Data Structures (5)

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).

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).