Decode Ways II

To solve this problem, you can use dynamic programming to keep track of the number of ways to decode the string at each position. The following Python code does just that:

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class Solution:
    def numDecodings(self, s: str) -> int:
        MOD = 10**9 + 7
        n = len(s)
        dp = [0] * (n + 1)
        dp[0] = 1

        # Function to calculate the number of ways to decode a single character
        def one_char(ch):
            return 9 if ch == '*' else (1 if ch != '0' else 0)
        
        # Function to calculate the number of ways to decode two characters
        def two_chars(ch1, ch2):
            if ch1 == '*' and ch2 == '*':
                return 15  # 9 combinations have 2 ways, 6 combinations have 1 way
            elif ch1 == '*':
                return 2 if ch2 <= '6' else 1
            elif ch2 == '*':
                if ch1 == '1':
                    return 9
                elif ch1 == '2':
                    return 6
                else:
                    return 0
            else:
                return int(ch1 + ch2) <= 26 and ch1 != '0'

        for i in range(1, n + 1):
            dp[i] = dp[i - 1] * one_char(s[i - 1]) % MOD
            if i > 1:
                dp[i] += dp[i - 2] * two_chars(s[i - 2], s[i - 1]) % MOD
                dp[i] %= MOD

        return dp[n]

This code uses dynamic programming to calculate the number of ways to decode the given string. It considers both single characters and pairs of characters, taking into account the special rules for the ‘*’ character. By working through the string and applying these rules at each step, the function computes the total number of valid decodings.

Identifying Problem Isomorphism

“Decode Ways II” is approximately isomorphic to “Distinct Subsequences II”.

In “Decode Ways II”, you are given a string s consisting of digits and “” characters. You need to count the number of ways to decode it where a digit from 1-9 can be mapped to A-I, a digit from 10-26 can be mapped to J-Z, and ‘’ can be mapped to 1-9 when it stands alone or 10-26 when paired with a preceding digit. The number of mappings can be large, so return it modulo 10^9 + 7.

“Distinct Subsequences II” presents a similar scenario where you are given a string s, and you need to count the number of distinct non-empty subsequences of s. Since the result may be large, return it modulo 10^9 + 7.

The reasoning behind this mapping is that both problems deal with counting different interpretations of a string under given rules. In both problems, a character or a combination of characters can be interpreted in different ways, and we need to count all possible interpretations. The problems require understanding of dynamic programming to maintain a count of the number of ways to decode or form subsequences up to each position in the string.

“Distinct Subsequences II” is more complex due to the additional requirement of the subsequences being distinct, which requires tracking of the last occurrence of each character. So, if we were to arrange them in terms of complexity, “Decode Ways II” would be simpler, followed by “Distinct Subsequences II”.

10 Prerequisite LeetCode Problems

This is related to dynamic programming and string processing. Here are some problems to build the skills necessary to tackle it:

91. Decode Ways: This is a simpler version of the problem that doesn’t involve the wildcard character. It’s the perfect starting point.
70. Climbing Stairs: While this problem is not about string processing, the dynamic programming approach is similar to “Decode Ways”.
198. House Robber: This problem helps practice dynamic programming techniques in a different context.
62. Unique Paths: This problem requires counting distinct ways to reach a destination, which is similar in nature to counting decoding options.
72. Edit Distance: This problem is a good way to practice dynamic programming with strings.
139. Word Break: This problem requires breaking a string into valid words, somewhat similar to decoding a string into valid characters.
10. Regular Expression Matching: This problem will help in handling the wildcard character in the “Decode Ways II” problem.
416. Partition Equal Subset Sum: This problem is another example of dynamic programming.
322. Coin Change: This problem involves finding ways to make up an amount, which can help you understand similar combinatorial problems.
647. Palindromic Substrings: This problem helps practice dynamic programming with strings.

These cover dynamic programming, handling strings, and dealing with the wildcard character, which are necessary to solve the “Decode Ways II” problem.

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class Solution:
    def numDecodings(self, s: str):
        if s[0] == '0': return 0
        cMap = {'0':0, '*': 9, '**': 15, '1*': 9, '2*': 6}
        for i in range(1, 27): cMap[str(i)] = 1
        for i in range(0, 7): cMap['*'+str(i)] = 2
        for i in range(7, 10): cMap['*'+str(i)] = 1
        
        dp = [0]*(len(s)+1)
        dp[0], dp[-1] = cMap[s[0]] , 1
        
        for i in range(1, len(s)):
            dp[i] += (cMap[s[i]]*dp[i-1] + cMap.get(s[i-1:i+1],0)*dp[i-2])%(10**9 + 7)
            if not dp[i]: return 0
            
        return dp[-2]

Problem Classification

Domain Categorization: This problem falls under the domain of Dynamic Programming and String Manipulation in Computer Science.

