在无序算术级数中找到缺失的数字

给定一个由 N 个整数组成的未排序数组arr[] ，它们属于算术级数，任务是打印给定系列中缺失的元素。

例子：

Input: arr[] = {12, 3, 6, 15, 18} 
Output: 9 
Explanation: 
The elements given in order are: 3, 6, 12, 15, 18. 
Hence missing element is 9.

Input: arr[] = {2, 8, 6, 10} 
Output: 4 
Explanation: 
The elements given in order are: 2, 6, 8, 10. 
Hence missing element is 4.

朴素的方法：这个想法是使用二分搜索。对给定的数组进行排序，然后数组将按 AP 系列的排序顺序排列，我们可以应用二分搜索（如本文所述），如下所述：

1. 找到中间元素并检查中间元素和中间元素的下一个元素之间的差异是否等于公共差异，如果不是，则缺少的元素位于mid和mid + 1 之间。
2. 如果中间元素等于给定算术级数中的^第(N/2) 项，则缺少的元素位于中间元素的右侧。
3. 否则元素位于中间元素的左侧。

下面是上述方法的实现：

C++

// C++ program for the above approach
 
#include 
using namespace std;
 
// Function to find the missing element
int findMissing(int arr[], int left,
                int right, int diff)
{
 
    // Fix left and right boundary
    // for binary search
    if (right <= left)
        return INT_MAX;
 
    // Find index of middle element
    int mid = left + (right - left) / 2;
 
    // Check if the element just after
    // the middle element is missing
    if (arr[mid + 1] - arr[mid] != diff)
        return (arr[mid] + diff);
 
    // Check if the element just
    // before mid is missing
    if (mid > 0
        && arr[mid] - arr[mid - 1] != diff)
        return (arr[mid - 1] + diff);
 
    // Check if the elements till mid
    // follow the AP, then recur
    // for right half
    if (arr[mid] == arr[0] + mid * diff)
        return findMissing(arr, mid + 1,
                           right, diff);
 
    // Else recur for left half
    return findMissing(arr, left,
                       mid - 1, diff);
}
 
// Function to find the missing
// element in AP series
int missingElement(int arr[], int n)
{
 
    // Sort the array arr[]
    sort(arr, arr + n);
 
    // Calculate Common Difference
    int diff = (arr[n - 1] - arr[0]) / n;
 
    // Binary search for the missing
    return findMissing(arr, 0, n - 1, diff);
}
 
// Driver Code
int main()
{
    // Given array arr[]
    int arr[] = { 2, 8, 6, 10 };
    int n = sizeof(arr) / sizeof(arr[0]);
 
    // Function Call
    cout << missingElement(arr, n);
    return 0;
}

Java

// Java program for the above approach
import java.util.Arrays;
class GFG{
     
// Function to find the missing element
static int findMissing(int arr[], int left,
                       int right, int diff)
{
 
    // Fix left and right boundary
    // for binary search
    if (right <= left)
        return 0;
 
    // Find index of middle element
    int mid = left + (right - left) / 2;
 
    // Check if the element just after
    // the middle element is missing
    if (arr[mid + 1] - arr[mid] != diff)
        return (arr[mid] + diff);
 
    // Check if the element just
    // before mid is missing
    if (mid > 0 &&
        arr[mid] - arr[mid - 1] != diff)
        return (arr[mid - 1] + diff);
 
    // Check if the elements till mid
    // follow the AP, then recur
    // for right half
    if (arr[mid] == arr[0] + mid * diff)
        return findMissing(arr, mid + 1,
                           right, diff);
 
    // Else recur for left half
    return findMissing(arr, left,
                       mid - 1, diff);
}
 
// Function to find the missing
// element in AP series
static int missingElement(int arr[], int n)
{
 
    // Sort the array arr[]
    Arrays.sort(arr);
 
    // Calculate Common Difference
    int diff = (arr[n - 1] - arr[0]) / n;
 
