1147 Heaps

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

题意

给定 M 个完全二叉树的层序遍历结果，每个结果包含 N 个结点，需要我们判断这 M 个树是否为堆，并输出其后序遍历的结果。

思路

本题可以利用完全二叉树的性质，完全二叉树的层序遍历结果在数组中有如下性质（下标从 1 开始）：

下标为 u 的根结点的左孩子下标为 u*2

下标为 u 的根结点的右孩子下标为 u*2+1

具体思路如下：

先输入层序遍历的数组，注意是从下标 1 开始输入，方便后续使用其性质。

判断是否为堆，用 lt 和 gt 变量来标记，如果出现根结点要小于其孩子结点，则标记 lt 为 true ；反之，标记 gt 为 true 。

根据 lt 和 gt 标记结果输出对应答案，如果 lt 和 gt 都为 true ，说明该二叉树并不是堆；如果 lt 为 true ，说明该二叉树是小根堆；如果 gt 为 true ，说明该二叉树是大根堆。

输出该二叉树的后序遍历结果，同样可以利用上述提到的下标性质进行递归遍历。

代码

#include<bits/stdc++.h>
using namespace std;
const int N = 1010;
int h[N];
int m, n;
//输出后序遍历结果遍历
void dfs(int u)
{
    if (u > n)    return;
    dfs(u * 2);
    dfs(u * 2 + 1);
    cout << h[u];
    if (u != 1)    cout << " ";
}
int main()
{
    cin >> m >> n;
    while (m--)
    {
        //输入层序遍历数组
        for (int i = 1; i <= n; i++)   cin >> h[i];
        //判断是否为堆
        bool lt = false, gt = false;
        for (int i = 1; i <= n; i++)
            for (int j = 0; j < 2; j++)
                if (i * 2 + j <= n)
                {
                    int a = h[i], b = h[i * 2 + j];
                    if (a < b) lt = true;
                    else    gt = true;
                }
        //根据判断结果输出对应类型
        if (lt && gt)  puts("Not Heap");
        else if (lt) puts("Min Heap");
        else        puts("Max Heap");
        //输出后序遍历结果
        dfs(1);
        puts("");
    }
    return 0;
}