圖解堆排序，帶你徹底瞭解清楚！

{"type":"doc","content":[{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","marks":[{"type":"color","attrs":{"color":"#773098","name":"user"}},{"type":"strong","attrs":{}}],"text":"寫在前面：","attrs":{}}]},{"type":"blockquote","content":[{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","marks":[{"type":"color","attrs":{"color":"#616161","name":"user"}}],"text":"大家好，我是時光。","attrs":{}}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","marks":[{"type":"color","attrs":{"color":"#616161","name":"user"}}],"text":"今天給大家帶來的是排序算法中的堆排序，這種排序跟二叉樹相關。我採用圖解方式講解，爭取寫透徹。話不多說，開始！","attrs":{}}]}],"attrs":{}},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","marks":[{"type":"color","attrs":{"color":"#773098","name":"user"}},{"type":"strong","attrs":{}}],"text":"思維導圖：","attrs":{}}]},{"type":"image","attrs":{"src":"https://static001.geekbang.org/infoq/36/3672a13f483e65c17c9b97f926f7cc26.png","alt":"堆排序導圖","title":null,"style":[{"key":"width","value":"75%"},{"key":"bordertype","value":"none"}],"href":null,"fromPaste":true,"pastePass":true}},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","marks":[{"type":"color","attrs":{"color":"#595959","name":"user"}}],"text":"堆排序導圖","attrs":{}}]},{"type":"heading","attrs":{"align":null,"level":2},"content":[{"type":"text","marks":[{"type":"color","attrs":{"color":"#773098","name":"user"}}],"text":"1，堆排序概念","attrs":{}}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","text":"堆排序","attrs":{}},{"type":"codeinline","content":[{"type":"text","text":"（Heapsort）","attrs":{}}],"marks":[{"type":"color","attrs":{"color":"#9654B5","name":"user"}}],"attrs":{}},{"type":"text","text":"是指利用堆這種數據結構所設計的一種排序算法。堆積是一個近似完全二叉樹的結構，並同時滿足堆積的性質：即子結點的鍵值或索引總是小於（或者大於）它的父節點。","attrs":{}}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","text":"相關概念：","attrs":{}}]},{"type":"heading","attrs":{"align":null,"level":3},"content":[{"type":"text","marks":[{"type":"color","attrs":{"color":"#773098","name":"user"}}],"text":"1.1，二叉樹","attrs":{}}]},{"type":"image","attrs":{"src":"https://static001.geekbang.org/infoq/51/51cd55cc60aab46dc095975d1b497b04.png","alt":"二叉樹","title":null,"style":[{"key":"width","value":"75%"},{"key":"bordertype","value":"none"}],"href":null,"fromPaste":true,"pastePass":true}},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","marks":[{"type":"color","attrs":{"color":"#595959","name":"user"}}],"text":"二叉樹","attrs":{}}]},{"type":"blockquote","content":[{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","marks":[{"type":"color","attrs":{"color":"#616161","name":"user"}}],"text":"特徵：每個節點最多隻有2個子節點（不存在度大於2的節點）","attrs":{}}]}],"attrs":{}},{"type":"heading","attrs":{"align":null,"level":3},"content":[{"type":"text","marks":[{"type":"color","attrs":{"color":"#773098","name":"user"}}],"text":"1.2，滿二叉樹","attrs":{}}]},{"type":"image","attrs":{"src":"https://static001.geekbang.org/infoq/b7/b7a3823ab5d08f86b05970848f3c7ea3.png","alt":"滿二叉樹","title":null,"style":[{"key":"width","value":"75%"},{"key":"bordertype","value":"none"}],"href":null,"fromPaste":true,"pastePass":true}},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","marks":[{"type":"color","attrs":{"color":"#595959","name":"user"}}],"text":"滿二叉樹","attrs":{}}]},{"type":"blockquote","content":[{