题目

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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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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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

题解

思路

  • 堆其实是一种很简单的数据结构
  • 是一种二叉完全树，所以可以用数组表示，为了方便我在前面加了一个dummy head
  • 这就意味着，所有节点下标的*2即它的左孩子下标,*2+1即它的右孩子的下标。
  • 这题就分为两步
    • 判断是大顶堆小顶堆还是不是堆
      • 就遍历每个节点，判断它和它父亲的关系即可
    • 后序遍历
      • 递归即可

数据结构

  • heap 是我们的堆
  • isMax 判断是否是大顶堆
  • 呃，好像没了

算法

  • 就照着思路里的写就行了
  • 后序遍历是先左后右后根，照着这样写就可以了。处理一下边界条件即可。

代码

  • 由于使用Python可以AC，因此只放了Python的解。
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def postorder(index):
    if index > num_nodes:
        return
    postorder(2 * index)
    postorder(2 * index + 1)
    path.append(heap[index])


num_test, num_nodes = list(map(int, input().split()))
for _ in range(num_test):
    heap = [0] + list(map(int, input().split()))
    isMax = heap[1] > heap[2]
    if isMax:
        for i in range(3, num_nodes + 1):
            if heap[i] > heap[i // 2]:
                print("Not Heap")
                break
        else:
            print("Max Heap")
    else:
        for i in range(3, num_nodes + 1):
            if heap[i] < heap[i // 2]:
                print("Not Heap")
                break
        else:
            print("Min Heap")
    path = []
    postorder(1)
    print(" ".join(list(map(str, path))))