(PAT)1147Heaps(判断是大顶堆还是小顶堆)

末蓝、 2022-04-18 06:38 222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]))

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

解题思路,给出一堆数列,判断是大顶堆,小顶堆还不是堆

我解这道题的初步思路是,利用堆的下调算法,依次对每个结点进行判断,如果结点与左子或右子比较有违背小顶堆或大顶堆的性质

(如在小顶堆中,一个结点大于其子节点),这个序列就不是小顶堆或者大顶堆

#include<iostream>
    #include <algorithm>
    using namespace std;
    const int maxn = 1010;
    const int manco = 101;
    int nheap[manco][maxn];
    bool ismax = true, ismin = true;
    
    void isMaxHeap(int a[],int pos,int n) {
    	int current = pos;
    	int onechild = pos * 2;
    	
    	if (onechild <= n) {
    		if (onechild + 1 <= n && a[onechild] < a[onechild + 1]) {  //左子小于右子的话
    			onechild = onechild + 1;
    		}
    		if (a[current] >= a[onechild]) {
    			isMaxHeap(a,onechild, n);       //右子递归寻找
    			isMaxHeap(a,onechild - 1, n);   //左子递归寻找
    		}
    		else {
    			ismax = false;
    		}
    	}
    }
    
    void isMinHeap(int a[],int pos, int n) {
    	int current = pos;
    	int onechild = pos * 2;
    
    	if (onechild <= n) {
    		if (onechild + 1 <= n && a[onechild] > a[onechild + 1]) {  //左子小于右子的话
    			onechild = onechild + 1;
    		}
    		if (a[current] <= a[onechild]) {
    			isMinHeap(a,onechild, n);  //左子递归寻找
    			isMinHeap(a,onechild-1, n);      //右子递归寻找
    		}
    		else {
    			ismin = false;
    		}
    	}
    }
    
    void postOrd(int a[],int node,int size) {
    	if (node * 2 <= size) {
    		postOrd(a,node * 2,size);
    	}
    	if (node * 2 + 1 <= size) {
    		postOrd(a,node * 2 + 1,size);
    	}
    
    	if (node == 1) {
    		cout << a[node];
    	}
    	else {
    		cout << a[node] << " ";
    	}
    }
    
    int main() {
    
    	int n, m;
    	cin >> n >> m;
    	for (int i = 1; i <= n; ++i) {
    		for (int j = 1; j <= m; ++j) {
    			cin >> nheap[i][j];
    		}
    	}
    
    	for (int i = 1; i <= n; ++i) {
    		isMaxHeap(nheap[i], 1, m);
    		if (ismax) {
    			cout << "Max Heap" << endl;
    			postOrd(nheap[i], 1, m);
    			cout << endl;
    			ismax = true;
    			ismin = true;
    			continue;
    		}
    
    		isMinHeap(nheap[i], 1, m);
    		if (ismin) {
    			cout << "Min Heap" << endl;
    			postOrd(nheap[i], 1, m);
    			cout << endl;
    			ismax = true;
    			ismin = true;
    			continue;
    		}
    		
    		cout << "Not Heap" << endl;
    		postOrd(nheap[i], 1, m);
    		cout << endl;
    		ismax = true;
    		ismin = true;
    	}
    
    	system("PAUSE");
    	return 0;
    }

上述代码用到了递归,所以5组数据只过了2组(剩下3组超时)

然后我换了一种思路,我们可以通过判断头结点和其子节点的大小关系初步得出这是小顶堆或是大顶堆的结论

然后判断每一个结点是否违背小顶堆(或是大顶堆),如有违背,那么这个序列不是堆

#include <iostream>
    #include <algorithm>
    using namespace std;
    
    const int maxc = 101;
    const int maxn = 1001;
    int mmheap[maxc][maxn];
    
    //后序遍历
    void postOrder(int a[],int root,int n) {
    	if (root * 2 <= n) {
    		postOrder(a, root * 2, n);
    	}
    	if (root * 2 + 1 <= n) {
    		postOrder(a, root * 2 + 1, n);
    	}
    
    	if (root == 1) {
    		cout << a[root];
    	}
    	else {
    		cout << a[root] << " ";
    	}
    }
    
    
    int main() {
    
    	int n, m;
    	cin >> n >> m;
    	for (int i = 1; i <= n; ++i) {
    		for (int j = 1; j <= m; ++j) {
    			cin >> mmheap[i][j];
    		}
    	}
    
    	for (int i = 1; i <= n; ++i) {
    		int flag;
    		flag = mmheap[i][1] > mmheap[i][2] ? 1 : -1;   //1是大顶堆,-1是小顶堆
    		for (int j = 1; j <= m / 2; ++j) {
    			int lchild = j * 2;
    			int rchild = j * 2 + 1;
    			if (flag == 1 && (mmheap[i][j] <= mmheap[i][lchild] || (rchild <= m && mmheap[i][j] <= mmheap[i][rchild])))
    			{
    				flag = 0;  //判断得出这不是堆
    			}
    			if (flag == -1 && (mmheap[i][j] >= mmheap[i][lchild] || (rchild <= m && mmheap[i][j] >= mmheap[i][rchild])))
    			{ 
    				flag = 0;  //判断得出这不是堆
    			}
    		}
    		if (flag == 0) cout << "Not Heap" << endl;
    		else if (flag == 1) cout << "Max Heap" << endl;
    		else if (flag == -1) cout << "Min Heap" << endl;
    		postOrder(mmheap[i], 1, m);
    		cout << endl;
    	}
    
    	system("PAUSE");
    	return 0;
    }

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