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","marks":[{"type":"color","attrs":{"color":"#616161","name":"user"}}],"text":"滿二叉樹：葉子節點全部都在最底層；除葉子節點外，每個節點都有左右孩子；","attrs":{}}]}],"attrs":{}},{"type":"heading","attrs":{"align":null,"level":3},"content":[{"type":"text","marks":[{"type":"color","attrs":{"color":"#773098","name":"user"}}],"text":"1.3，完全二叉樹","attrs":{}}]},{"type":"image","attrs":{"src":"https://static001.geekbang.org/infoq/4c/4c4f3f8f299314ff1bea892fe76bad7f.png","alt":"完全二叉樹","title":null,"style":[{"key":"width","value":"75%"},{"key":"bordertype","value":"none"}],"href":null,"fromPaste":true,"pastePass":true}},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","marks":[{"type":"color","attrs":{"color":"#595959","name":"user"}}],"text":"完全二叉樹","attrs":{}}]},{"type":"blockquote","content":[{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","marks":[{"type":"color","attrs":{"color":"#616161","name":"user"}}],"text":"完全二叉樹：葉子節點全部都在最底的兩層；最後一層只缺少右邊的葉子節點，左邊一定有葉子節點；除了最後一層，其他層的節點個數均達到最大值；","attrs":{}}]}],"attrs":{}},{"type":"heading","attrs":{"align":null,"level":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:[{"type":"color","attrs":{"color":"#616161","name":"user"}}],"text":"第一次交換，我們直接將元素9與元素0交換，此時堆頂元素爲0，設當前節點","attrs":{}},{"type":"codeinline","content":[{"type":"text","text":"index","attrs":{}}],"marks":[{"type":"color","attrs":{"color":"#9654B5","name":"user"}}],"attrs":{}},{"type":"text","text":"=0；","attrs":{}}]}],"attrs":{}},{"type":"image","attrs":{"src":"https://static001.geekbang.org/infoq/a2/a223543ae4c743741760df8bb55e7470.png","alt":"0與9交換","title":null,"style":[{"key":"width","value":"75%"},{"key":"bordertype","value":"none"}],"href":null,"fromPaste":true,"pastePass":true}},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","marks":[{"type":"color","attrs":{"color":"#595959","name":"user"}}],"text":"0與9交換","attrs":{}}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","marks":[{"type":"color","attrs":{"color":"#595959","name":"user"}}],"text":"這時，我們需要將剩下的元素（排除元素9）進行堆排序，直到下面這個結果：","attrs":{}}]},{"type":"image","attrs":{"src":"https://static001.geekbang.org/infoq/0a/0a9b0d3f8272d85eba25685cbcf8e746.png","alt":"除元素9以外剩下的元素排序","title":null,"style":[{"key":"width","value":"75%"},{"key":"bordertype","value":"none"}],"href":null,"fromPaste":true,"pastePass":true}},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","marks":[{"type":"color","attrs":{"color":"#595959","name":"user"}}],"text":"除元素9以外剩下的元素排序","attrs":{}}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","marks":[{"type":"color","attrs":{"color":"#595959","name":"user"}}],"text":"代碼：","attrs":{}}]},{"type":"codeblock","attrs":{"lang":"java"},"content":[{"type":"text","text":"/**\n * 第二步，交換堆頂（最大）元素和二叉堆的最後一個葉子節點元素。目的是交換元素\n * i從二叉堆的最後一個葉子節點元素開始，也就是len-1開始(數組下標從0開始)\n */\nfor(int i=len-1;i>=0;i--){\n    int temp=arr[i];\n    arr[i]=arr[0];\n    arr[0]=temp;\n    //交換完之後需要重新調整二叉堆，從頭開始調整，此時Index=0\n    