Maximum Sum Path in Two Arrays

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Given two sorted arrays having some elements in common. Find the sum of the maximum sum path to reach from the beginning of any array to the end of any of the two arrays. We can switch from one array to another array only at common elements. 

Note: The common elements do not have to be at the same indexes. And individual arrays have distinct elements only (no repetition within the array).

Examples: 

Input: ar1[] = {2, 3, 7, 10, 12}, ar2[] = {1, 5, 7, 8}
Output: 35
Explanation: 35 is sum of 1 + 5 + 7 + 10 + 12.
Start from the first element of ar2 which is 1, then move to 5, then 7.  From 7 switch to ar1 (as 7 is common) and traverse 10 and 12.

Input: ar1[] = {10, 12}, ar2 = {5, 7, 9}
Output: 22
Explanation: 22 is the sum of 10 and 12. 
Since there is no common element, take all elements from the array with more sum.

Input: ar1[] = {2, 3, 7, 10, 12, 15, 30, 34}, ar2[] = {1, 5, 7, 8, 10, 15, 16, 19}
Output: 122
Explanation: 122 is sum of 1, 5, 7, 8, 10, 12, 15, 30, 34
Start from the first element of ar2 which is 1, then move to 5, then 7. From 7 switch to ar1 (as 7 is common), then traverse the remaining ar1.

Maximum Sum Path in Two Arrays using Merge: This approach is based on the following idea:

The idea is to do something similar to the merge process of merge sort. This involves calculating the sum of elements between all common points of both arrays. Whenever there is a common point, compare the two sums and add the maximum of two to the result.

Illustration:

Input: arr1[] = {2, 3, 7, 10, 12}

           arr2[] = {1, 5, 7, 8}

          Initialise i = 0, j = 0, sum1 = 0, sum2 = 0, result = 0

Step - 1: arr1[i] > arr2[j] 
sum2 = sum2 + arr2[j] = 0 + 1 = 1
j = j + 1 = 1

Step - 2: arr1[i] < arr2[j] 
sum1 = sum1 + arr1[i] = 0 + 2 = 2
i = i + 1 = 1

Step - 3: arr1[i] < arr2[j] 
sum1 = sum1 + arr1[i] = 2 + 3 = 5
i = i + 1 = 2

Step - 4: arr1[i] > arr2[j] 
sum2 = sum2 + arr2[j] = 1 + 5 = 6
j = j + 1 = 2

Step - 5: arr1[i] == arr2[j] 
result = result + maximum(sum1, sum2) + arr1[i] = 0 + max(5, 6) + 7 = 13
sum1 = 0, sum2 = 0
i = i + 1 = 3
j = j + 1 = 3

Step - 6: arr1[i] > arr2[j] 
sum2 = sum2 + arr2[j] = 0 + 8 = 8
j = j + 1 = 4

Step - 7: sum1 = sum1 + arr1[i] = 0 + 10 = 10
i = i + 1 = 4

Step - 8: sum1 = sum1 + arr1[i] = 10 + 12 = 22
i = i + 1 = 5

Step - 9: result = result + max(sum1, sum2) = 13 + max(10, 22) = 35 

Hence, maximum sum path is 35.

Follow the steps below to solve the given problem:

• Initialize variables, result, sum1, sum2. Initialize result as 0. Also initialize two variables sum1 and sum2 as 0. Here sum1 and sum2 are used to store sum of element in ar1[] and ar2[] respectively. These sums are between two common points.
• Now run a loop to traverse elements of both arrays. While traversing compare current elements of array 1 and array 2 in the following order.
  
  • If current element of array 1 is smaller than current element of array 2, then update sum1, else if current element of array 2 is smaller, then update sum2.
  • If the current element of array 1 and array 2are same, then take the maximum of sum1 and sum2 and add it to the result. Also add the common element to the result.
  • This step can be compared to the merging of two sorted arrays, If the smallest element of the two current array indices is processed then it is guaranteed that if there is any common element it will be processed together. So the sum of elements between two common elements can be processed.

