A1155 Heap Paths (30 point(s))

二叉查找树的共同祖先

1. 原文

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))

One thing for sure is that all the keys along any path from the root to a leaf in a max/min heap must be in non-increasing/non-decreasing order.

Your job is to check every path in a given complete binary tree, in order to tell if it is a heap or not.

Input Specification:

Each input file contains one test case. For each case, the first line gives a positive integer N (1<N≤1,000), the number of keys in the tree. Then the next line 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, first print all the paths from the root to the leaves. Each path occupies a line, with all the numbers separated by a space, and no extra space at the beginning or the end of the line. The paths must be printed in the following order: for each node in the tree, all the paths in its right subtree must be printed before those in its left subtree.

Finally 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.

Sample Input 1:

8
98 72 86 60 65 12 23 50

Sample Output 1:

98 86 23
98 86 12
98 72 65
98 72 60 50
Max Heap

Sample Input 2:

8
8 38 25 58 52 82 70 60

Sample Output 2:

8 25 70
8 25 82
8 38 52
8 38 58 60
Min Heap

Sample Input 3:

8
10 28 15 12 34 9 8 56

Sample Output 3:

10 15 8
10 15 9
10 28 34
10 28 12 56
Not Heap

2. 解析思路

遍历：根右左遍历

堆判断： 大根堆 i > 2i && i> 2i +1 ， 小根堆 i < 2i && i< 2i +1

遍历方式直接可用递归实现

3. AC代码

#include<cstdio>
#include<vector>
using namespace std;
const int maxn=1010;
int level[maxn]={};
vector<int> v;
int n;
void dfs(int root)
{
	if (2*root>n&&2*root+1>n)
	{
		if (root<=n)
		{
			for (int i = 0; i < v.size(); ++i)
			{
				if (i!=0)
				{
					printf(" ");
				}
				printf("%d",level[v[i]]);
			}
			printf("\n");
		}
	}
	else{
		v.push_back(2*root+1);
		dfs(2*root+1);
		v.pop_back();
		v.push_back(2*root);
		dfs(2*root);
		v.pop_back();
	}
}
int main()
{
	scanf("%d",&n);
	for (int i = 1; i <= n; ++i)
	{
		scanf("%d",&level[i]);
	}
	v.push_back(1);
	dfs(1);
	bool max=true; bool min=true;
	for (int i = 1; i < n/2; ++i)
	{
		if(level[i]<level[2*i]||level[i]<level[2*i+1])
		{
			max=false;
		}
		if(level[i]>level[2*i]||level[i]>level[2*i+1])
		{
			min=false;
		}
	}
	if (max)
	{
		printf("Max Heap\n");
	}else if (min)
	{
		printf("Min Heap\n");
	}
	else{
		printf("Not Heap\n");
	}
	return 0;
}