一文学懂递归和动态规划！

原創

2020-09-22 12:33

"}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null}},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null}},{"type":"heading","attrs":{"align":null,"level":3},"content":[{"type":"text","text":"时间复杂度分析"}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null}},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","text":">如何评价一个算法的好坏？"}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null}},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","text":"很多问题都有多种解法，毕竟条条大路通罗马。但如何评价每种方法的优劣，我们一般是用"},{"type":"text","marks":[{"type":"strong"}],"text":"大 O 表达式"},{"type":"text","text":"来衡量"},{"type":"text","marks":[{"type":"strong"}],"text":"时间和空间"},{"type":"text","text":"复杂度。"}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null}},{"type":"blockquote","content":[{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","text":"时间复杂度：随着自变量的增长，所需时间的增长情况。"}]}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null}},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null}},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null}},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","text":"这里大 O 表示的是一个算法在 "},{"type":"text","marks":[{"type":"strong"}],"text":"worst case"},{"type":"text","text":" 的表现情况，这就是我们最关心的，不然春运抢车票的时候系统 hold 不住了，你跟我说这个算法很优秀？"}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null}},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","text":"当然还有其他衡量时间和空间的方式，比如"}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null}},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","text":">Theta: 描述的是 tight bound "}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","text":">Omega(n): 这个描述的是 best case，最好的情况，没啥意义"}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null}},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","text":"这也给我们了些许启发，不要说你平时表现有多好，没有意义；面试衡量的是你在 worst case 的水平；不要说面试没有发挥出你的真实水平，扎心的是那就是我们的真实水平。"}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null}},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null}},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null}},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","text":">那对于这个题来说，时间复杂度是多少呢？"}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null}},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","text":"答：因为我们每个节点都走了一遍，所以是把"},{"type":"text","marks":[{"type":"strong"}],"text":"所有节点的时间加起来"},{"type":"text","text":"就是总的时间。"}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null}},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","text":"在这里，我们在每个节点上做的事情就是相加求和，是 O(1) 的操作，且每个节点的时间都是一样的，所以："}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null}},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"codeinline","content":[{"type":"text","text":"总时间 = 节点个数 * 每个节点的时间"}]}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null}},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null}},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","text":"那就变成了求"},{"type":"codeinline","content":[{"type":"text","text":"节点个数"}]},{"type":"text","text":"的数学题："}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null}},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","text":"在 N = 5 时， "}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null}},{"type":"image","attrs":{"src":"https://static001.geekbang.org/infoq/63/630ff89ff52100930c68fa5bd70f451d.jpeg","alt":null,"title":null,"style":null,"href":null,"fromPaste":true,"pastePass":true}},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null}},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null}},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null}},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null}},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","text":"最上面一层有1个节点， "}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","text":"第二层 2 个， "}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","text":"第三层 4 个， "}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","text":"第四层 8 个， "}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","text":"第五层 16 个，如果填满的话，想象成一颗很大的树:) "}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null}},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","text":"这里就不要在意这个没填满的地方了，肯定是会有差这么几个 node，但是大 O 表达的时间复杂度我们刚说过了，求的是 worst case. "}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null}},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","text":"那么总的节点数就是： "}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","text":" "},{"type":"codeinline","content":[{"type":"text","text":"1 + 2 + 4 + 8 + 16"}]},{"type":"text","text":" "}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null}},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","text":"这就是一个等比数列求和了，当然你可以用数学公式来算，但还有个"},{"type":"codeinline","content":[{"type":"text","text":"小技巧"}]},{"type":"text","text":"可以帮助你快速计算："}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null}},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","text":" 其实前面每一层的节点相加起来的个数都不会超过最后一层的节点的个数，总的节点数最多也就是最后一层节点数 * 2，然后在大 O 