Check if an Original String Exists Given Two Encoded Strings

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class Solution:
    def __init__(self):
        self.s1 = ""
        self.s2 = ""
        self.memo = [[[None] * 2000 for _ in range(41)] for _ in range(41)]

    def possiblyEquals(self, s1: str, s2: str) -> bool:
        self.s1 = s1
        self.s2 = s2
        return self.dfs(0, 0, 0)

    def dfs(self, i: int, j: int, diff: int) -> bool:
        if i >= len(self.s1) and j >= len(self.s2) and diff == 0:
            return True
        if self.memo[i][j][diff + 1000] is not None:
            return self.memo[i][j][diff + 1000]

        res = False
        if i < len(self.s1):
            if self.s1[i].isdigit():
                count, value = 0, 0
                while i + count < len(self.s1) and count < 3 and self.s1[i + count].isdigit():
                    value = value * 10 + int(self.s1[i + count])
                    count += 1
                    if self.dfs(i + count, j, diff - value):
                        res = True
            else:
                if diff > 0:
                    if self.dfs(i + 1, j, diff - 1):
                        res = True
                elif diff == 0 and j < len(self.s2) and self.s1[i] == self.s2[j]:
                    if self.dfs(i + 1, j + 1, diff):
                        res = True

        if j < len(self.s2):
            if self.s2[j].isdigit():
                count, value = 0, 0
                while j + count < len(self.s2) and count < 3 and self.s2[j + count].isdigit():
                    value = value * 10 + int(self.s2[j + count])
                    count += 1
                    if self.dfs(i, j + count, diff + value):
                        res = True
            elif diff < 0 and self.dfs(i, j + 1, diff + 1):
                res = True

        self.memo[i][j][diff + 1000] = res
        return res

The problem “2060. Check if an Original String Exists Given Two Encoded Strings” involves understanding of strings, hash maps, and sliding window technique. Here are 10 problems to prepare for it:

  1. Two Sum: This problem introduces the usage of hash maps.

  2. Longest Substring Without Repeating Characters: This problem introduces the sliding window technique.

  3. Permutation in String: This problem combines the usage of hash maps and the sliding window technique.

  4. Find All Anagrams in a String: This problem uses both hash maps and sliding window technique on strings.

  5. First Unique Character in a String: This problem uses hash maps in the context of strings.

  6. Subarray Sum Equals K: This problem involves the sliding window technique.

  7. Minimum Window Substring: This problem is a harder version of sliding window problems and also involves working with strings and hash maps.

  8. Longest Substring with At Most Two Distinct Characters: This problem uses the sliding window technique with a variation.

  9. Contains Duplicate II: This problem involves the usage of hash maps to track the indices of elements in an array.

  10. Find Anagram Mappings: This problem involves finding mappings between two arrays using hash maps.

These cover how to manipulate strings and use data structures like hash maps, and the sliding window technique, which are crucial for solving “2060. Check if an Original String Exists Given Two Encoded Strings”.

Clarification Questions

What are the clarification questions we can ask about this problem?

Identifying Problem Isomorphism

shitgpt

Problem Classification

String Manipulation and Encoding

What

  1. Input: Two encoded strings s1 and s2 consisting of digits 1-9 (inclusive) and lowercase English letters.

  2. Operation: Encoding mechanism that allows for substring splits, and replacing these substrings with their lengths.

  3. Constraints:

    • Length of s1 and s2 is between 1 and 40.
    • Number of consecutive digits in s1 and s2 does not exceed 3.

  4. Goal: To determine if there exists a single original string that can be encoded into both s1 and s2.

  5. Output: Boolean value - true if such an original string exists, otherwise false.

  6. Type: This is a decision problem.

  7. Complexity:

    • Logical Complexity: Medium, as you need to think through possible split and replace combinations.
    • Computational Complexity: Not explicitly stated, but based on the constraints, the solution should likely be efficient.

The problem falls under the domain of string manipulation and encoding. The problem asks you to work backwards from an encoded string to see if there could be a common original string for both s1 and s2 using a specific encoding mechanism. The ‘What’ components clearly indicate the different elements you need to consider - the input strings, the operation (encoding), constraints, the goal, and the output.

Given the logical steps involved in decoding and comparison, this problem can be categorized as a decision problem with medium logical complexity. Computational efficiency is implied but not directly addressed in the problem statement.

Visual Model of the Problem

Visualizing this problem can be helpful for understanding its complexities. Consider the following ways to visualize it:

1. Flowchart

Create a flowchart that starts with the original string and demonstrates the encoding steps for both s1 and s2. This can visually show how different splits and replacements can lead to the same or different encoded strings.

2. Tree Diagram

For a given encoded string (either s1 or s2), imagine a tree where each node represents a possible original string, and branches represent the choices at each encoding step. This can quickly show the multiplicity of possible original strings for a single encoded string.

3. State Transition Diagram

You can visualize the problem as a state transition diagram, where each state represents a partial original string, and transitions represent applying an encoding step. This can clarify how you might move from one possible original string to another.

4. Side-by-side Comparison

Take s1 and s2 and lay them out side by side. Start decoding them in a parallel fashion, writing down all possible substrings they could represent. This can make it easier to see if they can converge to a single original string.

5. Color Coding

Use colors to highlight corresponding segments between s1 and s2. This can be helpful to quickly identify if a set of substrings in s1 could potentially match a set of substrings in s2.

6. Grid or Matrix

You can use a grid to compare all possible original strings for s1 against all possible original strings for s2. Each cell in the grid would represent a match or mismatch between potential original strings.

Key Takeaways

  • A visual representation can break down the complex steps into manageable parts.
  • It will help you to grasp the full range of possibilities for the original string from each of s1 and s2.
  • Using one or a combination of these methods can aid in devising an algorithmic solution for the problem.

