Check If Word Is Valid After Substitutions

To determine if a string s is valid, one approach is to iteratively remove all occurrences of the substring “abc” from s until no more such removals can be made. If s becomes an empty string, it is valid; otherwise, it is not.

The process can be visualized as a stack. Every time we encounter a character, we push it onto the stack. When the top three elements of the stack form the sequence “abc”, we pop these three elements out, effectively removing this substring from the string.

Here’s the approach:

Initialize an empty stack.
Iterate over each character in the string s.
- Push the character onto the stack.
- If the top three elements of the stack form the sequence “abc”, pop these three elements out.
After processing all characters, check if the stack is empty. If it is, the string is valid; otherwise, it’s not.

Python solution:

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class Solution:
    def isValid(self, s: str) -> bool:
        stack = []

        for char in s:
            stack.append(char)
            while len(stack) >= 3 and stack[-3:] == ['a', 'b', 'c']:
                stack.pop()
                stack.pop()
                stack.pop()

        return len(stack) == 0

This approach ensures we check and eliminate all substrings “abc” as soon as they form, ensuring that the string is being built correctly.

10 Prerequisite LeetCode Problems

For this, the following are a good preparation:

“20. Valid Parentheses” - This problem provides the concept of using stack to check validity based on order, which is a key approach in the main problem.
“155. Min Stack” - This problem is about implementing and using a stack, a structure you’ll use to solve problem 1003.
“316. Remove Duplicate Letters” - This problem introduces the concept of removing characters to form a valid string.
“394. Decode String” - This problem involves manipulating a string based on certain conditions.
“1021. Remove Outermost Parentheses” - It teaches how to remove unnecessary parts in a string, a concept relevant to the current problem.
“1047. Remove All Adjacent Duplicates In String” - This problem practices identifying and removing adjacent duplicates which is a core concept of the main problem.
“1209. Remove All Adjacent Duplicates in String II” - This problem builds upon problem 1047 with an additional condition for removing duplicates.
“682. Baseball Game” - Though the problem context is different, it also uses a stack to solve the problem, which is the main approach of the target problem.
“224. Basic Calculator” - This problem involves complex string manipulation and calculation, providing a good foundation for string operation skills.
“227. Basic Calculator II” - This problem adds more complexity to problem 224, thus helps to improve problem-solving skills.

These cover how to use stacks, manipulate strings, and validate strings based on certain conditions, which are key skills when solving the “1003. Check If Word Is Valid After Substitutions” problem.

Clarification Questions

Here are some potential clarification questions that could be asked about this problem:

Can “abc” be inserted multiple times consecutively, like “aaaabcabcabc”? Or must insertions be separated?
Can s contain characters other than ‘a’, ‘b’ and ‘c’?
What should the behavior be for an empty input string s? Should this be considered valid or invalid?
Can the insertions be done in any order to generate s? Or must it follow a specific sequence?
Is it guaranteed that s will only contain valid characters ‘a’, ‘b’ and ‘c’?
For very large input strings s, what size limits should we assume?
Are we returning the full sequence of insertions, or just a boolean of whether s is valid?
Does the case (upper vs lower) of characters matter for validity?
Can we assume the input string s is ASCII or Unicode?
Is performance an important consideration? What constraints exist?

Asking questions like these would help clarify ambiguity, reveal hidden assumptions, and illuminate corer aspects of the problem. This enables us to design a more robust, correct, and optimized solution within the problem’s constraints.

Problem Analysis and Key Insights

Here are some key insights gained from analyzing the problem statement:

This problem involves incrementally building up a string by inserting a fixed substring, rather than just checking the final string itself.
We can make use of the fixed inserted substring pattern “abc” to our advantage.
The core challenge is determining the reachability of s from empty string t by repeated insertions.
Order and sequence of insertions does not matter as long as s is achievable.
We only care about validity, not the actual sequence of insertions or intermediate strings.
The constraints allow us to make assumptions about character set (‘a’, ‘b’, ‘c’) and string size limits.
Checking partial matches of the “abc” pattern within s can help determine validity.
Certain strings like “ac” can be preemptively identified as invalid since “abc” can never create them.
We can disregard duplicate insertions or insertion ordering due to commutativity.

Overall, these insights let us simplify the problem space by focusing just on the end validity rather than tracking sequence. We can leverage insights around pattern matching and incremental construction to efficiently determine validity.

