如何检查给定数组是否代表二叉堆？

给定一个数组，如何检查给定的数组是否代表一个二元最大堆。
例子：

Input:  arr[] = {90, 15, 10, 7, 12, 2} 
Output: True
The given array represents below tree
       90
     /    \
   15      10
  /  \     /
 7    12  2 
The tree follows max-heap property as every
node is greater than all of its descendants.

Input:  arr[] = {9, 15, 10, 7, 12, 11} 
Output: False
The given array represents below tree
       9
     /    \
   15      10
  /  \     /
 7    12  11
The tree doesn't follows max-heap property 9 is 
smaller than 15 and 10, and 10 is smaller than 11. 

一个简单的解决方案是首先检查 root 是否大于其所有后代。然后检查根的孩子。此解决方案的时间复杂度为 O(n ² )
一个有效的解决方案是只比较根和它的孩子（不是所有的后代），如果根大于它的孩子并且对所有节点都是如此，那么树是最大堆（这个结论是基于 >运算符的传递属性，即如果 x > y 且 y > z，则 x > z）。
假设索引从 0 开始，最后一个内部节点出现在索引 (n-2)/2 处。
下面是这个解决方案的实现。

C++

// C program to check whether a given array
// represents a max-heap or not
#include 
#include 
 
// Returns true if arr[i..n-1] represents a
// max-heap
bool isHeap(int arr[], int i, int n)
{
    // If a leaf node
    if (i >= (n - 2) / 2)
        return true;
 
    // If an internal node and is
    // greater than its children,
    // and same is recursively
    // true for the children
    if (arr[i] >= arr[2 * i + 1] &&
        arr[i] >= arr[2 * i + 2]
        && isHeap(arr, 2 * i + 1, n)
        && isHeap(arr, 2 * i + 2, n))
        return true;
 
    return false;
}
 
// Driver program
int main()
{
    int arr[] = { 90, 15, 10, 7, 12, 2, 7, 3 };
    int n = sizeof(arr) / sizeof(int) - 1;
 
    isHeap(arr, 0, n) ? printf("Yes") : printf("No");
 
    return 0;
}

Java

// Java program to check whether a given array
// represents a max-heap or not
class GFG
{
 
    // Returns true if arr[i..n-1]
    // represents a max-heap
    static boolean isHeap(int arr[],
                          int i, int n)
    {
        // If a leaf node
        if (i >= (n - 2) / 2)
        {
            return true;
        }
 
        // If an internal node and
        // is greater than its
        // children, and same is
        // recursively true for the
        // children
        if (arr[i] >= arr[2 * i + 1]
            && arr[i] >= arr[2 * i + 2]
            && isHeap(arr, 2 * i + 1, n)
            && isHeap(arr, 2 * i + 2, n))
        {
            return true;
        }
 
        return false;
    }
 
    // Driver program
    public static void main(String[] args)
    {
        int arr[] = { 90, 15, 10, 7, 12, 2, 7, 3 };
        int n = arr.length - 1;
        if (isHeap(arr, 0, n)) {
            System.out.println("Yes");
        }
        else {
            System.out.println("No");
        }
    }
}
 
// This code contributed by 29AjayKumar

Python3

# Python3 program to check whether a
# given array represents a max-heap or not
 
# Returns true if arr[i..n-1]
# represents a max-heap
def isHeap(arr, i, n):
     
# If a leaf node
    if i >= int((n - 2) / 2):
        return True
     
    # If an internal node and is greater
    # than its children, and same is
    # recursively true for the children
    if(arr[i] >= arr[2 * i + 1] and
       arr[i] >= arr[2 * i + 2] and
       isHeap(arr, 2 * i + 1, n) and
       isHeap(arr, 2 * i + 2, n)):
        return True
     
    return False
 
# Driver Code
if __name__ == '__main__':
    arr = [90, 15, 10, 7, 12, 2, 7, 3]
    n = len(arr) - 1
 
    if isHeap(arr, 0, n):
        print("Yes")
    else:
        print("No")
 
