B+樹的定義：
1.任意非葉子結點最多有M個子節點；且M>2；
2.除根結點以外的非葉子結點至少有 M/2個子節點；
3.根結點至少有2個子節點；
4.除根節點外每個結點存放至少M/2和至多M個關鍵字；（至少2個關鍵字）
5.非葉子結點的子樹指針與關鍵字個數相同；
6.所有結點的關鍵字：K[1], K[2], …, K[M]；且K[i] < K[i+1]；
7.非葉子結點的子樹指針P[i]，指向關鍵字值屬於[K[i], K[i+1])的子樹；
8.所有葉子結點位於同一層；
5.爲所有葉子結點增加一個鏈指針；
6.所有關鍵字都在葉子結點出現；
Java實現：
接口：
package com.meidusa.test;
public interface B {
public Object get(Comparable key); //查詢
public void remove(Comparable key); //移除
public void insertOrUpdate(Comparable key, Object obj); //插入或者更新，如果已經存在，就更新，否則插入
}B+樹：
package com.meidusa.test;
import java.util.Random;
public class BplusTree implements B {
/** 根節點 */
protected Node root;
/** 階數，M值 */
protected int order;
/** 葉子節點的鏈表頭*/
protected Node head;
public Node getHead() {
return head;
}
public void setHead(Node head) {
this.head = head;
}
public Node getRoot() {
return root;
}
public void setRoot(Node root) {
this.root = root;
}
public int getOrder() {
return order;
}
public void setOrder(int order) {
this.order = order;
}
@Override
public Object get(Comparable key) {
return root.get(key);
}
@Override
public void remove(Comparable key) {
root.remove(key, this);
}
@Override
public void insertOrUpdate(Comparable key, Object obj) {
root.insertOrUpdate(key, obj, this);
}
public BplusTree(int order){
if (order < 3) {
System.out.print("order must be greater than 2");
System.exit(0);
}
this.order = order;
root = new Node(true, true);
head = root;
}
//測試
public static void main(String[] args) {
BplusTree tree = new BplusTree(6);
Random random = new Random();
long current = System.currentTimeMillis();
for (int j = 0; j < 100000; j++) {
for (int i = 0; i < 100; i++) {
int randomNumber = random.nextInt(1000);
tree.insertOrUpdate(randomNumber, randomNumber);
}
for (int i = 0; i < 100; i++) {
int randomNumber = random.nextInt(1000);
tree.remove(randomNumber);
}
}
long duration = System.currentTimeMillis() - current;
System.out.println("time elpsed for duration: " + duration);
int search = 80;
System.out.print(tree.get(search));
}
}
節點：
package com.meidusa.test;
import java.util.AbstractMap.SimpleEntry;
import java.util.ArrayList;
import java.util.List;
import java.util.Map.Entry;
public class Node {
/** 是否爲葉子節點 */
protected boolean isLeaf;
/** 是否爲根節點*/
protected boolean isRoot;
/** 父節點 */
protected Node parent;
/** 葉節點的前節點*/
protected Node previous;
/** 葉節點的後節點*/
protected Node next;
/** 節點的關鍵字 */
protected List<Entry<Comparable, Object>> entries;
/** 子節點 */
protected List<Node> children;
public Node(boolean isLeaf) {
this.isLeaf = isLeaf;
entries = new ArrayList<Entry<Comparable, Object>>();
if (!isLeaf) {
children = new ArrayList<Node>();
}
}
public Node(boolean isLeaf, boolean isRoot) {
this(isLeaf);
this.isRoot = isRoot;
}
public Object get(Comparable key) {
//如果是葉子節點
if (isLeaf) {
for (Entry<Comparable, Object> entry : entries) {
if (entry.getKey().compareTo(key) == 0) {
//返回找到的對象
return entry.getValue();
}
}
//未找到所要查詢的對象
return null;
//如果不是葉子節點
}else {
//如果key小於等於節點最左邊的key，沿第一個子節點繼續搜索
if (key.compareTo(entries.get(0).getKey()) <= 0) {
return children.get(0).get(key);
