Decode Ways II

Problem Statement

In this challenge, we are given a string s that represents a number-encoded message where each character ranges from '1' to '9' or can be a wildcard character '*'. This wildcard represents any number between '1' and '9'. Each character or combination of two characters in the string can potentially correspond to a letter in the alphabet according 'A' to 'Z' with 'A' mapped to '1' and 'Z' mapped to '26'. For example, the string "12" could correspond to 'AB' (1,2) or 'L' (12).

The task is to determine the total number of ways the string s can be decoded considering all possible interpretations of the wildcard character. Valid encodings are those that map to letters without leading zeroes, and it's crucial to consider that combinations like single and grouped digits can yield different respective characters.

Given that the computed number could be large, we must return the result modulo 10^9 + 7.

Examples

Example 1

Input:

s = "*"

Output:

9

Explanation:

The encoded message can represent any of the encoded messages "1", "2", "3", "4", "5", "6", "7", "8", or "9".
Each of these can be decoded to the strings "A", "B", "C", "D", "E", "F", "G", "H", and "I" respectively.
Hence, there are a total of 9 ways to decode "*".

Example 2

Input:

s = "1*"

Output:

18

Explanation:

The encoded message can represent any of the encoded messages "11", "12", "13", "14", "15", "16", "17", "18", or "19".
Each of these encoded messages have 2 ways to be decoded (e.g. "11" can be decoded to "AA" or "K").
Hence, there are a total of 9 * 2 = 18 ways to decode "1*".

Example 3

Input:

s = "2*"

Output:

15

Explanation:

The encoded message can represent any of the encoded messages "21", "22", "23", "24", "25", "26", "27", "28", or "29".
"21", "22", "23", "24", "25", and "26" have 2 ways of being decoded, but "27", "28", and "29" only have 1 way.
Hence, there are a total of (6 * 2) + (3 * 1) = 12 + 3 = 15 ways to decode "2*".

Constraints

• 1 <= s.length <= 105
• s[i] is a digit or '*'.

Approach and Intuition

The problem revolves around understanding and applying dynamic programming on strings, mixed with combinatorial logic given the wildcard capability. Here's a breakdown of the thought process:

1. Basic Mapping Understanding: Firstly, grasp how digits and combinations lead to character mappings based on the rules provided ('1' to 'Z' -> '26').

2. Handling of Wildcards:

   • A single '*' can represent any digit from '1' to '9', influencing individual character counts.
   • Pairings like '1*' indicate two character positions that can enhance possible encodings. 1* could represent '11' to '19', translating to either one or two alphabet letters depending upon grouping.

3. Dynamic Programming Setup:

   • Create a DP array where dp[i] represents the number of ways to decode the substring of s up to index i.
   • Initialize with dp[0] = 1 (an empty string has one way of decoding - doing nothing).

4. Filling DP Table:

   • For each index, consider if the character is a digit, a '', or a part of a pair including ''.
   • Update your dp values based on single digit interpretations and possible dual character pairings both involving numbers and wildcards.

5. Calculating Combinations:

   • Utilize combinatorial counts for wildcard expansions (for instance, '1*' contributing to multiple dual-character possibilities).
   • Sum up valid combinations by involving previous substrings' decoding counts.

6. Modulo Operation: Every update todp[i] should be taken modulo 10^9 + 7 to handle large numbers and prevent integer overflows.

By following these steps in a careful and structured manner, it becomes feasible to navigate through potentially massive combinatorial spaces efficiently, using dynamic programming's overlapping subproblems and optimal substructure properties. Each step builds on prior results, ensuring that all potential decodings are considered up to any index i in string s.

Solutions

public class Solution {
    int MOD = 1000000007;
    public int decodeWays(String str) {
        long prev = 1, curr = str.charAt(0) == '*' ? 9 : str.charAt(0) == '0' ? 0 : 1;
        for (int i = 1; i < str.length(); i++) {
            long temp = curr;
            if (str.charAt(i) == '*') {
                curr = 9 * curr % MOD;
                if (str.charAt(i - 1) == '1')
                    curr = (curr + 9 * prev) % MOD;
                else if (str.charAt(i - 1) == '2')
                    curr = (curr + 6 * prev) % MOD;
                else if (str.charAt(i - 1) == '*')
                    curr = (curr + 15 * prev) % MOD;
            } else {
                curr = str.charAt(i) != '0' ? curr : 0;
                if (str.charAt(i - 1) == '1')
                    curr = (curr + prev) % MOD;
                else if (str.charAt(i - 1) == '2' && str.charAt(i) <= '6')
                    curr = (curr + prev) % MOD;
                else if (str.charAt(i - 1) == '*')
                    curr = (curr + (str.charAt(i) <= '6' ? 2 : 1) * prev) % MOD;
            }
            prev = temp;
        }
        return (int) curr;
    }
}