Count possible decodings of a given Digit Sequence | Set 2

• Difficulty Level : Expert
• Last Updated : 05 Jan, 2022

Given an encoded string str consisting of digits and * which can be filled by any digit 1 – 9, the task is to find the number of ways to decode that string into a sequence of alphabets A-Z.

Note: The input string contains number from 0-9 and character ‘*’ only.

Examples:

Input: str = “1*”
Output: 18
Explanation:
Since * can be replaced by any value from (1-9),
The given string can be decoded as A[A-I] + [J-R] = 9 + 9 ways

Input: str = “12*3”
Output: 28

Naive Approach: A simple solution is to solve the problem using recursion considering all the possible decodings of the string.

Below is the recursion tree for the problem:

12*3
/             \
12*(3)           12(*3)
/     \            /      \
12(*)(3)  1(2*)(3)  1(2)(*3)   ""
/    \      \       /
1(2)(*)(3) ""     ""     ""
/
""

Efficient Approach: The idea is to solve the problem using Dynamic Programming using the optimal substructure for considering all the cases of the current and the previous digits of the string and their number of ways to decode the string.

Definition of the DP State: In this problem the denotes the number of ways to decode the string upto the index.

Initial DP States: Initially the value of the DP states are defined as below:

// Number of ways to decode
// an empty string
dp = 1

// Number of ways to decode the
// first character
if (s == '*')
dp = 9  // 9 ways
else
dp = 1

Optimal Sub-structure: There are generally two ways to decode the current character of the string:

• Including the current character as single-digit to decode: If the current character is used as single-digit then there can be generally two cases for the character to be:
• Case 1: If current character is equal to the , Then there are 9 possible ways to take any digit from [1-9] and decode it as any character from [A-Z].
if (current == "*")
dp[i] += 9 * dp[i-1]
• Case 2: If current character is equal to any other digit from [0-9], Then the number of possible ways to decode is equal to the number of way to decode the string upto index.
if (current != "*")
dp[i] += dp[i-1]
• Including the current character as two-digits to decode: If the current character is to be decoded as two-digits then there are two possible cases:
• Case 1: If the previous character is equal to the or , then there can be two more possible subcases which will depend upon the current character:
• Case 1.1: If the current character is equal to the , then total possible ways to decode are 9 if the previous character is 1, otherwise 6 if previous character is 2.
• Case 1.2: If the current character is less than equal to 6, Then the total number of possible way to decode the string will depend only on the number of way to decode upto the previous character. That is • Case 2: If the previous character is , then there can be two more possible subcases which will depend upon the current character:
• Case 2.1: If the current character is also , Then the total number of cases will be , because the single digits of decoding ways of previous character must be already included.
• Case 2.2: If the current character is less than 6, Then the total number of ways will be , because then the number of ways to choose the digits of the first character is 2. That is [1, 2].
• Case 2.3: If the current character is any digits, Then the total number of ways will be the number of ways to decode uptill the previous digit only. That is .

C++

 // C++ implementation to count the // possible decodings of the given // digit sequence   #include using namespace std;   // Function to count the number of // ways to decode the given digit // sequence int waysToDecode2(string s) {     int n = s.size();           // Array to store the dp states     vector dp(n+1,0);           // Case of empty string     dp=1;           // Condition to check if the     // first character of string is 0     if(s=='0')         return 0;               // Base case for single length     // string     dp= ((s=='*')? 9 : 1);           // Bottom-up dp for the string     for(int i=2;i<=n;i++)     {         // Previous character         char first= s[i-2];                   // Current character         char second= s[i-1];                   // Case to include the Current         // digit as a single digit for         // decoding the string         if(second=='*')         {             dp[i]+= 9*dp[i-1];         }         else if(second>'0')             dp[i]+=dp[i-1];                   // Case to include the current         // character as two-digit for         // decoding the string         if(first=='1'|| first=='2')         {                           // Condition to check if the             // current character is "*"             if(second=='*')             {                 if(first=='1')                     dp[i]+= 9 * dp[i-2];                 else if(first=='2')                     dp[i]+= 6 * dp[i-2];             }                           // Condition to check if the             // current character is less than             // or equal to 26             else if(((first-'0')* 10 +                     (second-'0'))<= 26)                 dp[i]+=dp[i-2];         }                   // Condition to check if the         // Previous digit is equal to "*"         else if(first=='*')         {             if(second=='*')             {                 dp[i]+= 15 * dp[i-2];             }             else if(second<='6')                 dp[i]+= 2* dp[i-2];             else                 dp [i]+= dp[i-2];         }     }     return dp[n]; }   // Driver Code int main() {     string str = "12*3";           // Function Call     cout << waysToDecode2(str) << endl;     return 0; }

