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Sum of Digit Differences of All Pairs

Difficulty: Medium

Problem Description

You are given an array nums consisting of positive integers where all integers have the same number of digits. The digit difference between two integers is the count of different digits that are in the same position in the two integers. Return the sum of the digit differences between all pairs of integers in nums.

Key Insights

  • The digit difference can be calculated by comparing each digit of two integers at corresponding positions.
  • For an array of n integers, the number of pairs is n * (n - 1) / 2.
  • We can optimize the counting of digit differences by iterating through each digit position across all numbers rather than comparing each pair directly.

Space and Time Complexity

Time Complexity: O(n * d) where n is the number of integers and d is the number of digits in each integer.
Space Complexity: O(1) since we are using only a constant amount of extra space.

Solution

To solve this problem, we can iterate through each digit position of the integers in the array and count how many times each digit appears at that position. For each digit position, we compute the number of digit differences based on the frequency of each digit. The algorithm proceeds as follows:

  1. For each digit position (from least significant to most significant):
    • Count occurrences of each digit (0-9).
    • Calculate the contribution of this digit position to the total digit difference using the counts.
  2. Sum the contributions from all digit positions to get the final result.

Code Solutions

def countDigitDifferences(nums):
    n = len(nums)
    num_digits = len(str(nums[0]))
    total_difference = 0

    for d in range(num_digits):
        count = [0] * 10  # Count for digits 0-9
        for num in nums:
            digit = (num // (10 ** d)) % 10
            count[digit] += 1
        
        # Calculate the contribution of this digit position
        for c in count:
            if c > 0:
                total_difference += c * (n - c)  # Each pair of different digits contributes

    return total_difference

# Example usage:
print(countDigitDifferences([13, 23, 12]))  # Output: 4
print(countDigitDifferences([10, 10, 10, 10]))  # Output: 0
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