利用Java实现红黑树
目录
- 1、红黑树的属性
- 2、旋转
- 3、插入
- 4、删除
- 5、所有代码
- 6、演示
1、红黑树的属性 红黑树是一种二分查找树,与普通的二分查找树不同的一点是,红黑树的每个节点都有一个颜色(color)属性。该属性的值要么是红色,要么是黑色。
通过限制从根到叶子的任何简单路径上的节点颜色,红黑树确保没有比任何其他路径长两倍的路径,从而使树近似平衡。
假设红黑树节点的属性有键(
key
)、颜色(color
)、左子节点(left
)、右子节点(right
),父节点(parent
)。一棵红黑树必须满足下面有下面这些特性( 红黑树特性 ):
- 树中的每个节点要么是红色,要么是黑色;
- 根节点是黑色;
- 每个叶子节点(null)是黑色;
- 如果某节点是红色的,它的两个子节点都是黑色;
- 对于每个节点到后面任一叶子节点(null)的所有路径,都有相同数量的黑色节点。
tree
,哨兵变量RedBlackTree.NULL
(下文代码中)是一个和其它节点有同样属性的节点,它的颜色(color
)属性是黑色,其它属性可以任意取值。我们使用哨兵变量是因为我们可以把一个节点
node
的子节点null
当成一个普通节点。在这里,我们使用哨兵变量
RedBlackTree.NULL
代替树中所有的null
(所有的叶子节点及根节点的父节点)。我们把从一个节点n(不包括)到任一叶子节点路径上的黑色节点的个数称为 黑色高度 ,用bh(n)表示。一棵红黑树的黑色高度是其根节点的黑色高度。
关于红黑树的搜索,求最小值,求最大值,求前驱,求后继这些操作的代码与二分查找树的这些操作的代码基本一致。可以在用
java
实现二分查找树查看。结合上文给出下面的代码。
用一个枚举类Color表示颜色:
public enum Color {Black("黑色"), Red("红色"); private String color; private Color(String color) {this.color = color; }@Overridepublic String toString() {return color; }}
类Node表示节点:
public class Node {public int key; public Color color; public Node left; public Node right; public Node parent; public Node() {}public Node(Color color) {this.color = color; }public Node(int key) {this.key = key; this.color = Color.Red; }public int height() {return Math.max(left != RedBlackTree.NULL ? left.height() : 0, right != RedBlackTree.NULL ? right.height() : 0) + 1; }public Node minimum() {Node pointer = this; while (pointer.left != RedBlackTree.NULL)pointer = pointer.left; return pointer; }@Overridepublic String toString() {String position = "null"; if (this.parent != RedBlackTree.NULL)position = this.parent.left == this ? "left" : "right"; return "[key: " + key + ", color: " + color + ", parent: " + parent.key + ", position: " + position + "]"; }}
类RedTreeNode表示红黑树:
public class RedBlackTree {// 表示哨兵变量public final static Node NULL = new Node(Color.Black); public Node root; public RedBlackTree() {this.root = NULL; }}
2、旋转 红黑树的插入和删除操作,能改变红黑树的结构,可能会使它不再有前面所说的某些特性性。为了维持这些特性,我们需要改变树中某些节点的颜色和位置。
我们可以通过旋转改变节点的结构。主要有
左旋转
和 右旋转
两种方式。具体如下图所示。左旋转:把一个节点n的右子节点right变为它的父节点,n变为right的左子节点,所以right不能为null。这时n的右指针空了出来,right的左子树被n挤掉,所以right原来的左子树称为n的右子树。
右旋转:把一个节点n的左子节点left变为它的父节点,n变为left的右子节点,所以left不能为null。这时n的左指针被空了出来,left的右子树被n挤掉,所以left原来的右子树被称为n的左子树。
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可在RedTreeNode类中,加上如下实现代码:
public void leftRotate(Node node) {Node rightNode = node.right; node.right = rightNode.left; if (rightNode.left != RedBlackTree.NULL)rightNode.left.parent = node; rightNode.parent = node.parent; if (node.parent == RedBlackTree.NULL)this.root = rightNode; else if (node.parent.left == node)node.parent.left = rightNode; elsenode.parent.right = rightNode; rightNode.left = node; node.parent = rightNode; }public void rightRotate(Node node) {Node leftNode = node.left; node.left = leftNode.right; if (leftNode.right != RedBlackTree.NULL)leftNode.right.parent = node; leftNode.parent = node.parent; if (node.parent == RedBlackTree.NULL) {this.root = leftNode; } else if (node.parent.left == node) {node.parent.left = leftNode; } else {node.parent.right = leftNode; }leftNode.right = node; node.parent = leftNode; }
3、插入 红黑树的插入代码与二分查找树的插入代码非常相似。只不过红黑树的插入操作会改变红黑树的结构,使其不在有该有的特性。
在这里,新插入的节点默认是红色。
所以在插入节点之后,要有维护红黑树特性的代码。