//如果key大於節點最右邊的key，沿最後一個子節點繼續搜索
}else if (key.compareTo(entries.get(entries.size()-1).getKey()) >= 0) {
return children.get(children.size()-1).get(key);
//否則沿比key大的前一個子節點繼續搜索
}else {
for (int i = 0; i < entries.size(); i++) {
if (entries.get(i).getKey().compareTo(key) <= 0 && entries.get(i+1).getKey().compareTo(key) > 0) {
return children.get(i).get(key);
}
}
}
}
return null;
}
public void insertOrUpdate(Comparable key, Object obj, BplusTree tree){
//如果是葉子節點
if (isLeaf){
//不需要分裂，直接插入或更新
if (contains(key) || entries.size() < tree.getOrder()){
insertOrUpdate(key, obj);
if (parent != null) {
//更新父節點
parent.updateInsert(tree);
}
//需要分裂
}else {
//分裂成左右兩個節點
Node left = new Node(true);
Node right = new Node(true);
//設置鏈接
if (previous != null){
previous.setNext(left);
left.setPrevious(previous);
}
if (next != null) {
next.setPrevious(right);
right.setNext(next);
}
if (previous == null){
tree.setHead(left);
}
left.setNext(right);
right.setPrevious(left);
previous = null;
next = null;
//左右兩個節點關鍵字長度
int leftSize = (tree.getOrder() + 1) / 2 + (tree.getOrder() + 1) % 2;
int rightSize = (tree.getOrder() + 1) / 2;
//複製原節點關鍵字到分裂出來的新節點
insertOrUpdate(key, obj);
for (int i = 0; i < leftSize; i++){
left.getEntries().add(entries.get(i));
}
for (int i = 0; i < rightSize; i++){
right.getEntries().add(entries.get(leftSize + i));
}
//如果不是根節點
if (parent != null) {
//調整父子節點關係
int index = parent.getChildren().indexOf(this);
parent.getChildren().remove(this);
left.setParent(parent);
right.setParent(parent);
parent.getChildren().add(index,left);
parent.getChildren().add(index + 1, right);
setEntries(null);
setChildren(null);
//父節點插入或更新關鍵字
parent.updateInsert(tree);
setParent(null);
//如果是根節點
}else {
isRoot = false;
Node parent = new Node(false, true);
tree.setRoot(parent);
left.setParent(parent);
right.setParent(parent);
parent.getChildren().add(left);
parent.getChildren().add(right);
setEntries(null);
setChildren(null);
//更新根節點
parent.updateInsert(tree);
}
}
//如果不是葉子節點
}else {
//如果key小於等於節點最左邊的key，沿第一個子節點繼續搜索
if (key.compareTo(entries.get(0).getKey()) <= 0) {
children.get(0).insertOrUpdate(key, obj, tree);
//如果key大於節點最右邊的key，沿最後一個子節點繼續搜索
}else if (key.compareTo(entries.get(entries.size()-1).getKey()) >= 0) {
children.get(children.size()-1).insertOrUpdate(key, obj, tree);
//否則沿比key大的前一個子節點繼續搜索
}else {
for (int i = 0; i < entries.size(); i++) {
if (entries.get(i).getKey().compareTo(key) <= 0 && entries.get(i+1).getKey().compareTo(key) > 0) {
children.get(i).insertOrUpdate(key, obj, tree);
break;
}
}
}
}
}
/** 插入節點後中間節點的更新 */
protected void updateInsert(BplusTree tree){
validate(this, tree);
//如果子節點數超出階數，則需要分裂該節點
if (children.size() > tree.getOrder()) {
//分裂成左右兩個節點
Node left = new Node(false);
Node right = new Node(false);
//左右兩個節點關鍵字長度
int leftSize = (tree.getOrder() + 1) / 2 + (tree.getOrder() + 1) % 2;
int rightSize = (tree.getOrder() + 1) / 2;
//複製子節點到分裂出來的新節點，並更新關鍵字
for (int i = 0; i < leftSize; i++){
left.getChildren().add(children.get(i));
left.getEntries().add(new SimpleEntry(children.get(i).getEntries().get(0).getKey(), null));