    // Binary search for the missing
    return findMissing(arr, 0, n - 1, diff);
}
 
// Driver Code
public static void main(String[] args)
{
    // Given array arr[]
    int arr[] = new int[]{ 2, 8, 6, 10 };
    int n = arr.length;
 
    // Function Call
    System.out.println(missingElement(arr, n));
}
}
 
// This code is contributed by rock_cool

Python3

# Python3 program for the above approach
import sys
 
# Function to find the missing element
def findMissing(arr, left, right, diff):
 
    # Fix left and right boundary
    # for binary search
    if (right <= left):
        return sys.maxsize
 
    # Find index of middle element
    mid = left + (right - left) // 2
 
    # Check if the element just after
    # the middle element is missing
    if (arr[mid + 1] - arr[mid] != diff):
        return (arr[mid] + diff)
 
    # Check if the element just
    # before mid is missing
    if (mid > 0 and
        arr[mid] - arr[mid - 1] != diff):
        return (arr[mid - 1] + diff)
 
    # Check if the elements till mid
    # follow the AP, then recur
    # for right half
    if (arr[mid] == arr[0] + mid * diff):
        return findMissing(arr, mid + 1,
                           right, diff)
 
    # Else recur for left half
    return findMissing(arr, left,
                       mid - 1, diff)
 
# Function to find the missing
# element in AP series
def missingElement(arr, n):
 
    # Sort the array arr[]
    arr.sort()
 
    # Calculate Common Difference
    diff = (arr[n - 1] - arr[0]) // n
 
    # Binary search for the missing
    return findMissing(arr, 0, n - 1, diff)
 
# Driver Code
 
# Given array arr[]
arr = [ 2, 8, 6, 10 ]
n = len(arr)
 
# Function call
print(missingElement(arr, n))
 
# This code is contributed by sanjoy_62

C#

// C# program for the above approach
using System;
class GFG{
     
// Function to find the missing element
static int findMissing(int []arr, int left,
                       int right, int diff)
{
 
    // Fix left and right boundary
    // for binary search
    if (right <= left)
        return 0;
 
    // Find index of middle element
    int mid = left + (right - left) / 2;
 
    // Check if the element just after
    // the middle element is missing
    if (arr[mid + 1] - arr[mid] != diff)
        return (arr[mid] + diff);
 
    // Check if the element just
    // before mid is missing
    if (mid > 0 &&
        arr[mid] - arr[mid - 1] != diff)
        return (arr[mid - 1] + diff);
 
    // Check if the elements till mid
    // follow the AP, then recur
    // for right half
    if (arr[mid] == arr[0] + mid * diff)
        return findMissing(arr, mid + 1,
                           right, diff);
 
    // Else recur for left half
    return findMissing(arr, left,
                       mid - 1, diff);
}
 
// Function to find the missing
// element in AP series
static int missingElement(int []arr, int n)
{
 
    // Sort the array []arr
    Array.Sort(arr);
 
    // Calculate Common Difference
    int diff = (arr[n - 1] - arr[0]) / n;
 
    // Binary search for the missing
    return findMissing(arr, 0, n - 1, diff);
}
 
// Driver Code
public static void Main(String[] args)
{
    // Given array []arr
    int []arr = new int[]{ 2, 8, 6, 10 };
    int n = arr.Length;
 
    // Function Call
    Console.WriteLine(missingElement(arr, n));
}
}
 
// This code is contributed by Rohit_ranjan

Javascript

 
// Javascript program for the above approach
 
// Function to find the missing element
function findMissing(arr, left, right, diff)
{
     
    // Fix left and right boundary
    // for binary search
    if (right <= left)
        return 0;
 
    // Find index of middle element
    let mid = left + parseInt((right - left) / 2, 10);
 
    // Check if the element just after
    // the middle element is missing
    if (arr[mid + 1] - arr[mid] != diff)
        return (arr[mid] + diff);
 
    // Check if the element just
    // before mid is missing
    if (mid > 0 &&
        arr[mid] - arr[mid - 1] != diff)
        return (arr[mid - 1] + diff);
 