# This code is contributed by PranchalK

C#

// C# program to check whether a given 
// array represents a max-heap or not
using System;
 
class GFG
{
 
// Returns true if arr[i..n-1] represents a
// max-heap
static bool isHeap(int []arr, int i, int n)
{
    // If a leaf node
    if (i >= (n - 2) / 2)
    {
        return true;
    }
 
    // If an internal node and is greater
    // than its children, and same is
    // recursively true for the children
    if (arr[i] >= arr[2 * i + 1] &&
        arr[i] >= arr[2 * i + 2] &&
        isHeap(arr, 2 * i + 1, n) &&
        isHeap(arr, 2 * i + 2, n))
    {
        return true;
    }
 
    return false;
}
 
// Driver Code
public static void Main(String[] args)
{
    int []arr = {90, 15, 10, 7, 12, 2, 7, 3};
    int n = arr.Length-1;
    if (isHeap(arr, 0, n))
    {
        Console.Write("Yes");
    }
     
    else
    {
        Console.Write("No");
    }
}
}

PHP

= ($n - 2) / 2)
    return true;
 
// If an internal node and is greater
// than its children, and same is
// recursively true for the children
if ($arr[$i] >= $arr[2 * $i + 1] &&
    $arr[$i] >= $arr[2 * $i + 2] &&
    isHeap($arr, 2 * $i + 1, $n) &&
    isHeap($arr, 2 * $i + 2, $n))
    return true;
 
return false;
}
 
// Driver Code
$arr = array(90, 15, 10, 7, 12, 2, 7, 3);
$n = sizeof($arr);
 
if(isHeap($arr, 0, $n))
    echo "Yes";
else
    echo "No";
 
// This code is contributed
// by Akanksha Rai
?>

Javascript

// Javascript program to check whether a given array
// represents a max-heap or not
 
// Returns true if arr[i..n-1]
    // represents a max-heap
function isHeap(arr,i,n)
{
    // If a leaf node
        if (i >= (n - 2) / 2)
        {
            return true;
        }
  
        // If an internal node and
        // is greater than its
        // children, and same is
        // recursively true for the
        // children
        if (arr[i] >= arr[2 * i + 1]
            && arr[i] >= arr[2 * i + 2]
            && isHeap(arr, 2 * i + 1, n)
            && isHeap(arr, 2 * i + 2, n))
        {
            return true;
        }
  
        return false;
}
 
// Driver program
let arr=[ 90, 15, 10, 7, 12, 2, 7, 3 ];
let n = arr.length - 1;
if (isHeap(arr, 0, n)) {
    document.write("Yes<br>");
}
else {
    document.write("No<br>");
}
 
 
// This code is contributed by rag2127

输出：

Yes

该解决方案的时间复杂度为 O(n)。解决方法类似于二叉树的先序遍历。
感谢 Utkarsh Trivedi 提出上述解决方案。
迭代解决方案是遍历所有内部节点并检查 id 节点是否大于其子节点。

C++

// C program to check whether a given array
// represents a max-heap or not
#include 
#include 
 
// Returns true if arr[i..n-1] represents a
// max-heap
bool isHeap(int arr[],  int n)
{
    // Start from root and go till the last internal
    // node
    for (int i=0; i<=(n-2)/2; i++)
    {
        // If left child is greater, return false
        if (arr[2*i +1] > arr[i])
                return false;
 
        // If right child is greater, return false
        if (2*i+2 < n && arr[2*i+2] > arr[i])
                return false;
    }
    return true;
}
 
// Driver program
int main()
{
    int arr[] = {90, 15, 10, 7, 12, 2, 7, 3};
    int n = sizeof(arr) / sizeof(int);
 
    isHeap(arr, n)? printf("Yes"): printf("No");
 
    return 0;
}

Java

// Java program to check whether a given array
// represents a max-heap or not
 
class GFG {
 
// Returns true if arr[i..n-1] represents a
// max-heap
    static boolean isHeap(int arr[], int n) {
        // Start from root and go till the last internal
        // node
        for (int i = 0; i <= (n - 2) / 2; i++) {
            // If left child is greater, return false
            if (arr[2 * i + 1] > arr[i]) {
                return false;
            }
 
