用Java实现二叉堆、大顶堆和小顶堆

先了解了解

什么是二叉堆

二叉堆就是完全二叉树，或者是靠近完全二叉树结构的二叉树。在二叉树建树时采取前序建树就是建立的完全二叉树。也就是二叉堆。所以二叉堆的建堆过程理论上讲和前序建树一样。

什么是大顶堆、小顶堆

二叉堆本质上是一棵近完全的二叉树，那么大顶堆和小顶堆必然也是满足这个结构要求的。在此之上，大顶堆要求对于一个节点来说，它的左右节点都比它小；小顶堆要求对于一个节点来说，它的左右节点都比它大。

建堆

二叉堆建堆本质上和前序建堆差不多，只不过需要考虑的一点就是大小关系，这一点和二叉搜索树建树有点相似，所以可以得出结论，建树，本质上都是递归建树，只不过因为数据结构的大小要求不一样，需要的判断函数不一样，节点进入哪个位置也不一样。

大顶堆和小顶堆也分为稳定和不稳定的堆。稳定和不稳定指如果具备相同的值，那么他们的插入顺序应该和节点顺序一致。

程序实现

首先，定义出基本的堆结构

public class BinaryHeap {private Integer value;private BinaryHeap leftChild;private BinaryHeap rightChild;
}

建堆过程与建二叉树过程一致

public static BinaryHeap buildHeap(BinaryHeap binaryHeap, Integer value) {if (Objects.isNull(binaryHeap)) binaryHeap = new BinaryHeap();if (Objects.isNull(binaryHeap.getValue())) {binaryHeap.setValue(value);return binaryHeap;}if (Objects.isNull(binaryHeap.getLeftChild())) {BinaryHeap binaryHeap1 = new BinaryHeap();binaryHeap1.setValue(value);binaryHeap.setLeftChild(binaryHeap1);} else if (Objects.nonNull(binaryHeap.getLeftChild())) {if (Objects.isNull(binaryHeap.getRightChild())) {BinaryHeap binaryHeap1 = new BinaryHeap();binaryHeap1.setValue(value);binaryHeap.setRightChild(binaryHeap1);} else {// TODO: 2022/1/14 左右节点两种都不为nullif (checkNull(binaryHeap.getLeftChild())) buildHeap(binaryHeap.getLeftChild(), value);else if (checkNull(binaryHeap.getRightChild())) buildHeap(binaryHeap.getRightChild(), value);else buildHeap(binaryHeap.getLeftChild(), value);}}return binaryHeap;
}

主要原理就是如果当前节点的左节点为空，则把当前值放到左节点，如果左节点不为空，右节点为空，则把值放到右节点。如果左右节点都不为空，就将建树过程转移到下一层，如果左节点有为空的子节点，就转移给左节点，如果左节点没有为空的子节点，且右节点有为空的子节点，那么转移给右节点。如果左右节点都没有为空的子节点，那么也转移给左节点。

以序列2,3,4,5,9,6,8,7为例，按照该算法建立出来的二叉堆结构如下：

{"value": 2,"left_child": {"value": 3,"left_child": {"value": 4,"left_child": {"value": 8,"left_child": null,"right_child": null},"right_child": {"value": 7,"left_child": null,"right_child": null}},"right_child": {"value": 5,"left_child": null,"right_child": null}},"right_child": {"value": 1,"left_child": {"value": 9,"left_child": null,"right_child": null},"right_child": {"value": 6,"left_child": null,"right_child": null}}
}

建立大顶堆

大顶堆在建堆的基础上，有一个要求，根节点比左右子树的任何节点的值都大。那么建树的过程可以分为两步，对于每一个值，首先按照建树过程，会到二叉堆的最底部，然后通过不断的让自己与自己的根节点做比较，如果自己大于根节点，就交换自己与根节点的位置，递归回溯即可。

逻辑过程

假设现在红色节点组成的已经是一个大顶堆，现在新增了一个节点到这个二叉堆中，而且是比任意节点都大，那么黑色箭头将是该节点的行动路线，它会反复与父级比较，如果大于父级，则交换和父级的关系。

程序实现

public static BinaryHeap up(BinaryHeap father) {if (Objects.nonNull(father.getLeftChild())) {if (father.getValue() < father.getLeftChild().getValue()) {int c = father.getValue();father.setValue(father.getLeftChild().getValue());father.getLeftChild().setValue(c);}up(father.getLeftChild());}if (Objects.nonNull(father.getRightChild())) {if (father.getValue() < father.getRightChild().getValue()) {int c = father.getValue();father.setValue(father.getRightChild().getValue());father.getRightChild().setValue(c);}up(father.getRightChild());}return father;
}

该方法放在普通建树方法之后，就是大顶堆的建树方法了，总的方法如下：

public static BinaryHeap bigPush(BinaryHeap binaryHeap, Integer value) {binaryHeap = buildHeap(binaryHeap, value);up(binaryHeap);return binaryHeap;
}