public void insert(Node node) {Node parentPointer = RedBlackTree.NULL; Node pointer = this.root; while (this.root != RedBlackTree.NULL) {parentPointer = pointer; pointer = node.key < pointer.key ? pointer.left : pointer.right; }node.parent = parentPointer; if(parentPointer == RedBlackTree.NULL) {this.root = node; }else if(node.key < parentPointer.key) {parentPointer.left = node; }else {parentPointer.right = node; }node.left = RedBlackTree.NULL; node.right = RedBlackTree.NULL; node.color = Color.Red; // 维护红黑树属性的方法this.insertFixUp(node); }
用上述方法插入一个新节点的时候,有两类情况会违反红黑树的特性。
- 当树中没有节点时,此时插入的节点称为根节点,而此节点的颜色为红色。
- 当新插入的节点成为一个红色节点的子节点时,此时存在一个红色结点有红色子节点的情况。
具体代码如下:
public void insertFixUp(Node node) {// 当node不是根结点,且node的父节点颜色为红色while (node.parent.color == Color.Red) {// 先判断node的父节点是左子节点,还是右子节点,这不同的情况,解决方案也会不同if (node.parent == node.parent.parent.left) {Node uncleNode = node.parent.parent.right; if (uncleNode.color == Color.Red) {// 如果叔叔节点是红色,则父父一定是黑色// 通过把父父节点变成红色,父节点和兄弟节点变成黑色,然后在判断父父节点的颜色是否合适uncleNode.color = Color.Black; node.parent.color = Color.Black; node.parent.parent.color = Color.Red; node = node.parent.parent; } else if (node == node.parent.right) {node = node.parent; this.leftRotate(node); } else {node.parent.color = Color.Black; node.parent.parent.color = Color.Red; this.rightRotate(node.parent.parent); }} else {Node uncleNode = node.parent.parent.left; if (uncleNode.color == Color.Red) {uncleNode.color = Color.Black; node.parent.color = Color.Black; node.parent.parent.color = Color.Red; node = node.parent.parent; } else if (node == node.parent.left) {node = node.parent; this.rightRotate(node); } else {node.parent.color = Color.Black; node.parent.parent.color = Color.Red; this.leftRotate(node.parent.parent); }}}// 如果之前树中没有节点,那么新加入的点就成了新结点,而新插入的结点都是红色的,所以需要修改。this.root.color = Color.Black; }
下面的图分别对应第二类情况中的六种及相应处理结果。
情况1:
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情况2:
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情况3:
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情况4:
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情况5:
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情况6:
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4、删除 红黑树中节点的删除会使一个结点代替另外一个节点。所以先要实现这样的代码:
public void transplant(Node n1, Node n2) {if(n1.parent == RedBlackTree.NULL){this.root = n2; }else if(n1.parent.left == n1) {n1.parent.left = n2; }else {n1.parent.right = n2; }n2.parent = n1.parent; }
红黑树的删除节点代码是基于二分查找树的删除节点代码而写的。
删除结点代码:
public void delete(Node node) {Node pointer1 = node; // 用于记录被删除的颜色,如果是红色,可以不用管,但如果是黑色,可能会破坏红黑树的属性Color pointerOriginColor = pointer1.color; // 用于记录问题的出现点Node pointer2; if (node.left == RedBlackTree.NULL) {pointer2 = node.right; this.transplant(node, node.right); } else if (node.right == RedBlackTree.NULL) {pointer2 = node.left; this.transplant(node, node.left); } else {// 如要删除的字节有两个子节点,则找到其直接后继(右子树最小值),直接后继节点没有非空左子节点。pointer1 = node.right.minimum(); // 记录直接后继的颜色和其右子节点pointerOriginColor = pointer1.color; pointer2 = pointer1.right; // 如果其直接后继是node的右子节点,不用进行处理if (pointer1.parent == node) {pointer2.parent = pointer1; } else {// 否则,先把直接后继从树中提取出来,用来替换nodethis.transplant(pointer1, pointer1.right); pointer1.right = node.right; pointer1.right.parent = pointer1; }// 用node的直接后继替换node,继承node的颜色this.transplant(node, pointer1); pointer1.left = node.left; pointer1.left.parent = pointer1; pointer1.color = node.color; }if (pointerOriginColor == Color.Black) {this.deleteFixUp(pointer2); }}
当被删除节点的颜色是黑色时需要调用方法维护红黑树的特性。
主要有两类情况:
- 当node是红色时,直接变成黑色即可。
- 当node是黑色时,需要调整红黑树结构。,
