Count Possible Decodings of a given Digit Sequence - GeeksforGeeks

28 Feb, 2022
Let 1 represent ‘A’, 2 represents ‘B’, etc. Given a digit sequence, count the number of possible decodings of the given digit sequence. 
Examples: 
Input:  digits[] = "121"
Output: 3
// The possible decodings are "ABA", "AU", "LA"

Input: digits[] = "1234"
Output: 3
// The possible decodings are "ABCD", "LCD", "AWD"
An empty digit sequence is considered to have one decoding. It may be assumed that the input contains valid digits from 0 to 9 and there are no leading 0’s, no extra trailing 0’s, and no two or more consecutive 0’s.
This problem is recursive and can be broken into sub-problems. We start from the end of the given digit sequence. We initialize the total count of decodings as 0. We recur for two subproblems. 1) If the last digit is non-zero, recur for the remaining (n-1) digits and add the result to the total count. 2) If the last two digits form a valid character (or smaller than 27), recur for remaining (n-2) digits and add the result to the total count.
Following is the implementation of the above approach.  
C++
Java
Python3
C#
PHP
Javascript
// C++ implementation to count number of// decodings that can be formed from a// given digit sequence#include <cstring>#include <iostream>using namespace std; // recuring function to find// ways in how many ways a// string can be decoded of length// greater than 0 and starting with// digit 1 and greater.int countDecoding(char* digits, int n){    // base cases    if (n == 0 || n == 1)        return 1;    if (digits[0] == '0')        return 0;     // for base condition "01123" should return 0    // Initialize count    int count = 0;     // If the last digit is not 0,    // then last digit must add    // to the number of words    if (digits[n - 1] > '0')        count = countDecoding(digits, n - 1);     // If the last two digits form a number smaller    // than or equal to 26, then consider    // last two digits and recur    if (digits[n - 2] == '1'        || (digits[n - 2] == '2'        && digits[n - 1] < '7'))        count += countDecoding(digits, n - 2);     return count;} // Given a digit sequence of length n,// returns count of possible decodings by// replacing 1 with A, 2 with B, ... 26 with Zint countWays(char* digits, int n){    if (n == 0 || (n == 1 && digits[0] == '0'))        return 0;    return countDecoding(digits, n);} // Driver codeint main(){    char digits[] = "1234";    int n = strlen(digits);    cout << "Count is " << countWays(digits, n);    return 0;}// Modified by Atanu Sen
// A naive recursive Java implementation// to count number of decodings that// can be formed from a given digit sequence class GFG {     // recuring function to find    // ways in how many ways a    // string can be decoded of length    // greater than 0 and starting with    // digit 1 and greater.    static int countDecoding(char[] digits, int n)    {        // base cases        if (n == 0 || n == 1)            return 1;         // for base condition "01123" should return 0        if (digits[0] == '0')            return 0;         // Initialize count        int count = 0;         // If the last digit is not 0, then        // last digit must add to        // the number of words        if (digits[n - 1] > '0')            count = countDecoding(digits, n - 1);         // If the last two digits form a number        // smaller than or equal to 26,        // then consider last two digits and recur        if (digits[n - 2] == '1'            || (digits[n - 2] == '2'                && digits[n - 1] < '7'))            count += countDecoding(digits, n - 2);         return count;    }     // Given a digit sequence of length n,    // returns count of possible decodings by    // replacing 1 with A, 2 with B, ... 26 with Z    static int countWays(char[] digits, int n)    {        if (n == 0 || (n == 1 && digits[0] == '0'))            return 0;        return countDecoding(digits, n);    }     // Driver code    public static void main(String[] args)    {        char digits[] = { '1', '2', '3', '4' };        int n = digits.length;        System.out.printf("Count is %d",                          countWays(digits, n));    }} // This code is contributed by Smitha Dinesh Semwal.// Modified by Atanu Sen
