Solution Explanation:

This problem asks to count the occurrences of a digit d within the range [low, high]. A naive approach would iterate through each number and count the occurrences, which is highly inefficient for a large range. The provided solution employs a more efficient dynamic programming approach with memoization.

Core Idea: The solution breaks down the counting process into smaller subproblems. It recursively counts the occurrences of d in numbers with a given number of digits. The dfs function (or its equivalent in different languages) is the heart of this recursive approach.

dfs Function (Recursive Counting):

The dfs function takes the following arguments:

• pos: The current digit position being processed (from the most significant digit to the least significant).
• cnt: The count of occurrences of d found so far.
• lead: A boolean indicating whether the current digit is the leading digit (leading zeros are treated differently).
• limit: A boolean indicating whether the current digit can reach its upper bound. This is crucial for handling upper limits correctly.

The function works as follows:

1. Base Case: If pos reaches 0, it means all digits have been processed, so it returns the current count cnt.

2. Recursive Step: It iterates through possible digits (0 to 9 or up to a limit).

3. Leading Zero Handling: If lead is true and the current digit is 0, it recursively calls dfs to continue processing without incrementing the count.

4. Digit Count Update: If the current digit is equal to d, the count cnt is incremented.

5. Limit Check: The limit parameter ensures that the upper bound is not exceeded.

6. Memoization: The dp table (or @cache in Python) stores the results of subproblems to avoid redundant computations. This significantly improves performance.

f Function (Range Handling):

The f function processes a single integer n. It converts the integer into an array of digits (a), calls dfs to count the occurrences of d in n, and returns the count.

Main Function (digitsCount):

The digitsCount function calculates the difference between the count of d in high and the count in low - 1. This difference gives the total count of d within the range [low, high].

Time Complexity: O(log10(high)), because the depth of recursion is at most the number of digits in high. The memoization ensures that each subproblem is solved only once.

Space Complexity: O(log10(high)), primarily due to the recursive call stack and the memoization table (dp).

Code Differences Across Languages:

The core logic remains consistent across the Python, Java, C++, and Go implementations. The primary differences lie in syntax, data structure implementations (arrays vs. lists), and memoization techniques (@cache in Python, dp array in other languages). Go's implementation has a slightly different structure to handle closures and function declarations. Each implementation handles the base cases, recursive steps, and memoization effectively to solve the problem with optimized complexity.