算法導論 第十八章；B 樹

       B樹是爲磁盤或其他直接存取輔助存儲設備而設計的一種平衡查找樹。B樹的”分支因子“可能很大，即每個節點可以有很多子女。這一因子由所用磁盤特性所決定，並且可以降低磁盤I/O操作次數。許多數據庫系統都使用B樹或B樹的變形來存儲信息。

B樹結構形式如下:

其特點：

1）每個節點x有以下域：

a)  x.n:當前存儲在節點x中的關鍵字

b) x.n 個key值，以非降序順序存放，即 x.key(1) ≤ x.key(2) ≤ ... ≤ x.key(x.n)

c) x.leaf：bool型，若爲葉子節點 x.leaf=TRUE,反之爲FALUSE

2） 每個節點x包含x.n+1個指向其子女的指針x.c(1),x.c(2),...x.c(x.n+1)。（葉子節點無此域）

3）各關鍵字x.key(i)對存儲在各子樹中的關鍵字範圍加以分隔：如果k(i)爲存儲在x.c(i)爲根的子樹中的關鍵字，則：

        

4） 每個葉子節點具有相同的深度，即樹的高度 h

5） 每個節點包含的key數有一個上界和下界，這些界可以用一個稱爲B樹的最小度數的固定整數 t ≥ 2 來表示：

a) 每個非根的節點必須至少有t-1個 key. 每個非根內節點至少有t個子女。如果樹非空，則根節點至少包含一個key.

b) 每個節點可包含至多2t-1個key。所以一個內節點至多可能有2t個子女。如果一個節點恰好有2t-1個key,則稱該點是滿的。

6）如果n ≥ 1，則對任意的一棵包含n個關鍵字，高度爲h,最小度t ≥ 2的B樹T，有：

             

B樹插入和搜索操作的完整代碼如下：

#include<iostream>
#include<cstdlib>
#include<cstring>
#define Disk_Write(x)
#define Disk_Read(x)
#define t 2
#define N 2*t
using namespace std;

typedef struct BTNode{
	int n;           //the number of keys storing in the current node
	char key[N];     //N keys storing in nondecreasing order
	bool leaf;       //TRUE:left;FALSE:internal node
	BTNode *c[N+1];  //point n+1 children
	}BTNode;

typedef struct BTree{
	BTNode *root;
	}BTree;

void BTree_Print(BTNode *x)
{
	for(int i=1;i<=x->n;i++)
	{
		if(x->leaf == false)
			BTree_Print(x->c[i]);
		cout<< x->key[i]<<"  ";
	 	} 
	if(x->leaf == false)
		BTree_Print(x->c[x->n+1]);
	}

void BTree_SplitChild(BTNode *x,int i)
{
	BTNode *z=new BTNode();
	BTNode *y=x->c[i];   //split y (2t-1 keys) into y (t-1 keys) and  z(t-1 keys)
	z->leaf=y->leaf;
	z->n=t-1;
	for(int j=1 ; j<=t-1 ; j++)
		z->key[j]=y->key[t+j];

	if(y->leaf==false)//if y has children ,copy its children to z
		for(int j=1; j<=t; j++)
			z->c[j]=y->c[j+t];
	y->n=t-1;

	//let z become the (i+1)th child of x
	for(int j=x->n+1; j>=i+1; j--)
		x->c[j+1]=x->c[j];
	x->c[i+1]=z;
	
	//insert the (t)th key of y into (i)th index of x
	for(int j=x->n; j>=i; j--)
	   x->key[j+1]=x->key[j];
	x->key[i]=y->key[t];
	x->n++;