    // Check if the elements till mid
    // follow the AP, then recur
    // for right half
    if (arr[mid] == arr[0] + mid * diff)
        return findMissing(arr, mid + 1,
                           right, diff);
 
    // Else recur for left half
    return findMissing(arr, left,
                       mid - 1, diff);
}
 
// Function to find the missing
// element in AP series
function missingElement(arr, n)
{
 
    // Sort the array []arr
    arr.sort(function(a, b){return a - b});
 
    // Calculate Common Difference
    let diff = parseInt((arr[n - 1] -
                         arr[0]) / n, 10);
 
    // Binary search for the missing
    return findMissing(arr, 0, n - 1, diff);
}
 
// Driver code
 
// Given array []arr
let arr = [ 2, 8, 6, 10 ];
let n = arr.length;
 
// Function Call
document.write(missingElement(arr, n));
 
// This code is contributed by rameshtravel07
 

输出：

4

时间复杂度： O(N*log ₂ N)
有效的方法：为了优化上述方法，我们将使用 XOR 的以下属性，即a ^ a = 0 ，因此， (a ^ b ^ c) ^ (a ^ c) = b 。以下是步骤：

• 从给定数组中查找最大和最小元素。
• 找出共同的差异：

$Common Difference = \frac {(max Element - min Element)}{N}$

• 计算数组中所有元素的异或。
• 对实际系列的所有元素执行异或以找到结果值是缺失的元素。

下面是上述方法的实现：

C++

// C++ program for the above approach
#include 
using namespace std;
 
// Function to get the missing element
int missingElement(int arr[], int n)
{
    // For maximum Element in the array
    int max_ele = arr[0];
 
    // For minimum Element in the array
    int min_ele = arr[0];
 
    // For xor of all elements
    int x = 0;
 
    // Common difference of AP series
    int d;
 
    // find maximum and minimum element
    for (int i = 0; i < n; i++) {
        if (arr[i] > max_ele)
            max_ele = arr[i];
 
        if (arr[i] < min_ele)
            min_ele = arr[i];
    }
 
    // Calculating common difference
    d = (max_ele - min_ele) / n;
 
    // Calculate the XOR of all elements
    for (int i = 0; i < n; i++) {
        x = x ^ arr[i];
    }
 
    // Perform XOR with actual AP series
    // resultant x will be the ans
    for (int i = 0; i <= n; i++) {
        x = x ^ (min_ele + (i * d));
    }
 
    // Return the missing element
    return x;
}
 
// Driver Code
int main()
{
    // Given array
    int arr[] = { 12, 3, 6, 15, 18 };
    int n = sizeof(arr) / sizeof(arr[0]);
 
    // Function Call
    int element = missingElement(arr, n);
 
    // Print the missing element
    cout << element;
}

Java

// Java program for the above approach
class GFG{
     
// Function to get the missing element
static int missingElement(int arr[], int n)
{
     
    // For maximum Element in the array
    int max_ele = arr[0];
 
    // For minimum Element in the array
    int min_ele = arr[0];
 
    // For xor of all elements
    int x = 0;
 
    // Common difference of AP series
    int d;
 
    // find maximum and minimum element
    for(int i = 0; i < n; i++)
    {
       if (arr[i] > max_ele)
           max_ele = arr[i];
            
       if (arr[i] < min_ele)
           min_ele = arr[i];
    }
 
    // Calculating common difference
    d = (max_ele - min_ele) / n;
 
    // Calculate the XOR of all elements
    for(int i = 0; i < n; i++)
    {
       x = x ^ arr[i];
    }
 
    // Perform XOR with actual AP series
    // resultant x will be the ans
    for(int i = 0; i <= n; i++)
    {
       x = x ^ (min_ele + (i * d));
    }
 
    // Return the missing element
    return x;
}
 
// Driver code
public static void main(String[] args)
{
     
    // Given array
    int arr[] = new int[]{ 12, 3, 6, 15, 18 };
    int n = arr.length;
     
    // Function Call
    int element = missingElement(arr, n);
 
