18年春季第四题 PAT甲级 1147 Heaps (30分) 可以不建树

女爷i 2023-07-07 05:47 123阅读 0赞

# 汇总贴 #

[2020年3月PAT甲级满分必备刷题技巧][2020_3_PAT]

# 题目 #

来源：[https://pintia.cn/problem-sets/994805342720868352/problems/994805342821531648][https_pintia.cn_problem-sets_994805342720868352_problems_994805342821531648]

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

# 题目大意 #

给定堆的层序，要求判断堆的种类（大顶堆、小顶堆、不是堆）和输出后序

# 相关知识点整理 #

1.堆属于完全二叉树。可以用数组存，不用建树。如果用数组存树，应该想到的就是**深度优先遍历能生成后序遍历序列**。

2.虽然《算法笔记》的9.7有介绍堆，1098题也是跟堆有关。但是在PAT甲级近三年的题目里，没考过向下调整、建堆写法，只了解到堆的概念就行。目前的考纲也没有明确地提到“堆”。近三年也就18年春季和18年冬季考过“堆”。

# 题目分析 #

用数组存就行了，不建树，DFS生成后序遍历序列，堆的类型就直接用定义排除掉明显不对的可能。

# 满分代码 #

#include<iostream>
    #include<vector>
    using namespace std;
    const int maxn=1005;
    int n,m;
    vector<int> v(maxn);
    void dfs(int i){
        if(i>m)return;
        dfs(2*i);
        dfs(2*i+1);
        printf("%d%s",v[i],i==1?"\n":" ");
    }
    int main(){
        cin>>n>>m;
        for(int i=0;i<n;i++){
            int minheap=1,maxheap=1;
            for(int j=1;j<=m;j++){
                scanf("%d",&v[j]);
                if(j>1 && v[j/2]>v[j])minheap=0;
                if(j>1 && v[j/2]<v[j])maxheap=0;
            }
            if(minheap==1)printf("Min Heap\n");
            else printf("%s\n",maxheap==1?"Max Heap":"Not Heap");
            dfs(1);
        }
        
        return 0;
    }

[2020_3_PAT]: https://blog.csdn.net/allisonshing/article/details/104213959#comments
[https_pintia.cn_problem-sets_994805342720868352_problems_994805342821531648]: https://pintia.cn/problem-sets/994805342720868352/problems/994805342821531648
[https_en.wikipedia.org_wiki_Heap_data_structure]: https://en.wikipedia.org/wiki/Heap_%28data_structure