一文學懂遞歸和動態規劃！

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":"時間複雜度分析"}]},{"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":">如何評價一個算法的好壞？"}]},{"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":"很多問題都有多種解法，畢竟條條大路通羅馬。但如何評價每種方法的優劣，我們一般是用"},{"type":"text","marks":[{"type":"strong"}],"text":"大 O 表達式"},{"type":"text","text":"來衡量"},{"type":"text","marks":[{"type":"strong"}],"text":"時間和空間"},{"type":"text","text":"複雜度。"}]},{"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":"時間複雜度：隨着自變量的增長，所需時間的增長情況。"}]}]},{"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":"這裏大 O 表示的是一個算法在 "},{"type":"text","marks":[{"type":"strong"}],"text":"worst case"},{"type":"text","text":" 的表現情況，這就是我們最關心的，不然春運搶車票的時候系統 hold 不住了，你跟我說這個算法很優秀？"}]},{"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":"當然還有其他衡量時間和空間的方式，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): 這個描述的是 best case，最好的情況，沒啥意義"}]},{"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":"這也給我們了些許啓發，不要說你平時表現有多好，沒有意義；面試衡量的是你在 worst case 的水平；不要說面試沒有發揮出你的真實水平，扎心的是那就是我們的真實水平。"}]},{"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":">那對於這個題來說，時間複雜度是多少呢？"}]},{"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：因爲我們每個節點都走了一遍，所以是把"},{"type":"text","marks":[{"type":"strong"}],"text":"所有節點的時間加起來"},{"type":"text","text":"就是總的時間。"}]},{"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":"在這裏，我們在每個節點上做的事情就是相加求和，是 O(1) 的操作，且每個節點的時間都是一樣的，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":"總時間 = 節點個數 * 每個節點的時間"}]}]},{"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":"那就變成了求"},{"type":"codeinline","content":[{"type":"text","text":"節點個數"}]},{"type":"text","text":"的數學題："}]},{"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 時， "}]},{"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":"最上面一層有1個節點， "}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","text":"第二層 2 個， "}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","text":"第三層 4 個， "}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","text":"第四層 8 個， "}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","text":"第五層 16 個，如果填滿的話，想象成一顆很大的樹:) "}]},{"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":"這裏就不要在意這個沒填滿的地方了，肯定是會有差這麼幾個 node，但是大 O 表達的時間複雜度我們剛說過了，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":"那麼總的節點數就是： "}]},{"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":"這就是一個等比數列求和了，當然你可以用數學公式來算，但還有個"},{"type":"codeinline","content":[{"type":"text","text":"小技巧"}]},{"type":"text","text":"可以幫助你快速計算："}]},{"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":" 其實前面每一層的節點相加起來的個數都不會超過最後一層的節點的個數，總的節點數最多也就是最後一層節點數 * 2，然後在大 O 的時間複雜度裏面常數項也是無所謂的，所以這個總的時間複雜度就是："}]},{"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":"最後一層節點的個數：2^n"}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null}},{"type":"heading","attrs":{"align":null,"level":3},"content":[{"type":"text","text":"空間複雜度分析"}]},{"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":" 一般書上寫的空間複雜度是指："}]},{"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":" >算法運行期間所需佔用的"},{"type":"text","marks":[{"type":"strong"}],"text":"所有"},{"type":"text","text":"內存空間"}]},{"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":" 但是在公司裏大家常用的，也是面試時問的指的是 "}]},{"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":" > 運行算法時所需佔用的"},{"type":"text","marks":[{"type":"strong"}],"text":"額外"},{"type":"text","text":"空間。"}]},{"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":" 舉例說明區別：比如結果讓你輸出一個長度爲 n 的數組，那麼這 O(n) 的空間是不算在算法的空間複雜度裏的，因爲這個空間是跑不掉的，不是取決於你的算法的。"}]},{"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":"那空間複雜度怎麼分析呢？"}]},{"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":"我們剛剛說到了馮諾伊曼體系，從圖中也很容易看出來，是"},{"type":"codeinline","content":[{"type":"text","text":"最左邊這條路線"}]},{"type":"text","text":"佔用 stack 的空間最多，一直不斷的"},{"type":"codeinline","content":[{"type":"text","text":"壓棧"}]},{"type":"text","text":"，也就是從 5 到 4 到 3 到 2 一直壓到 1，纔到 base case 返回，每個節點佔用的空間複雜度是 O(1)，所以加起來總的空間複雜度就是 "},{"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👆的視頻裏也提到了，不懂的同學往上翻看視頻哦～"}]},{"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":"優化算法"}]},{"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":"那我們就想了，爲什麼這麼一個簡簡單單的運算竟然要指數級的時間複雜度？到底是爲什麼讓時間如此之大。"}]},{"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":"那也不難看出來，在這棵 "},{"type":"codeinline","content":[{"type":"text","text":"Recursion Tree"}]},{"type":"text","text":" 裏，有太多的"},{"type":"codeinline","content":[{"type":"text","text":"重複計算"}]},{"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":"比如一個 F(2) 在這裏都被計算了 3 次，F(3) 被計算了 2 次，每次還都要再重新算，這不就是"},{"type":"codeinline","content":[{"type":"text","text":"狗熊掰棒子"}]},{"type":"text","text":"嗎，真的是一把辛酸淚。"}]},{"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":"那找到了原因之後，爲了解決這種重複計算，計算機採用的方法其實和我們人類是一樣的："},{"type":"codeinline","content":[{"type":"text","text":"記筆記"}]},{"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":"對很多職業來說，比如醫生、律師、以及我們工程師，爲什麼越老經驗值錢？