Below is the implementation of the above code: 

// C++ program to find maximum sum path
#include <iostream>
using namespace std;

// Utility function to find maximum of two integers
int max(int x, int y) { return (x > y) ? x : y; }

// This function returns the sum of elements on maximum path
// from beginning to end
int maxPathSum(int ar1[], int ar2[], int m, int n)
{
    // initialize indexes for ar1[] and ar2[]
    int i = 0, j = 0;

    // Initialize result and current sum through ar1[] and
    // ar2[].
    int result = 0, sum1 = 0, sum2 = 0;

    // Below 3 loops are similar to merge in merge sort
    while (i < m && j < n) {
        // Add elements of ar1[] to sum1
        if (ar1[i] < ar2[j])
            sum1 += ar1[i++];

        // Add elements of ar2[] to sum2
        else if (ar1[i] > ar2[j])
            sum2 += ar2[j++];

        else // we reached a common point
        {
            // Take the maximum of two sums and add to
            // result
            // Also add the common element of array, once
            result += max(sum1, sum2) + ar1[i];

            // Update sum1 and sum2 for elements after this
            // intersection point
            sum1 = 0;
            sum2 = 0;

            // update i and j to move to next element of
            // each array
            i++;
            j++;
        }
    }

    // Add remaining elements of ar1[]
    while (i < m)
        sum1 += ar1[i++];

    // Add remaining elements of ar2[]
    while (j < n)
        sum2 += ar2[j++];

    // Add maximum of two sums of remaining elements
    result += max(sum1, sum2);

    return result;
}

// Driver code
int main()
{
    int ar1[] = { 2, 3, 7, 10, 12, 15, 30, 34 };
    int ar2[] = { 1, 5, 7, 8, 10, 15, 16, 19 };
    int m = sizeof(ar1) / sizeof(ar1[0]);
    int n = sizeof(ar2) / sizeof(ar2[0]);

    // Function call
    cout << "Maximum sum path is "
         << maxPathSum(ar1, ar2, m, n);
    return 0;
}

// JAVA program to find maximum sum path
class MaximumSumPath {
    // Utility function to find maximum of two integers
    int max(int x, int y) { return (x > y) ? x : y; }

    // This function returns the sum of elements on maximum
    // path from beginning to end
    int maxPathSum(int ar1[], int ar2[], int m, int n)
    {
        // initialize indexes for ar1[] and ar2[]
        int i = 0, j = 0;

        // Initialize result and current sum through ar1[]
        // and ar2[].
        int result = 0, sum1 = 0, sum2 = 0;

        // Below 3 loops are similar to merge in merge sort
        while (i < m && j < n) {
            // Add elements of ar1[] to sum1
            if (ar1[i] < ar2[j])
                sum1 += ar1[i++];

            // Add elements of ar2[] to sum2
            else if (ar1[i] > ar2[j])
                sum2 += ar2[j++];

            // we reached a common point
            else {
                // Take the maximum of two sums and add to
                // result
                // Also add the common element of array,
                // once
                result += max(sum1, sum2) + ar1[i];

                // Update sum1 and sum2 for elements after
                // this intersection point
                sum1 = 0;
                sum2 = 0;

                // update i and j to move to next element of
                // each array
                i++;
                j++;
            }
        }

        // Add remaining elements of ar1[]
        while (i < m)
            sum1 += ar1[i++];

        // Add remaining elements of ar2[]
        while (j < n)
            sum2 += ar2[j++];

        // Add maximum of two sums of remaining elements
        result += max(sum1, sum2);

        return result;
    }

    // Driver code
    public static void main(String[] args)
    {
        MaximumSumPath sumpath = new MaximumSumPath();
        int ar1[] = { 2, 3, 7, 10, 12, 15, 30, 34 };
        int ar2[] = { 1, 5, 7, 8, 10, 15, 16, 19 };
        int m = ar1.length;
        int n = ar2.length;

        // Function call
        System.out.println(
            "Maximum sum path is :"
            + sumpath.maxPathSum(ar1, ar2, m, n));
    }
}

