… 原谅它罢，它的作用就是当例子，一次正面的，一次负面的，作为例子算是功成圆满了！" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "刚开始的时候，初学者好不容易搞明白递归函数究竟是怎么回事之后，就不由自主地想 “我如何才能学会递归式思考呢？” —— 其实吧，这种想法本身可能并不是太正确或者准确。\n", "\n", "准确地讲，递归是一种解决问题的方式。当我们需要解决的问题，可以被逐步拆分成很多越来越小的模块，然后每个小模块还都能用同一种算法处理的时候，用递归函数最简洁有效。所以，只不过是在遇到可以用递归函数解决问题的时候，才需要去写递归函数。\n", "\n", "从这个意义上来看，递归函数是程序员为了自己方便而使用的，并不是为了计算机方便而使用 —— 计算机么，你给它的任务多一点或者少一点，对它来讲无所谓，反正有电就能运转，它自己又不付电费……\n", "\n", "理论上来讲，所有用递归函数能完成的任务，不用递归函数也能完成，只不过代码多一点，啰嗦一点，看起来没有那么优美而已。" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "还有，递归，不像 “序列类型” 那样，是某个编程语言的特有属性。它其实是一种特殊算法，也是一种编程技巧，任何编程语言，都可以使用递归算法，都可以通过编写递归函数巧妙地解决问题。" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "但是，学习递归函数本身就很烧脑啊！这才是最大的好事。从迷惑，到不太迷惑，到清楚，到很清楚，再到特别清楚 —— 这是个非常有趣，非常有成就感的过程。\n", "\n", "这种过程锻炼的是脑力 —— 在此之后，再遇到大多数人难以理解的东西，你就可以使用这一次积累的经验，应用你已经磨炼过的脑力。有意思。" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "至此，封面上的那个 “伪代码” 应该很好理解了：\n", "\n", "```python\n", "def teach_yourself(anything):\n", " while not create():\n", " learn()\n", " practice()\n", " return teach_yourself(another)\n", "\n", "teach_yourself(coding)\n", "```\n", "\n", "自学还真的就是递归函数呢……" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## 思考与练习" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "普林斯顿大学的一个网页，有很多递归的例子\n", "\n", "https://introcs.cs.princeton.edu/java/23recursion/\n", " " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "-----\n", "**脚注**\n", "\n", "[1]：参见 Stackoverflow 上的讨论：[Boolean identity == True vs is True](https://stackoverflow.com/questions/27276610/boolean-identity-true-vs-is-true)\n", "\n", "↑Back to Content↑\n", "\n", "\n", "[2]：关于[阿西莫夫三铁律](https://zh.wikipedia.org/wiki/%E6%9C%BA%E5%99%A8%E4%BA%BA%E4%B8%89%E5%AE%9A%E5%BE%8B)（Three Laws of Robotics）的类比，来自著名的 Python 教程，[Think Python: How to Think Like a Computer Scientist](http://greenteapress.com/thinkpython2/html/index.html)\n", "\n", "↑Back to Content↑" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Next Page" ] } ], "metadata": { "kernelspec": { "display_name": "Python 3", "language": "python", "name": "python3" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.8.3" } }, "nbformat": 4, "nbformat_minor": 2 }