Count Possible Decoding of a given Digit Sequence with Hidden Characters

Count possible decoding of a given digit sequence with hidden characters is a fascinating problem in the realm of string decoding. In this tutorial, we delve into the challenge of decoding a sequence of digits that may contain hidden characters denoted by asterisks ('*').

The task at hand is to determine the number of ways these hidden characters can be decoded, taking into account a specific mapping of letters from A to Z to the digits 1 to 26. We present an efficient solution using the power of C++ programming language and dynamic programming techniques.

By employing a bottom-up approach, we develop a C++ program that traverses the digit sequence, analyzes the hidden characters, and computes the total count of possible decodings. Throughout the tutorial, we discuss the problem statement, illustrate the solution algorithm, and provide a step-by-step implementation in C++, empowering readers to understand and apply this decoding technique to their own scenarios. So let’s get started!

Problem Statement

Consider a string ‘S’ consisting of digits and a special character '*', which represents a hidden character. The task is to find the number of possible decodings for this string, taking into account the hidden character.

The final result should be returned modulo ‘10^9 + 7’ to handle the potentially large number of decodings.

Each character in the string can be mapped to a corresponding digit based on the given mapping, where 'A' represents "1", 'B' represents "2", and so on until 'Z' represents "26". It's important to note that characters representing digits greater than 26 are not considered valid mappings (e.g., 'J' is not mapped to 10).

The objective is to calculate the total count of valid decodings, considering the presence of the hidden character represented by '*'. Let's explore this problem further through examples.

Sample Example 1

Input

String: "*"

Output

Number of possible decodings: 9

Explanation: In this example, the input string is "", which represents a hidden character. The hidden character can be replaced by any digit from 1 to 9. Each digit corresponds to a unique letter from 'A' to 'I'. Hence, the encoded message has the potential to represent any of the following encoded messages: "1", "2", "3", "4", "5", "6", "7", "8", or "9". Each of these encoded messages can be decoded into the corresponding letters "A", "B", "C", "D", "E", "F", "G", "H", and "I" respectively. Hence, there are a total of 9 unique ways to decode the string "", considering all possible replacements of the hidden character.

Sample Example 2

Input

String: "1*"

Output

Number of possible decodings: 18

Explanation: In this example, the input string is "1*". The hidden character represented by '*' can be replaced by any digit from 0 to 9. This leads to multiple possible encoded messages for this string, such as "10", "11", "12", "13", "14", "15", "16", "17", "18", or "19". Each of these encoded messages has 2 unique ways to be decoded. For example, the encoded message "11" can be decoded into either "AA" or "K". Therefore, there are a total of 9 possibilities for the first digit, and each of them has 2 decoding options, resulting in a total of 9 * 2 = 18 unique ways to decode the string "1".

In both examples, the output represents the total count of valid decodings for the given input string, considering all possible replacements of the hidden character.

Algorithm

1. Start with the given digit sequence as input.

2. Initialize a dynamic programming table ‘dp’ of size ‘n + 1’, where ‘n’ is the length of the sequence.

3. Set ‘dp[0] = 1’ since there is only one way to decode an empty sequence.

4. Check the first digit of the sequence:

• If it is '0', return 0, as it cannot be decoded on its own.

• If it is '*', set ‘dp[1] = 9’ since it can represent any digit from 1 to 9.

• Otherwise, set ‘dp[1] = 1’ since the first digit can be decoded on its own.

5. Iterate from the second digit to the end of the sequence:

• If the current digit is '*':

• Multiply the count by 9 to consider the possibility of it being any digit from 1 to 9.

• Check the previous digit to calculate additional combinations:

• If the previous digit is '1', add ‘9 * dp[i - 2]’ to the count.

• If the previous digit is '2', add ‘6 * dp[i - 2]’ to the count.

• If the previous digit is '*', add ‘15 * dp[i - 2]’ to the count (considering all possibilities).

• Otherwise (the current digit is not '*'):

• If the current digit is not '0', set ‘dp[i] = dp[i - 1]’ since it can be decoded on its own.

• Check the previous digit to calculate additional combinations:

• If the previous digit is '1', add ‘dp[i - 2]’ to the count.

• If the previous digit is '2' and the current digit is less than or equal to '6', add ‘dp[i - 2]’ to the count.

• If the previous digit is '*', add ‘(2 or 1) * dp[i - 2]’ to the count, depending on the value of the current digit.

6. Return ‘dp[n]’, which represents the number of possible decodings of the given sequence.

The algorithm uses dynamic programming to build the count of decodings iteratively, considering the constraints and possibilities for each digit.

Example

Implementation of the above algorithm using C++

The below C++ program calculates the number of ways to decode a given string based on a set of decoding rules. It uses a bottom-up dynamic programming approach to efficiently compute the number of decodings at each position in the string. The program iterates over the characters of the string, applies the decoding rules, and stores the results in a dynamic programming array. Finally, it returns the number of possible decodings for the entire string.

Input

"1*"

Output

Number of possible decodings: 18

Example

#include <iostream>
#include <vector>
const int MOD = 1000000007;
int countDecodings(const std::string& sequence) {
   int n = sequence.length();
   // Base cases
   if (n == 0 || sequence[0] == '0') {
      return 0;
   }
   // Initialize the dynamic programming table
   std::vector<int> dp(n + 1, 0);
   dp[0] = 1;
   dp[1] = (sequence[0] != '0');
   // Fill the dynamic programming table
   for (int i = 2; i <= n; ++i) {
      // If the current digit is '*', multiply the count by 9
      if (sequence[i - 1] == '*') {
         dp[i] = (9 * dp[i - 1]) % MOD;
         // Consider the previous digit for additional combinations
         if (sequence[i - 2] == '1') {
            dp[i] = (dp[i] + 9 * dp[i - 2]) % MOD;
         } else if (sequence[i - 2] == '2') {
            dp[i] = (dp[i] + 6 * dp[i - 2]) % MOD;
         } else if (sequence[i - 2] == '*') {
            dp[i] = (dp[i] + 15 * dp[i - 2]) % MOD;
         }
      } else {
         // If the current digit is not '*', check if it is valid on its own
         dp[i] = (sequence[i - 1] != '0' ? dp[i - 1] : 0);
         // Consider the previous digit for additional combinations
         if (sequence[i - 2] == '1') {
            dp[i] = (dp[i] + dp[i - 2]) % MOD;
         } else if (sequence[i - 2] == '2' && sequence[i - 1] <= '6') {
            dp[i] = (dp[i] + dp[i - 2]) % MOD;
         } else if (sequence[i - 2] == '*') {
            dp[i] = (dp[i] + (sequence[i - 1] <= '6' ? 2 : 1) * dp[i - 2]) % MOD;
         }
      }
   }
   return dp[n];
}
int main() {
   std::string sequence = "1*";
   int numDecodings = countDecodings(sequence);
   std::cout << "Number of possible decodings: " << numDecodings << std::endl;
   return 0;
}

Output

Number of possible decodings: 18

Conclusion

To sum up, we have explored the problem of decoding a digit sequence with hidden characters and presented an effective solution using C++ and dynamic programming. By leveraging the power of C++ programming language and employing a bottom-up approach, we have developed a program that efficiently counts the possible decodings of such sequences. Our solution takes into account the specific mapping of letters to digits and handles the presence of hidden characters denoted by asterisks. By following the step-by-step implementation and understanding the underlying algorithm, readers can now tackle similar decoding challenges in their own projects.