Find the smallest positive number missing from an unsorted array

You are given an unsorted array with both positive and negative elements. You have to find the smallest positive number missing from the array in O(n) time using constant extra space. You can modify the original array.

Examples

 Input:  {2, 3, 7, 6, 8, -1, -10, 15}
 Output: 1

 Input:  { 2, 3, -7, 6, 8, 1, -10, 15 }
 Output: 4

 Input: {1, 1, 0, -1, -2}
 Output: 2

Recommended: Please solve it on “PRACTICE” first, before moving on to the solution.

A naive method to solve this problem is to search all positive integers, starting from 1 in the given array. We may have to search at most n+1 numbers in the given array. So this solution takes O(n^2) in worst case.

We can use sorting to solve it in lesser time complexity. We can sort the array in O(nLogn) time. Once the array is sorted, then all we need to do is a linear scan of the array. So this approach takes O(nLogn + n) time which is O(nLogn).

We can also use hashing. We can build a hash table of all positive elements in the given array. Once the hash table is built. We can look in the hash table for all positive integers, starting from 1. As soon as we find a number which is not there in hash table, we return it. This approach may take O(n) time on average, but it requires O(n) extra space.

A O(n) time and O(1) extra space solution:
The idea is similar to this post. We use array elements as index. To mark presence of an element x, we change the value at the index x to negative. But this approach doesn’t work if there are non-positive (-ve and 0) numbers. So we segregate positive from negative numbers as first step and then apply the approach.

Following is the two step algorithm.
1) Segregate positive numbers from others i.e., move all non-positive numbers to left side. In the following code, segregate() function does this part.
2) Now we can ignore non-positive elements and consider only the part of array which contains all positive elements. We traverse the array containing all positive numbers and to mark presence of an element x, we change the sign of value at index x to negative. We traverse the array again and print the first index which has positive value. In the following code, findMissingPositive() function does this part. Note that in findMissingPositive, we have subtracted 1 from the values as indexes start from 0 in C.

C

/* C program to find the smallest positive missing number */
#include <stdio.h> 
#include <stdlib.h> 
  
/* Utility to swap to integers */
void swap(int *a, int *b) 
{ 
    int temp; 
    temp = *a; 
    *a = *b; 
    *b = temp; 
} 
  
/* Utility function that puts all 
non-positive (0 and negative) numbers on left  
side of arr[] and return count of such numbers */
int segregate (int arr[], int size) 
{ 
    int j = 0, i; 
    for(i = 0; i < size; i++) 
    { 
    if (arr[i] <= 0)  
    { 
        swap(&arr[i], &arr[j]); 
        j++; // increment count of non-positive integers 
    } 
    } 
  
    return j; 
} 
  
/* Find the smallest positive missing number  
in an array that contains all positive integers */
int findMissingPositive(int arr[], int size) 
{ 
int i; 
  
// Mark arr[i] as visited by making arr[arr[i] - 1] negative.  
// Note that 1 is subtracted because index start  
// from 0 and positive numbers start from 1 
for(i = 0; i < size; i++) 
{ 
    if(abs(arr[i]) - 1 < size && arr[abs(arr[i]) - 1] > 0) 
    arr[abs(arr[i]) - 1] = -arr[abs(arr[i]) - 1]; 
} 
  
// Return the first index value at which is positive 
for(i = 0; i < size; i++) 
    if (arr[i] > 0) 
    // 1 is added becuase indexes start from 0 
    return i+1;  
  
return size+1; 
} 
  
/* Find the smallest positive missing  
number in an array that contains 
both positive and negative integers */
int findMissing(int arr[], int size) 
{ 
// First separate positive and negative numbers 
int shift = segregate (arr, size); 
  
// Shift the array and call findMissingPositive for 
// positive part 
return findMissingPositive(arr+shift, size-shift); 
} 
  
int main() 
{ 
int arr[] = {0, 10, 2, -10, -20}; 
int arr_size = sizeof(arr)/sizeof(arr[0]); 
int missing = findMissing(arr, arr_size); 
printf("The smallest positive missing number is %d ", missing); 
getchar(); 
return 0; 
} 

Java

// Java program to find the smallest 
// positive missing number 
import java.util.*; 
  
class Main 
{    
       
    /* Utility function that puts all non-positive  
       (0 and negative) numbers on left side of  
       arr[] and return count of such numbers */
    static int segregate (int arr[], int size) 
    { 
        int j = 0, i; 
        for(i = 0; i < size; i++) 
        { 
           if (arr[i] <= 0)   
           { 
               int temp; 
               temp = arr[i]; 
               arr[i] = arr[j]; 
               arr[j] = temp; 
               // increment count of non-positive  
               // integers 
               j++;   
           } 
        } 
       
        return j; 
    } 
       
    /* Find the smallest positive missing  
       number in an array that contains 
       all positive integers */
    static int findMissingPositive(int arr[], int size) 
    { 
      int i; 
       
      // Mark arr[i] as visited by making  
      // arr[arr[i] - 1] negative. Note that  
      // 1 is subtracted because index start  
      // from 0 and positive numbers start from 1 
      for(i = 0; i < size; i++) 
      { 
        int x = Math.abs(arr[i]); 
        if(x - 1 < size && arr[x - 1] > 0) 
          arr[x - 1] = -arr[x - 1]; 
      } 
       
