Data Structures (6)

最新推荐文章于 2025-12-05 14:45:56 发布

原创最新推荐文章于 2025-12-05 14:45:56 发布 · 137 阅读

0 ·

CC 4.0 BY-SA版权

文章标签：

#数据结构 #链表

数据结构专栏收录该内容

9 篇文章

订阅专栏

本文介绍了二叉堆的数据结构及其在优先级队列中的应用。二叉堆分为最小堆和最大堆，它们都是完全二叉树。插入操作通过比较新节点与其父节点的大小进行调整，删除操作通常替换根节点并重新调整堆。插入和删除的时间复杂度为O(logN)，全元素插入后再调整的时间复杂度为O(nlogn)。此外，文章还探讨了数组和链表等不同数据结构实现堆的优缺点。

堆
【二叉堆】
要求：会构建堆插入删除；画图，手动执行中间数据；

Piority Queue(Heap)

Can find the element with the highest \ lowest priority.

	Insertion	Deletion
Array	O(1)	find:O(n) delete:O(n)
Linked List	O(1)	find:O(n) delete:O(n)
Ordered Array	find:O(1) insert:O(n)	O(1)
Ordered Linked List	find:O(1) insert:O(n)	O(1)
## complete binary tree
Its nodes correspond to the nodes numbered from 1 to n in the perfect binary tree of height h.
A complete binary tree of height h has between 2^h and 2^{h+1}-1 nodes.
Array representation: BT[n+1](BT[0]is not used)
For node i:
- parent: i/2
- left child: 2*i
- right child: 2*i+1

A min tree is a tree in which the key value in each node is no larger than the key values in its children.
A min heap :a complete binary tree + a min tree. And the max heap is visa versa.

Insertion(Min Heap)

Because a heap is a complete binary tree, the nth node only has one possible position to insert. Then, if nth node is smaller than its parent:exchange upward will its location is proper.

for ( i = ++H->Size; H->Elements[ i / 2 ] > X; i /= 2 ) 
	H->Elements[ i ] = H->Elements[ i / 2 ]; 

     H->Elements[ i ] = X;

Another kind of Insertion is: insert all nodes first, then adjust their location. We need to check from n/2 node. If its children node is smaller than it, choose the smallest node and exchange them. n/2 node go downward until reach the proper location. Then do the same thing to the n/2-1 node until the first node.

Deletion

We only need to delete the root node——the smallest node.

Exchange the root node with the last node——the nth node.
do the same thing mentioned above in the Italic text: re-arrange the new node.
Iteratively delete root node until delete the whole heap.
*Those 2 operations’ time complexity are O(log N). Insertion all node is O(nlogn).

For the perfect binary tree of height h containing 2h+1 - 1 nodes, the sum of the heights of the nodes is 2h+1 - 1 - (h + 1).