B樹的插入、刪除與遍歷
阿新 • • 發佈:2018-12-25
B樹的插入
定義:
1、根節點至少有兩個分支
2、除了根節點以外,所有節點的關鍵字個數至少為M/2個,最多為M-1
3、每個節點的度數均是關鍵字數加一
4、所有的葉子節點都在同一層
插入:
我們設計節點的結構如下:
#define M 5
#define MAX M - 1
#define MIN M/2
typedef char KeyType;
typedef struct {}Record;
typedef struct ElemType
{
KeyType key;
Record *recptr;
}ElemType;
typedef struct BNode
{
int num;
BNode *parent;
ElemType data[M+1];
BNode*sub[M+1];
}BNode,*BTree;
//查詢函式的返回值型別
typedef struct Result
{
bool tag;
BNode*pnode;
int index;
}Result;
插入程式碼:
當以個節點的個數大於MAX時就分裂,如果根節點分裂會產生新根,否則就將分裂出來的節點插入到雙親中,如果雙親又大於MAX就繼續分裂,這樣就能保證B樹的定義的正確性,程式碼如下:
ElemType MoveItem(BNode*ptr , BNode *s, int pos)
{
int tmp = ptr->num;
for (int i = pos + 1, j = 0; i <= tmp; i++, j++)
{
s->data[j] = ptr->data[i];
s->sub[j] = ptr->sub[i];
if (ptr->sub[i] != NULL)
{
s->sub[j]->parent = s;
}
}
s->parent = ptr->parent;
s ->num = ptr->num = MIN;
return s->data[0];
}
BNode * MakeRoot(ElemType x, BNode *left, BNode *right)
{
BNode *s = BuyNode();
s->num = 1;
s->data[1] = x;
s->sub[0] = left;
if (left != NULL)
left->parent = s;
s->sub[1] = right;
if (right != NULL)
right->parent = s;
return s;
}
bool InsertItem(BNode*ptr, int pos, ElemType e, BNode*right);
BNode *Splice(BNode*ptr)
{
BNode *s = BuyNode();
ElemType e = MoveItem(ptr, s, MIN);
if (ptr->parent == NULL)
{
return MakeRoot(e,ptr, s);
}
else
{
ptr = ptr->parent;
int i = ptr->num;
ptr->data[0] = e;//這句很關鍵,如果ptr->data[0]未設定就會和0位置比較還沒有結果,插入位置就會出錯
while (ptr->data[i].key > s->data[0].key) --i;
InsertItem(ptr, i + 1, s->data[0], s);
if (ptr->num > MAX)
{
return Splice(ptr);
}
return NULL;
}
}
bool InsertItem(BNode*ptr, int pos, ElemType e, BNode*right)//BNode&node
{
for (int i = ptr->num; i >= pos; --i)
{
ptr->data[i + 1] = ptr->data[i];
ptr->sub[i + 1] = ptr->sub[i];
}
//
ptr->data[pos] = e;
ptr->sub[pos] = right;
if (right != NULL)
{
right->parent = ptr;
}//
ptr->num += 1;
return true;
}
bool Insert(BTree *ptr, ElemType e)
{
if (ptr == NULL)
return false;
if (*ptr == NULL)
{
*ptr = MakeRoot(e, NULL, NULL);
return true;
}
Result res=FindValue(*ptr, e.key);
if (res.pnode == NULL || res.tag) return false;
InsertItem(res.pnode, res.index+1, e, NULL);
if (res.pnode->num > MAX)
{
BNode*p = Splice(res.pnode);
if (p != NULL)
{
*ptr = p;
}
}
return true;
}
輔助函式:
BNode* BuyNode()
{
BNode *node = new BNode();
if (node == NULL)
exit(-1);
memset(node, 0, sizeof(BNode));
return node;
}
Result FindValue(BNode*ptr, KeyType e)
{
Result res = { false, NULL, -1 };
while (ptr != NULL)
{
int i = ptr->num;
ptr->data[0].key = e;
while (ptr->data[i].key > e) --i;
res.pnode = ptr;
res.index = i;
if (i != 0 && ptr->data[i].key == e)
{
res.tag = true;
break;
}
else
ptr = ptr->sub[i];
}
return res;
}
B樹的刪除
B樹的刪除,我們將帶有分支的節點中的關鍵碼刪除,用他的前驅和後繼替換掉這個被刪除的關鍵碼,然後刪除前驅或者後繼,刪除前驅或者後繼之後,會出現與B樹定義不相符的情況,比如關鍵碼個數小於MIN的情況,這個時候就要做相應的旋轉,如過旋轉不了就只有進行節點的合併,合併有可能會產生新根,程式碼如下:
//找前驅
BNode *FindPre(BNode*ptr)
{
while (ptr!=NULL&&ptr->sub[ptr->num] != NULL)
{
ptr = ptr->sub[ptr->num];
}
return ptr;
}
//找後繼
BNode *FindNext(BNode*ptr)
{
while (ptr != NULL&&ptr->sub[0] != NULL)
{
ptr = ptr->sub[0];
}
return ptr;
}
//刪除葉子結點
void DelLeafItem(BNode *ptr, int pos)
{
for (int i = pos; i < ptr->num; i++)
{
ptr->data[i] = ptr->data[i + 1];
ptr->sub[i] = ptr->sub[i + 1];
}