还是以序列2,3,4,5,9,6,8,7为例，按照该算法建立出来的大顶堆结构如下：

{"value": 9,"left_child": {"value": 8,"left_child": {"value": 7,"left_child": {"value": 2,"left_child": null,"right_child": null},"right_child": {"value": 4,"left_child": null,"right_child": null}},"right_child": {"value": 3,"left_child": null,"right_child": null}},"right_child": {"value": 6,"left_child": {"value": 1,"left_child": null,"right_child": null},"right_child": {"value": 5,"left_child": null,"right_child": null}}
}

建立小顶堆

小顶堆与大顶堆类似

逻辑过程

过程与大顶堆一致，不过此时是比父级小就和父级交换。

程序实现

public static BinaryHeap down(BinaryHeap father) {if (Objects.nonNull(father.getLeftChild())) {if (father.getValue() > father.getLeftChild().getValue()) {int c = father.getValue();father.setValue(father.getLeftChild().getValue());father.getLeftChild().setValue(c);}down(father.getLeftChild());}if (Objects.nonNull(father.getRightChild())) {if (father.getValue() > father.getRightChild().getValue()) {int c = father.getValue();father.setValue(father.getRightChild().getValue());father.getRightChild().setValue(c);}down(father.getRightChild());}return father;
}

这个是向下走的过程，最终代码为：

public static BinaryHeap smallPush(BinaryHeap binaryHeap, Integer value) {binaryHeap = buildHeap(binaryHeap, value);down(binaryHeap);return binaryHeap;
}

以序列2,3,4,5,9,6,8,7为例，按照该算法建立出来的小顶堆结构如下：

{"value": 1,"left_child": {"value": 3,"left_child": {"value": 4,"left_child": {"value": 8,"left_child": null,"right_child": null},"right_child": {"value": 7,"left_child": null,"right_child": null}},"right_child": {"value": 5,"left_child": null,"right_child": null}},"right_child": {"value": 2,"left_child": {"value": 9,"left_child": null,"right_child": null},"right_child": {"value": 6,"left_child": null,"right_child": null}}
}

从堆顶取数据并重构大小顶堆

public static Integer bigPop(BinaryHeap binaryHeap) {Integer val = binaryHeap.getValue();if (binaryHeap.getLeftChild().getValue() >= binaryHeap.getRightChild().getValue()) {binaryHeap.setValue(binaryHeap.getLeftChild().getValue());BinaryHeap binaryHeap1 = mergeTree(binaryHeap.getLeftChild().getLeftChild(), binaryHeap.getLeftChild().getRightChild());up(binaryHeap1);binaryHeap.setLeftChild(binaryHeap1);} else {binaryHeap.setValue(binaryHeap.getRightChild().getValue());BinaryHeap binaryHeap1 = mergeTree(binaryHeap.getRightChild().getLeftChild(), binaryHeap.getRightChild().getRightChild());up(binaryHeap1);binaryHeap.setRightChild(binaryHeap1);}return val;
}public static Integer smallPop(BinaryHeap binaryHeap) {Integer val = binaryHeap.getValue();if (binaryHeap.getLeftChild().getValue() <= binaryHeap.getRightChild().getValue()) {binaryHeap.setValue(binaryHeap.getLeftChild().getValue());BinaryHeap binaryHeap1 = mergeTree(binaryHeap.getLeftChild().getLeftChild(), binaryHeap.getLeftChild().getRightChild());down(binaryHeap1);binaryHeap.setLeftChild(binaryHeap1);} else {binaryHeap.setValue(binaryHeap.getRightChild().getValue());BinaryHeap binaryHeap1 = mergeTree(binaryHeap.getRightChild().getLeftChild(), binaryHeap.getRightChild().getRightChild());down(binaryHeap1);binaryHeap.setRightChild(binaryHeap1);}return val;}

取出来之后，需要重新调用down或者up函数。以构建小顶堆，取出五次后的结果

public static void main(String[] args) {int[] a = new int[]{2, 3, 1, 4, 5, 9, 6, 8, 7};BinaryHeap binaryHeap = new BinaryHeap();for (int i = 0; i < a.length; i++) {binaryHeap = smallPush(binaryHeap, a[i]);}System.out.println(Json.toJson(smallPop(binaryHeap)));System.out.println(Json.toJson(smallPop(binaryHeap)));System.out.println(Json.toJson(smallPop(binaryHeap)));System.out.println(Json.toJson(smallPop(binaryHeap)));System.out.println(Json.toJson(smallPop(binaryHeap)));System.out.println(Json.toJson(binaryHeap));}

![image.png](https://img-blog.csdnimg.cn/img_convert/69ab1242179011c09e8d8ad17169340d.png#clientId=u3bc32b9f-4cff-4&crop=0&crop=0&crop=1&crop=1&from=paste&height=287&id=u92c998f3&margin=[object Object]&name=image.png&originHeight=287&originWidth=2347&originalType=binary&ratio=1&rotation=0&showTitle=false&size=24421&status=done&style=none&taskId=u39d79f5d-2fda-492e-89bb-36a5fdcb79a&title=&width=2347)
取完后的小顶堆为：

{"value": 6,"left_child": {"value": 7,"left_child": {"value": 8,"left_child": null,"right_child": null},"right_child": null},"right_child": {"value": 9,"left_child": null,"right_child": null}
}