What Components:

We are given an encoded string s, consisting of digits (0-9) and ‘*’ characters.
Each digit can be mapped to a character, where ‘A’ maps to “1”, ‘B’ maps to “2”, …, ‘Z’ maps to “26”.
The ‘*’ character can represent any digit from ‘1’ to ‘9’.
We need to find the total number of ways to decode the string s, where decoding involves grouping the digits and mapping them back to characters. Note that the grouping (1 11 06) is invalid because “06” cannot be mapped into ‘F’ since “6” is different from “06”.
The final answer could be very large, hence we are required to return the answer modulo 10^9 + 7.

Problem Classification: This problem can be classified as a Counting Problem, where we need to find the number of ways to perform a certain operation (decoding in this case). Additionally, it’s a Dynamic Programming problem as the solution of a subproblem will help in solving larger problems. In essence, the number of ways to decode the string up to a certain index will depend on the number of ways to decode the string up to previous indices. Lastly, it can also be seen as a string parsing problem where we need to interpret and make decisions based on characters in the input string.

Combinatorics/Counting Problem: The main goal is to determine the total number of valid ways to decode the given string, which involves exploring all potential groupings that conform to the stated rules.
Dynamic Programming Problem: The problem is solved using a dynamic programming approach, where the solution to the current problem depends on the solution to sub-problems. This is typically used when counting possibilities, as with this problem.
String Processing Problem: The input is a string that represents an encoded message. This string needs to be processed and interpreted according to certain rules. This involves both interpreting individual characters (‘digits’) and pairs of characters.
Number Theory Problem: The string is essentially a coded number and needs to be decoded into letters according to a specific mapping. This involves understanding and applying the rules of the number-letter mapping.
Modular Arithmetic Problem: The solution involves performing operations modulo 1,000,000,007 to manage the large numbers that may be produced by the problem.

These classifications help to indicate what strategies and techniques might be relevant when solving the problem.

Language Agnostic Coding Drills

Here are the units of learning identified in this code snippet:

String Manipulation: Understanding how to create, index, and manipulate strings. In this case, the string represents a sequence of encoded numbers, so being able to correctly handle and process it is key to solving the problem.
Data Structures (Dictionary/Map): Creating and using dictionaries/maps to associate keys with values. This concept is used in this problem to map encoded strings to the number of ways they can be decoded.
Conditionals: Using conditional statements to perform different actions based on certain conditions. Here, they’re used to handle edge cases and to determine how to count the number of decodings.
Loops: Looping over sequences of data. In this case, you’re looping over the characters in a string and performing actions for each one.
Dynamic Programming: This problem-solving approach is used when a problem can be broken down into smaller, overlapping sub-problems. The solutions to these smaller problems can be stored and reused to solve the larger problem.

In this problem, the approach could be summarized as follows:

First, you initialize a dictionary that maps each valid encoding to the number of ways it can be decoded. This includes single-character encodings (‘1’ through ‘9’, and ‘’) as well as two-character encodings (combinations of digits and ‘’).
Next, you create an array to store the number of ways to decode the string up to each position. The base cases are initialized according to the first character(s) in the string.
Then, for each subsequent character in the string, you calculate the number of ways to decode the string up to that position by considering the current character and the previous one. This involves looking up the number of decodings in the dictionary and multiplying by the number of decodings for the previous position(s).
Finally, you return the number of ways to decode the entire string, which is stored in the last position of the array.

Practice the individual skills listed above separately (string manipulation, using dictionaries, writing conditionals and loops, understanding dynamic programming), then solve similar problems that involve decoding strings or sequences. This problem involves a good amount of edge case handling and error checking, so practicing those skills could be useful as well.

Targeted Drills in Python

Here are some Python coding drills that break down the problem into smaller learning units. Remember to import any necessary modules at the beginning of your Python scripts.

Working with Strings: Practice accessing and manipulating strings, as these skills are critical to decoding the message.

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s = "12345*"
# Accessing elements of the string
print(s[0])  # Expected output: "1"
# Slicing the string
print(s[1:3])  # Expected output: "23"

Dictionary Usage: Familiarize yourself with Python dictionaries. You’ll use these to map encoded characters to their possible counts.