# Recursive implementation of numDecodingsdef numDecodings(s: str) -> int:    if len(s) == 0    or (len(s) == 1        and s[0] == '0'):        return 0    return numDecodingsHelper(s, len(s))  def numDecodingsHelper(s: str, n: int) -> int:    if n == 0 or n == 1:        return 1    count = 0    if s[n-1] > "0":        count = numDecodingsHelper(s, n-1)    if (s[n - 2] == '1'        or (s[n - 2] == '2'            and s[n - 1] < '7')):        count += numDecodingsHelper(s, n - 2)    return count  # Driver codedigits = "1234"print("Count is ", numDecodings(digits))# This code is contributed by Frank Hu
// A naive recursive C# implementation// to count number of decodings that// can be formed from a given digit sequenceusing System; class GFG {     // recuring function to find    // ways in how many ways a    // string can be decoded of length    // greater than 0 and starting with    // digit 1 and greater.    static int countDecoding(char[] digits, int n)    {         // base cases        if (n == 0 || n == 1)            return 1;         // Initialize count        int count = 0;         // If the last digit is not 0, then        // last digit must add to        // the number of words        if (digits[n - 1] > '0')            count = countDecoding(digits, n - 1);         // If the last two digits form a number        // smaller than or equal to 26, then        // consider last two digits and recur        if (digits[n - 2] == '1'            || (digits[n - 2] == '2'                && digits[n - 1] < '7'))            count += countDecoding(digits, n - 2);         return count;    }     // Given a digit sequence of length n,    // returns count of possible decodings by    // replacing 1 with A, 2 with B, ... 26 with Z    static int countWays(char[] digits, int n)    {        if (n == 0 || (n == 1 && digits[0] == '0'))            return 0;        return countDecoding(digits, n);    }     // Driver code    public static void Main()    {        char[] digits = { '1', '2', '3', '4' };        int n = digits.Length;        Console.Write("Count is ");        Console.Write(countWays(digits, n));    }} // This code is contributed by nitin mittal.
<?php// A naive recursive PHP implementation// to count number of decodings that can// be formed from a given digit sequence //recuring function to find//ways in how many ways a//string can be decoded of length//greater than 0 and starting with//digit 1 and greater.function countDecoding(&$digits, $n){    // base cases    if ($n == 0 || $n == 1)        return 1;     $count = 0; // Initialize count     // If the last digit is not 0, then last    // digit must add to the number of words    if ($digits[$n - 1] > '0')        $count = countDecoding($digits, $n - 1);     // If the last two digits form a number    // smaller than or equal to 26, then    // consider last two digits and recur    if ($digits[$n - 2] == '1' ||       ($digits[$n - 2] == '2' &&        $digits[$n - 1] < '7') )        $count += countDecoding($digits, $n - 2);     return $count;} // Given a digit sequence of length n,// returns count of possible decodings by// replacing 1 with A, 2 with B, ... 26 with Zfunction countWays(&$digits, $n){  if($n==0 || ($n == 1 && $digits[0] == '0'))     return 0;  return countDecoding($digits, $n);} // Driver Code$digits = "1234";$n = strlen($digits);echo "Count is " . countWays($digits, $n); // This code is contributed by ita_c?>
<script> // A naive recursive JavaScript implementation// to count number of decodings that// can be formed from a given digit sequence            // recuring function to find    // ways in how many ways a    // string can be decoded of length    // greater than 0 and starting with    // digit 1 and greater.    function countDecoding(digits, n)    {        // base cases        if (n == 0 || n == 1)        {            return 1;        }        // for base condition "01123" should return 0        if (digits[0] == '0')        {            return 0;        }                 // Initialize count        let count = 0;                 // If the last digit is not 0, then        // last digit must add to        // the number of words        if (digits[n - 1] > '0')        {            count = countDecoding(digits, n - 1);        }        // If the last two digits form a number        // smaller than or equal to 26,        // then consider last two digits and recur        if (digits[n - 2] == '1'            || (digits[n - 2] == '2'                && digits[n - 1] < '7'))        {            count += countDecoding(digits, n - 2);        }        return count;    }    // Given a digit sequence of length n,    // returns count of possible decodings by    // replacing 1 with A, 2 with B, ... 26 with Z    function countWays(digits, n)    {        if (n == 0 || (n == 1 && digits[0] == '0'))        {            return 0;        }        return countDecoding(digits, n);    }         // Driver code    digits=['1', '2', '3', '4'];    let n = digits.length;    document.write("Count is ",countWays(digits, n));         // This code is contributed by avanitrachhadiya2155     </script>
Output: 
Count is 3
The time complexity of above the code is exponential. If we take a closer look at the above program, we can observe that the recursive solution is similar to Fibonacci Numbers. Therefore, we can optimize the above solution to work in O(n) time using Dynamic Programming. 