因爲我們見得多積累的多，下次再遇到類似的問題時，能夠很快的給出解決方案，哪怕一時解決不了，也避免了一些盲目的試錯，我們會"},{"type":"codeinline","content":[{"type":"text","text":"站在過去的高度不斷進步"}]},{"type":"text","text":"，而不是每次都從零開始。"}]},{"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":"回到優化算法上來，那計算機如何記筆記呢？"}]},{"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":"我們要想求 F(n)，無非也就是要 "}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","text":" "},{"type":"codeinline","content":[{"type":"text","text":"記錄 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":" 那選取一個合適的數據結構來存儲就好了。 "}]},{"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":"那這裏很明顯了，用一個數組來存："}]},{"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":"那有了這個 cheat sheet，我們就可以從前到後得到結果了，這樣每一個點就只算了一遍，用一個 for loop 就可以寫出來，代碼也非常簡單。"}]},{"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":"這個速度就是 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":"但是我們可以看到，空間應該還有優化的餘地。"}]},{"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":"那仔細想想，其實我們"},{"type":"codeinline","content":[{"type":"text","text":"記筆記的時候需要記錄這麼多嗎"}]},{"type":"text","text":"？需要從幼兒園到小學到初中到高中的筆記都留着嗎？"}]},{"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":"那其實每項的計算"},{"type":"codeinline","content":[{"type":"text","text":"只取決於它前面的兩項"}]},{"type":"text","text":"，所以只用保留這兩個就好了。"}]},{"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":"那我們可以用一個長度爲 2 的數組來計算，或者就用 2 個變量。"}]},{"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":"更新代碼："}]},{"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":"這樣我們就把空間複雜度優化到了 "},{"type":"codeinline","content":[{"type":"text","text":"O(1)"}]},{"type":"text","text":"，時間複雜度和用數組記錄一樣都是 "},{"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":"這種方法其實就是"},{"type":"codeinline","content":[{"type":"text","text":"動態規劃"}]},{"type":"text","text":" "},{"type":"codeinline","content":[{"type":"text","text":"Dynamic Programming"}]},{"type":"text","text":"，寫出來的代碼非常簡單。"}]},{"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":"那我們比較一下 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":" 是從大到小，層層分解，直到 base case 分解不了了再組合返回上去； "}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","marks":[{"type":"strong"}],"text":"DP"},{"type":"text","text":" 是從小到大，記好筆記，不斷進步。 "}]}]},{"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":"如何記錄這個筆記，如何高效的記筆記，這是 DP 的難點。"}]},{"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":"有人說 DP 是拿空間換時間，但我不這麼認爲，這道題就是一個很好的例證。"}]},{"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":"在用遞歸解題時，我們可以看到，空間是 O(n) 在棧上的，但是用 DP 我們可以把空間優化到 O(1)，DP 可以做到時間空間的雙重優化。"}]},{"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":"其實呢，斐波那契數列在現實生活中也有很多應用。"}]},{"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裏，每個任務要給分值，1分表示大概需要花1天時間完成，然後分值只有1，2，3，5，8這5種，（如果有大於8分的任務，就需要把它 break down 成8分以內的，以便大家在兩週內能完成。） "}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","text":">因爲任務是永遠做不完的而每個人的時間是有限的，所以每次小組會開會，挑出最重要的任務讓大家來做，然後每個人根據自己的 available 的天數去 pick up 相應的任務。"}]},{"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":"那有同學可能會想，這題這麼簡單，這都 2020 年了，面試還會考麼？"}]},{"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":"真的會。"}]},{"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":"只是不能以這麼直白的方式給你了。"}]},{"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":"爬樓梯"},{"type":"text","text":"問題："}]},{"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":">一個 N 階的樓梯，每次能走一層或者兩層，問一共有多少種走法。"}]},{"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":"這個題這麼想："}]},{"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":" 站在當前位置，只能是從前一層，或者前兩層上來的，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":"這題是我當年面試時真實被問的，那時我還在寫 python，爲了炫技，還用了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":"遞歸的寫法時間複雜度太高，所以又寫了一個 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":"然後還寫了個 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":"還順便和麪試官聊了下 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 尾遞歸：就是遞歸的這句話是整個方法的最後一句話。"}]},{"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":" 那這個有什麼特別之處呢？"}]},{"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":" > 尾遞歸的特點就是我們可以"},{"type":"text","marks":[{"type":"strong"}],"text":"很容易的把它轉成 iterative 的寫法"},{"type":"text","text":"，當然有些智能的編譯器會自動幫我們做了（不是說顯性的轉化，而是在運行時按照 iterative 的方式去運行，實際消耗的空間是 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":" 那爲什麼呢？"}]},{"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":" > 因爲回來的時候不需要 backtrack，遞歸這裏就是最後一步了，不需要再往上一層返值。"}]},{"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":"最終，拿出了我的殺手鐗：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":"看到面試官滿意的表情後，就開始繼續深入的聊了..."}]},{"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":"所以說，不要以爲它簡單，同一道題可以用七八種方法來解，分析好每個方法的優缺點，引申到你可以引申的地方，展示自己紮實的基本功，這場面試其實就是你 show off 的機會，這樣才能騙過面試官啊～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":"如果你喜歡這篇文章，記得給我點贊留言哦～你們的支持和認可，就是我創作的最大動力，我們下篇文章見！"}]},{"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":"我是小齊，紐約程序媛，終生學習者，每天晚上 9 點，雲自習室裏不見不散！"}]},{"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":"更多幹貨文章見我的 Github: https://github.com/xiaoqi6666/NYCSDE"}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null}}]}