1147 Heaps (30 point(s))

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最新推荐文章于 2024-03-13 13:51:37 发布

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本文深入探讨了堆数据结构的定义及其在计算机科学中的应用。详细解释了如何判断一个完全二叉树是否为最大堆或最小堆，并提供了后序遍历的实现方法。通过示例输入输出，帮助读者理解堆的特性和遍历过程。

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1147 Heaps (30 point(s))

In computer science, a heap is a specialized tree-based data structure that satisfies the heap property: if P is a parent node of C, then the key (the value) of P is either greater than or equal to (in a max heap) or less than or equal to (in a min heap) the key of C. A common implementation of a heap is the binary heap, in which the tree is a complete binary tree. (Quoted from Wikipedia at https://en.wikipedia.org/wiki/Heap_(data_structure))

Your job is to tell if a given complete binary tree is a heap.

Input Specification:

Each input file contains one test case. For each case, the first line gives two positive integers: M (≤ 100), the number of trees to be tested; and N (1 < N ≤ 1,000), the number of keys in each tree, respectively. Then M lines follow, each contains N distinct integer keys (all in the range of int), which gives the level order traversal sequence of a complete binary tree.

Output Specification:

For each given tree, print in a line Max Heap if it is a max heap, or Min Heap for a min heap, or Not Heap if it is not a heap at all. Then in the next line print the tree's postorder traversal sequence. All the numbers are separated by a space, and there must no extra space at the beginning or the end of the line.

Sample Input:

3 8
98 72 86 60 65 12 23 50
8 38 25 58 52 82 70 60
10 28 15 12 34 9 8 56

Sample Output:

Max Heap
50 60 65 72 12 23 86 98
Min Heap
60 58 52 38 82 70 25 8
Not Heap
56 12 34 28 9 8 15 10

堆的定义和判断、后序遍历

堆的定义和判断

堆是具有以下性质的完全二叉树：每个结点的值都大于或等于其左右孩子结点的值，称为大顶堆；或者每个结点的值都小于或等于其左右孩子结点的值，称为小顶堆。如下图：

同时，我们对堆中的结点按层进行编号，将这种逻辑结构映射到数组中就是下面这样：

该数组从逻辑上讲就是一个堆结构，我们用简单的公式来描述一下堆的定义就是：

大顶堆：arr[i] >= arr[2i+1] && arr[i] >= arr[2i+2]

小顶堆：arr[i] <= arr[2i+1] && arr[i] <= arr[2i+2]

注意点

1. 注意结点范围；

2. 注意后序遍历输出格式。

#include<iostream>
#include<cstring>
#include<vector>
using namespace std;
int M,N;
int levelOrder[1007];
bool isMaxHeap(){
	for(int i=1;i<=N/2;i++){
		if(2*i<=N){
			if(!(levelOrder[i]>=levelOrder[2*i])) return false;
		}
		if(2*i+1<=N){
			if(!(levelOrder[i]>=levelOrder[2*i+1])) return false;
		}
	}
	return true;
}
bool isMinHeap(){
	for(int i=1;i<=N/2;i++){
		if(2*i<=N){
			if(!(levelOrder[i]<levelOrder[2*i])) return false;
		}
		if(2*i+1<=N){
			if(!(levelOrder[i]<levelOrder[2*i+1])) return false;
		}
	}
	return true;
}
vector<int> v;
void postOrder(int root){
	if(2*root<=N) postOrder(2*root);
	if(2*root+1<=N) postOrder(2*root+1);
	v.push_back(levelOrder[root]);
}
int main(void){
	cin>>M>>N;
	while(M--){
		for(int i=1;i<=N;i++) cin>>levelOrder[i];
		if(isMaxHeap()) puts("Max Heap");
		else if(isMinHeap()) puts("Min Heap");
		else puts("Not Heap");
		v.clear();
		postOrder(1);
		for(int i=0;i<v.size();i++){
			if(i==0) cout<<v[i];
			else cout<<" "<<v[i];
		}
		cout<<endl;
	}
	return 0;
}