            // If right child is greater, return false
            if (2 * i + 2 < n && arr[2 * i + 2] > arr[i]) {
                return false;
            }
        }
        return true;
    }
 
// Driver program
    public static void main(String[] args) {
        int arr[] = {90, 15, 10, 7, 12, 2, 7, 3};
        int n = arr.length;
        if (isHeap(arr, n)) {
            System.out.println("Yes");
        } else {
            System.out.println("No");
        }
    }
}
// This code is contributed by 29AjayKumar

蟒蛇3

# Python3 program to check whether a
# given array represents a max-heap or not
 
# Returns true if arr[i..n-1]
# represents a max-heap
def isHeap(arr, n):
     
    # Start from root and go till
    # the last internal node
    for i in range(int((n - 2) / 2) + 1):
         
        # If left child is greater,
        # return false
        if arr[2 * i + 1] > arr[i]:
                return False
 
        # If right child is greater,
        # return false
        if (2 * i + 2 < n and
            arr[2 * i + 2] > arr[i]):
                return False
    return True
 
# Driver Code
if __name__ == '__main__':
    arr = [90, 15, 10, 7, 12, 2, 7, 3]
    n = len(arr)
 
    if isHeap(arr, n):
        print("Yes")
    else:
        print("No")
         
# This code is contributed by PranchalK

C＃

// C# program to check whether a given array
// represents a max-heap or not
using System;
 
class GFG
{
 
// Returns true if arr[i..n-1]
// represents a max-heap
static bool isHeap(int []arr, int n)
{
    // Start from root and go till
    // the last internal node
    for (int i = 0; i <= (n - 2) / 2; i++)
    {
        // If left child is greater,
        // return false
        if (arr[2 * i + 1] > arr[i])
        {
            return false;
        }
 
        // If right child is greater,
        // return false
        if (2 * i + 2 < n && arr[2 * i + 2] > arr[i])
        {
            return false;
        }
    }
    return true;
}
 
// Driver Code
public static void Main()
{
    int []arr = {90, 15, 10, 7, 12, 2, 7, 3};
    int n = arr.Length;
    if (isHeap(arr, n))
    {
        Console.Write("Yes");
    }
    else
    {
        Console.Write("No");
    }
}
}
 
// This code is contributed
// by 29AjayKumar

PHP

 $arr[$i])
                return False;
 
        // If right child is greater,
        // return false
        if (2 * $i + 2 < $n &&
                $arr[2 * $i + 2] > $arr[$i])
                return False;
     
    return True;
    }
}
 
// Driver Code
$arr = array(90, 15, 10, 7, 12, 2, 7, 3);
$n = sizeof($arr);
 
if(isHeap($arr, 0, $n))
    echo "Yes";
else
    echo "No";
     
// This code is contributed by Princi Singh
?>

Javascript

 
// Javascript program to check
// whether a given array
// represents a max-heap or not
 
// Returns true if arr[i..n-1]
// represents a max-heap
function isHeap( arr, n)
{
    // Start from root and go till
    // the last internal node
    for (let i=0; i<=Math.floor((n-2)/2); i++)
    {
        // If left child is greater,
        // return false
        if (arr[2*i +1] > arr[i])
                return false;
 
        // If right child is greater,
        // return false
        if (2*i+2 < n && arr[2*i+2] > arr[i])
                return false;
    }
    return true;
}
    // driver code
     
    let arr = [90, 15, 10, 7, 12, 2, 7, 3];
    let n = arr.length;
    if (isHeap(arr, n)) {
        document.write("Yes");
        }
    else {
        document.write("No");
        }
     

输出：

Yes

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