// This code has been contributed by Mayank Jaiswal

# Python program to find maximum sum path

# This function returns the sum of elements on maximum path from
# beginning to end


def maxPathSum(ar1, ar2, m, n):

    # initialize indexes for ar1[] and ar2[]
    i, j = 0, 0

    # Initialize result and current sum through ar1[] and ar2[]
    result, sum1, sum2 = 0, 0, 0

    # Below 3 loops are similar to merge in merge sort
    while (i < m and j < n):

        # Add elements of ar1[] to sum1
        if ar1[i] < ar2[j]:
            sum1 += ar1[i]
            i += 1

        # Add elements of ar2[] to sum2
        elif ar1[i] > ar2[j]:
            sum2 += ar2[j]
            j += 1

        else:   # we reached a common point

            # Take the maximum of two sums and add to result
            result += max(sum1, sum2) + ar1[i]
            # update sum1 and sum2 to be considered fresh for next elements
            sum1 = 0
            sum2 = 0
            # update i and j to move to next element in each array
            i += 1
            j += 1

    # Add remaining elements of ar1[]
    while i < m:
        sum1 += ar1[i]
        i += 1
    # Add remaining elements of b[]
    while j < n:
        sum2 += ar2[j]
        j += 1

    # Add maximum of two sums of remaining elements
    result += max(sum1, sum2)

    return result


# Driver code
ar1 = [2, 3, 7, 10, 12, 15, 30, 34]
ar2 = [1, 5, 7, 8, 10, 15, 16, 19]
m = len(ar1)
n = len(ar2)

# Function call
print("Maximum sum path is", maxPathSum(ar1, ar2, m, n))

# This code is contributed by __Devesh Agrawal__

// C# program for Maximum Sum Path in
// Two Arrays
using System;

class GFG {

    // Utility function to find maximum
    // of two integers
    static int max(int x, int y) { return (x > y) ? x : y; }

    // This function returns the sum of
    // elements on maximum path from
    // beginning to end
    static int maxPathSum(int[] ar1, int[] ar2, int m,
                          int n)
    {

        // initialize indexes for ar1[]
        // and ar2[]
        int i = 0, j = 0;

        // Initialize result and current
        // sum through ar1[] and ar2[].
        int result = 0, sum1 = 0, sum2 = 0;

        // Below 3 loops are similar to
        // merge in merge sort
        while (i < m && j < n) {
            // Add elements of ar1[] to sum1
            if (ar1[i] < ar2[j])
                sum1 += ar1[i++];

            // Add elements of ar2[] to sum2
            else if (ar1[i] > ar2[j])
                sum2 += ar2[j++];

            // we reached a common point
            else {

                // Take the maximum of two sums and add to
                // result
                // Also add the common element of array,
                // once
                result += max(sum1, sum2) + ar1[i];

                // Update sum1 and sum2 for elements after
                // this intersection point
                sum1 = 0;
                sum2 = 0;

                // update i and j to move to next element of
                // each array
                i++;
                j++;
            }
        }

        // Add remaining elements of ar1[]
        while (i < m)
            sum1 += ar1[i++];

        // Add remaining elements of ar2[]
        while (j < n)
            sum2 += ar2[j++];

        // Add maximum of two sums of
        // remaining elements
        result += max(sum1, sum2);

        return result;
    }

    // Driver code
    public static void Main()
    {
        int[] ar1 = { 2, 3, 7, 10, 12, 15, 30, 34 };
        int[] ar2 = { 1, 5, 7, 8, 10, 15, 16, 19 };
        int m = ar1.Length;
        int n = ar2.Length;

        // Function call
        Console.Write("Maximum sum path is :"
                      + maxPathSum(ar1, ar2, m, n));
    }
}

// This code is contributed by nitin mittal.

