转自http://dsqiu.iteye.com/blog/1714961

1.Pairing Heap简介

斐波那契堆主要有两个缺点：编程实现难度较大和实际效率没有理论的那么快（由于它的存储结构和四个指针）。Pairing Heap的提出就是弥补斐波那契堆的两个缺点——编程简单操作的时间复杂度和斐波那契堆一样。

Pairing Heap其实就是一个具有堆（最大堆或最小堆）性质的树，它的特性不是由它的结构决定的，而是由于它的操作（插入，合并，减小一个关键字等）决定的。

1.1Pairing Heap的ADT

PairingHeapNode
{
	int							key;
	struct	PairingHeapNode*	child;
	struct	PairingHeapNode*	sibling;
	struct	PairingHeapNode*	prev;

}PairHeap;

2.Pairing Heap 的操作

注意：图解过程是以最大堆来演示的，但是代码是以最小堆来写的，见谅！

2.1.合并两个子堆

static PairHeap* merge_subheaps(PairHeap *p, PairHeap *q)
{
	if(q == NULL)
		return p;
	else if(p->key <= q->key)
	{
		q->prev = p;
		p->sibling = q->sibling;
		if(p->sibling != NULL)
			p->sibling->prev = p;

		q->sibling = p->child;
		if(q->sibling != NULL)
			q->sibling->prev = q;

		p->child = q;
		return p;
	}
	else
	{
		q->prev = p->prev;
		p->prev = q;
		p->sibling = q->child;
		if(p->sibling != NULL)
			p->sibling->prev = p;

		q->child = p;
		return q;
	}
}

2.2. 插入一个结点

PairHeap* PairHeap_insert(PairHeap *p, int key)
{
	PairHeap *node;

	node = (PairHeap*)malloc(sizeof(*node));
	if(node == NULL)
		return NULL;

	node->key = key;
	node->child = node->sibling = node->prev = NULL;

	if(p == NULL)
		return node;
	else
		return merge_subheaps(p, node);
}

2.3.增加（减小）一个关键字

PairHeap* PairHeap_DecreaseKey(PairHeap *p, PairHeap *pos, int d)
{
	if(d < 0)
		return p;

	pos->key = pos->key - d;
	if(p == pos)
		return p;

	if(pos->sibling != NULL)
		pos->sibling->prev = pos->prev;

	if(pos->prev->child = pos)
		pos->prev->child = pos->sibling;
	else
		pos->prev->sibling = p->sibling;

	p->sibling = NULL;
	return merge_subheaps(p, pos);
} 

2.5.删除最小结点

3.Pairing Heap完整代码实现

#include <stdlib.h>

typedef struct PairingHeapNode
{
	int							key;
	struct	PairingHeapNode*	child;
	struct	PairingHeapNode*	sibling;
	struct	PairingHeapNode*	prev;

}PairHeap;

static PairHeap* merge_subheaps(PairHeap *p, PairHeap *q);
static PairHeap* combine_siblings(PairHeap *p);

PairHeap* PairHeap_insert(PairHeap *p, int key)
{
	PairHeap *node;

	node = (PairHeap*)malloc(sizeof(*node));
	if(node == NULL)
		return NULL;

	node->key = key;
	node->child = node->sibling = node->prev = NULL;

	if(p == NULL)
		return node;
	else
		return merge_subheaps(p, node);
}

PairHeap* PairHeap_DecreaseKey(PairHeap *p, PairHeap *pos, int d)
{
	if(d < 0)
		return p;

	pos->key = pos->key - d;
	if(p == pos)
		return p;

	if(pos->sibling != NULL)
		pos->sibling->prev = pos->prev;

	if(pos->prev->child = pos)
		pos->prev->child = pos->sibling;
	else
		pos->prev->sibling = p->sibling;

	p->sibling = NULL;
	return merge_subheaps(p, pos);
}

PairHeap* PairHeap_DeleteMin(int *key, PairHeap *p)
{
	PairHeap *new_root;

	if(p == NULL)
		return NULL;
	else
	{
		*key = p->key;
		if(p->child != NULL)
			new_root = combine_siblings(p->child);

		free(p);
	}
	return new_root;
}

static PairHeap* combine_siblings(PairHeap *p)
{
	PairHeap *tree_array[1024];
	int i, count;

	if(p->sibling == NULL)
		return p;

	for(count = 0; p != NULL; count++)
	{
		tree_array[count] = p;
		p->prev->sibling = NULL;
		p = p->sibling;
	}
	tree_array[count] = NULL;

	for(i = 1; i < count; i++)
		tree_array[i] = merge_subheaps(tree_array[i-1], tree_array[i]);

	return tree_array[count-1];
}

static PairHeap* merge_subheaps(PairHeap *p, PairHeap *q)
{
	if(q == NULL)
		return p;
	else if(p->key <= q->key)
	{
		q->prev = p;
		p->sibling = q->sibling;
		if(p->sibling != NULL)
			p->sibling->prev = p;

		q->sibling = p->child;
		if(q->sibling != NULL)
			q->sibling->prev = q;

		p->child = q;
		return p;
	}
	else
	{
		q->prev = p->prev;
		p->prev = q;
		p->sibling = q->child;
		if(p->sibling != NULL)
			p->sibling->prev = p;

		q->child = p;
		return q;
	}
}

小结

终于到小结了，这篇文章写得比较吃力，如斐波那契堆的“减小一个关键字”的操作我就对使用级联剪切还没有找到解释，还有还没有找到Pairing Heap的定义，唯独只有自己理解和揣测（比如pairing heap没有明确的定义，本人个人理解pairing heap的独特性一定在他的操作，即行为决定特征）但总归还是相对完整，希望能对你有说帮助。如果你有任何建议、批评或补充，还请你不吝提出，不甚感激。更多参考请移步互联网。

参考：

①酷~行天下：

http://mindlee.net/2011/09/26/binomial-heaps/

②Björn B. Brandenburg：http://www.cs.unc.edu/~bbb/#binomial_heaps

③酷~行天下：
http://mindlee.net/2011/09/29/fibonacci-heaps/

④Adoo's blog ：http://www.roading.org/algorithm/introductiontoalgorithm/%E6%96%90%E6%B3%A2%E9%82%A3%E5%A5%91%E5%A0%86fibonacci-heaps.html

⑤Golden_Shadow：http://blog.csdn.net/golden_shadow/article/details/6216921

⑥Sartaj Sahni：http://www.cise.ufl.edu/~sahni/dsaaj/enrich/c13/pairing.htm

⑦C++ template Fibonacci heap, with demonstration：http://ideone.com/9jYnv

⑧"The pairing heap: a new form of self-adjusting heap"：http://www.cs.cmu.edu/afs/cs.cmu.edu/user/sleator/www/papers/pairing-heaps.pdf

⑨Vikas Tandi ：http://programmingpraxis.com/2009/08/14/pairing-heaps/

⑩http://www.cise.ufl.edu/~sahni/cop5536/slides/lec156.pdf

⑩+1： 算法导论和维基百科