children.get(i).setParent(left);
}
for (int i = 0; i < rightSize; i++){
right.getChildren().add(children.get(leftSize + i));
right.getEntries().add(new SimpleEntry(children.get(leftSize + i).getEntries().get(0).getKey(), null));
children.get(leftSize + i).setParent(right);
}
//如果不是根節點
if (parent != null) {
//調整父子節點關係
int index = parent.getChildren().indexOf(this);
parent.getChildren().remove(this);
left.setParent(parent);
right.setParent(parent);
parent.getChildren().add(index,left);
parent.getChildren().add(index + 1, right);
setEntries(null);
setChildren(null);
//父節點更新關鍵字
parent.updateInsert(tree);
setParent(null);
//如果是根節點
}else {
isRoot = false;
Node parent = new Node(false, true);
tree.setRoot(parent);
left.setParent(parent);
right.setParent(parent);
parent.getChildren().add(left);
parent.getChildren().add(right);
setEntries(null);
setChildren(null);
//更新根節點
parent.updateInsert(tree);
}
}
}
/** 調整節點關鍵字*/
protected static void validate(Node node, BplusTree tree) {
// 如果關鍵字個數與子節點個數相同
if (node.getEntries().size() == node.getChildren().size()) {
for (int i = 0; i < node.getEntries().size(); i++) {
Comparable key = node.getChildren().get(i).getEntries().get(0).getKey();
if (node.getEntries().get(i).getKey().compareTo(key) != 0) {
node.getEntries().remove(i);
node.getEntries().add(i, new SimpleEntry(key, null));
if(!node.isRoot()){
validate(node.getParent(), tree);
}
}
}
// 如果子節點數不等於關鍵字個數但仍大於M / 2並且小於M，並且大於2
} else if (node.isRoot() && node.getChildren().size() >= 2
||node.getChildren().size() >= tree.getOrder() / 2
&& node.getChildren().size() <= tree.getOrder()
&& node.getChildren().size() >= 2) {
node.getEntries().clear();
for (int i = 0; i < node.getChildren().size(); i++) {
Comparable key = node.getChildren().get(i).getEntries().get(0).getKey();
node.getEntries().add(new SimpleEntry(key, null));
if (!node.isRoot()) {
validate(node.getParent(), tree);
}
}
}
}
/** 刪除節點後中間節點的更新*/
protected void updateRemove(BplusTree tree) {
validate(this, tree);
// 如果子節點數小於M / 2或者小於2，則需要合併節點
if (children.size() < tree.getOrder() / 2 || children.size() < 2) {
if (isRoot) {
// 如果是根節點並且子節點數大於等於2，OK
if (children.size() >= 2) {
return;
// 否則與子節點合併
} else {
Node root = children.get(0);
tree.setRoot(root);
root.setParent(null);
root.setRoot(true);
setEntries(null);
setChildren(null);
}
} else {
//計算前後節點
int currIdx = parent.getChildren().indexOf(this);
int prevIdx = currIdx - 1;
int nextIdx = currIdx + 1;
Node previous = null, next = null;
if (prevIdx >= 0) {
previous = parent.getChildren().get(prevIdx);
}
if (nextIdx < parent.getChildren().size()) {
next = parent.getChildren().get(nextIdx);
}
// 如果前節點子節點數大於M / 2並且大於2，則從其處借補
if (previous != null
&& previous.getChildren().size() > tree.getOrder() / 2
&& previous.getChildren().size() > 2) {
//前葉子節點末尾節點添加到首位
int idx = previous.getChildren().size() - 1;
Node borrow = previous.getChildren().get(idx);
previous.getChildren().remove(idx);
borrow.setParent(this);
children.add(0, borrow);
validate(previous, tree);
validate(this, tree);
parent.updateRemove(tree);
// 如果後節點子節點數大於M / 2並且大於2，則從其處借補
