图解堆排序，带你彻底了解清楚！

{"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":"写在前面：","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，我是时光。","attrs":{}}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","marks":[{"type":"color","attrs":{"color":"#616161","name":"user"}}],"text":"今天给大家带来的是排序算法中的堆排序，这种排序跟二叉树相关。我采用图解方式讲解，争取写透彻。话不多说，开始！","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":"思维导图：","attrs":{}}]},{"type":"image","attrs":{"src":"https://static001.geekbang.org/infoq/36/3672a13f483e65c17c9b97f926f7cc26.png","alt":"堆排序导图","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":"堆排序导图","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":"是指利用堆这种数据结构所设计的一种排序算法。堆积是一个近似完全二叉树的结构，并同时满足堆积的性质：即子结点的键值或索引总是小于（或者大于）它的父节点。","attrs":{}}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","text":"相关概念：","attrs":{}}]},{"type":"heading","attrs":{"align":null,"level":3},"content":[{"type":"text","marks":[{"type":"color","attrs":{"color":"#773098","name":"user"}}],"text":"1.1，二叉树","attrs":{}}]},{"type":"image","attrs":{"src":"https://static001.geekbang.org/infoq/51/51cd55cc60aab46dc095975d1b497b04.png","alt":"二叉树","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":"二叉树","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":"特征：每个节点最多只有2个子节点（不存在度大于2的节点）","attrs":{}}]}],"attrs":{}},{"type":"heading","attrs":{"align":null,"level":3},"content":[{"type":"text","marks":[{"type":"color","attrs":{"color":"#773098","name":"user"}}],"text":"1.2，满二叉树","attrs":{}}]},{"type":"image","attrs":{"src":"https://static001.geekbang.org/infoq/b7/b7a3823ab5d08f86b05970848f3c7ea3.png","alt":"满二叉树","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":"满二叉树","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":"满二叉树：叶子节点全部都在最底层；除叶子节点外，每个节点都有左右孩子；","attrs":{}}]}],"attrs":{}},{"type":"heading","attrs":{"align":null,"level":3},"content":[{"type":"text","marks":[{"type":"color","attrs":{"color":"#773098","name":"user"}}],"text":"1.3，完全二叉树","attrs":{}}]},{"type":"image","attrs":{"src":"https://static001.geekbang.org/infoq/4c/4c4f3f8f299314ff1bea892fe76bad7f.png","alt":"完全二叉树","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":"完全二叉树","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":"完全二叉树：叶子节点全部都在最底的两层；最后一层只缺少右边的叶子节点，左边一定有叶子节点；除了最后一层，其他层的节点个数均达到最大值；","attrs":{}}]}],"attrs":{}},{"type":"heading","attrs":{"align":null,"level":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:[{"type":"color","attrs":{"color":"#616161","name":"user"}}],"text":"第一次交换，我们直接将元素9与元素0交换，此时堆顶元素为0，设当前节点","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与9交换","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与9交换","attrs":{}}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","marks":[{"type":"color","attrs":{"color":"#595959","name":"user"}}],"text":"这时，我们需要将剩下的元素（排除元素9）进行堆排序，直到下面这个结果：","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":"代码：","attrs":{}}]},{"type":"codeblock","attrs":{"lang":"java"},"content":[{"type":"text","text":"/**\n * 第二步，交换堆顶（最大）元素和二叉堆的最后一个叶子节点元素。目的是交换元素\n * i从二叉堆的最后一个叶子节点元素开始，也就是len-1开始(数组下标从0开始)\n */\nfor(int i=len-1;i>=0;i--){\n    int temp=arr[i];\n    arr[i]=arr[0];\n    arr[0]=temp;\n    //交换完之后需要重新调整二叉堆，从头开始调整，此时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：这里有个小细节问题，前面我们写的初始化堆方法有三个参数，分别是数组","attrs":{}},{"type":"codeinline","content":[{"type":"text","text":"arr","attrs":{}}],"marks":[{"type":"color","attrs":{"color":"#9654B5","name":"user"}}],"attrs":{}},{"type":"text","text":"，当前节点","attrs":{}},{"type":"codeinline","content":[{"type":"text","text":"index","attrs":{}}],"marks":[{"type":"color","attrs":{"color":"#9654B5","name":"user"}}],"attrs":{}},{"type":"text","text":"以及数组长度","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":"那么，为何不直接传入两个参数即可，数组长度直接用","attrs":{}},{"type":"codeinline","content":[{"type":"text","text":"arr.length","attrs":{}}],"marks":[{"type":"color","attrs":{"color":"#9654B5","name":"user"}}],"attrs":{}},{"type":"text","text":"表示不就行了吗？