Problem Boundary

Based on the problem statement and analysis, the scope of this problem is:

Input space - A string s containing only characters ‘a’, ‘b’, ‘c’ and no more than 20,000 characters
Output - A boolean value indicating if s is valid or not
Rules - s is valid if it can be constructed by inserting “abc” into an initially empty string
Objective - Determine validity, not the actual sequence of insertions or intermediate strings
Non-goals - Finding optimal insertion order, tracking full sequence, handling arbitrary strings or insertions

So in summary, the scope is limited to:

Checking only final validity of s rather than full generation steps
Handling strings with a fixed limited character set
Determining if “abc” insertions can construct s, without tracking details
Returning a boolean for overall validity

Anything beyond a simple validity check like optimal construction order, sequence tracking, complex strings, etc. is out of scope. The core focus is on efficient reachability determination for the given string and operation.

Here are some ways to establish the boundaries and scope of this problem:

Inputs:

The input is restricted to strings containing only ‘a’, ‘b’ and ‘c’ characters. No other inputs are supported.
Insertion operation is limited to only the fixed string “abc”. No other operations allowed.
Maximum input string length is 20,000 characters based on constraints.

Preprocessing:

Input string can be preprocessed to optimize validity checks.

Processing:

Only allowed operation is inserting “abc” into current string.
Insertion ordering and number of times does not matter.
We only care about final validity, not the steps.

Outputs:

The only output is a boolean validity indicator.
No need to return or reconstruct the actual insertion sequence.

State:

No additional state or parameters are involved. Each input string can be processed independently.

So in summary, the problem is bounded by the fixed input character set, single allowed operation, focus on final validity over sequences, and simple boolean output. The core scope lies in efficient reachability determination within these set constraints.

Problem Classification

This is a string validation problem where we need to check if a given string can be generated by repeatedly inserting a specific substring.

The key ‘What’ components are:

Input string s to validate
Allowed operation of inserting “abc” into any position of current string
Transform empty string t to s by repeated insertions
Determine if s is reachable from t using the insertion operation

Based on these aspects, we can categorize this problem as:

String operations - Involves manipulating and building up strings
Validation - Need to check if string is valid based on rules
State sequence - Transforming string t through a sequence of operations
Reachability - Determining if s is reachable from t via insertions

So in summary, this is a string validation problem involving checking reachability from an initial empty string to a final string by applying a repeating insertion operation. It relies heavily on concepts of strings, state transformation through operations, and validation. The core challenge is determining if the input string can be created by repeated insertions.

Distilling the Problem to Its Core Elements

This problem is based on the fundamental concept of state reachability by applying valid operations.

At its core, it involves determining if one string can be constructed from another using a fixed insertion operation. I would describe it simply as:

“Given a start string, target string, and an insertion operation, can you get from the start to target by repeating the insertion?”

The core problem is checking reachability, not actually performing the insertions. We can simplify it to:

“Is the target string reachable from the start string by repeated insertions?”

The key components are:

The start string
The target string
The valid insertion operation
Determining reachability

The minimum operations are:

Initialize start string
Define allowed insertion operation
Check if target is achievable by repeated insertions
Return whether target is reachable or not

So in essence, this problem focuses on using an allowed operation to transform one string into another and checking reachability, rather than actually constructing the sequence.

Visual Model of the Problem

Here are some ways we could visualize this problem statement:

Show the starting empty string and final target string as two states, with an arrow indicating the insertion operation transforming start to end.
Use a flow chart to demonstrate applying the insertion operation repeatedly to transition from start to end string.
Show strings that are valid vs invalid endings based on the allowed insertions.
Animate the process of constructing the final string from empty by performing insertions one by one until target is reached.
Visualize the strings as states in a graph and insertions as directed edges between them. Highlight valid paths.
For invalid endings, illustrate the failures or roadblocks when trying to construct the string through allowed insertions.
Label parts of the final string that can be created or not created using “abc” insertions.
Use visualization libraries like D3.js to animate the string construction interactively.

The goal is to make the core concepts of incremental state transformation and validity checking intuitive through animations, graphs, examples and flow charts. Interactivity also aids understanding.

Problem Restatement

Here’s how I would paraphrase the problem statement in my own words:

We are given an input string s containing only characters ‘a’, ‘b’ and ‘c’. Our goal is to determine if s can be constructed by starting with an empty string t, and repeatedly inserting the substring “abc” into t.

Specifically, we need to check if s is achievable by applying the insertion operation of t = tleft + “abc” + tright multiple times, where tleft and tright represent the parts of t before and after the inserted “abc”.

