Sum of all possible expressions of a numeric string possible by inserting addition operators

Given a numeric string str of length N, the task is to find the sum of all possible expressions by inserting the ‘+’ operator between the characters of the string any number of times.

Examples:

Input: str = “125” Output: 176 Explanation: Inserting “+” after 1st index modifies str to “1+25” and value = 26 Inserting “+” after 2nd index modifies str to “12+5” and value = 17 Inserting “+” after both 1st and 2nd index modifies str to “1+2+5” and value = 8 Therefore, the total sum of all possible expression is 125 + 26 + 17 + 8 = 176

Input: str = “9999999999”
Output: 12656242944

Approach: The idea is to insert the ‘+’ operator at all possible index of the string in all possible ways and calculate the sum. Finally, print the total sum obtained. Follow the steps below to solve the problem:

• Initialize a variable say, sumOfExp to store the sum of all possible expression by inserting the ‘+’ operator at all possible indices of the string.
• Generate all possible subset of indices of the string iteratively. For every subset of indices inserts the ‘+’ operator at elements of the subset and increment sumOfExp by the sum of the current expression.
• Finally, print the value of sumOfExp.

Below is the implementation of the above approach:

Python3

# Python program to implement
# the above approach

# Function to find sum of all expressions by
# inserting '+' operator at all possible indices
def findSumOfExpressions(S, N):

    # Stores sum of all expressions by inserting
    # '+' operator at all possible indices
    sumOfExp = 0

    # Generate all possible subset
    # of indices iteratively
    for i in range(2 ** (N - 1)):

        # Stores sum of
        # current expressions
        ans_sub = 0

        # Stores numbers of
        # current expressions
        subst = S[0]

        # Traverse the string at insert + at
        # current subset of indices
        for j in range(N - 1):

            # If current index exists
            # in the current subset
            if (i >> j) & 1:

                # Update ans_sub
                ans_sub += int(subst)

                # Update subset
                subst = S[j + 1]
            else:

                # Update subset
                subst += S[j + 1]

            # + can't be inserted after
            # the last index
            if j == N - 2:
                ans_sub += int(subst)

        # Update ans
        sumOfExp += ans_sub

    # Base case
    if N == 1:
        print(int(S))
    else:

        # Print answer
        print(sumOfExp)

# Driver Code
if __name__ == '__main__':

    # Given string
    S = "9999999999"

    # Length of the string
    N = len(S)

    # Function call
    findSumOfExpressions(S, N)

Output:

12656242944

Time Complexity: O(2^N * N)
Auxiliary Space: O(1)