的时间复杂度里面常数项也是无所谓的，所以这个总的时间复杂度就是："}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null}},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","text":" "}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null}},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null}},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null}},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","text":"最后一层节点的个数：2^n"}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null}},{"type":"heading","attrs":{"align":null,"level":3},"content":[{"type":"text","text":"空间复杂度分析"}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null}},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","text":" 一般书上写的空间复杂度是指："}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null}},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","text":" >算法运行期间所需占用的"},{"type":"text","marks":[{"type":"strong"}],"text":"所有"},{"type":"text","text":"内存空间"}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null}},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","text":" 但是在公司里大家常用的，也是面试时问的指的是 "}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","text":" "},{"type":"codeinline","content":[{"type":"text","text":"Auxiliary space complexity"}]},{"type":"text","text":"："}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null}},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","text":" > 运行算法时所需占用的"},{"type":"text","marks":[{"type":"strong"}],"text":"额外"},{"type":"text","text":"空间。"}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null}},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","text":" 举例说明区别：比如结果让你输出一个长度为 n 的数组，那么这 O(n) 的空间是不算在算法的空间复杂度里的，因为这个空间是跑不掉的，不是取决于你的算法的。"}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null}},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","text":"那空间复杂度怎么分析呢？"}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null}},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","text":"我们刚刚说到了冯诺伊曼体系，从图中也很容易看出来，是"},{"type":"codeinline","content":[{"type":"text","text":"最左边这条路线"}]},{"type":"text","text":"占用 stack 的空间最多，一直不断的"},{"type":"codeinline","content":[{"type":"text","text":"压栈"}]},{"type":"text","text":"，也就是从 5 到 4 到 3 到 2 一直压到 1，才到 base case 返回，每个节点占用的空间复杂度是 O(1)，所以加起来总的空间复杂度就是 "},{"type":"codeinline","content":[{"type":"text","text":"O(n)"}]},{"type":"text","text":". "}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null}},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","text":" 我在上面👆的视频里也提到了，不懂的同学往上翻看视频哦～"}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null}},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null}},{"type":"heading","attrs":{"align":null,"level":3},"content":[{"type":"text","text":"优化算法"}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null}},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","text":"那我们就想了，为什么这么一个简简单单的运算竟然要指数级的时间复杂度？到底是为什么让时间如此之大。"}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null}},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","text":"那也不难看出来，在这棵 "},{"type":"codeinline","content":[{"type":"text","text":"Recursion Tree"}]},{"type":"text","text":" 里，有太多的"},{"type":"codeinline","content":[{"type":"text","text":"重复计算"}]},{"type":"text","text":"了。"}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null}},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","text":"比如一个 F(2) 在这里都被计算了 3 次，F(3) 被计算了 2 次，每次还都要再重新算，这不就是"},{"type":"codeinline","content":[{"type":"text","text":"狗熊掰棒子"}]},{"type":"text","text":"吗，真的是一把辛酸泪。"}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null}},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","text":"那找到了原因之后，为了解决这种重复计算，计算机采用的方法其实和我们人类是一样的："},{"type":"codeinline","content":[{"type":"text","text":"记笔记"}]},{"type":"text","text":"。"}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null}},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","text":"对很多职业来说，比如医生、律师、以及我们工程师，为什么越老经验值钱？因为我们见得多积累的多，下次再遇到类似的问题时，能够很快的给出解决方案，哪怕一时解决不了，也避免了一些盲目的试错，我们会"},{"type":"codeinline","content":[{"type":"text","text":"站在过去的高度不断进步"}]},{"type":"text","text":"，而不是每次都从零开始。"}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null}},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","text":"回到优化算法上来，那计算机如何记笔记呢？"}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null}},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","text":"我们要想求 F(n)，无非也就是要 "}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","text":" "},{"type":"codeinline","content":[{"type":"text","text":"记录 F(0) ~ F(n-1) 的值"}]},{"type":"text","text":"， "}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","text":" 那选取一个合适的数据结构来存储就好了。 "}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null}},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","text":"那这里很明显了，用一个数组来存："}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null}},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","text":"| Index | 0 | 1 | 2 | 3 | 4 | 5 |"}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","text":"| ------ | ----- | ------ | ------ |----- | ------ | ------ |"}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","text":"| F(n) | 0 | 1 | 1 |2|3|5"}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null}},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null}},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","text":"那有了这个 cheat sheet，我们就可以从前到后得到结果了，这样每一个点就只算了一遍，用一个 for loop 就可以写出来，代码也非常简单。"