Following is the implementation for the same. 
C++
Java
Python3
C#
PHP
Javascript
// A Dynamic Programming based C++// implementation to count decodings#include <iostream>#include <cstring>using namespace std; // A Dynamic Programming based function// to count decodingsint countDecodingDP(char *digits, int n){    // A table to store results of subproblems    int count[n+1];    count[0] = 1;    count[1] = 1;    //for base condition "01123" should return 0    if(digits[0]=='0')          return 0;    for (int i = 2; i <= n; i++)    {        count[i] = 0;         // If the last digit is not 0,        // then last digit must add to the number of words        if (digits[i-1] > '0')            count[i] = count[i-1];         // If second last digit is smaller        // than 2 and last digit is smaller than 7,        // then last two digits form a valid character        if (digits[i-2] == '1' ||              (digits[i-2] == '2' && digits[i-1] < '7') )            count[i] += count[i-2];    }    return count[n];} // Driver program to test above functionint main(){    char digits[] = "1234";    int n = strlen(digits);    cout << "Count is " << countDecodingDP(digits, n);    return 0;}// Modified by Atanu Sen
// A Dynamic Programming based Java// implementation to count decodingsimport java.io.*; class GFG{     // A Dynamic Programming based// function to count decodingsstatic int countDecodingDP(char digits[],                           int n){    // A table to store results of subproblems    int count[] = new int[n + 1];    count[0] = 1;    count[1] = 1;    if(digits[0]=='0')   //for base condition "01123" should return 0          return 0;    for (int i = 2; i <= n; i++)    {        count[i] = 0;         // If the last digit is not 0,        // then last digit must add to        // the number of words        if (digits[i - 1] > '0')            count[i] = count[i - 1];         // If second last digit is smaller        // than 2 and last digit is smaller        // than 7, then last two digits        // form a valid character        if (digits[i - 2] == '1' ||           (digits[i - 2] == '2' &&            digits[i - 1] < '7'))            count[i] += count[i - 2];    }    return count[n];} // Driver Codepublic static void main (String[] args){    char digits[] = {'1','2','3','4'};    int n = digits.length;    System.out.println("Count is " +               countDecodingDP(digits, n));}} // This code is contributed by anuj_67// Modified by Atanu Sen
# A Dynamic Programming based Python3# implementation to count decodings # A Dynamic Programming based function# to count decodingsdef countDecodingDP(digits, n):     count = [0] * (n + 1); # A table to store                           # results of subproblems    count[0] = 1;    count[1] = 1;     for i in range(2, n + 1):         count[i] = 0;         # If the last digit is not 0, then last        # digit must add to the number of words        if (digits[i - 1] > '0'):            count[i] = count[i - 1];         # If second last digit is smaller than 2        # and last digit is smaller than 7, then        # last two digits form a valid character        if (digits[i - 2] == '1' or           (digits[i - 2] == '2' and            digits[i - 1] < '7') ):            count[i] += count[i - 2];     return count[n]; # Driver Codedigits = "1234";n = len(digits);print("Count is" ,       countDecodingDP(digits, n)); # This code is contributed by mits
// A Dynamic Programming based C#// implementation to count decodingsusing System; class GFG{     // A Dynamic Programming based// function to count decodingsstatic int countDecodingDP(char[] digits,                           int n){    // A table to store results of subproblems    int[] count = new int[n + 1];    count[0] = 1;    count[1] = 1;     for (int i = 2; i <= n; i++)    {        count[i] = 0;         // If the last digit is not 0,        // then last digit must add to        // the number of words        if (digits[i - 1] > '0')            count[i] = count[i - 1];         // If second last digit is smaller        // than 2 and last digit is smaller        // than 7, then last two digits        // form a valid character        if (digits[i - 2] == '1' ||           (digits[i - 2] == '2' &&            digits[i - 1] < '7'))            count[i] += count[i - 2];    }    return count[n];} // Driver Codepublic static void Main(){    char[] digits = {'1','2','3','4'};    int n = digits.Length;    Console.WriteLine("Count is " +            countDecodingDP(digits, n));}} // This code is contributed// by Akanksha Rai// Modified by Atanu Sen