} else if (next != null
&& next.getChildren().size() > tree.getOrder() / 2
&& next.getChildren().size() > 2) {
//後葉子節點首位添加到末尾
Node borrow = next.getChildren().get(0);
next.getChildren().remove(0);
borrow.setParent(this);
children.add(borrow);
validate(next, tree);
validate(this, tree);
parent.updateRemove(tree);
// 否則需要合併節點
} else {
// 同前面節點合併
if (previous != null
&& (previous.getChildren().size() <= tree.getOrder() / 2 || previous.getChildren().size() <= 2)) {
for (int i = previous.getChildren().size() - 1; i >= 0; i--) {
Node child = previous.getChildren().get(i);
children.add(0, child);
child.setParent(this);
}
previous.setChildren(null);
previous.setEntries(null);
previous.setParent(null);
parent.getChildren().remove(previous);
validate(this, tree);
parent.updateRemove(tree);
// 同後面節點合併
} else if (next != null
&& (next.getChildren().size() <= tree.getOrder() / 2 || next.getChildren().size() <= 2)) {
for (int i = 0; i < next.getChildren().size(); i++) {
Node child = next.getChildren().get(i);
children.add(child);
child.setParent(this);
}
next.setChildren(null);
next.setEntries(null);
next.setParent(null);
parent.getChildren().remove(next);
validate(this, tree);
parent.updateRemove(tree);
}
}
}
}
}
public void remove(Comparable key, BplusTree tree){
//如果是葉子節點
if (isLeaf){
//如果不包含該關鍵字，則直接返回
if (!contains(key)){
return;
}
//如果既是葉子節點又是跟節點，直接刪除
if (isRoot) {
remove(key);
}else {
//如果關鍵字數大於M / 2，直接刪除
if (entries.size() > tree.getOrder() / 2 && entries.size() > 2) {
remove(key);
}else {
//如果自身關鍵字數小於M / 2，並且前節點關鍵字數大於M / 2，則從其處借補
if (previous != null
&& previous.getEntries().size() > tree.getOrder() / 2
&& previous.getEntries().size() > 2
&& previous.getParent() == parent) {
int size = previous.getEntries().size();
Entry<Comparable, Object> entry = previous.getEntries().get(size - 1);
previous.getEntries().remove(entry);
//添加到首位
entries.add(0, entry);
remove(key);
//如果自身關鍵字數小於M / 2，並且後節點關鍵字數大於M / 2，則從其處借補
}else if (next != null
&& next.getEntries().size() > tree.getOrder() / 2
&& next.getEntries().size() > 2
&& next.getParent() == parent) {
Entry<Comparable, Object> entry = next.getEntries().get(0);
next.getEntries().remove(entry);
//添加到末尾
entries.add(entry);
remove(key);
//否則需要合併葉子節點
}else {
//同前面節點合併
if (previous != null
&& (previous.getEntries().size() <= tree.getOrder() / 2 || previous.getEntries().size() <= 2)
&& previous.getParent() == parent) {
for (int i = previous.getEntries().size() - 1; i >=0; i--) {
//從末尾開始添加到首位
entries.add(0, previous.getEntries().get(i));
}
remove(key);
previous.setParent(null);
previous.setEntries(null);
parent.getChildren().remove(previous);
//更新鏈表
if (previous.getPrevious() != null) {
Node temp = previous;
temp.getPrevious().setNext(this);
previous = temp.getPrevious();
temp.setPrevious(null);
temp.setNext(null);
}else {
tree.setHead(this);
previous.setNext(null);
previous = null;
}
//同後面節點合併
}else if(next != null
&& (next.getEntries().size() <= tree.getOrder() / 2 || next.getEntries().size() <= 2)
&& next.getParent() == parent){
for (int i = 0; i < next.getEntries().size(); i++) {
