Maximum difference between two elements such that larger element appears after the smaller number

2.7

Given an array arr[] of integers, find out the difference between any two elements such that larger element appears after the smaller number in arr[].

Examples: If array is [2, 3, 10, 6, 4, 8, 1] then returned value should be 8 (Diff between 10 and 2). If array is [ 7, 9, 5, 6, 3, 2 ] then returned value should be 2 (Diff between 7 and 9)

Method 1 (Simple)
Use two loops. In the outer loop, pick elements one by one and in the inner loop calculate the difference of the picked element with every other element in the array and compare the difference with the maximum difference calculated so far.

C

#include<stdio.h>

/* The function assumes that there are at least two
   elements in array.
   The function returns a negative value if the array is
   sorted in decreasing order. 
   Returns 0 if elements are equal */
int maxDiff(int arr[], int arr_size)
{     
  int max_diff = arr[1] - arr[0];
  int i, j;
  for (i = 0; i < arr_size; i++)
  {
    for (j = i+1; j < arr_size; j++)
    {        
      if (arr[j] - arr[i] > max_diff)   
         max_diff = arr[j] - arr[i];
    }    
  }          
  return max_diff;
}    

/* Driver program to test above function */
int main()
{
  int arr[] = {1, 2, 90, 10, 110};
  printf("Maximum difference is %d",  maxDiff(arr, 5));
  getchar();
  return 0;
}

Java

class MaximumDiffrence 
{
    /* The function assumes that there are at least two
       elements in array.
       The function returns a negative value if the array is
       sorted in decreasing order. 
       Returns 0 if elements are equal */
    int maxDiff(int arr[], int arr_size) 
    {
        int max_diff = arr[1] - arr[0];
        int i, j;
        for (i = 0; i < arr_size; i++) 
        {
            for (j = i + 1; j < arr_size; j++) 
            {
                if (arr[j] - arr[i] > max_diff)
                    max_diff = arr[j] - arr[i];
            }
        }
        return max_diff;
    }

    /* Driver program to test above functions */
    public static void main(String[] args) 
    {
        MaximumDiffrence maxdif = new MaximumDiffrence();
        int arr[] = {1, 2, 90, 10, 110};
        System.out.println("Maximum differnce is " + 
                                maxdif.maxDiff(arr, 5));
    }
}

// This code has been contributed by Mayank Jaiswal

Python3

# Python 3 code to find Maximum difference
# between two elements such that larger 
# element appears after the smaller number

# The function assumes that there are at 
# least two elements in array. The function 
# returns a negative value if the array is
# sorted in decreasing order. Returns 0 
# if elements are equal
def maxDiff(arr, arr_size):
    max_diff = arr[1] - arr[0]
    
    for i in range( 0, arr_size ):
        for j in range( i+1, arr_size ):
            if(arr[j] - arr[i] > max_diff): 
                max_diff = arr[j] - arr[i]
    
    return max_diff
    
# Driver program to test above function 
arr = [1, 2, 90, 10, 110]
size = len(arr)
print ("Maximum difference is", maxDiff(arr, size))

# This code is contributed by Swetank Modi


Time Complexity: O(n^2)
Auxiliary Space: O(1)

Method 2 (Tricky and Efficient)
In this method, instead of taking difference of the picked element with every other element, we take the difference with the minimum element found so far. So we need to keep track of 2 things:
1) Maximum difference found so far (max_diff).
2) Minimum number visited so far (min_element).

C

#include<stdio.h>

/* The function assumes that there are at least two
   elements in array.
   The function returns a negative value if the array is
   sorted in decreasing order.
   Returns 0 if elements are equal  */
int maxDiff(int arr[], int arr_size)
{
  int max_diff = arr[1] - arr[0];
  int min_element = arr[0];
  int i;
  for(i = 1; i < arr_size; i++)
  {       
    if (arr[i] - min_element > max_diff)                               
      max_diff = arr[i] - min_element;
    if (arr[i] < min_element)
         min_element = arr[i];                     
  }
  return max_diff;
}    

/* Driver program to test above function */
int main()
{
  int arr[] = {1, 2, 6, 80, 100};
  int size = sizeof(arr)/sizeof(arr[0]);
  printf("Maximum difference is %d",  maxDiff(arr, size));
  getchar();
  return 0;
}

Java

class MaximumDiffrence 
{
    /* The function assumes that there are at least two
       elements in array.
       The function returns a negative value if the array is
       sorted in decreasing order.
       Returns 0 if elements are equal  */
    int maxDiff(int arr[], int arr_size) 
    {
        int max_diff = arr[1] - arr[0];
        int min_element = arr[0];
        int i;
        for (i = 1; i < arr_size; i++) 
        {
            if (arr[i] - min_element > max_diff)
                max_diff = arr[i] - min_element;
            if (arr[i] < min_element)
                min_element = arr[i];
        }
        return max_diff;
    }

    /* Driver program to test above functions */
    public static void main(String[] args) 
    {
        MaximumDiffrence maxdif = new MaximumDiffrence();
        int arr[] = {1, 2, 90, 10, 110};
        int size = arr.length;
        System.out.println("MaximumDifference is " + 
                                maxdif.maxDiff(arr, size));
    }
}

// This code has been contributed by Mayank Jaiswal

Python3

# Python 3 code to find Maximum difference
# between two elements such that larger 
# element appears after the smaller number

