Problem Description

You are given an integer array nums.

The task is to replace each element in the array with the sum of its digits. For example, if an element is 123, you would replace it with 1 + 2 + 3 = 6.

After performing this replacement operation on all elements in the array, you need to return the minimum element among all the replaced values.

Example walkthrough:

• If nums = [123, 45, 67]
• After replacement: [6, 9, 13] (since 1+2+3=6, 4+5=9, 6+7=13)
• The minimum element would be 6

The solution iterates through each number in the array, converts it to a string to access individual digits, sums those digits, and then finds the minimum among all these sums.

Intuition

The problem asks us to find the minimum value after transforming each number into the sum of its digits. This is a straightforward transformation problem where we need to:

1. Transform each element independently
2. Find the minimum among the transformed values

The key insight is that we can compute the digit sum of a number by converting it to a string and then summing up each character (digit) after converting it back to an integer. For a number like 245, we can access each digit by converting it to string "245", then iterate through each character '2', '4', '5' and convert them back to integers to sum them up: 2 + 4 + 5 = 11.

Since we need the minimum value after all transformations, we can either:

• Store all transformed values and then find the minimum
• Or more efficiently, compute the minimum on-the-fly as we transform each element

The solution uses Python's generator expression with the min() function to elegantly combine both steps - it transforms each number to its digit sum and finds the minimum in a single pass through the array. The expression sum(int(b) for b in str(x)) handles the digit sum calculation for each number x, and the outer min() finds the smallest among all these sums.

Learn more about Math patterns.

Solution Approach

The solution follows a simulation approach where we process each element in the array and transform it according to the problem requirements.

Implementation Steps:

1. Iterate through the array: We traverse each element x in the nums array.

2. Calculate digit sum for each element: For each number x:

   • Convert the number to a string using str(x). This allows us to access individual digits.
   • Iterate through each character b in the string representation.
   • Convert each character back to an integer using int(b).
   • Sum all these digits using sum(int(b) for b in str(x)).

3. Find the minimum: Apply the min() function to find the smallest digit sum among all elements.

The solution combines all these steps into a single line using a generator expression:

return min(sum(int(b) for b in str(x)) for x in nums)

Breaking down the expression:

• for x in nums: Iterates through each element in the input array
• str(x): Converts the current number to a string (e.g., 123 → "123")
• for b in str(x): Iterates through each character/digit in the string
• int(b): Converts each character back to an integer (e.g., '1' → 1)
• sum(...): Calculates the sum of all digits for the current number
• min(...): Finds the minimum value among all the digit sums

Time Complexity: O(n * m) where n is the length of the array and m is the average number of digits in each element.

Space Complexity: O(1) as we only use constant extra space (not counting the space used by the generator expression which processes one element at a time).

Example Walkthrough

Let's walk through the solution with a concrete example: nums = [38, 192, 5]

Step 1: Process first element (38)

• Convert 38 to string: "38"
• Extract and sum digits: '3' → 3, '8' → 8
• Digit sum: 3 + 8 = 11

Step 2: Process second element (192)

• Convert 192 to string: "192"
• Extract and sum digits: '1' → 1, '9' → 9, '2' → 2
• Digit sum: 1 + 9 + 2 = 12

Step 3: Process third element (5)

• Convert 5 to string: "5"
• Extract and sum digits: '5' → 5
• Digit sum: 5

Step 4: Find minimum

• We have transformed values: [11, 12, 5]
• The minimum is 5

Let's trace through the actual code execution:

return min(sum(int(b) for b in str(x)) for x in nums)

The generator expression produces:

• For x=38: sum(int(b) for b in "38") = sum([3, 8]) = 11
• For x=192: sum(int(b) for b in "192") = sum([1, 9, 2]) = 12
• For x=5: sum(int(b) for b in "5") = sum([5]) = 5

The min() function then evaluates: min(11, 12, 5) = 5

Therefore, the function returns 5.

Solution Implementation

Time and Space Complexity

Time Complexity: O(n × log M)

The algorithm iterates through each element in the array (n elements). For each element x, it converts the number to a string and sums its digits. Converting a number to string takes O(log x) time (proportional to the number of digits), and summing the digits also takes O(log x) time. In the worst case, x can be as large as M (the maximum value in the array), so the time per element is O(log M). Therefore, the total time complexity is O(n × log M).

Space Complexity: O(log M)

While the algorithm uses a generator expression which doesn't store all transformed values at once, the string conversion str(x) creates a temporary string for each number. The largest string created would be for the maximum value M, which requires O(log M) space to store its digits as a string. The min() function only needs to track the minimum value seen so far, requiring O(1) additional space. Therefore, the overall space complexity is O(log M).

Note: The reference answer states O(1) space complexity, which would be accurate if we consider the string conversion as not using additional space or if the implementation used arithmetic operations to extract digits instead of string conversion.

Learn more about how to find time and space complexity quickly.

Common Pitfalls

1. Handling Negative Numbers Incorrectly

The most critical pitfall is not accounting for negative numbers in the input array. When converting a negative number to a string, the minus sign becomes part of the string, which will cause an error when trying to convert it back to an integer.

Problem Example:

nums = [-15, 23, -8]
# str(-15) returns "-15"
# Attempting int('-') will raise ValueError

Solution: Use the absolute value before processing digits:

def minElement(self, nums: List[int]) -> int:
    return min(
        sum(int(digit) for digit in str(abs(number)))
        for number in nums
    )

2. Integer Overflow in Other Languages

While Python handles large integers gracefully, implementing this in languages like C++ or Java requires consideration of integer overflow when summing digits of very large numbers.

Solution: In other languages, ensure the sum variable can handle the maximum possible digit sum (9 × number of digits in the largest possible integer).

3. Empty Array Input

If the input array is empty, calling min() on an empty sequence will raise a ValueError.

Problem Example:

nums = []
# min() with empty sequence raises ValueError

Solution: Add a guard clause or handle edge cases:

def minElement(self, nums: List[int]) -> int:
    if not nums:
        return 0  # or raise an appropriate exception
  
    return min(
        sum(int(digit) for digit in str(abs(number)))
        for number in nums
    )

4. Inefficient String Conversion for Mathematical Approach

While string conversion is intuitive, it's not the most efficient approach for extracting digits.

Alternative Mathematical Solution:

def minElement(self, nums: List[int]) -> int:
    def digit_sum(n):
        n = abs(n)
        total = 0
        while n > 0:
            total += n % 10
            n //= 10
        return total
  
    return min(digit_sum(num) for num in nums)

This avoids string conversion overhead and can be more efficient for large datasets.