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cMap = {'*': 9, '**': 15}
# Accessing elements
print(cMap['*'])  # Expected output: 9
# Adding elements
cMap['1*'] = 9
print(cMap)  # Expected output: {'*': 9, '**': 15, '1*': 9}

Dynamic Programming: Learn to work with dynamic programming by creating an array and filling it up with values based on previous values.

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dp = [0]*10
dp[0] = 1
dp[1] = 2
for i in range(2, 10):
    dp[i] = dp[i-1] + dp[i-2]
print(dp)  # Expected output: [1, 2, 3, 5, 8, 13, 21, 34, 55, 89]

Modulo Operations: Get comfortable with modulo operations, which are essential when working with large numbers in Python.
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big_number = 1000000008 print(big_number % (10**9 + 7)) # Expected output: 1
Working with ‘*’: ‘*’ character can represent any digit from ‘1’ to ‘9’. Practice working with such strings.
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s = "1*" # Check if '*' exists in the string print('*' in s) # Expected output: True

Problem Specific Drill: Understand how to find number of ways to decode a single character and a pair of characters (with and without ‘*’).

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# Single character
s1 = '1'
# Pair of characters
s2 = '12'
s3 = '1*'
s4 = '2*'
s5 = '26'
# Dictionary to keep track of ways to decode
cMap = {'0':0, '*': 9, '**': 15, '1*': 9, '2*': 6}
for i in range(1, 27): cMap[str(i)] = 1
for i in range(0, 7): cMap['*'+str(i)] = 2
for i in range(7, 10): cMap['*'+str(i)] = 1
# Print ways to decode
print(cMap[s1])  # Expected output: 1
print(cMap[s2])  # Expected output: 1
print(cMap[s3])  # Expected output: 9
print(cMap[s4])  # Expected output: 6
print(cMap[s5])  # Expected output: 1

After going through these exercises, you should be ready to tackle the complete problem.

Problem Analysis and Key Insights

What are the key insights from analyzing the problem statement?

Problem Boundary

What is the scope of this problem?

How to establish the boundary of this problem?

Problem Classification

Problem Statement: A message containing letters from A-Z can be encoded into numbers using the following mapping:

‘A’ -> “1” ‘B’ -> “2” … ‘Z’ -> “26” To decode an encoded message, all the digits must be grouped then mapped back into letters using the reverse of the mapping above (there may be multiple ways). For example, “11106” can be mapped into:

“AAJF” with the grouping (1 1 10 6) “KJF” with the grouping (11 10 6) Note that the grouping (1 11 06) is invalid because “06” cannot be mapped into ‘F’ since “6” is different from “06”.

In addition to the mapping above, an encoded message may contain the ‘’ character, which can represent any digit from ‘1’ to ‘9’ (‘0’ is excluded). For example, the encoded message “1” may represent any of the encoded messages “11”, “12”, “13”, “14”, “15”, “16”, “17”, “18”, or “19”. Decoding “1*” is equivalent to decoding any of the encoded messages it can represent.

Given a string s consisting of digits and ‘*’ characters, return the number of ways to decode it.

Since the answer may be very large, return it modulo 109 + 7.

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 ‘*’.

Analyze the provided problem statement. Categorize it based on its domain, ignoring ‘How’ it might be solved. Identify and list out the ‘What’ components. Based on these, further classify the problem. Explain your categorizations.

Distilling the Problem to Its Core Elements

Can you identify the fundamental concept or principle this problem is based upon? Please explain. What is the simplest way you would describe this problem to someone unfamiliar with the subject? What is the core problem we are trying to solve? Can we simplify the problem statement? Can you break down the problem into its key components? What is the minimal set of operations we need to perform to solve this problem?

Visual Model of the Problem

How to visualize the problem statement for this problem?

Problem Restatement

Could you start by paraphrasing the problem statement in your own words? Try to distill the problem into its essential elements and make sure to clarify the requirements and constraints. This exercise should aid in understanding the problem better and aligning our thought process before jumping into solving it.

Abstract Representation of the Problem

Could you help me formulate an abstract representation of this problem?

Given this problem, how can we describe it in an abstract way that emphasizes the structure and key elements, without the specific real-world details?

Terminology

Are there any specialized terms, jargon, or technical concepts that are crucial to understanding this problem or solution? Could you define them and explain their role within the context of this problem?