    // Print the missing element
    System.out.print(element);
}
}
 
// This code is contributed by Pratima Pandey

蟒蛇3

# Python3 program for the above approach
 
# Function to get the missing element
def missingElement(arr, n):
     
    # For maximum element in the array
    max_ele = arr[0]
 
    # For minimum Element in the array
    min_ele = arr[0]
 
    # For xor of all elements
    x = 0
 
    # Common difference of AP series
    d = 0
 
    # Find maximum and minimum element
    for i in range(n):
        if (arr[i] > max_ele):
            max_ele = arr[i]
 
        if (arr[i] < min_ele):
            min_ele = arr[i]
 
    # Calculating common difference
    d = (max_ele - min_ele) // n
 
    # Calculate the XOR of all elements
    for i in range(n):
        x = x ^ arr[i]
 
    # Perform XOR with actual AP series
    # resultant x will be the ans
    for i in range(n + 1):
        x = x ^ (min_ele + (i * d))
 
    # Return the missing element
    return x
 
# Driver Code
if __name__ == '__main__':
     
    # Given array
    arr = [ 12, 3, 6, 15, 18 ]
    n = len(arr)
 
    # Function Call
    element = missingElement(arr, n)
 
    # Print the missing element
    print(element)
 
# This code is contributed by mohit kumar 29

C＃

// C# program for the above approach
using System;
class GFG{
     
// Function to get the missing element
static int missingElement(int[] arr, int n)
{
     
    // For maximum Element in the array
    int max_ele = arr[0];
 
    // For minimum Element in the array
    int min_ele = arr[0];
 
    // For xor of all elements
    int x = 0;
 
    // Common difference of AP series
    int d;
 
    // find maximum and minimum element
    for(int i = 0; i < n; i++)
    {
       if (arr[i] > max_ele)
           max_ele = arr[i];
            
       if (arr[i] < min_ele)
           min_ele = arr[i];
    }
 
    // Calculating common difference
    d = (max_ele - min_ele) / n;
 
    // Calculate the XOR of all elements
    for(int i = 0; i < n; i++)
    {
       x = x ^ arr[i];
    }
 
    // Perform XOR with actual AP series
    // resultant x will be the ans
    for(int i = 0; i <= n; i++)
    {
       x = x ^ (min_ele + (i * d));
    }
 
    // Return the missing element
    return x;
}
 
// Driver code
public static void Main()
{
     
    // Given array
    int[] arr = new int[]{ 12, 3, 6, 15, 18 };
    int n = arr.Length;
     
    // Function Call
    int element = missingElement(arr, n);
 
    // Print the missing element
    Console.Write(element);
}
}
 
// This code is contributed by Ritik Bansal

Javascript

 
    // Javascript program for the above approach
     
    // Function to get the missing element
    function missingElement(arr, n)
    {
        // For maximum Element in the array
        let max_ele = arr[0];
 
        // For minimum Element in the array
        let min_ele = arr[0];
 
        // For xor of all elements
        let x = 0;
 
        // Common difference of AP series
        let d;
 
        // find maximum and minimum element
        for (let i = 0; i < n; i++) {
            if (arr[i] > max_ele)
                max_ele = arr[i];
 
            if (arr[i] < min_ele)
                min_ele = arr[i];
        }
 
        // Calculating common difference
        d = parseInt((max_ele - min_ele) / n, 10);
 
        // Calculate the XOR of all elements
        for (let i = 0; i < n; i++) {
            x = x ^ arr[i];
        }
 
        // Perform XOR with actual AP series
        // resultant x will be the ans
        for (let i = 0; i <= n; i++) {
            x = x ^ (min_ele + (i * d));
        }
 
        // Return the missing element
        return x;
    }
 
    // Given array
    let arr = [ 12, 3, 6, 15, 18 ];
    let n = arr.length;
   
    // Function Call
    let element = missingElement(arr, n);
   
    // Print the missing element
    document.write(element);
     

输出：

9

时间复杂度： O(N)
辅助空间： O(1)