}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null}},{"type":"codeblock","attrs":{"lang":"java"},"content":[{"type":"text","text":"class Solution {\n public int fib(int N) {\n if (N == 0) {\n return 0;\n }\n if (N== 1) {\n return 1;\n }\n int[] notes = new int[N+1];\n notes[0] = 0;\n notes[1] = 1;\n for(int i = 2; i <= N; i++) {\n notes[i] = notes[i-1] + notes[i-2];\n }\n return notes[N];\n }\n}"}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null}},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","text":"这个速度就是 100% 了～"}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null}},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null}},{"type":"image","attrs":{"src":"https://static001.geekbang.org/infoq/65/654745223ad41fb191fc5f5f957053ee.jpeg","alt":null,"title":null,"style":null,"href":null,"fromPaste":true,"pastePass":true}},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null}},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null}},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","text":"但是我们可以看到，空间应该还有优化的余地。"}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null}},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","text":"那仔细想想，其实我们"},{"type":"codeinline","content":[{"type":"text","text":"记笔记的时候需要记录这么多吗"}]},{"type":"text","text":"？需要从幼儿园到小学到初中到高中的笔记都留着吗？"}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null}},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","text":"那其实每项的计算"},{"type":"codeinline","content":[{"type":"text","text":"只取决于它前面的两项"}]},{"type":"text","text":"，所以只用保留这两个就好了。"}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null}},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","text":"那我们可以用一个长度为 2 的数组来计算，或者就用 2 个变量。"}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null}},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","text":"更新代码："}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null}},{"type":"codeblock","attrs":{"lang":"java"},"content":[{"type":"text","text":"class Solution {\n public int fib(int N) {\n int a = 0;\n int b = 1;\n if(N == 0) {\n return a;\n }\n if(N == 1) {\n return b;\n }\n for(int i = 2; i <= N; i++) {\n int tmp = a + b;\n a = b;\n b = tmp;\n }\n return b;\n }\n}"}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null}},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null}},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","text":"这样我们就把空间复杂度优化到了 "},{"type":"codeinline","content":[{"type":"text","text":"O(1)"}]},{"type":"text","text":"，时间复杂度和用数组记录一样都是 "},{"type":"codeinline","content":[{"type":"text","text":"O(n)"}]},{"type":"text","text":"."}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null}},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","text":"这种方法其实就是"},{"type":"codeinline","content":[{"type":"text","text":"动态规划"}]},{"type":"text","text":" "},{"type":"codeinline","content":[{"type":"text","text":"Dynamic Programming"}]},{"type":"text","text":"，写出来的代码非常简单。"}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null}},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","text":"那我们比较一下 Recursion 和 DP："}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null}},{"type":"blockquote","content":[{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","marks":[{"type":"strong"}],"text":"Recursion"},{"type":"text","text":" 是从大到小，层层分解，直到 base case 分解不了了再组合返回上去； "}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","marks":[{"type":"strong"}],"text":"DP"},{"type":"text","text":" 是从小到大，记好笔记，不断进步。 "}]}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null}},{"type":"blockquote","content":[{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","text":"也就是 "},{"type":"text","marks":[{"type":"strong"}],"text":"Recursion + Cache = DP"}]}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null}},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","text":"如何记录这个笔记，如何高效的记笔记，这是 DP 的难点。"}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null}},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","marks":[{"type":"strong"}],"text":"有人说 DP 是拿空间换时间，但我不这么认为，这道题就是一个很好的例证。"}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null}},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","text":"在用递归解题时，我们可以看到，空间是 O(n) 在栈上的，但是用 DP 我们可以把空间优化到 O(1)，DP 可以做到时间空间的双重优化。"}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null}},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null}},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null}},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null}},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","text":"其实呢，斐波那契数列在现实生活中也有很多应用。"}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null}},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","text":">比如在我司以及很多大公司里，每个任务要给分值，1分表示大概需要花1天时间完成，然后分值只有1，2，3，5，8这5种，（如果有大于8分的任务，就需要把它 break down 成8分以内的，以便大家在两周内能完成。） "}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","text":">因为任务是永远做不完的而每个人的时间是有限的，所以每次小组会开会，挑出最重要的任务让大家来做，然后每个人根据自己的 available 的天数去 pick up 相应的任务。"}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null}},{"type":"heading","attrs":{"align":null,"level":2},"content":[{"type":"text","text":"披着羊皮的狼"}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null}},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","text":"那有同学可能会想，这题这么简单，这都 2020 年了，面试还会考么？"