HeapAdjust(arr,0,i);\n}\n","attrs":{}}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","marks":[{"type":"color","attrs":{"color":"#595959","name":"user"}}],"text":"注意：這裏有個小細節問題，前面我們寫的初始化堆方法有三個參數，分別是數組","attrs":{}},{"type":"codeinline","content":[{"type":"text","text":"arr","attrs":{}}],"marks":[{"type":"color","attrs":{"color":"#9654B5","name":"user"}}],"attrs":{}},{"type":"text","text":"，當前節點","attrs":{}},{"type":"codeinline","content":[{"type":"text","text":"index","attrs":{}}],"marks":[{"type":"color","attrs":{"color":"#9654B5","name":"user"}}],"attrs":{}},{"type":"text","text":"以及數組長度","attrs":{}},{"type":"codeinline","content":[{"type":"text","text":"len","attrs":{}}],"marks":[{"type":"color","attrs":{"color":"#9654B5","name":"user"}}],"attrs":{}},{"type":"text","text":"，如下：","attrs":{}}]},{"type":"codeblock","attrs":{"lang":"java"},"content":[{"type":"text","text":"HeapAdjust(int[] arr,int index,int len)\n","attrs":{}}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","marks":[{"type":"color","attrs":{"color":"#595959","name":"user"}}],"text":"那麼，爲何不直接傳入兩個參數即可，數組長度直接用","attrs":{}},{"type":"codeinline","content":[{"type":"text","text":"arr.length","attrs":{}}],"marks":[{"type":"color","attrs":{"color":"#9654B5","name":"user"}}],"attrs":{}},{"type":"text","text":"表示不就行了嗎？初始化堆的時候是可以，但是後面在交換堆頂元素和末尾的葉子節點時，在對剩下的元素進行排序時，此時的數組長度可不是","attrs":{}},{"type":"codeinline","content":[{"type":"text","text":"arr.length","attrs":{}}],"marks":[{"type":"color","attrs":{"color":"#9654B5","name":"user"}}],"attrs":{}},{"type":"text","text":"了，應該是變化的參數","attrs":{}},{"type":"codeinline","content":[{"type":"text","text":"i","attrs":{}}],"marks":[{"type":"color","attrs":{"color":"#9654B5","name":"user"}}],"attrs":{}},{"type":"text","text":"，因爲交換之後的元素（比如9）就不計入堆中排序了，所以需要3個參數。","attrs":{}}]},{"type":"blockquote","content":[{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","marks":[{"type":"color","attrs":{"color":"#616161","name":"user"}}],"text":"我們進行第二次交換，我們直接將元素8與元素2交換，此時堆頂元素爲2，設當前節點","attrs":{}},{"type":"codeinline","content":[{"type":"text","text":"index","attrs":{}}],"marks":[{"type":"color","attrs":{"color":"#9654B5","name":"user"}}],"attrs":{}},{"type":"text","text":"=2；","attrs":{}}]}],"attrs":{}},{"type":"image","attrs":{"src":"https://static001.geekbang.org/infoq/e5/e51aa1aab63f70c8a589aecb087be57f.png","alt":"8與2交換","title":null,"style":[{"key":"width","value":"75%"},{"key":"bordertype","value":"none"}],"href":null,"fromPaste":true,"pastePass":true}},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","marks":[{"type":"color","attrs":{"color":"#595959","name":"user"}}],"text":"8與2交換","attrs":{}}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","marks":[{"type":"color","attrs":{"color":"#595959","name":"user"}}],"text":"這時，我們需要將剩下的元素（排除元素9和8）進行堆排序，直到下面這個結果：","attrs":{}}]},{"type":"image","attrs":{"src":"https://static001.geekbang.org/infoq/7f/7f7a8d2a714a44cbcf6f421f00b75470.png","alt":"除元素9和8以外剩下的元素排序","title":null,"style":[{"key":"width","value":"75%"},{"key":"bordertype","value":"none"}],"href":null,"fromPaste":true,"pastePass":true}},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","marks":[{"type":"color","attrs":{"color":"#595959","name":"user"}}],"text":"除元素9和8以外剩下的元素排序","attrs":{}}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","marks":[{"type":"color","attrs":{"color":"#595959","name":"user"}}],"text":"到這個時候，我們再重複上述步驟，不斷調換堆頂和最末尾的節點元素即可，再不斷地對剩下的元素進行排序，最後就能得到從小到大排序好的堆了，如下圖所示，這就是我們想要的結果：","attrs":{}}]},{"type":"image","attrs":{"src":"https://static001.geekbang.org/infoq/fd/fd8fbbbbc0bfbaba6d6b65cbb8b2b560.