# The function assumes that there are 
# at least two elements in array.
# The function returns a negative 
# value if the array is sorted in 
# decreasing order. Returns 0 if 
# elements are equal
def maxDiff(arr, arr_size):
    max_diff = arr[1] - arr[0]
    min_element = arr[0]
    
    for i in range( 1, arr_size ):
        if (arr[i] - min_element > max_diff):
            max_diff = arr[i] - min_element
    
        if (arr[i] < min_element):
            min_element = arr[i]
    return max_diff
    
# Driver program to test above function 
arr = [1, 2, 6, 80, 100]
size = len(arr)
print ("Maximum difference is", 
        maxDiff(arr, size))

# This code is contributed by Swetank Modi

Time Complexity: O(n)
Auxiliary Space: O(1)

Like min element, we can also keep track of max element from right side. See below code suggested by Katamaran

int maxDiff(int arr[], int n)
{
    int maxDiff = -1; // Initialize Result

    int maxRight = arr[n-1]; // Initialize max element from right side

    for (int i = n-2; i >= 0; i--)
    {
        if (arr[i] > maxRight)
            maxRight = arr[i];
        else
        {
            int diff = maxRight - arr[i];
            if (diff > maxDiff)
            {
                maxDiff = diff;
            }
        }
    }
    return maxDiff;
}

Method 3 (Another Tricky Solution)
First find the difference between the adjacent elements of the array and store all differences in an auxiliary array diff[] of size n-1. Now this problems turns into finding the maximum sum subarray of this difference array.
Thanks to Shubham Mittal for suggesting this solution.

C

#include<stdio.h>

int maxDiff(int arr[], int n)
{
    // Create a diff array of size n-1. The array will hold
    //  the difference of adjacent elements
    int diff[n-1];
    for (int i=0; i < n-1; i++)
        diff[i] = arr[i+1] - arr[i];

    // Now find the maximum sum subarray in diff array
    int max_diff = diff[0];
    for (int i=1; i<n-1; i++)
    {
        if (diff[i-1] > 0)
            diff[i] += diff[i-1];
        if (max_diff < diff[i])
            max_diff = diff[i];
    }
    return max_diff;
}

/* Driver program to test above function */
int main()
{
    int arr[] = {80, 2, 6, 3, 100};
    int size = sizeof(arr)/sizeof(arr[0]);
    printf("Maximum difference is %d",  maxDiff(arr, size));
    return 0;
}

Java


class MaximumDiffrence 
{
    int maxDiff(int arr[], int n) 
    {
        // Create a diff array of size n-1. The array will hold
        //  the difference of adjacent elements
        int diff[] = new int[n - 1];
        for (int i = 0; i < n - 1; i++)
            diff[i] = arr[i + 1] - arr[i];

        // Now find the maximum sum subarray in diff array
        int max_diff = diff[0];
        for (int i = 1; i < n - 1; i++) 
        {
            if (diff[i - 1] > 0) 
                diff[i] += diff[i - 1];
            if (max_diff < diff[i])
                max_diff = diff[i];
        }
        return max_diff;
    }

    // Driver program to test above functions
    public static void main(String[] args) 
    {
        MaximumDiffrence mxdif = new MaximumDiffrence();
        int arr[] = {80, 2, 6, 3, 100};
        int size = arr.length;
        System.out.println(mxdif.maxDiff(arr, size));
    }
}
// This code has been contributed by Mayank Jaiswal

Python3

# Python 3 code to find Maximum difference
# between two elements such that larger 
# element appears after the smaller number

def maxDiff(arr, n):
    diff = [0] * (n - 1)
    for i in range (0, n-1):
        diff[i] = arr[i+1] - arr[i]
        
    # Now find the maximum sum 
    # subarray in diff array    
    max_diff = diff[0]
    for i in range(1, n-1):
        if (diff[i-1] > 0):
            diff[i] += diff[i-1]
    
        if (max_diff < diff[i]):
            max_diff = diff[i]
    
    return max_diff

# Driver program to test above function 
arr = [80, 2, 6, 3, 100]
size = len(arr)
print ("Maximum difference is", 
       maxDiff(arr, size))

# This code is contributed by Swetank Modi


Output:
98

This method is also O(n) time complexity solution, but it requires O(n) extra space

Time Complexity: O(n)
Auxiliary Space: O(n)

We can modify the above method to work in O(1) extra space. Instead of creating an auxiliary array, we can calculate diff and max sum in same loop. Following is the space optimized version.

int maxDiff (int arr[], int n)
{
    // Initialize diff, current sum and max sum
    int diff = arr[1]-arr[0];
    int curr_sum = diff;
    int max_sum = curr_sum;

    for(int i=1; i<n-1; i++)
    {
        // Calculate current diff
        diff = arr[i+1]-arr[i];

        // Calculate current sum
        if (curr_sum > 0)
           curr_sum += diff;
        else
           curr_sum = diff;

        // Update max sum, if needed
        if (curr_sum > max_sum)
           max_sum = curr_sum;
    }

    return max_sum;
}

Time Complexity: O(n)
Auxiliary Space: O(1)

Below is a variation of this problem:
Maximum difference of sum of elements in two rows in a matrix

Please write comments if you find any bug in above codes/algorithms, or find other ways to solve the same problem

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