<?php// A Dynamic Programming based// php implementation to count decodings // A Dynamic Programming based function to count decodingsfunction countDecodingDP($digits, $n){    // A table to store results of subproblems    $count[$n+1]=array();    $count[0] = 1;    $count[1] = 1;     for ($i = 2; $i <= $n; $i++)    {        $count[$i] = 0;         // If the last digit is not 0, then last digit must add to        // the number of words        if ($digits[$i-1] > '0')            $count[$i] = $count[$i-1];         // If second last digit is smaller than 2 and last digit is        // smaller than 7, then last two digits form a valid character        if ($digits[$i-2] == '1' || ($digits[$i-2] == '2' && $digits[$i-1] < '7') )            $count[$i] += $count[$i-2];    }    return $count[$n];} // Driver program to test above function    $digits = "1234";    $n = strlen($digits);    echo  "Count is " , countDecodingDP($digits, $n); #This code is contributed by ajit.?>
<script> // A Dynamic Programming based Javascript// implementation to count decodings // A Dynamic Programming based// function to count decodingsfunction countDecodingDP(digits, n){         // A table to store results of subproblems    let count = new Array(n + 1);    count[0] = 1;    count[1] = 1;         // For base condition "01123" should return 0    if (digits[0] == '0')            return 0;               for(let i = 2; i <= n; i++)    {        count[i] = 0;          // If the last digit is not 0,        // then last digit must add to        // the number of words        if (digits[i - 1] > '0')            count[i] = count[i - 1];          // If second last digit is smaller        // than 2 and last digit is smaller        // than 7, then last two digits        // form a valid character        if (digits[i - 2] == '1' ||           (digits[i - 2] == '2' &&            digits[i - 1] < '7'))            count[i] += count[i - 2];    }    return count[n];} // Driver Codelet digits = [ '1','2','3','4' ];let n = digits.length; document.write("Count is " +               countDecodingDP(digits, n));                // This code is contributed by rag2127     </script>
Output: 
Count is 3
Time Complexity of the above solution is O(n) and it requires O(n) auxiliary space. We can reduce auxiliary space to O(1) by using the space-optimized version discussed in the Fibonacci Number Post. Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above
Method 3 : ( Top Down DP )
Approach :  
The above problem can be solved using Top down DP in the following way . One of the basic intuition is that we need to find the total number of  ways to decode the given string such that each and every number in the string must lie in between the range of [ 1 , 26 ] both inclusive and without any leading 0’s . Let us consider an example string .
str = “123”
If we observe carefully we can observe a pattern over here i.e., the number of ways a particular substring can be decoded depends on the number of ways the remaining string is going to be decoded . For example , we want the number of ways to decode the string with “1” as a prefix the result depends on the number of ways the remaining string, i.e., “23” can be decoded . The number of ways the string “23” can be decoded are “2” , “3” and “23” there are 2 ways in both of these cases we can just append “1” to get the number of ways the given string can be decoded with “1” as a prefix i.e., “1” , “2” , “3” and “1” , “23” . Now we have found the number of ways we can decode the given string with “1” as a prefix but “12” also lies in between the range of [ 1 , 26 ] both inclusive the number of ways to decode the given string with “12” as a prefix depends on the result on how the remaining string is decoded . Here the remaining string is “3” it can be decoded in only 1 way so we can just append “12” in front of the string “3” to get it i.e., “12” , “3” . So the total number of ways the given string can be decoded are 3 ways .
But we can see some of the overlapping of subproblems over here i.e., when we are computing the total number of ways to decode the string “23” we are computing the number of ways the string “3” can be decoded as well as when we are computing the number of ways the string “12” can be decoded we are again computing the number of ways the string “3” can be decoded . So we can avoid this by storing the result of every substring . Here we can identify each and every sub problem through the index of the string . So , if at any point of time if we have already computed the number of ways the substring can be decoded we can directly return the result and that leads to a lot of optimization .
