左偏树leftistTree和斜堆skewHeap的分析和实现

最近学习了很多东西(深入学习操作系统中), 博客的更新也搁置了一段时间, 主要是学习占用了大量的时间 (学的越多, 也发现自己不会的越多, 就像苏格拉底说的"我只知道我一无所知");

今天主要分析一下左偏树和斜堆, 这两个数据结构是二叉堆的进化版, 一方面能够实现与二叉堆相同的功能, 代码结构也比较简单; 另一方面它们的合并操作的时间复杂度为O(logn), 而普通二叉堆实现合并操作的时间复杂度为O(n);

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, 取最小操作都是基于合并操作);

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;

左偏树模板如下(只有新建/合并/插入/取堆顶元素/销毁等操作, 没有加入删除功能):

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;
}

斜堆模板如下(只有新建/合并/插入/取堆顶元素/销毁等操作, 没有加入删除功能):

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;
}

最近应该会多发一些学习总结性的内容, 巩固一下知识, 有空把我的学习笔记也发上来供大家参考;

如果算法或分析有问题, 请指正, 谢谢