Problem Restatement

You’re given two encoded strings, s1 and s2, made up of numbers and lowercase English letters. These strings are created through a specific encoding process that starts with an original string. This original string is broken into several non-empty substrings. Then, some of these substrings can be replaced by their lengths. Finally, the remaining substrings and numbers are concatenated to form the encoded string.

Your task is to figure out if there could be one original string that, when subjected to this encoding process, could yield both s1 and s2.

Constraints:

  • The length of each encoded string will be between 1 and 40 characters.
  • The strings contain only numbers between 1-9 and lowercase English letters.
  • Consecutive numbers in the encoded strings will not be longer than 3 digits.

The goal is to return true if such an original string exists, and false otherwise.

Abstract Representation of the Problem

Let’s formulate an abstract representation of this problem:

Definitions:

  • Original String (O): A sequence consisting of lowercase alphabets.
  • Encoded String (E): A sequence consisting of lowercase alphabets and digits. It is derived from an Original String (O) through a specific transformation process.
  • Transformation Function (T): A function that takes an Original String (O) and produces an Encoded String (E). The function performs the following operations:
    1. Partition (O) into non-empty substrings.
    2. Optionally replace some substrings with their length.
    3. Concatenate the substrings and numbers to produce (E).

Problem Statement:

Given two Encoded Strings ( E_1 ) and ( E_2 ), determine if there exists an Original String ( O ) such that ( T(O) = E_1 ) and ( T(O) = E_2 ).

Constraints:

  • ( |E_1|, |E_2| \leq 40 )
  • Elements of ( E_1 ) and ( E_2 ) are from the set ( {1, 2, …, 9, a, b, …, z} )
  • The number of consecutive digits in ( E_1 ) and ( E_2 ) does not exceed 3.

Goal:

Find if such an ( O ) exists. If yes, return true. If not, return false.

This abstract representation strips away the specific context and focuses on the structural elements of the problem: the Original String, the Encoded Strings, the Transformation Function, and the Constraints.

Terminology

Understanding some specialized terms can help clarify this problem:

  1. String Encoding: The process of transforming data into a different format. Here, it refers to the transformation of an original string into an encoded string through specific operations (substring partition and replacement by length).

  2. Substring: A contiguous sequence of characters within a string. Understanding how to partition a string into substrings is crucial for solving this problem.

  3. Concatenation: The operation of linking things together. In this context, it refers to the action of combining substrings and/or their lengths to form the encoded string.

  4. Isomorphism: A concept where two structures can become identical under certain conditions. This problem is an investigation into whether two encoded strings could be isomorphic, meaning they can be mapped to a common original string.

  5. Decision Problem: A problem that results in a yes/no answer. In this case, the problem asks whether a common original string exists for two given encoded strings, so it’s a decision problem.

  6. Constraint Programming: The practice of solving problems by specifying constraints or conditions that the solution must adhere to. The constraints on string length and character set are crucial here.

  7. Boolean Value: A type of value that can be either true or false. The solution to this problem is a Boolean value indicating the existence or non-existence of a common original string.

  8. Backtracking: A general algorithm for finding solutions by exploring each possibility until finding a solution or until the possibility has been ascertained to not work. This concept might be part of the solution, as you may need to backtrack to explore different partitioning and replacement options.

Understanding these terms clarifies what steps are involved in encoding the strings, what the constraints are, and what kind of problem you are trying to solve.

Problem Simplification and Explanation

Let’s break down this problem into its core elements and make it relatable with an analogy.

Core Elements:

  1. Original String: Think of this as the “real story” that you start with.
  2. Encoded String: This is like a “summary” of the real story. You create it by taking some parts of the story and either keeping them as is or summarizing them into numbers (the length of those parts).
  3. Transformation: This is the act of turning your real story into its summary.

Key Concepts:

  • Splitting: Breaking the original story into smaller chunks or scenes.
  • Replacement: Summarizing some of those scenes by stating how long they are.
  • Concatenation: Putting those chunks and summaries back together to form the summary.

Interaction:

  1. You start with a full story (Original String).
  2. You break it into smaller scenes (Splitting).
  3. You decide to summarize some scenes by stating their length (Replacement).
  4. You compile these scenes and summaries to create a complete summary (Encoded String).

Analogy:

Imagine you have a jigsaw puzzle (Original String). Each piece of the puzzle is a part of the story.

  1. Splitting: You group some puzzle pieces together based on a theme, much like chapters in a story.
  2. Replacement: Instead of showing the grouped pieces, you put a label stating how many pieces are in each group.
  3. Concatenation: You put these labeled groups and individual pieces in a line to create a summary of the puzzle.

Your task is like having two different summaries (Encoded Strings) and figuring out if they could be summaries of the same original jigsaw puzzle (Original String).

By understanding these key concepts and their interactions, you can tackle the problem of whether two encoded strings can represent the same original string.

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.

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?

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

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

Stepwise Refinement

  1. Could you please provide a stepwise refinement of our approach to solving this problem?

  2. How can we take the high-level solution approach and distill it into more granular, actionable steps?

  3. Could you identify any parts of the problem that can be solved independently?

  4. Are there any repeatable patterns within our solution?

Solution Approach and Analysis

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.

Thought Process

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.

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?

Coding Constructs

  1. What are the high-level problem-solving strategies or techniques being used by this code?

  2. If you had to explain the purpose of this code to a non-programmer, what would you say?

  3. Can you identify the logical elements or constructs used in this code, independent of any programming language?

  4. Could you describe the algorithmic approach used by this code in plain English?

  5. What are the key steps or operations this code is performing on the input data, and why?

  6. 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.

  1. 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.

  2. 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.

  3. 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.

  1. 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.

  2. 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.

  3. 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.

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