	Disk_Write(y);
	Disk_Write(z);
	Disk_Write(x);		
	}
void BTree_Insert_Nonfull(BTNode *x,char k)
{
	int i=x->n;
	if(x->leaf)
	{//x node is leaf
		while(i>=1 && k < x->key[i])  //search for the insert index
		{
			x->key[i+1]=x->key[i];
			i--;
		 	}
		//insert key
		x->key[i+1]=k;
		x->n++;
		Disk_Write(x);
		} 
	else   // x node is not leaf
	{ 
		while(i>=1 && k < x->key[i])
			i--;
		i++;
	//Read its child,and insert the key into its child node
	Disk_Read(x->c[i]);
	//case 1: the child is full
	if(x->c[i]->n == 2*t-1)
	{
		BTree_SplitChild(x,i);
		if(k > x->key[i])
			i++;
		}
	//case 2:the child is not full
	BTree_Insert_Nonfull(x->c[i],k);
	}
}
void BTree_Insert(BTree *T,char k)
{
	BTNode *r=T->root;
	if(r->n == 2*t-1)
	{//root node is full
		//a new node s becomes the root
		BTNode *s=new BTNode(); 
		T->root=s;
		s->leaf=false;
		s->n=0;
		s->c[1]=r;
		//split the original root into two chilren of s
		BTree_SplitChild(s,1);
		//insert the key into the nonfull node
		BTree_Insert_Nonfull(s,k);
		}
		else//root node is not full
			BTree_Insert_Nonfull(r,k); //insert key into root node directly
	}
BTNode *BTree_Search(BTNode *x,char k,int &i)
{//return pair(y,i)consisting of a node y and an index i such that y.keyi=k
	 i=1;
	while(i <= x->n && k > x->key[i])
		i++;
	if(i <= x->n && k == x->key[i])
		return x;
	else if(x->leaf)
	{     
		i=0;
		return NULL;
		}
	else
	{  
		Disk_Read(x->c[i]);
		return BTree_Search(x->c[i],k,i);
	 	} 
	}

void BTree_Create(BTree *T,string ch)
{
	//first,create an empty root node
	BTNode *x=new BTNode();
	x->leaf=true;
	x->n=0;

	Disk_Write(x);
	T->root=x;


	//second,add new keys into T by calling Insert method
	for(int i=0;i<ch.length();i++)     
		BTree_Insert(T,ch[i]);      	
	}

void BTree_PrintDetail(BTNode *x)
{
	
	cout<<"The root is:";
	for(int i=1;i<=x->n;i++)
		cout<<x->key[i]<<"  ";
	cout<<endl;
	
	cout<<"The root's child is:"<<endl;
	for(int j=1;j<=x->n+1;j++)
	  {
		BTNode *child=x->c[j];
		for(int i=1;i<=child->n;i++)
			cout<<child->key[i]<<"   ";
		cout<<endl;
		}
	
	for(int i=1;i<=x->n+1;i++)
	{ 
		cout<<"The "<<i<<" child"<<endl;
		BTNode *child0=x->c[i];
		int m=child0->n+1;
		for(int j=1;j<=m;j++)
	 	{
			BTNode *c1=child0->c[j];
			for(int jj=1;jj<=c1->n;jj++)
				cout<<c1->key[jj]<<"  ";
			cout<<endl;
			}
		}
}

int main()
{	
	//string test_ch={'F','S','Q','K','C','L','H','T','V','W','M','R','N','P','A','B','X','Y','D','Z','E'};
	string test_ch="FSQKCLHTVWMRNPABXYDZE";
	cout<<"/*----------------------------Create B-tree---------------------------*/"<<endl;
	BTree *T=new BTree();
	BTree_Create(T,test_ch);       
	cout<<"After creating ,the B-tree(its degree is "<<t<<"):"<<endl;	
	BTree_Print(T->root);
	cout<<endl;
	cout<<"The detail B-tree is:"<<endl;
	BTree_PrintDetail(T->root);
	cout<<"/*--------------------------------------------------------------------*/"<<endl;
	
	cout<<"/*---------------------------Insert B-tree----------------------------*/"<<endl;
	char ich;
	cout<<"Please input the inserting char:";
	cin>>ich;
	BTree_Insert(T,ich);
	cout<<"After inserting ,the B-tree:"<<endl;
	BTree_Print(T->root);
	cout<<endl;

	cout<<"The detail of B-tree is:"<<endl;
	BTree_PrintDetail(T->root);
	cout<<"/*---------------------------------------------------------------------*/"<<endl;
	
	cout<<"/*--------------------------Search B-tree------------------------------*/"<<endl;
	char sch;
	BTNode *sNode=NULL;
	int index;
	cout<<"Please input the searching char:";
	cin>>sch;
	sNode=BTree_Search(T->root,sch,index);
	if(sNode==NULL)
		cout<<"The key doesn't exist in the B-tree."<<endl;
	else
	{ 
		cout<<"The key in the Node:";
		for(int i=1;i<=sNode->n;i++)
			cout<<sNode->key[i]<<"  ";
		cout<<endl;
		cout<<"The index of the key in the node is:"<<index<<endl;
		} 
	cout<<"/*---------------------------------------------------------------------*/"<<endl;


	return 0;
	}

運行結果：

【注：若有錯誤，請指正~~~】