<script>

// Javascript program to find maximum sum path

// Utility function to find maximum of two integers
function max(x, y) 
{ 
    return (x > y) ? x : y; 
}

// This function returns the sum of elements
// on maximum path from beginning to end
function maxPathSum(ar1, ar2, m, n)
{
    
    // Initialize indexes for ar1[] and ar2[]
    let i = 0, j = 0;

    // Initialize result and current sum
    // through ar1[] and ar2[].
    let result = 0, sum1 = 0, sum2 = 0;

    // Below3 loops are similar to 
    // merge in merge sort
    while (i < m && j < n)
    {
        
        // Add elements of ar1[] to sum1
        if (ar1[i] < ar2[j])
            sum1 += ar1[i++];

        // Add elements of ar2[] to sum2
        else if (ar1[i] > ar2[j])
            sum2 += ar2[j++];
            
        // We reached a common point
        else 
        {
            // Take the maximum of two sums and add to
            // result
              //Also add the common element of array, once
            result += Math.max(sum1, sum2) + ar1[i];

            // Update sum1 and sum2 for elements after this
            // intersection point
            sum1 = 0;
              sum2 = 0;
          
              //update i and j to move to next element of each array
              i++;
              j++;
            

        }
    }

    // Add remaining elements of ar1[]
    while (i < m)
        sum1 += ar1[i++];

    // Add remaining elements of ar2[]
    while (j < n)
        sum2 += ar2[j++];

    // Add maximum of two sums of
    // remaining elements
    result += Math.max(sum1, sum2);

    return result;
}

// Driver code
let ar1 = [ 2, 3, 7, 10, 12, 15, 30, 34 ];
let ar2 = [ 1, 5, 7, 8, 10, 15, 16, 19 ];
let m = ar1.length;
let n = ar2.length;

// Function call
document.write("Maximum sum path is " + 
   maxPathSum(ar1, ar2, m, n));
    
// This code is contributed by Mayank Tyagi

</script>

<?php
// PHP Program to find Maximum Sum 
// Path in Two Arrays

// This function returns the sum of
// elements on maximum path
// from beginning to end
function maxPathSum($ar1, $ar2, $m, $n)
{
    
    // initialize indexes for 
    // ar1[] and ar2[]
    $i = 0; 
    $j = 0;

    // Initialize result and 
    // current sum through ar1[] 
    // and ar2[].
    $result = 0;
    $sum1 = 0; 
    $sum2 = 0;

    // Below 3 loops are similar
    // to merge in merge sort
    while ($i < $m and $j < $n)
    {
        
        // Add elements of
        // ar1[] to sum1
        if ($ar1[$i] < $ar2[$j])
            $sum1 += $ar1[$i++];

        // Add elements of 
        // ar2[] to sum2
        else if ($ar1[$i] > $ar2[$j])
            $sum2 += $ar2[$j++];

        // we reached a 
        // common point
        else 
        {
            
            // Take the maximum of two sums and add to
            // result
              //Also add the common element of array, once
            $result += max($sum1, $sum2) + $ar1[$i];

            // Update sum1 and sum2 for elements after this
            // intersection point
            $sum1 = 0; 
            $sum2 = 0;
          
              //update i and j to move to next element of each array
            $i++;
            $j++;
          

        }
    }

    // Add remaining elements of ar1[]
    while ($i < $m)
        $sum1 += $ar1[$i++];

    // Add remaining elements of ar2[]
    while ($j < $n)
        $sum2 += $ar2[$j++];

    // Add maximum of two sums 
    // of remaining elements
    $result += max($sum1, $sum2);

    return $result;
}

    // Driver Code
    $ar1 = array(2, 3, 7, 10, 12, 15, 30, 34);
    $ar2 = array(1, 5, 7, 8, 10, 15, 16, 19);
    $m = count($ar1);
    $n = count($ar2);

    // Function call
    echo "Maximum sum path is "
        , maxPathSum($ar1, $ar2, $m, $n);
        
// This code is contributed by anuj_67.
?>

Maximum sum path is 122

Time complexity: O(m+n). In every iteration of while loops, an element from either of the two arrays is processed. There are total m + n elements. Therefore, the time complexity is O(m+n).
Auxiliary Space: O(1), Any extra space is not required, so the space complexity is constant.