B树的实现

B树：1970年，R.Bayer和E.mccreight提出了一种适合外查找的树，它是一种平衡的多叉树，称为B树。（有些地方写的是B-树，注意不要误读 成"B减树"） 一棵M阶(M>2)的B树，是一棵平衡的M路平衡搜索树，可以是空树或者满足以下性质：
1. 根节点至少有两个孩子

2. 每个非根节点有[ (M/2)向上取整 ,M]个孩子

3. 每个非根节点有[ (M/2)向上取整 -1,M-1]个关键字，并且以升序排列

4. key[i]和key[i+1]之间的孩子节点的值介于key[i]、key[i+1]之间

5. 所有的叶子节点都在同一层

实现代码：

#include<iostream>
using namespace std;

template<class K, int M = 3>
struct BTreeNode
{
	K _keys[M];
	BTreeNode<K, M>* _subs[M + 1];
	size_t _size;
	BTreeNode<K, M>* _parent;

	BTreeNode()
		:_size(0)
		, _parent(NULL)
	{
		for (size_t i = 0; i < M + 1; ++i)
		{
			_subs[i] = NULL;
		}
	}
};

template<class K, class V>
struct Pair
{
	K _first;
	V _second;

	Pair(const K& k = K(), const V& v = V())
		:_first(k)
		, _second(v)
	{}
};

template<class K, int M = 3>
class BTree
{
	typedef BTreeNode<K, M> Node;
public:
	BTree()
		:_root(NULL)
	{}

	Pair<Node*, int> Find(const K& key)
	{
		Node* parent = NULL;
		Node* cur = _root;
		while (cur)
		{
			int i = 0;
			while (i < cur->_size && cur->_keys[i] < key)
			{
				++i;
			}
			if (cur->_keys[i] == key)
			{
				return	Pair<Node*, int>(cur, i);
			}
			parent = cur;
			cur = cur->_subs[i];
		}
		return Pair<Node*, int>(parent, -1);
	}
	
	bool Insert(const K& key)
	{
		if (_root == NULL)
		{
			_root = new Node;
			_root->_keys[0] = key;
			++_root->_size;
			return true;
		}

		Pair<Node*, int> ret = Find(key);
		if (ret._second != -1)
		{
			return false;
		}
		K k = key;
		Node *cur = ret._first;
		Node *sub = NULL;
		while (1)
		{
			_InsertKey(cur, k, sub);
			if (cur->_size < M)
			{
				return true;
			}

			int boundary = M / 2;
			Node *tmp = new Node;
			size_t index = 0; 
			size_t size = cur->_size;

			for (int i = boundary + 1; i < size; ++i)
			{
				tmp->_keys[index++] = cur->_keys[i];
				tmp->_size++;
				cur->_size--;
			}

			index = 0;
			for (int i = boundary + 1; i <= size; ++i)
			{
				tmp->_subs[index] = cur->_subs[i];
				if (tmp->_subs[index])
				{
					tmp->_subs[index]->_parent = tmp;
				}
				++index;
			}

			k = cur->_keys[boundary];
			cur->_size--;

			if (cur->_parent == NULL)
			{
				_root = new Node;
				_root->_keys[0] = k;
				_root->_subs[0] = cur;
				_root->_subs[1] = tmp;
				_root->_size = 1;

				tmp->_parent = _root;
				cur->_parent = _root;

				return true;
			}

			cur = cur->_parent;
			sub = tmp;

		}
	}

	void InOrder()
	{
		_InOrder(_root);
		cout << endl;
	}

protected:

	void _InsertKey(Node* cur, const K& k, Node* sub)
	{
		int i = cur->_size - 1;
		while (i >= 0)
		{
			if (cur->_keys[i] > k)
			{
				cur->_keys[i + 1] = cur->_keys[i];
				cur->_subs[i + 2] = cur->_subs[i + 1];
				--i;
			}
			else
			{
				break;
			}
		}
		cur->_keys[i + 1] = k;
		cur->_subs[i + 2] = sub;
		if (sub != NULL)
		{
			sub->_parent = cur;
		}
		cur->_size++;
	}

	void _InOrder(Node* root)
	{
		if (root == NULL)
		{
			return;
		}
		for (size_t i = 0; i < root->_size; ++i)
		{
			_InOrder(root->_subs[i]);
			cout << root->_keys[i] << " ";
		}
		_InOrder(root->_subs[root->_size]);
	}

private:
	Node* _root;
};

int main()
{
	int a[] = { 53, 75, 139, 49, 145, 36, 101 };
	BTree<int> bt;
	for (int i = 0; i < sizeof(a) / sizeof(a[0]); i++)
	{
		bt.Insert(a[i]);
	}
	bt.InOrder();
	return 0;
}