左偏树的性质
- 本节点的键值key小于其左右子节点键值key(与二叉堆相同);
- 本节点的左子节点的距离大于等于本节点的右子节点(这意味着每个节点中除了要存储键值外, 还需要一个额外的dist存储距离);
- 节点的距离是其右子节点的距离+1(这意味着, 一个节点的dist是从它出发到达最近终端节点的距离);
斜堆的性质
- 本节点的键值key小于其左右子节点键值key;
- 斜堆节点不存储距离dist值, 取而代之的是在每次合并操作时都做swap处理(节省了存储空间);
核心操作
- 合并操作
- 插入操作
- 取最小操作
实现
左偏树(堆)merge函数具体实现
- 采用递归实现;
- 每层递归中, 当
roota->val > rootb->val
时, 交换roota
和rootb
; - 向下递归;
- 如左子节点距离小于右子节点距离, 交换左右子节点;
- 更新本节点距离值;
- 返回本节点指针;
斜堆merge函数具体实现
--采用递归实现(也有非递归算法);
--每层递归中, 当roota->val > rootb->val时, 交换roota和rootb;
--向下递归;
--交换左右子节点;
--返回本节点指针;
代码
左偏树
typedef int elemType;
struct leftistTreeNode {
elemType data;
unsigned int dist;
leftistTreeNode *lchild, *rchild;
leftistTreeNode(const elemType &val): data(val), dist(0), lchild(NULL), rchild(NULL) {}
};
template <typename type>
void swapPtr(type &x, type &y) {
type t = x;
x = y; y = t;
}
leftistTreeNode *createLeftistTree(const vector<elemType> &vec);
void destroyLeftistTree(leftistTreeNode *&root);
leftistTreeNode *mergeLeftistTree(leftistTreeNode *&roota, leftistTreeNode *&rootb);
void insertLeftistTreeNode(leftistTreeNode *&root, const elemType &dt);
leftistTreeNode *extractMinNode(leftistTreeNode *&root);
leftistTreeNode *createLeftistTree(const vector<elemType> &vec) {
leftistTreeNode *root = NULL;
for (int i = 0; i != vec.size(); ++i)
insertLeftistTreeNode(root, vec[i]);
return root;
}
void destroyLeftistTree(leftistTreeNode *&root) {
leftistTreeNode *left = root->lchild, *right = root->rchild;
delete(root); root = NULL;
if (left) destroyLeftistTree(left);
if (right) destroyLeftistTree(right);
}
leftistTreeNode *mergeLeftistTree(leftistTreeNode *&roota, leftistTreeNode *&rootb) {//核心部分
if (!roota || !rootb)
return roota ? roota : rootb;
if (roota->data > rootb->data)
swapPtr<leftistTreeNode*>(roota, rootb);//注意: 此处交换的是指针值
roota->rchild = mergeLeftistTree(roota->rchild, rootb);
if (!roota->lchild || roota->lchild->dist < roota->rchild->dist)
swapPtr<leftistTreeNode*>(roota->lchild, roota->rchild);
if (!roota->rchild)
roota->dist = 0;
else
roota->dist = roota->rchild->dist + 1;
return roota;
}
void insertLeftistTreeNode(leftistTreeNode *&root, const elemType &dt) {
leftistTreeNode *cur = new leftistTreeNode(dt);
root = mergeLeftistTree(root, cur);
}
leftistTreeNode *extractMinNode(leftistTreeNode *&root) {
leftistTreeNode *min = root;
root = mergeLeftistTree(root->lchild, root->rchild);
return min;
}
斜堆
typedef int elemType;
struct skewHeapNode {
elemType data;
skewHeapNode *lchild, *rchild;
skewHeapNode(const elemType &val): data(val), lchild(NULL), rchild(NULL) {}
};
template <typename type>
void swapPtr(type &x, type &y) {
type t = x;
x = y; y = t;
}
skewHeapNode *createskewHeap(const vector<elemType> &vec);
void destroyskewHeap(skewHeapNode *&root);
skewHeapNode *mergeskewHeap(skewHeapNode *&roota, skewHeapNode *&rootb);
void insertskewHeapNode(skewHeapNode *&root, const elemType &dt);
skewHeapNode *extractMinNode(skewHeapNode *&root);
skewHeapNode *createskewHeap(const vector<elemType> &vec) {
skewHeapNode *root = NULL;
for (int i = 0; i != vec.size(); ++i)
insertskewHeapNode(root, vec[i]);
return root;
}
void destroyskewHeap(skewHeapNode *&root) {
skewHeapNode *left = root->lchild, *right = root->rchild;
delete(root); root = NULL;
if (left) destroyskewHeap(left);
if (right) destroyskewHeap(right);
}
skewHeapNode *mergeskewHeap(skewHeapNode *&roota, skewHeapNode *&rootb) {//此处与左偏堆不同, 不判断左右子节点距离
if (!roota || !rootb)
return roota ? roota : rootb;
if (roota->data > rootb->data)
swapPtr<skewHeapNode*>(roota, rootb);
roota->rchild = mergeskewHeap(roota->rchild, rootb);
swapPtr(roota->lchild, rootb->rchild);
return roota;
}
void insertskewHeapNode(skewHeapNode *&root, const elemType &dt) {
skewHeapNode *cur = new skewHeapNode(dt);
root = mergeskewHeap(root, cur);
}
skewHeapNode *extractMinNode(skewHeapNode *&root) {
skewHeapNode *min = root;
root = mergeskewHeap(root->lchild, root->rchild);
return min;
}
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