//從首位開始添加到末尾
entries.add(next.getEntries().get(i));
}
remove(key);
next.setParent(null);
next.setEntries(null);
parent.getChildren().remove(next);
//更新鏈表
if (next.getNext() != null) {
Node temp = next;
temp.getNext().setPrevious(this);
next = temp.getNext();
temp.setPrevious(null);
temp.setNext(null);
}else {
next.setPrevious(null);
next = null;
}
}
}
}
parent.updateRemove(tree);
}
//如果不是葉子節點
}else {
//如果key小於等於節點最左邊的key，沿第一個子節點繼續搜索
if (key.compareTo(entries.get(0).getKey()) <= 0) {
children.get(0).remove(key, tree);
//如果key大於節點最右邊的key，沿最後一個子節點繼續搜索
}else if (key.compareTo(entries.get(entries.size()-1).getKey()) >= 0) {
children.get(children.size()-1).remove(key, tree);
//否則沿比key大的前一個子節點繼續搜索
}else {
for (int i = 0; i < entries.size(); i++) {
if (entries.get(i).getKey().compareTo(key) <= 0 && entries.get(i+1).getKey().compareTo(key) > 0) {
children.get(i).remove(key, tree);
break;
}
}
}
}
}
/** 判斷當前節點是否包含該關鍵字*/
protected boolean contains(Comparable key) {
for (Entry<Comparable, Object> entry : entries) {
if (entry.getKey().compareTo(key) == 0) {
return true;
}
}
return false;
}
/** 插入到當前節點的關鍵字中*/
protected void insertOrUpdate(Comparable key, Object obj){
Entry<Comparable, Object> entry = new SimpleEntry<Comparable, Object>(key, obj);
//如果關鍵字列表長度爲0，則直接插入
if (entries.size() == 0) {
entries.add(entry);
return;
}
//否則遍歷列表
for (int i = 0; i < entries.size(); i++) {
//如果該關鍵字鍵值已存在，則更新
if (entries.get(i).getKey().compareTo(key) == 0) {
entries.get(i).setValue(obj);
return;
//否則插入
}else if (entries.get(i).getKey().compareTo(key) > 0){
//插入到鏈首
if (i == 0) {
entries.add(0, entry);
return;
//插入到中間
}else {
entries.add(i, entry);
return;
}
}
}
//插入到末尾
entries.add(entries.size(), entry);
}
/** 刪除節點*/
protected void remove(Comparable key){
int index = -1;
for (int i = 0; i < entries.size(); i++) {
if (entries.get(i).getKey().compareTo(key) == 0) {
index = i;
break;
}
}
if (index != -1) {
entries.remove(index);
}
}
public Node getPrevious() {
return previous;
}
public void setPrevious(Node previous) {
this.previous = previous;
}
public Node getNext() {
return next;
}
public void setNext(Node next) {
this.next = next;
}
public boolean isLeaf() {
return isLeaf;
}
public void setLeaf(boolean isLeaf) {
this.isLeaf = isLeaf;
}
public Node getParent() {
return parent;
}
public void setParent(Node parent) {
this.parent = parent;
}
public List<Entry<Comparable, Object>> getEntries() {
return entries;
}
public void setEntries(List<Entry<Comparable, Object>> entries) {
this.entries = entries;
}
public List<Node> getChildren() {
return children;
}
public void setChildren(List<Node> children) {
this.children = children;
}
public boolean isRoot() {
return isRoot;
}
public void setRoot(boolean isRoot) {
this.isRoot = isRoot;
}
public String toString(){
StringBuilder sb = new StringBuilder();
sb.append("isRoot: ");
sb.append(isRoot);
sb.append(", ");
sb.append("isLeaf: ");
sb.append(isLeaf);
sb.append(", ");
sb.append("keys: ");
for (Entry entry : entries){
sb.append(entry.getKey());
sb.append(", ");
}
sb.append(", ");
return sb.toString();
}
}