private void deleteFixUp(Node node) {// 如果node不是根节点,且是黑色while (node != this.root && node.color == Color.Black) {// 如果node是其父节点的左子节点if (node == node.parent.left) {// 记录node的兄弟节点Node pointer1 = node.parent.right; // 如果他兄弟节点是红色if (pointer1.color == Color.Red) {pointer1.color = Color.Black; node.parent.color = Color.Red; leftRotate(node.parent); pointer1 = node.parent.right; }if (pointer1.left.color == Color.Black && pointer1.right.color == Color.Black) {pointer1.color = Color.Red; node = node.parent; } else if (pointer1.right.color == Color.Black) {pointer1.left.color = Color.Black; pointer1.color = Color.Red; rightRotate(pointer1); pointer1 = node.parent.right; } else {pointer1.color = node.parent.color; node.parent.color = Color.Black; pointer1.right.color = Color.Black; leftRotate(node.parent); node = this.root; }} else {// 记录node的兄弟节点Node pointer1 = node.parent.left; // 如果他兄弟节点是红色if (pointer1.color == Color.Red) {pointer1.color = Color.Black; node.parent.color = Color.Red; rightRotate(node.parent); pointer1 = node.parent.left; }if (pointer1.right.color == Color.Black && pointer1.left.color == Color.Black) {pointer1.color = Color.Red; node = node.parent; } else if (pointer1.left.color == Color.Black) {pointer1.right.color = Color.Black; pointer1.color = Color.Red; leftRotate(pointer1); pointer1 = node.parent.left; } else {pointer1.color = node.parent.color; node.parent.color = Color.Black; pointer1.left.color = Color.Black; rightRotate(node.parent); node = this.root; }}}node.color = Color.Black; }
对第二类情况,有下面8种:
情况1:
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情况2:
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情况3:
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情况4:
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情况5:
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情况6:
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情况7:
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情况8:
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5、所有代码
public enum Color {Black("黑色"), Red("红色"); private String color; private Color(String color) {this.color = color; }@Overridepublic String toString() {return color; }}public class Node {public int key; public Color color; public Node left; public Node right; public Node parent; public Node() {}public Node(Color color) {this.color = color; }public Node(int key) {this.key = key; this.color = Color.Red; }/*** 求在树中节点的高度* * @return*/public int height() {return Math.max(left != RedBlackTree.NULL ? left.height() : 0, right != RedBlackTree.NULL ? right.height() : 0) + 1; }/*** 在以该节点为根节点的树中,求最小节点* * @return*/public Node minimum() {Node pointer = this; while (pointer.left != RedBlackTree.NULL)pointer = pointer.left; return pointer; }@Overridepublic String toString() {String position = "null"; if (this.parent != RedBlackTree.NULL)position = this.parent.left == this ? "left" : "right"; return "[key: " + key + ", color: " + color + ", parent: " + parent.key + ", position: " + position + "]"; }}import java.util.LinkedList; import java.util.Queue; public class RedBlackTree {public final static Node NULL = new Node(Color.Black); public Node root; public RedBlackTree() {this.root = NULL; }/*** 左旋转* * @param node*/public void leftRotate(Node node) {Node rightNode = node.right; node.right = rightNode.left; if (rightNode.left != RedBlackTree.NULL)rightNode.left.parent = node; rightNode.parent = node.parent; if (node.parent == RedBlackTree.NULL)this.root = rightNode; else if (node.parent.left == node)node.parent.left = rightNode; elsenode.parent.right = rightNode; rightNode.left = node; node.parent = rightNode; }/*** 右旋转* * @param node*/public void rightRotate(Node node) {Node leftNode = node.left; node.left = leftNode.right; if (leftNode.right != RedBlackTree.NULL)leftNode.right.parent = node; leftNode.parent = node.parent; if (node.parent == RedBlackTree.NULL) {this.root = leftNode; } else if (node.parent.left == node) {node.parent.left = leftNode; } else {node.parent.right = leftNode; }leftNode.right = node; node.parent = leftNode; }public void insert(Node node) {Node parentPointer = RedBlackTree.NULL; Node pointer = this.root; while (pointer != RedBlackTree.NULL) {parentPointer = pointer; pointer = node.key < pointer.key ? pointer.left : pointer.right; }node.parent = parentPointer; if (parentPointer == RedBlackTree.NULL) {this.root = node; } else if (node.key < parentPointer.key) {parentPointer.left = node; } else {parentPointer.right = node; }node.left = RedBlackTree.NULL; node.right = RedBlackTree.NULL; node.color = Color.Red; this.insertFixUp(node); }private void insertFixUp(Node node) {// 当node不是根结点,且node的父节点颜色为红色while (node.parent.color == Color.Red) {// 先判断node的父节点是左子节点,还是右子节点,这不同的情况,解决方案也会不同if (node.parent == node.parent.parent.left) {Node uncleNode = node.parent.parent.right; if (uncleNode.color == Color.Red) { // 如果叔叔节点是红色,则父父一定是黑色// 通过把父父节点变成红色,父节点和兄弟节点变成黑色,然后在判断父父节点的颜色是否合适uncleNode.color = Color.Black; node.parent.color = Color.Black; node.parent.parent.color = Color.Red; node = node.parent.parent; } else if (node == node.parent.right) { // node是其父节点的右子节点,且叔叔节点是黑色// 对node的父节点进行左旋转node = node.parent; this.leftRotate(node); } else { // node如果叔叔节点是黑色,node是其父节点的左子节点,父父节点是黑色// 把父节点变成黑色,父父节点变成红色,对父父节点进行右旋转node.parent.color = Color.Black; node.parent.parent.color = Color.Red; this.rightRotate(node.parent.parent); }} else {Node uncleNode = node.parent.parent.left; if (uncleNode.color == Color.Red) {uncleNode.color = Color.Black; node.parent.color = Color.Black; node.parent.parent.color = Color.Red; node = node.parent.parent; } else if (node == node.parent.left) {node = node.parent; this.rightRotate(node); } else {node.parent.color = Color.Black; node.parent.parent.color = Color.Red; this.leftRotate(node.parent.parent); }}}// 如果之前树中没有节点,那么新加入的点就成了新结点,而新插入的结点都是红色的,所以需要修改。this.root.color = Color.Black; }/*** n2替代n1* * @param n1* @param n2*/private void transplant(Node n1, Node n2) {if (n1.parent == RedBlackTree.NULL) { // 如果n1是根节点this.root = n2; } else if (n1.parent.left == n1) { // 如果n1是其父节点的左子节点n1.parent.left = n2; } else { // 如果n1是其父节点的右子节点n1.parent.right = n2; }n2.parent = n1.parent; }/*** 删除节点node* * @param node*/public void delete(Node node) {Node pointer1 = node; // 用于记录被删除的颜色,如果是红色,可以不用管,但如果是黑色,可能会破坏红黑树的属性Color pointerOriginColor = pointer1.color; // 用于记录问题的出现点Node pointer2; if (node.left == RedBlackTree.NULL) {pointer2 = node.right; this.transplant(node, node.right); } else if (node.right == RedBlackTree.NULL) {pointer2 = node.left; this.transplant(node, node.left); } else {// 如要删除的字节有两个子节点,则找到其直接后继(右子树最小值),直接后继节点没有非空左子节点。