ptr->num -= 1;
}
//右旋轉
void RightRotateLeaf(BNode *leftbro, BNode*ptr, BNode *parent, int pos)
{
ptr->data[0] = parent->data[pos];
for (int i = ptr->num; i >= 0; i--)
{
ptr->data[i + 1] = ptr->data[i];
ptr->sub[i + 1] = ptr->sub[i];
}
ptr->num += 1;
ptr->sub[0] = leftbro->sub[leftbro->num];
if (ptr->sub[0] != NULL)//
{
ptr->sub[0]->parent = ptr;
}
parent->data[pos] = leftbro->data[leftbro->num];
leftbro->num -= 1;
}
//左旋轉
void LeftRotateLeaf(BNode *rightbro,BNode *ptr,BNode *parent,int pos)
{
ptr ->data[ptr->num+1] = parent->data[pos + 1];
ptr->sub[ptr->num + 1] = rightbro->sub[0];
if (ptr->sub[ptr->num+1]!=NULL)
{
ptr->sub[ptr->num + 1]->parent = ptr;
}
ptr->num += 1;
parent->data[pos + 1] = rightbro->data[1];
for (int i =0; i < rightbro->num; i++)
{
rightbro->data[i] = rightbro->data[i + 1];
rightbro->sub[i] = rightbro->sub[i + 1];
}
rightbro->num -= 1;
}
//向左合併
void LeftMerge(BNode*leftbro, BNode*ptr, BNode*parent, int pos)
{
ptr->data[0] = parent->data[pos];
for (int i = 0,j=leftbro->num+1; i <= ptr->num; i++,j++)
{
leftbro->data[j] = ptr->data[i];
leftbro->sub[j] = ptr->sub[i];
if (leftbro->sub[j] != NULL)
{
leftbro->sub[j]->parent = leftbro;
}
}
leftbro->num = leftbro->num + ptr->num + 1;
free(ptr);
DelLeafItem(parent, pos);
}
//向右合併
void RightMerge(BNode *ptr, BNode *rightbro, BNode *parent, int pos)
{
LeftMerge(ptr, rightbro, parent, pos+1);
}
//出現小於MIN的情況的調整函式
BNode *AdjusLeaf(BNode*ptr)
{
BNode*parent = ptr->parent;
int pos = 0;
while (parent->sub[pos] != ptr) ++pos;
BNode*leftbro = pos-1<0?NULL:parent->sub[pos-1];
BNode*rightbro = pos+1>=MAX?NULL:parent->sub[pos+1];
if (leftbro!=NULL&&leftbro->num>MIN)
{
RightRotateLeaf(leftbro,ptr,parent,pos);
}
else if (rightbro!=NULL&&rightbro->num>MIN)
{
LeftRotateLeaf(rightbro, ptr,parent, pos);
}
else if(leftbro!=NULL)
{
LeftMerge(leftbro, ptr, parent, pos);
ptr = leftbro;
}
else if (rightbro != NULL)
{
RightMerge(ptr, rightbro, parent, pos);
// ptr = rightbro;
}
if (parent->parent != NULL&&parent->num < MIN)
{
return AdjusLeaf(parent);
}
if (parent->parent == NULL&&parent->num <= 0)
{
free(parent);
ptr->parent = NULL;
return ptr;
}
return NULL;
}
//刪除函式
void ReMove(BNode*&root, KeyType e)
{
if (root == NULL)
return;
Result res = FindValue(root, e);
if (res.pnode == NULL || res.tag==false) return;
BNode *ptr = res.pnode;
int pos = res.index;
BNode*pre = FindPre(ptr->sub[pos-1]);
BNode*next = FindNext(ptr->sub[pos]);
if (pre != NULL&&pre->num > MIN)
{
ptr->data[pos] = pre->data[pre->num];
ptr = pre;
pos = pre->num;
}
else if (next != NULL&&next->num > MIN)
{
ptr->data[pos] = next->data[1];
ptr = next;
pos = 1;
}
else if (pre != NULL)
{
ptr->data[pos] = pre->data[pre->num];
ptr = pre;
pos = pre->num;
}
else if (next != NULL)
{
ptr->data[pos] = next->data[1];
ptr = next;
pos = 1;
}
DelLeafItem(ptr, pos);//
if (ptr->parent != NULL&&ptr->num < MIN)
{
BNode*newroot = AdjusLeaf(ptr);
if (newroot != NULL)
{
root = newroot;
}
}
else if (ptr->parent == NULL&&ptr->num <= 0)
{
free(root);
root = NULL;
}
}
B樹的遍歷
利用遞迴的特性 ,程式碼十分簡潔,先遞迴到最左邊,然後列印一個關鍵碼就遍歷一個分支,程式碼如下:
void InOder(BNode*root)
{
if (root != NULL)
{
InOder(root->sub[0]);
for (int i = 1; i <= root->num; i++)
{
cout << root->data[i].key;
InOder(root->sub[i]);
}
}
}