Problem Simplification and Explanation

Could you please break down this problem into simpler terms? What are the key concepts involved and how do they interact? Can you also provide a metaphor or analogy to help me understand the problem better?

Constraints

Given the problem statement and the constraints provided, identify specific characteristics or conditions that can be exploited to our advantage in finding an efficient solution. Look for patterns or specific numerical ranges that could be useful in manipulating or interpreting the data.

What are the key insights from analyzing the constraints?

Case Analysis

Could you please provide additional examples or test cases that cover a wider range of the input space, including edge and boundary conditions? In doing so, could you also analyze each example to highlight different aspects of the problem, key constraints and potential pitfalls, as well as the reasoning behind the expected output for each case? This should help in generating key insights about the problem and ensuring the solution is robust and handles all possible scenarios.

Provide names by categorizing these cases

What are the edge cases?

What are the key insights from analyzing the different cases?

Identification of Applicable Theoretical Concepts

Can you identify any mathematical or algorithmic concepts or properties that can be applied to simplify the problem or make it more manageable? Think about the nature of the operations or manipulations required by the problem statement. Are there existing theories, metrics, or methodologies in mathematics, computer science, or related fields that can be applied to calculate, measure, or perform these operations more effectively or efficiently?

Simple Explanation

Can you explain this problem in simple terms or like you would explain to a non-technical person? Imagine you’re explaining this problem to someone without a background in programming. How would you describe it? If you had to explain this problem to a child or someone who doesn’t know anything about coding, how would you do it? In layman’s terms, how would you explain the concept of this problem? Could you provide a metaphor or everyday example to explain the idea of this problem?

Problem Breakdown and Solution Methodology

Given the problem statement, can you explain in detail how you would approach solving it? Please break down the process into smaller steps, illustrating how each step contributes to the overall solution. If applicable, consider using metaphors, analogies, or visual representations to make your explanation more intuitive. After explaining the process, can you also discuss how specific operations or changes in the problem’s parameters would affect the solution? Lastly, demonstrate the workings of your approach using one or more example cases.

Inference of Problem-Solving Approach from the Problem Statement

Can you identify the key terms or concepts in this problem and explain how they inform your approach to solving it? Please list each keyword and how it guides you towards using a specific strategy or method.

How did you infer from the problem statement that this problem can be solved using ?

Simple Explanation of the Proof

I’m having trouble understanding the proof of this algorithm. Could you explain it in a way that’s easy to understand?

Could you please provide a stepwise refinement of our approach to solving this problem?
How can we take the high-level solution approach and distill it into more granular, actionable steps?
Could you identify any parts of the problem that can be solved independently?
Are there any repeatable patterns within our solution?

Solution Approach and Analysis

Identify Invariant

What is the invariant in this problem?

Identify Loop Invariant

What is the loop invariant in this problem?

Thought Process

Can you explain the basic thought process and steps involved in solving this type of problem?

Explain the thought process by thinking step by step to solve this problem from the problem statement and code the final solution. Write code in Python3. What are the cues in the problem statement? What direction does it suggest in the approach to the problem? Generate insights about the problem statement.

Establishing Preconditions and Postconditions

Parameters:
- What are the inputs to the method?
- What types are these parameters?
- What do these parameters represent in the context of the problem?
Preconditions:
- Before this method is called, what must be true about the state of the program or the values of the parameters?
- Are there any constraints on the input parameters?
- Is there a specific state that the program or some part of it must be in?
Method Functionality:
- What is this method expected to do?
- How does it interact with the inputs and the current state of the program?
Postconditions:
- After the method has been called and has returned, what is now true about the state of the program or the values of the parameters?
- What does the return value represent or indicate?
- What side effects, if any, does the method have?
Error Handling:
- How does the method respond if the preconditions are not met?
- Does it throw an exception, return a special value, or do something else?

Problem Decomposition

Problem Understanding:
- Can you explain the problem in your own words? What are the key components and requirements?
Initial Breakdown:
- Start by identifying the major parts or stages of the problem. How can you break the problem into several broad subproblems?
Subproblem Refinement:
- For each subproblem identified, ask yourself if it can be further broken down. What are the smaller tasks that need to be done to solve each subproblem?
Task Identification:
- Within these smaller tasks, are there any that are repeated or very similar? Could these be generalized into a single, reusable task?
Task Abstraction:
- For each task you’ve identified, is it abstracted enough to be clear and reusable, but still makes sense in the context of the problem?
Method Naming:
- Can you give each task a simple, descriptive name that makes its purpose clear?
Subproblem Interactions:
- How do these subproblems or tasks interact with each other? In what order do they need to be performed? Are there any dependencies?

