PAT甲级：A1147 Heaps (30分)

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:

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2
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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:

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3
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5
6
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：给出一颗完全二叉树顺序存储的序列，判断这棵树是大顶堆、小顶堆或不是堆，之后输出这完全二叉树的后序遍历序列。
  • 分析：因为是完全二叉树且是顺序存储的方式，所以从下标为1的位置开始依次读入到heap数组中，之后设定isMaxHeap、isMinHeap两变量，从下标为2的结点开始，遍历该数组，当前结点编号 i 除以2就是父结点编号，如果孩子结点值大于父结点值则一定不是大顶堆，将isMaxHeap标记为 false，小顶堆同理。最后递归输出后序遍历序列。

注意：输出后序遍历序列最后不能有多余空格，因此设置全局变量cnt判断是否输出空格。

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#include <bits/stdc++.h>
using namespace std;
int m, n, cnt = 0, heap[1010];
void postOrder(int i) {
    if (i > n) return;
    postOrder(i * 2);
    postOrder(i * 2 + 1);
    printf("%d", heap[i]);
    if (++cnt < n) printf(" ");
}
int main() {
    scanf("%d%d", &m, &n);
    while (m--) {
        for (int j = 1; j <= n; j++) scanf("%d", &heap[j]);
        bool isMaxHeap = true, isMinHeap = true;
        for (int i = 2; i <= n; i++) {
            if (heap[i] > heap[i / 2]) isMaxHeap = false;
            if (heap[i] < heap[i / 2]) isMinHeap = false;
        }
        if (!isMaxHeap && !isMinHeap) printf("Not Heap\n");
        else printf("%s\n", isMaxHeap ? "Max Heap" : "Min Heap");
        cnt = 0;
        postOrder(1);
        printf("\n");
    }
    return 0;
}