[Data structure and algorithm] Implementing red-black tree

1. Five rules of red-black trees

In addition to complying with the basic rules of binary search trees, red-black trees also add the following features:

• Rule 1: Nodes are red or black;
• Rule 2: The root node is black;
• Rule 3: Each leaf node is a black empty node (NIL node);
• Rule 4: The two child nodes of each red node are black (it is impossible to have two consecutive red nodes on all paths from each leaf to the root);
• Rule 5: All paths from any node to each of its leaf nodes contain the same number of black nodes;

Relative balance of red and black trees

The constraints of the previous five rules ensure the following key characteristics of red-black trees:

• The longest pathfromroot to leaf node , will not exceed the shortest pathtwice;
• The result is that this treeis basically balanced;
• Although it does not achieve absolute balance, it can ensure that the tree is still efficient in the worst case;

Why can it be achievedthat the longest path is no more than twice the shortest path?

• Property 4 determines that there cannot be two connected red nodes on the path;
• Therefore, the longest path must be formed by alternating red nodes and black nodes;
• Since the root node and leaf nodes are both black, the shortest path may all be black nodes, and the longest path must have more black nodes than red nodes;
• Property 5 determines that there are the same number of black nodes on all paths;
• This shows that no path is more than twice as long as any other path.

2. Three transformations of red-black trees

When a new node is inserted, it is possible that the tree is no longer balanced. The tree can be balanced in three ways:

• change color；
• Left rotation；
• Right rotation;

2.1.Discoloration

In order to re-comply with the rules of the red-black tree, it is necessary to change the red node into black a>, or change the black node to red;

Insertednew nodes are usuallyred nodes :

• When the inserted node isred, in most cases it is not violated Any rules for red-black trees;
• And inserting a black node, will inevitably lead to one more black node on a path. a>, this is difficult to adjust;
• Although red nodes may lead tored-red connections, this situation can be resolved throughcolor Transpose and rotate to adjust;

2.2.Left rotation

Taking node The position of the left child node is replaced by the parent node;

Detailed explanation:

As shown in the picture above, after left rotation:

• Node X replaces the original position of node a;
• Node Y replaces the original position of node X;
• Node X'sLeft subtree a is still node X'sLeft subtree (Here, the left subtree of
• node Y'sright subtree c is still node Y'sright subtree;
• Point Y directionZozishu b directionleft flat shiftachievement point

In addition, the binary search tree is still a binary search tree after left rotation:

2.3.Right rotation

Taking node The position of the right child node is replaced by the parent node;

Detailed explanation:

As shown in the picture above, after right rotation:

• Node X replaces the original position of node a;
• Node Y replaces the original position of node X;
• node X'sright subtree a is still node X'sright subtree (Although the right subtree of
• The left subtree of node YLeft subtree b is still the left subtree of node YLeft subtree;
• The right subtree **c of node Y is translated to the right ** and becomes the node XLeft subtree;

In addition, the binary search tree is still a binary search tree after right rotation:

3. Insertion operation of red-black tree

First of all, it is necessary to make it clear that the newly inserted node must bea red node while ensuring that the five rules of the red-black tree are met.

For the convenience of explanation, the following four nodes are specified: the newly inserted node is N (Node), and the parent node of N is (Grandpa), as shown in the figure below: G (Uncle), U’s The parent node is U (Parent), P’s sibling node isP

3.1.Case 1

When the new node N inserted is at the root of the tree, it has no parent node.

In this case, you only need to change the red node into a black node to satisfy rule 2.

3.2.Case 2

The parent node P of the new boundary point N is a black node, and no changes are required at this time.

At this time, both Rule 4 and Rule 5 are satisfied. Although the new node is red, the new node N has two black nodes NIL, so the number of black nodes on the path leading to it is still equal, so rule 5 is satisfied.

3.3.Case 3

Node P is red, and node U is also red. At this time, node G must be black, that is,father red, uncle red, ancestor black.

In this case you need:

• First, change the parent node P to black;
• Then change the uncle node U to black;
• Finally, the grandparent node G turns red;

becomesfather black uncle black ancestor red, as shown in the figure below:

Problems that may arise:

• The parent node of N's grandparent node G may also be red, which violates rule 4. In this case, the node color can be adjusted recursively;
• When the recursive adjustment reaches the root node, it needs to be rotated, as shown in node A and node B in the figure below. The specific situation will be introduced later;

3.4.Case 4

Node P is a red node, node U is a black node, and node N is the left child node of node P. At this time, node G It must be a black node, that is, father red, uncle black and ancestor black.