From Brute Force to Optimal Solution

Could you please begin by illustrating a brute force solution for this problem? After detailing and discussing the inefficiencies of the brute force approach, could you then guide us through the process of optimizing this solution? Please explain each step towards optimization, discussing the reasoning behind each decision made, and how it improves upon the previous solution. Also, could you show how these optimizations impact the time and space complexity of our solution?

Code Explanation and Design Decisions

Identify the initial parameters and explain their significance in the context of the problem statement or the solution domain.
Discuss the primary loop or iteration over the input data. What does each iteration represent in terms of the problem you’re trying to solve? How does the iteration advance or contribute to the solution?
If there are conditions or branches within the loop, what do these conditions signify? Explain the logical reasoning behind the branching in the context of the problem’s constraints or requirements.
If there are updates or modifications to parameters within the loop, clarify why these changes are necessary. How do these modifications reflect changes in the state of the solution or the constraints of the problem?
Describe any invariant that’s maintained throughout the code, and explain how it helps meet the problem’s constraints or objectives.
Discuss the significance of the final output in relation to the problem statement or solution domain. What does it represent and how does it satisfy the problem’s requirements?

Remember, the focus here is not to explain what the code does on a syntactic level, but to communicate the intent and rationale behind the code in the context of the problem being solved.

Coding Constructs

Consider the following piece of complex software code.

What are the high-level problem-solving strategies or techniques being used by this code?
If you had to explain the purpose of this code to a non-programmer, what would you say?
Can you identify the logical elements or constructs used in this code, independent of any programming language?
Could you describe the algorithmic approach used by this code in plain English?
What are the key steps or operations this code is performing on the input data, and why?
Can you identify the algorithmic patterns or strategies used by this code, irrespective of the specific programming language syntax?

Language Agnostic Coding Drills

Your mission is to deconstruct this code into the smallest possible learning units, each corresponding to a separate coding concept. Consider these concepts as unique coding drills that can be individually implemented and later assembled into the final solution.

Dissect the code and identify each distinct concept it contains. Remember, this process should be language-agnostic and generally applicable to most modern programming languages.
Once you’ve identified these coding concepts or drills, list them out in order of increasing difficulty. Provide a brief description of each concept and why it is classified at its particular difficulty level.
Next, describe the problem-solving approach that would lead from the problem statement to the final solution. Think about how each of these coding drills contributes to the overall solution. Elucidate the step-by-step process involved in using these drills to solve the problem. Please refrain from writing any actual code; we’re focusing on understanding the process and strategy.

Targeted Drills in Python

Now that you’ve identified and ordered the coding concepts from a complex software code in the previous exercise, let’s focus on creating Python-based coding drills for each of those concepts.

Begin by writing a separate piece of Python code that encapsulates each identified concept. These individual drills should illustrate how to implement each concept in Python. Please ensure that these are suitable even for those with a basic understanding of Python.
In addition to the general concepts, identify and write coding drills for any problem-specific concepts that might be needed to create a solution. Describe why these drills are essential for our problem.
Once all drills have been coded, describe how these pieces can be integrated together in the right order to solve the initial problem. Each drill should contribute to building up to the final solution.

Remember, the goal is to not only to write these drills but also to ensure that they can be cohesively assembled into one comprehensive solution.

Decode Ways II

Identifying Problem Isomorphism

10 Prerequisite LeetCode Problems

Problem Classification

Language Agnostic Coding Drills

Targeted Drills in Python

Problem Analysis and Key Insights

Problem Boundary

Problem Classification

Distilling the Problem to Its Core Elements

Visual Model of the Problem

Problem Restatement

Abstract Representation of the Problem

Terminology

Problem Simplification and Explanation

Constraints

Case Analysis

Identification of Applicable Theoretical Concepts

Simple Explanation

Problem Breakdown and Solution Methodology

Inference of Problem-Solving Approach from the Problem Statement

Simple Explanation of the Proof

Stepwise Refinement

Solution Approach and Analysis

Identify Invariant

Identify Loop Invariant

Thought Process

Establishing Preconditions and Postconditions

Problem Decomposition

From Brute Force to Optimal Solution

Code Explanation and Design Decisions

Coding Constructs

Language Agnostic Coding Drills

Targeted Drills in Python

Q&A

Similar Problems