png","alt":"最終排序結果：從小到大","title":null,"style":[{"key":"width","value":"75%"},{"key":"bordertype","value":"none"}],"href":null,"fromPaste":true,"pastePass":true}},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","marks":[{"type":"color","attrs":{"color":"#595959","name":"user"}}],"text":"最終排序結果：從小到大","attrs":{}}]},{"type":"heading","attrs":{"align":null,"level":2},"content":[{"type":"text","marks":[{"type":"color","attrs":{"color":"#773098","name":"user"}}],"text":"3，完整代碼","attrs":{}}]},{"type":"codeblock","attrs":{"lang":"java"},"content":[{"type":"text","text":"import java.util.Arrays;\n\npublic class Head_Sort {\n    public static void main(String[] args) {\n        int[] arr={4,2,8,0,5,7,1,3,9};\n        System.out.println(\"排序前:\"+Arrays.toString(arr));\n        System.out.println(\"排序後:\"+Arrays.toString(HeapSort(arr)));\n    }\n\n    //堆排序\n    public static int[] HeapSort(int[] arr){\n        int len=arr.length;\n        /**\n         * 第一步，初始化堆，最大堆，從小到大。目的是對元素排序\n         * i從完全二叉樹的第一個非葉子節點開始，也就是len/2-1開始(數組下標從0開始)\n         */\n        for(int i=len/2-1;i>=0;i--){\n            HeapAdjust(arr,i,len);\n        }\n        /**\n         * 第二步，交換堆頂（最大）元素和二叉堆的最後一個葉子節點元素。目的是交換元素\n         * i從二叉堆的最後一個葉子節點元素開始，也就是len-1開始(數組下標從0開始)\n         */\n        for(int i=len-1;i>=0;i--){\n            int temp=arr[i];\n            arr[i]=arr[0];\n            arr[0]=temp;\n            //交換完之後需要重新調整二叉堆，從頭開始調整，此時Index=0\n            HeapAdjust(arr,0,i);\n        }\n        return arr;\n    }\n\n    /**\n     *初始化堆\n     * @param arr 待調整的二叉樹(數組)\n     * @param index 待調整的節點下標，二叉樹的第一個非葉子節點\n     * @param len 待調整的數組長度\n     */\n    public static void HeapAdjust(int[] arr,int index,int len){\n        //先保存當前節點的下標\n        int max = index;\n        //保存左右孩子數組下標\n        int lchild = index*2+1;\n        int rchild = index*2+2;\n        //開始調整，左右孩子下標必須小於len，也就是確保index必須是非葉子節點\n        if(lchild arr[max]){\n            max=lchild;\n        }\n        if(rchild arr[max]){\n            max=rchild;\n        }\n        //若發生了交換，則max肯定不等於index了。此時max節點值需要與index節點值交換，上面交換索引值，這裏交換節點值\n        if(max!=index){\n            int temp=arr[index];\n            arr[index]=arr[max];\n            arr[max]=temp;\n            //交換完之後需要再次對max進行調整，因爲此時max有可能不滿足最大堆\n            HeapAdjust(arr,max,len);\n        }\n    }\n}\n\n","attrs":{}}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","text":"運行結果：","attrs":{}}]},{"type":"image","attrs":{"src":"https://static001.geekbang.org/infoq/03/03806df817263a5c283f7a22931001ab.png","alt":"運行結果","title":null,"style":[{"key":"width","value":"75%"},{"key":"bordertype","value":"none"}],"href":null,"fromPaste":true,"pastePass":true}},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","marks":[{"type":"color","attrs":{"color":"#595959","name":"user"}}],"text":"運行結果","attrs":{}}]},{"type":"heading","attrs":{"align":null,"level":2},"content":[{"type":"text","marks":[{"type":"color","attrs":{"color":"#773098","name":"user"}}],"text":"4，算法分析","attrs":{}}]},{"type":"heading","attrs":{"align":null,"level":3},"content":[{"type":"text","marks":[{"type":"color","attrs":{"color":"#773098","name":"user"}}],"text":"4.1，時間複雜度","attrs":{}}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","text":"建堆的時候初始化堆過程","attrs":{}},{"type":"codeinline","content":[{"type":"text","text":"(HeapAdjust)","attrs":{}}],"marks":[{"type":"color","attrs":{"color":"#9654B5","name":"user"}}],"attrs":{}},{"type":"text","text":"是堆排序的關鍵，時間複雜度爲","attrs":{}},{"type":"codeinline","content":[{"type":"text","text":"O(log