【PAT】A1155 Heap Paths【堆的判别】

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

题意

输出二叉堆的所有路径（优先输出右边的路径）并判断它是大根堆、小根堆或者不是堆。

思路

递归输出路径（注意优先右边），然后按照定义判断即可。

代码

#include <cstdio>
#include <vector>
#define MAX_N 1001
using namespace std;
int heap[MAX_N], N;
bool maxHeap = true, minHeap = true;
vector<int> path;
void dfs(int k){
    if(k > N) return;
    path.push_back(heap[k]);
    if(k * 2 > N){// 到达叶子结点，输出路径
        printf("%d", path[0]);
        for(int i = 1, l = (int)path.size(); i < l; i++){
            printf(" %d", path[i]);
            if(maxHeap && path[i] > path[i - 1]) maxHeap = false;// 发现路径上有不符合大根堆的偏序关系，则不是大根堆
            if(minHeap && path[i] < path[i - 1]) minHeap = false;// 同理
        }
        putchar('\n');
    }else{
        dfs(k * 2 + 1);
        dfs(k * 2);
    }
    path.pop_back();
    
}
int main() {
    scanf("%d", &N);
    for(int i = 1; i <= N; i++){
        scanf("%d", &heap[i]);
    }
    
    dfs(1);
    printf(maxHeap ? "Max Heap" : (minHeap ? "Min Heap" : "Not Heap"));
    return 0;
}