In this case you need:

• Change color first: change the parent node P to black, and change the grandparent node G to red;
• Post-rotation: perform right rotation with the grandparent node G as the root;

3.5.Case 5

Node P is a red node, node U is a black node, and node N is the right child node of node P. At this time, node G It must be a black node, that is, father red, uncle black and ancestor black.

In this case you need:

• First perform a left rotation with node P as the root. After the rotation, as shown in Figure b;
• will then rednodesP and Red, the newly inserted N1Node Red viewed as a wholeBNodeBlackN is seen asrednode P1 As shown in Figure c. At this point the whole situation changes to situation 4.

Then you can handle it according to situation 4:

• Change color first: change the parent node P1 of the N1 node to black, and change the grandparent node G to red;
• Post-rotation: Right rotation with the grandparent node G as the root, as shown in Figure e after rotation;
• Finally, nodes N1 and P1 are transformed back to complete the insertion of node N, as shown in Figure f;

3.6.Case

Insert nodes in the binary tree sequentially: 10, 9, 8, 7, 6, 5, 4, 3, 2, 1.

If you directly use an ordinary binary search tree, after all nodes are inserted, it will look like this:

is a seriousimbalanced tree, which is equivalent to a linked list and cannot reflect the high efficiency of a binary search tree. Inserting nodes according to the five rules of red-black trees can ensure that the search binary tree is a balanced tree to the greatest extent. The following is a detailed explanation of the process:For convenience of explanation, some leaf nodes (NIL) of the red-black tree are omitted

Insert 10

SignContext 1:

• insert node 10;
• Change the color of node 10 to black;

Insert 9

SignInformation 2：

• No changes are required;

Insert 8

How to quickly determine whether it is case 3 or case 4:

Starting from the newly inserted node N, the four nodes passed by the arrow as shown in the figure are red, red, black and red 3 red nodes It is case 3. If there are red, red, black and black two red nodes, it is case 4;

SignCircumstances 4：

• The parent node 9 turns black, and the grandparent node 10 turns red;
• Rotate right with the grandparent node as the root;

Insert 7

SignCircumstances 3：

• Parent node 8 and uncle node 10 turn black, and grandparent node 9 turns red;
• A problem will arise at this time: it does not comply with rule 2, that is, the root node is not black. At this time, the binary search tree with 9 as the root node can be regarded as a whole as a newly inserted node N, and at this time it is consistent with the situation. 1, just change 9 back to black.

Insert 6

SignCircumstances 4：

• Parent node 7 turns black, and grandparent node 8 turns red;
• Perform right rotation with grandparent node 8 as the root;

Insert 5

SignCircumstances 3：

• Parent node 6 and uncle node 8 turn black, and grandparent node 7 turns red;

Insert 4

SignCircumstances 4：

• Parent node 5 turns black, and grandparent node 6 turns red;
• Perform right rotation with grandparent node 6 as the root;

Insert 3

First transformation: consistent withCase 3:

• Parent node 4 and uncle node 6 turn black, and grandparent node 5 turns red;

After the transformation, it was found that 5 and 7 were two connected red nodes, so the entire subtree rooted at 5 was regarded as a newly inserted node N1, and the second transformation was performed.

Second change: SignInformation 4:

• Parent node 7 turns black, and grandparent node 9 turns red;
• Perform right rotation with grandparent node 9 as the root;

Finally, N1 is restored to obtain the transformed red-black tree:

Insert 2

SignCircumstances 4：

• Parent node 3 turns black, and grandparent node 4 turns red;
• Perform right rotation with grandparent node 4 as the root;

Insert 1

First transformation: consistent withCase 3:

• Parent node 2 and uncle node 4 turn black, and grandparent node 3 turns red;

After the transformation, it is found that 3 and 5 are two connected red nodes, so the entire subtree with 3 as the root is regarded as a newly inserted node N1, and the second transformation is performed.

Second change: SignInformation 3:

• Parent node 5 and uncle node 9 turn black, and grandparent node 7 turns red; that is, picture b -> picture c.

After the transformation, it is found that the red color of root node 7 does not comply with rule 2, so the red-black tree with 7 as the root node is regarded as a newly inserted node N2, and the third transformation is performed.

Third change: SignInformation 1:

• Just change root node 7 to black.

Thus, the insertion of nodes 1~10 is completed. Although case 5 is not encountered, case 5 can be converted to case 4 through a left rotation operation. The principle is the same. As shown in the figure below, after completing the NIL of the leaf nodes of this red-black tree, it has been tested to meet the five rules of the red-black tree and basically belongs to a balanced tree, higher efficiency.