Java

 // Java implementation to count the // possible decodings of the given // digit sequence class GFG{   // Function to count the number of // ways to decode the given digit // sequence static int waysToDecode2(char []s) {           int n = s.length;           // Array to store the dp states     int []dp = new int[n + 1];           // Case of empty String     dp = 1;           // Condition to check if the     // first character of String is 0     if(s == '0')         return 0;               // Base case for single length     // String     dp = ((s == '*') ? 9 : 1);           // Bottom-up dp for the String     for(int i = 2; i <= n; i++)     {               // Previous character     char first = s[i - 2];               // Current character     char second = s[i - 1];               // Case to include the Current     // digit as a single digit for     // decoding the String     if(second == '*')     {         dp[i] += 9 * dp[i - 1];     }     else if(second > '0')         dp[i] += dp[i - 1];               // Case to include the current     // character as two-digit for     // decoding the String     if(first == '1' || first == '2')     {                       // Condition to check if the         // current character is "*"         if(second == '*')         {             if(first == '1')                 dp[i] += 9 * dp[i - 2];             else if(first == '2')                 dp[i] += 6 * dp[i - 2];         }                       // Condition to check if the         // current character is less than         // or equal to 26         else if(((first - '0') * 10 +                 (second - '0')) <= 26)         {             dp[i] += dp[i - 2];         }     }               // Condition to check if the     // previous digit is equal to "*"     else if(first == '*')     {         if(second == '*')         {             dp[i] += 15 * dp[i - 2];         }         else if(second <= '6')         {             dp[i] += 2 * dp[i - 2];         }         else         {             dp[i] += dp[i - 2];         }     }     }     return dp[n]; }   // Driver Code public static void main(String[] args) {     String str = "12*3";           // Function Call     System.out.print(waysToDecode2(                     str.toCharArray()) + "\n"); } }   // This code is contributed by amal kumar choubey

C#

 // C# implementation to count the // possible decodings of the given // digit sequence using System;   class GFG{   // Function to count the number of // ways to decode the given digit // sequence static int waysToDecode2(char []s) {     int n = s.Length;           // Array to store the dp states     int []dp = new int[n + 1];           // Case of empty String     dp = 1;           // Condition to check if the     // first character of String is 0     if(s == '0')         return 0;               // Base case for single length     // String     dp = ((s == '*') ? 9 : 1);           // Bottom-up dp for the String     for(int i = 2; i <= n; i++)     {               // Previous character     char first = s[i - 2];               // Current character     char second = s[i - 1];               // Case to include the current     // digit as a single digit for     // decoding the String     if(second == '*')     {         dp[i] += 9 * dp[i - 1];     }     else if(second > '0')     {         dp[i] += dp[i - 1];     }               // Case to include the current     // character as two-digit for     // decoding the String     if(first == '1' || first == '2')     {                       // Condition to check if the         // current character is "*"         if(second == '*')         {             if(first == '1')             {                 dp[i] += 9 * dp[i - 2];             }             else if(first == '2')             {                 dp[i] += 6 * dp[i - 2];             }         }                       // Condition to check if the         // current character is less than         // or equal to 26         else if(((first - '0') * 10 +                 (second - '0')) <= 26)         {             dp[i] += dp[i - 2];         }     }               // Condition to check if the     // previous digit is equal to "*"     else if(first == '*')     {         if(second == '*')         {             dp[i] += 15 * dp[i - 2];         }         else if(second <= '6')         {             dp[i] += 2 * dp[i - 2];         }         else         {             dp[i] += dp[i - 2];         }     }     }     return dp[n]; }   // Driver Code public static void Main(String[] args) {     String str = "12*3";           // Function Call     Console.Write(waysToDecode2(                 str.ToCharArray()) + "\n"); } }   // This code is contributed by amal kumar choubey



Python3

 # Python3 implementation to count the # possible decodings of the given # digit sequence     # Function to count the number of # ways to decode the given digit # sequence def waysToDecode2(s):     n = len(s)       # Array to store the dp states     dp =  * (n + 1)       # Case of empty string     dp = 1       # Condition to check if the     # first character of string is 0     if s == "0":         return 0       # Base case for single length     # string     dp = 9 if (s == "*") else 1       # Bottom-up dp for the string     for i in range(2, n+1):         # Previous character         first = s[i - 2]           # Current character         second = s[i - 1]           # Case to include the Current         # digit as a single digit for         # decoding the string         if second == "*":             dp[i] += 9 * dp[i - 1]           elif second > "0":             dp[i] += dp[i - 1]           # Case to include the current         # character as two-digit for         # decoding the string         if first == "1" or first == "2":               # Condition to check if the             # current character is "*"             if second == "*":                 if first == "1":                     dp[i] += 9 * dp[i - 2]                 elif first == "2":                     dp[i] += 6 * dp[i - 2]               # Condition to check if the             # current character is less than             # or equal to 26             elif (ord(first) - ord("0")) * 10 + (ord(second) - ord("0")) <= 26:                 dp[i] += dp[i - 2]           # Condition to check if the         # Previous digit is equal to "*"         elif first == "*":             if second == "*":                 dp[i] += 15 * dp[i - 2]               elif second <= "6":                 dp[i] += 2 * dp[i - 2]             else:                 dp[i] += dp[i - 2]       return dp[n]     # Driver Code if __name__ == "__main__":     str = "12*3"       # Function Call     print(waysToDecode2(str))

Output:

28
• Time Complexity: O(N)
• Auxiliary Space: O(N)

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