      // Return the first index value at which  
      // is positive 
      for(i = 0; i < size; i++) 
        if (arr[i] > 0) 
          return i+1;  // 1 is added becuase indexes  
                       // start from 0 
       
      return size+1; 
    } 
       
    /* Find the smallest positive missing  
       number in an array that contains 
       both positive and negative integers */
    static int findMissing(int arr[], int size) 
    { 
       // First separate positive and  
       // negative numbers 
       int shift = segregate (arr, size); 
       int arr2[] = new int[size-shift]; 
       int j=0; 
       for(int i=shift;i<size;i++) 
           { 
               arr2[j] = arr[i]; 
               j++; 
           }     
       // Shift the array and call  
       // findMissingPositive for 
       // positive part 
       return findMissingPositive(arr2, j); 
    } 
    // main function 
    public static void main (String[] args)  
    { 
        int arr[] = {0, 10, 2, -10, -20}; 
        int arr_size = arr.length; 
        int missing = findMissing(arr, arr_size); 
        System.out.println("The smallest positive missing number is "+  
                            missing); 
    } 
} 

C#

// C# program to find the smallest 
// positive missing number 
using System; 
  
class main 
{    
        
    // Utility function that puts all 
    // non-positive (0 and negative) 
    // numbers on left side of arr[] 
    // and return count of such numbers 
    static int segregate (int []arr, int size) 
    { 
        int j = 0, i; 
        for(i = 0; i < size; i++) 
        { 
           if (arr[i] <= 0)   
           { 
               int temp; 
               temp = arr[i]; 
               arr[i] = arr[j]; 
               arr[j] = temp; 
                 
               // increment count of non-positive  
               // integers 
               j++;   
           } 
        } 
        
        return j; 
    } 
        
    // Find the smallest positive missing  
    // number in an array that contains 
    // all positive integers 
    static int findMissingPositive(int []arr, int size) 
    { 
      int i; 
        
      // Mark arr[i] as visited by making  
      // arr[arr[i] - 1] negative. Note that  
      // 1 is subtracted as index start from 
      // 0 and positive numbers start from 1 
      for(i = 0; i < size; i++) 
      { 
        if(Math.Abs(arr[i]) - 1 < size &&  
            arr[Math.Abs(arr[i]) - 1] > 0) 
          arr[Math.Abs(arr[i]) - 1] = -arr[Math.Abs(arr[i]) - 1]; 
      } 
        
      // Return the first index value at  
      // which is positive 
      for(i = 0; i < size; i++) 
        if (arr[i] > 0) 
        return i+1; 
          
        // 1 is added becuase indexes  
        // start from 0 
        return size+1; 
    } 
        
    // Find the smallest positive  
    // missing number in array that 
    // contains both positive and 
    // negative integers 
    static int findMissing(int []arr, int size) { 
      
       // First separate positive and  
       // negative numbers 
       int shift = segregate (arr, size); 
       int []arr2 = new int[size-shift]; 
       int j=0; 
         
       for(int i=shift;i<size;i++) 
           { 
               arr2[j] = arr[i]; 
               j++; 
           }   
             
       // Shift the array and call  
       // findMissingPositive for 
       // positive part 
       return findMissingPositive(arr2, j); 
    } 
      
    // main function 
    public static void Main ()  
    { 
        int []arr = {0, 10, 2, -10, -20}; 
        int arr_size = arr.Length; 
        int missing = findMissing(arr, arr_size); 
        Console.WriteLine("The smallest positive missing number is "+  
                            missing); 
    } 
} 
  
//This code is contributed by Anant Agarwal. 

Output:

The smallest positive missing number is 1

Note that this method modifies the original array. We can change the sign of elements in the segregated array to get the same set of elements back. But we still loose the order of elements. If we want to keep the original array as it was, then we can create a copy of the array and run this approach on the temp array.

Another approach:In this problem, we have created a list full of 0’s with size of the max() value of our given array. Now, whenever we encounter any positive value in our original array, we change the index value of our list to 1. So, after we are done, we simply iterate through our modified list, the first 0 we encounter, its (index value + 1) should be our answer since index in python starts from 0.

Below is the implementation of above approach:

Python 3

# Python Program to find the smallest 
# positive missing number 
  
def solution(A):#Our original array 
  
    m = max(A) #Storing maximum value 
    if m < 1: 
          
        # In case all values in our array are negative 
        return 1 
    if len(A) == 1: 
          
        #If it contains only one element 
        return 2 if A[0] == 1 else 1     
    l = [0] * m 
    for i in range(len(A)): 
        if A[i] > 0: 
            if l[A[i] - 1] != 1: 
                  
                #Changing the value status at the index of our list 
                l[A[i] - 1] = 1 
    for i in range(len(l)): 
          
        #Encountering first 0, i.e, the element with least value 
        if l[i] == 0:  
            return i+1
            #In case all values are filled between 1 and m 
    return i+2     
  
A = [0, 10, 2, -10, -20] 
print(solution(A)) 

PHP

<?php  
// PHP Program to find the smallest 
// positive missing number 
   
function solution($A){//Our original array 
   
    $m = max($A); //Storing maximum value 
    if ($m < 1) 
    {          
        // In case all values in our array are negative 
        return 1; 
    } 
    if (sizeof($A) == 1) 
    {   
        //If it contains only one element 
        if ($A[0] == 1) 
            return 2 ; 
        else 
            return 1 ; 
    }         
    $l = array_fill(0,$m,NULL); 
    for($i = 0; $i < sizeof($A); $i++) 
    {         
        if( $A[$i] > 0) 
        { 
            if ($l[$A[$i] - 1] != 1) 
            { 
                  
                //Changing the value status at the index of our list 
                $l[$A[$i] - 1] = 1; 
            } 
        } 
    } 
    for ($i = 0;$i < sizeof($l); $i++) 
    { 
           
        //Encountering first 0, i.e, the element with least value 
        if ($l[$i] == 0)  
            return $i+1; 
    } 
            //In case all values are filled between 1 and m 
    return $i+2;     
} 
  
$A = array(0, 10, 2, -10, -20); 
echo solution($A); 
return 0; 
?> 