}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null}},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","text":"答："},{"type":"text","marks":[{"type":"strong"}],"text":"真的会。"}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null}},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","text":"只是不能以这么直白的方式给你了。"}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null}},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","text":"比如很有名的"},{"type":"text","marks":[{"type":"strong"}],"text":"爬楼梯"},{"type":"text","text":"问题："}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null}},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","text":">一个 N 阶的楼梯，每次能走一层或者两层，问一共有多少种走法。"}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null}},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null}},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","text":"这个题这么想："}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null}},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","text":" 站在当前位置，只能是从前一层，或者前两层上来的，所以 f(n) = f(n-1) + f(n-2). "}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null}},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null}},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","text":"这题是我当年面试时真实被问的，那时我还在写 python，为了炫技，还用了lambda function: "}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null}},{"type":"codeblock","attrs":{"lang":"python"},"content":[{"type":"text","text":"f = lambda n: 1 if n in (1, 2) else f(n-1) + f(n-2)"}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null}},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","text":"递归的写法时间复杂度太高，所以又写了一个 for loop 的版本"}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null}},{"type":"codeblock","attrs":{"lang":"python"},"content":[{"type":"text","text":"def fib(n)\n a, b = 1, 1\n for i in range(n-1):\n\ta, b = b, a+b\n return a "}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null}},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","text":"然后还写了个 caching 的方法:"}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null}},{"type":"codeblock","attrs":{"lang":"python"},"content":[{"type":"text","text":"def cache(f):\n\tmemo = {}\n\tdef helper(x):\n\t\tif x not in memo:\n\t\t\tmemo[x] = f(x)\n\t\treturn memo[x]\n\treturn helper\n@cache\ndef fibR(n):\n\tif n==1 or n==2: return 1\n\treturn fibR(n-1) + fibR(n-2)"}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null}},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","text":"还顺便和面试官聊了下 tail recursion: "}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null}},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","text":" >tail recursion 尾递归：就是递归的这句话是整个方法的最后一句话。"}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null}},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","text":" 那这个有什么特别之处呢？"}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null}},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","text":" > 尾递归的特点就是我们可以"},{"type":"text","marks":[{"type":"strong"}],"text":"很容易的把它转成 iterative 的写法"},{"type":"text","text":"，当然有些智能的编译器会自动帮我们做了（不是说显性的转化，而是在运行时按照 iterative 的方式去运行，实际消耗的空间是 O(1)）"}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null}},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","text":" 那为什么呢？"}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null}},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","text":" > 因为回来的时候不需要 backtrack，递归这里就是最后一步了，不需要再往上一层返值。"}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null}},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","text":" "}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null}},{"type":"codeblock","attrs":{"lang":"python"},"content":[{"type":"text","text":"def fib(n, a=0, b=1):\n\tif n==0: return a\n\t if n==1: return b\n\treturn fib(n-1, b, a+b)"}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null}},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","text":"最终，拿出了我的杀手鐗：lambda and reduce"}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null}},{"type":"codeblock","attrs":{"lang":"python"},"content":[{"type":"text","text":"fibRe = lambda n: reduce(lambda x, n: [x[1], x[0]+x[1]], range(n), [0, 1])"}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null}},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","text":"看到面试官满意的表情后，就开始继续深入的聊了..."}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null}},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","text":"所以说，不要以为它简单，同一道题可以用七八种方法来解，分析好每个方法的优缺点，引申到你可以引申的地方，展示自己扎实的基本功，这场面试其实就是你 show off 的机会，这样才能骗过面试官啊～lol"}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null}},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","text":"如果你喜欢这篇文章，记得给我点赞留言哦～你们的支持和认可，就是我创作的最大动力，我们下篇文章见！"}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null}},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","marks":[{"type":"strong"}],"text":"我是小齐，纽约程序媛，终生学习者，每天晚上 9 点，云自习室里不见不散！"}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null}},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","marks":[{"type":"strong"}],"text":"更多干货文章见我的 Github: https://github.com/xiaoqi6666/NYCSDE"}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null}}]}