Below is the C++ implementation
C++
Python3
Javascript
#include<bits/stdc++.h>using namespace std; int mod = 1e9 + 7; // function which returns the number of ways to decode the messageint decodeMessage(vector<int> &dp,int s,string &str,int n){    // an empty string can also form 1 valid decoding    if(s >= n)        return 1;         /*        if we have already computed the number of        ways to decode the substring return the        answer directly    */    if(dp[s] != -1)        return dp[s];         int num,tc;    num = tc = 0;    for(int i=s;i<n;i++)    {        // generate the number        num = num*10 + (str[i] - '0');         // validate the number        if(num >= 1 and num <= 26)        {            /*                since the number of ways to decode any string                depends on the result of                how the remaining string is decoded so get the                number of ways how the rest of the string can                be decoded            */            int c = decodeMessage(dp,i+1,str,n);             // add all the ways that the substring            // from the current index can be decoded            tc = (tc%mod + c%mod)%mod;        }         // leading 0’s or the number        // generated so far is greater than 26        // we can just stop the process        // as it can never be a part of our solution        else            break;    }     // store all the possible decodings and return the result    return (dp[s] = tc);}int CountWays(string str){    int n = str.size();     // empty string can form 1 valid decoding    if(n == 0)        return 1;     // dp vector to store the  number of ways    // to decode each and every substring    vector<int> dp(n,-1);     // return the result    return decodeMessage(dp,0,str,n);}int main(){    string str = "1234";    cout << CountWays(str) << endl;    return 0;}
mod = 1e9 + 7 # function which returns the number of ways to decode the messagedef decodeMessage(dp, s, str, n):       # an empty string can also form 1 valid decoding    if(s >= n):        return 1         # if we have already computed the number of        # ways to decode the substring return the        # answer directly     if(dp[s] != -1):        return dp[s]         num = 0    tc = 0    for i in range(s,n):        # generate the number        num = num*10 + (ord(str[i]) - ord('0'))         # validate the number        if(num >= 1 and num <= 26):                             # since the number of ways to decode any string                # depends on the result of                # how the remaining string is decoded so get the                # number of ways how the rest of the string can                # be decoded            c = decodeMessage(dp, i + 1, str, n)             # add all the ways that the substring            # from the current index can be decoded            tc = int((tc%mod + c%mod)%mod)         # leading 0’s or the number        # generated so far is greater than 26        # we can just stop the process        # as it can never be a part of our solution        else:            break     # store all the possible decodings and return the result    dp[s] = tc    return dp[s] def CountWays(str):     n = len(str)     # empty string can form 1 valid decoding    if(n == 0):        return 1     # dp vector to store the  number of ways    # to decode each and every substring    dp = [-1]*(n)     # return the result    return decodeMessage(dp, 0, str, n)   # driver codeif __name__ == "__main__" :    str = "1234"   print(CountWays(str))   # This code is contributed by shinjanpatra.
<script> const mod = 1e9 + 7; // function which returns the number of ways to decode the messagefunction decodeMessage(dp,s,str,n){    // an empty string can also form 1 valid decoding    if(s >= n)        return 1;         /*        if we have already computed the number of        ways to decode the substring return the        answer directly    */    if(dp[s] != -1)        return dp[s];         let num,tc;    num = tc = 0;    for(let i=s;i<n;i++)    {        // generate the number        num = num*10 + (str.charCodeAt(i) - '0'.charCodeAt(0));         // validate the number        if(num >= 1 && num <= 26)        {            /*                since the number of ways to decode any string                depends on the result of                how the remaining string is decoded so get the                number of ways how the rest of the string can                be decoded            */            let c = decodeMessage(dp,i+1,str,n);             // add all the ways that the substring            // from the current index can be decoded            tc = (tc%mod + c%mod)%mod;        }         // leading 0’s or the number        // generated so far is greater than 26        // we can just stop the process        // as it can never be a part of our solution        else            break;    }     // store all the possible decodings and return the result    return (dp[s] = tc);}function CountWays(str){    let n = str.length;     // empty string can form 1 valid decoding    if(n == 0)        return 1;     // dp vector to store the  number of ways    // to decode each and every substring    let dp = new Array(n).fill(-1);     // return the result    return decodeMessage(dp,0,str,n);}// driver code let str = "1234";document.write(CountWays(str),"</br>"); // This code is contributed by shinjanpatra.</script>
Output : 
3
Time Complexity   :  O ( N ) where N is the length of the string . As we are solving each and every sub – problem only once .
Space Complexity :  O ( N ) as we are using vector to store the result of each and every substring .
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