pointer1 = node.right.minimum(); // 记录直接后继的颜色和其右子节点pointerOriginColor = pointer1.color; pointer2 = pointer1.right; // 如果其直接后继是node的右子节点,不用进行处理if (pointer1.parent == node) {pointer2.parent = pointer1; } else {// 否则,先把直接后继从树中提取出来,用来替换nodethis.transplant(pointer1, pointer1.right); pointer1.right = node.right; pointer1.right.parent = pointer1; }// 用node的直接后继替换node,继承node的颜色this.transplant(node, pointer1); pointer1.left = node.left; pointer1.left.parent = pointer1; pointer1.color = node.color; }if (pointerOriginColor == Color.Black) {this.deleteFixUp(pointer2); }}/*** The procedure RB-DELETE-FIXUP restores properties 1, 2, and 4* * @param node*/private void deleteFixUp(Node node) {// 如果node不是根节点,且是黑色while (node != this.root && node.color == Color.Black) {// 如果node是其父节点的左子节点if (node == node.parent.left) {// 记录node的兄弟节点Node pointer1 = node.parent.right; // 如果node兄弟节点是红色if (pointer1.color == Color.Red) {pointer1.color = Color.Black; node.parent.color = Color.Red; leftRotate(node.parent); pointer1 = node.parent.right; }if (pointer1.left.color == Color.Black && pointer1.right.color == Color.Black) {pointer1.color = Color.Red; node = node.parent; } else if (pointer1.right.color == Color.Black) {pointer1.left.color = Color.Black; pointer1.color = Color.Red; rightRotate(pointer1); pointer1 = node.parent.right; } else {pointer1.color = node.parent.color; node.parent.color = Color.Black; pointer1.right.color = Color.Black; leftRotate(node.parent); node = this.root; }} else {// 记录node的兄弟节点Node pointer1 = node.parent.left; // 如果他兄弟节点是红色if (pointer1.color == Color.Red) {pointer1.color = Color.Black; node.parent.color = Color.Red; rightRotate(node.parent); pointer1 = node.parent.left; }if (pointer1.right.color == Color.Black && pointer1.left.color == Color.Black) {pointer1.color = Color.Red; node = node.parent; } else if (pointer1.left.color == Color.Black) {pointer1.right.color = Color.Black; pointer1.color = Color.Red; leftRotate(pointer1); pointer1 = node.parent.left; } else {pointer1.color = node.parent.color; node.parent.color = Color.Black; pointer1.left.color = Color.Black; rightRotate(node.parent); node = this.root; }}}node.color = Color.Black; }private void innerWalk(Node node) {if (node != NULL) {innerWalk(node.left); System.out.println(node); innerWalk(node.right); }}/*** 中序遍历*/public void innerWalk() {this.innerWalk(this.root); }/*** 层次遍历*/public void print() {Queuequeue = new LinkedList<>(); queue.add(this.root); while (!queue.isEmpty()) {Node temp = queue.poll(); System.out.println(temp); if (temp.left != NULL)queue.add(temp.left); if (temp.right != NULL)queue.add(temp.right); }}// 查找public Node search(int key) {Node pointer = this.root; while (pointer != NULL && pointer.key != key) {pointer = pointer.key < key ? pointer.right : pointer.left; }return pointer; }}
6、演示 演示代码:
public class Test01 {public static void main(String[] args) {int[] arr = { 1, 2, 3, 4, 5, 6, 7, 8 }; RedBlackTree redBlackTree = new RedBlackTree(); for (int i = 0; i < arr.length; i++) {redBlackTree.insert(new Node(arr[i])); }System.out.println("树的高度: " + redBlackTree.root.height()); System.out.println("左子树的高度: " + redBlackTree.root.left.height()); System.out.println("右子树的高度: " + redBlackTree.root.right.height()); System.out.println("层次遍历"); redBlackTree.print(); // 要删除节点Node node = redBlackTree.search(4); redBlackTree.delete(node); System.out.println("树的高度: " + redBlackTree.root.height()); System.out.println("左子树的高度: " + redBlackTree.root.left.height()); System.out.println("右子树的高度: " + redBlackTree.root.right.height()); System.out.println("层次遍历"); redBlackTree.print(); }}
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