n)","attrs":{}}],"marks":[{"type":"color","attrs":{"color":"#9654B5","name":"user"}}],"attrs":{}},{"type":"text","text":"，下面看堆排序的兩個過程；","attrs":{}}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","text":"第一步，初始化堆，這一步時間複雜度是","attrs":{}},{"type":"codeinline","content":[{"type":"text","text":"O(n)","attrs":{}}],"marks":[{"type":"color","attrs":{"color":"#9654B5","name":"user"}}],"attrs":{}},{"type":"text","text":"；","attrs":{}}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","text":"第二步，交換堆頂元素過程，需要用到n-1次循環，且每一次都要用到","attrs":{}},{"type":"codeinline","content":[{"type":"text","text":"(HeapAdjust)","attrs":{}}],"marks":[{"type":"color","attrs":{"color":"#9654B5","name":"user"}}],"attrs":{}},{"type":"text","text":"，所以時間複雜度爲","attrs":{}},{"type":"codeinline","content":[{"type":"text","text":"((n-1)*log n)~O(nlog n)","attrs":{}}],"marks":[{"type":"color","attrs":{"color":"#9654B5","name":"user"}}],"attrs":{}},{"type":"text","text":"；","attrs":{}}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","text":"最終時間複雜度：","attrs":{}},{"type":"codeinline","content":[{"type":"text","text":"O(n)+O(nlog n)","attrs":{}}],"marks":[{"type":"color","attrs":{"color":"#9654B5","name":"user"}}],"attrs":{}},{"type":"text","text":"，後者複雜度高於前者，所以","attrs":{}},{"type":"text","marks":[{"type":"color","attrs":{"color":"#773098","name":"user"}},{"type":"strong","attrs":{}}],"text":"堆排序的時間複雜度爲O(nlog n)","attrs":{}},{"type":"text","text":"；","attrs":{}}]},{"type":"heading","attrs":{"align":null,"level":3},"content":[{"type":"text","marks":[{"type":"color","attrs":{"color":"#773098","name":"user"}}],"text":"4.2，空間複雜度","attrs":{}}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","text":"空間複雜度是","attrs":{}},{"type":"codeinline","content":[{"type":"text","text":"O(1)","attrs":{}}],"marks":[{"type":"color","attrs":{"color":"#9654B5","name":"user"}}],"attrs":{}},{"type":"text","text":"，因爲沒有用到額外開闢的集合空間。","attrs":{}}]},{"type":"heading","attrs":{"align":null,"level":3},"content":[{"type":"text","marks":[{"type":"color","attrs":{"color":"#773098","name":"user"}}],"text":"4.3，算法穩定性","attrs":{}}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","marks":[{"type":"color","attrs":{"color":"#773098","name":"user"}},{"type":"strong","attrs":{}}],"text":"堆排序是不穩定的","attrs":{}},{"type":"text","text":"，比方說二叉樹[6a，8，13，6b]，（這裏的6a和6b數值上都是6，只不過爲了區別6所以這樣）經過堆初始化後以及排序過後就變成[6b，6a，8，13]；可見堆排序是不穩定的。","attrs":{}}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null}},{"type":"horizontalrule","attrs":{}},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","text":"微信搜索公衆號《","attrs":{}},{"type":"text","marks":[{"type":"size","attrs":{"size":12}},{"type":"color","attrs":{"color":"#FF7021","name":"orange"}}],"text":"程序員的時光","attrs":{}},{"type":"text","text":"》","attrs":{}}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","text":"好了，今天就先分享到這裏了，下期繼續給大家帶來 排序算法面試內容！","attrs":{}}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","text":"更多幹貨、優質文章，歡迎關注我的原創技術公衆號~","attrs":{}}]}]}