初始化堆的时候是可以，但是后面在交换堆顶元素和末尾的叶子节点时，在对剩下的元素进行排序时，此时的数组长度可不是","attrs":{}},{"type":"codeinline","content":[{"type":"text","text":"arr.length","attrs":{}}],"marks":[{"type":"color","attrs":{"color":"#9654B5","name":"user"}}],"attrs":{}},{"type":"text","text":"了，应该是变化的参数","attrs":{}},{"type":"codeinline","content":[{"type":"text","text":"i","attrs":{}}],"marks":[{"type":"color","attrs":{"color":"#9654B5","name":"user"}}],"attrs":{}},{"type":"text","text":"，因为交换之后的元素（比如9）就不计入堆中排序了，所以需要3个参数。","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":"我们进行第二次交换，我们直接将元素8与元素2交换，此时堆顶元素为2，设当前节点","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与2交换","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与2交换","attrs":{}}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","marks":[{"type":"color","attrs":{"color":"#595959","name":"user"}}],"text":"这时，我们需要将剩下的元素（排除元素9和8）进行堆排序，直到下面这个结果：","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":"到这个时候，我们再重复上述步骤，不断调换堆顶和最末尾的节点元素即可，再不断地对剩下的元素进行排序，最后就能得到从小到大排序好的堆了，如下图所示，这就是我们想要的结果：","attrs":{}}]},{"type":"image","attrs":{"src":"https://static001.geekbang.org/infoq/fd/fd8fbbbbc0bfbaba6d6b65cbb8b2b560.png","alt":"最终排序结果：从小到大","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":"最终排序结果：从小到大","attrs":{}}]},{"type":"heading","attrs":{"align":null,"level":2},"content":[{"type":"text","marks":[{"type":"color","attrs":{"color":"#773098","name":"user"}}],"text":"3，完整代码","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(\"排序后:\"+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，从小到大。目的是对元素排序\n         * i从完全二叉树的第一个非叶子节点开始，也就是len/2-1开始(数组下标从0开始)\n         */\n        for(int i=len/2-1;i>=0;i--){\n            HeapAdjust(arr,i,len);\n        }\n        /**\n         * 第二步，交换堆顶（最大）元素和二叉堆的最后一个叶子节点元素。目的是交换元素\n         * i从二叉堆的最后一个叶子节点元素开始，也就是len-1开始(数组下标从0开始)\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            //交换完之后需要重新调整二叉堆，从头开始调整，此时Index=0\n            HeapAdjust(arr,0,i);\n        }\n        return arr;\n    }\n\n    /**\n     *初始化堆\n     * @param arr 待调整的二叉树(数组)\n     * @param index 待调整的节点下标，二叉树的第一个非叶子节点\n     * @param len 待调整的数组长度\n     */\n    public static void HeapAdjust(int[] arr,int index,int len){\n        //先保存当前节点的下标\n        int max = index;\n        //保存左右孩子数组下标\n        int lchild = index*2+1;\n        int rchild = index*2+2;\n        //开始调整，左右孩子下标必须小于len，也就是确保index必须是非叶子节点\n        if(lchild arr[max]){\n            max=lchild;\n        }\n        if(rchild arr[max]){\n            max=rchild;\n        }\n        //若发生了交换，则max肯定不等于index了。此时max节点值需要与index节点值交换，上面交换索引值，这里交换节点值\n        if(max!=index){\n            int temp=arr[index];\n            arr[index]=arr[max];\n            arr[max]=temp;\n            //交换完之后需要再次对max进行调整，因为此时max有可能不满足最大堆\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":"运行结果：","attrs":{}}]},{"type":"image","attrs":{"src":"https://static001.geekbang.org/infoq/03/03806df817263a5c283f7a22931001ab.png","alt":"运行结果","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":"运行结果","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，时间复杂度","attrs":{}}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","text":"建堆的时候初始化堆过程","attrs":{}},{"type":"codeinline","content":[{"type":"text","text":"(HeapAdjust)","attrs":{}}],"marks":[{"type":"color","attrs":{"color":"#9654B5","name":"user"}}],"attrs":{}},{"type":"text","text":"是堆排序的关键，时间复杂度为","attrs":{}},{"type":"codeinline","content":[{"type":"text","text":"O(log n)","attrs":{}}],"marks":[{"type":"color","attrs":{"color":"#9654B5","name":"user"}}],"attrs":{}},{"type":"text","text":"，下面看堆排序的两个过程；","attrs":{}}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","text":"第一步，初始化堆，这一步时间复杂度是","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，交换堆顶元素过程，需要用到n-1次循环，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":"，所以时间复杂度为","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":"最终时间复杂度：","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":"，后者复杂度高于前者，所以","attrs":{}},{"type":"text","marks":[{"type":"color","attrs":{"color":"#773098","name":"user"}},{"type":"strong","attrs":{}}],"text":"堆排序的时间复杂度为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，空间复杂度","attrs":{}}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","text":"空间复杂度是","attrs":{}},{"type":"codeinline","content":[{"type":"text","text":"O(1)","attrs":{}}],"marks":[{"type":"color","attrs":{"color":"#9654B5","name":"user"}}],"attrs":{}},{"type":"text","text":"，因为没有用到额外开辟的集合空间。","attrs":{}}]},{"type":"heading","attrs":{"align":null,"level":3},"content":[{"type":"text","marks":[{"type":"color","attrs":{"color":"#773098","name":"user"}}],"text":"4.3，算法稳定性","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":"堆排序是不稳定的","attrs":{}},{"type":"text","text":"，比方说二叉树[6a，8，13，6b]，（这里的6a和6b数值上都是6，只不过为了区别6所以这样）经过堆初始化后以及排序过后就变成[6b，6a，8，13]；可见堆排序是不稳定的。","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":"微信搜索公众号《","attrs":{}},{"type":"text","marks":[{"type":"size","attrs":{"size":12}},{"type":"color","attrs":{"color":"#FF7021","name":"orange"}}],"text":"程序员的时光","attrs":{}},{"type":"text","text":"》","attrs":{}}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","text":"好了，今天就先分享到这里了，下期继续给大家带来 排序算法面试内容！","attrs":{}}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","text":"更多干货、优质文章，欢迎关注我的原创技术公众号~","attrs":{}}]}]}