(PAT 1155) Heap Paths (堆+完全二叉树遍历)

冷不防 2022-04-23 09:36 146阅读 0赞

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][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

解题思路:

利用一个vector保留遍历时的路径(深度优先),每到达一个叶子节点就输出当前vector中的所有元素

至于大顶堆小顶堆,每遍历到一个结点就将这个结点和它的左右子结点比较,创建两个bool变量记录

#include <iostream>
    #include <vector>
    #include <algorithm>
    using namespace std;
    const int MAXN = 1010;
    int BinayTree[MAXN];
    int N;
    //root从0开始
    bool MAXHEAP, MINHEAP;
    vector<int> preorderTemp;
    void preOrderTRA(int root) {
    	preorderTemp.push_back(BinayTree[root]);  
    	if (root * 2 + 1 >= N && root * 2 + 2 >= N) {
    		for (auto itr = preorderTemp.begin(); itr != preorderTemp.end(); ++itr) {
    			cout << *itr;
    			if (itr < preorderTemp.end() - 1) cout << " ";
    		}
    		cout << endl;
    	}
    	if (root * 2 + 2 < N) { 
    		if (BinayTree[root * 2 + 2] > BinayTree[root]) {
    			MAXHEAP = false;
    		}
    		if (BinayTree[root * 2 + 2] < BinayTree[root]) {
    			MINHEAP = false;
    		}
    		preOrderTRA(root * 2 + 2);
    	}
    	if (root * 2 + 1 < N) { 
    		if (BinayTree[root * 2 + 1] > BinayTree[root]) {
    			MAXHEAP = false;
    		}
    		if (BinayTree[root * 2 + 1] < BinayTree[root]) {
    			MINHEAP = false;
    		}
    		preOrderTRA(root * 2 + 1); 
    	}
    	preorderTemp.pop_back();
    }
    
    int main() {
    	MAXHEAP = true;
    	MINHEAP = true;
    	cin >> N;
    	for (int i = 0; i < N; ++i) {
    		cin >> BinayTree[i];
    	}
    	preOrderTRA(0);
    
    	if (MAXHEAP && MINHEAP) {
    		cout << "Not Heap" << endl;
    		system("PAUSE");
    		return 0;
    	}
    
    	if (MAXHEAP || MINHEAP) {
    		if (MAXHEAP) {
    			cout << "Max Heap" << endl;
    		}
    		if (MINHEAP) {
    			cout << "Min Heap" << endl;
    		}
    	}
    	else {
    		cout << "Not Heap" << endl;
    	}
    
    	system("PAUSE");
    	return 0;
    }

[https_en.wikipedia.org_wiki_Heap_data_structure]: https://en.wikipedia.org/wiki/Heap_%28data_structure