1155 Heap Paths
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 • 1 • 2
Sample Output 1:
98 86 23 98 86 12 98 72 65 98 72 60 50 Max Heap
Sample Output 2:
8 25 70 8 25 82 8 38 52 8 38 58 60 Min Heap • 1 • 2 • 3 • 4 • 5
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
在计算机科学中,堆是一种的基于树的专用数据结构,它具有堆属性:
如果 P 是 C 的父结点,则在大顶堆中 P 结点的权值大于或等于 C 结点的权值,在小顶堆中 P 结点的权值小于或等于 C 结点的权值。
一种堆的常见实现是二叉堆,它是由完全二叉树来实现的。
可以肯定的是,在大顶/小顶堆中,任何从根到叶子的路径都必须按非递增/非递减顺序排列。
你的任务是检查给定完全二叉树中的每个路径并输出它们,以判断它是否是堆
思路
我们可以通过递归的方式去输出路径,每查找到一条路径就输出它。
在输出路径的时候,去判断该序列是递增还是递减还是非顺序,用两个变量去记录,一个是 lt 表示该堆是小根堆即如果该序列是递增的就标记 lt = true ,另一个是 gt 表示该堆是大根堆即如果该序列是递减的就标记 gt = true 。
遍历完所有路径后,判断 lt 和 gt ,如果两个变量都是真值,说明该树不是堆(只有满足大根堆或小根堆性质的完全二叉树才能被称为堆),否则输出对应的判断结果。
代码
#include<bits/stdc++.h> using namespace std; const int N = 1010; int n; int h[N]; vector<int> path; bool lt, gt; void dfs(int u) { path.push_back(h[u]); if (u * 2 > n) //如果已经到达叶子结点 { cout << path[0]; for (int i = 1; i < path.size(); i++) { cout << " " << path[i]; //判断该树是大根堆还是小根堆 if (path[i] > path[i - 1]) lt = true; else if (path[i] < path[i - 1]) gt = true; } cout << endl; } //先递归右子树,再递归左子树 if (u * 2 + 1 <= n) dfs(u * 2 + 1); if (u * 2 <= n) dfs(u * 2); path.pop_back(); } int main() { cin >> n; for (int i = 1; i <= n; i++) cin >> h[i]; dfs(1); if (gt && lt) puts("Not Heap"); else if (gt) puts("Max Heap"); else puts("Min Heap"); return 0; }