Restore The Array

class Solution {
    public int countArrays(String digits, int maxVal) {
        int len = digits.length(), maxDigits = String.valueOf(maxVal).length();
        int modulo = 1_000_000_007;
            
        // Tracks the number of possible decodings for the prefix of the string
        int[] dp = new int[maxDigits + 1];
            
        // Initialize with the base case of an empty prefix
        dp[0] = 1;
            
        for (int start = 0; start < len; ++start) {
            // Skip if the current digit is '0' since no valid number starts with '0'
            if (digits.charAt(start) == '0') {
                dp[start % (maxDigits + 1)] = 0;
                continue;
            }
                
            for (int end = start; end < len; ++end) {
                String subNum = digits.substring(start, end + 1);
                    
                if (Long.parseLong(subNum) > maxVal)
                    break;
                    
                // Update the count of ways to split at position end
                dp[(end + 1) % (maxDigits + 1)] = (dp[(end + 1) % (maxDigits + 1)] + dp[start % (maxDigits + 1)]) % modulo;
            }
    
            // Reset to zero for the next iteration
            dp[start % (maxDigits + 1)] = 0;
        }
        return dp[len % (maxDigits + 1)];
    }
}

The Java solution provided focuses on counting the number of ways to restore an array from a given string digits, such that each segmented integer does not exceed a specified maximum value maxVal. Here's how the solution achieves this:

• The string's length and the number of digits in maxVal are calculated.
• A dynamic programming approach is utilized with an array dp to store the count of possible restorations up to a given position in the digits string.
• The modulo 1_000_000_007 is used to avoid overflow issues.
• A loop iterates over the string. If the current digit is zero, the processing skips to the next iteration since numbers can't start with zero.
• Another nested loop checks all possible end positions for the current start position. It extracts the substring from start to end, converts it into a number, and checks if it is within the bounds of maxVal.
• If valid, the dp array at the position corresponding to end+1 is updated with the sum of its current value and the value from the start position, ensuring to take modulo 1_000_000_007.
• Once all possibilities for a starting index have been exhausted, that index in the dp array is reset to zero to prevent incorrect accumulations for subsequent iterations.
• The result is found in dp[len % (maxDigits + 1)], which represents the number of possible decodings that fit the length and value constraints.

This method ensures efficiency by leveraging dynamic programming to avoid recalculating results and by effectively managing constraints with modulo operations.

class Solution:
    def countValidCombinations(self, digits: str, max_val: int) -> int:
        length_digits, length_max_val = len(digits), len(str(max_val))
        modulus = 10 ** 9 + 7
        dp_array = [1] + [0] * length_max_val
    
        for idx_start in range(length_digits):
            if digits[idx_start] == '0':
                dp_array[idx_start % (length_max_val + 1)] = 0
                continue
    
            # Look for all digits sequences from idx_start to idx_end
            for idx_end in range(idx_start, length_digits):
                if int(digits[idx_start: idx_end + 1]) > max_val:
                    break
    
                dp_array[(idx_end + 1) % (length_max_val + 1)] += dp_array[idx_start % (length_max_val + 1)]
                dp_array[(idx_end + 1) % (length_max_val + 1)] %= modulus
    
            dp_array[idx_start % (length_max_val + 1)] = 0
    
        return dp_array[length_digits % (length_max_val + 1)]

The solution provided tackles the problem of "Restore The Array" by defining a method countValidCombinations in the Python class Solution. This method calculates the number of valid combinations by segmenting a string of digits such that each segment, when converted to an integer, does not exceed a specified maximum value (max_val).

The approach utilizes dynamic programming to efficiently achieve the solution. The core idea is based on building an array dp_array that keeps track of the count of valid combinations ending at each position in the input digits string. Here's a concise breakdown of the implementation process:

• Initialize two variables length_digits and length_max_val to store the lengths of the input digits string and the max_val string.

• Set modulus to (10^9 + 7), which is commonly used in competitive programming to ensure results fit within standard integer ranges after performing operations.

• Create a list dp_array initialized with 1 followed by zeros, sized based on length_max_val.

• Iterate through each starting index (idx_start) in the digits string. If the digit at idx_start is zero, skip further processing from this index, as numbers cannot start with zero.

• From each valid starting point, explore all potential ending indices (idx_end). If a number formed from digits[idx_start: idx_end+1] exceeds max_val, break out of the loop for subsequent positions.

• Update the appropriate position in dp_array to include combinations identified by extending up to idx_end and apply modulo operation to manage potential overflow.

• After processing numbers starting at idx_start, reset the count at idx_start % (length_max_val + 1) in dp_array to zero.

• Return the final result from dp_array[length_digits % (length_max_val + 1)].

This structured and carefully partitioned approach to solving the problem ensures that each segment of the input string is considered uniquely, adhering to the constraints provided by max_val, thereby efficiently counting valid number combinations within the input string.