The key points are:

We start with t as empty and want to build s
The only operation allowed is inserting “abc” into t
The insertions can be done in any order and any number of times
We only care about the final string s being achievable, not the specific steps
The input string contains only ‘a’, ‘b’ and ‘c’ characters

Our goal is to determine whether s is reachable from empty string t by repeating the “abc” insertion operation. We don’t need to find the actual sequence of insertions.

In summary, we must check if s can be constructed by arbitrary applications of the insertion rule, and return true if it’s possible, false otherwise. The focus is on validity, not the actual insertions.

Does this help summarize the essence of the problem? Let me know if any part needs more clarification or I missed anything important.

Abstract Representation of the Problem

Here is a way to formulate an abstract representation of this problem:

Let’s define:

S = the set of all possible strings made of characters ‘a’, ‘b’ and ‘c’
s = the target string we are given as input
t = the start string, initialized to empty ""
f = the insertion function, which inserts “abc” into a given string

The problem can then be represented as:

Given:

Set S
Target string s ∈ S
Start string t ∈ S, initially ""
Insertion function f: S → S that inserts “abc” into a string

Determine:

If repeated application of f to t can generate s

Return:

True if s is reachable from t by applying f
False otherwise

This frames the problem more generically as:

Given a set of states, start state, target state, and a state transition function, determine if the target is reachable from the start by applying the function.

The key aspects are the state set, start/target states, and the state-modifying function that transforms states.

Terminology

Here are some key terms and concepts for understanding this problem:

Reachability - The core problem of determining if one state is reachable from another via some operation. We want to check if the target string is reachable.

Valid string - A string that follows construction rules, i.e. can be created by repeated insertions. We must check validity.

Insertion operation - The allowed operation of inserting a substring into a string. Transforms strings by appending/inserting characters.

State space - All possible configurations a system can be in. Here, all possible strings represent the state space.

Transition function - An operation that transforms the current state to a new state. The insertion operation transitions between string states.

Commutativity - Order of operations does not affect end result. Insertions can be done in any order.

Incremental construction - Building up the target string step-by-step by repeatedly applying the insertion operation.

Regex pattern matching - Can leverage regular expression patterns like “abc” to help determine validity.

These concepts like reachability, state transitions, and commutativity help model and solve the problem by providing ways to represent insertions transforming string states. The other ideas guide the implementation.

Problem Simplification and Explanation

Key Concepts:

String - The input and output are strings of characters
Insertion - We have an operation that inserts a fixed substring
Transformation - Applying the insertion transforms an initial string into a new string
Validation - We need to check if the final string is valid based on the rules
Reachability - Determining if we can get from the initial to final string using the allowed insertions

Analogy:

Think of the initial empty string as a destination, and the final string as our target destination. The insertion operation gives us a fixed route we can take - inserting “abc”.

We want to see if we can reach the target destination if we start from the initial destination and repeatedly take the “abc” route. The specific path we take doesn’t matter as long as the route gets us there.

It’s like checking if I can travel from city A to city B given that the only route adds 3 stops - X, Y and Z. I don’t care about the order as long as repeatedly taking the route can get me to the target.

In essence, we just need to check if the target string is reachable by applying the insertion route, without worrying about the exact steps taken.

Constraints

Here are some characteristics of the problem that we can leverage to optimize the solution:

Fixed substring “abc” - We know exactly the pattern we need to match and insert. Can take advantage of this structure.
Small character set (‘a’, ‘b’, ‘c’) - Can encode or map characters to small integers to optimize string operations.
Insertions are commutative - Order doesn’t matter, so we don’t need to track full sequence.
Only valid characters are given - Don’t need to validate input, can make assumptions.
Output is binary - Can exit early once invalid string is detected.
Upper bound on string size - Can adapt solution for longer strings if needed but constraint allows simplifying logic.
Incremental construction - We can build up string rather than recreate it at end.
Overlapping matches - Inserted strings can overlap, don’t need to check for duplicate insertions.

These properties allow us to optimize our algorithms and data structures since we can leverage the fixed substring pattern, operate over a limited character set, avoid ordering and duplication checks, and incrementally validate the string rather than recreating it.

Here are some key insights gained by analyzing the constraints:

The fixed “abc” substring enables optimizations through pattern matching and assumptions.
Small character set allows techniques like bitmapping, integer encoding, etc. to optimize string operations.
Commutativity of insertions removes need to track or recreate sequences.
No input validation needed since only ‘a’, ‘b’, ‘c’ characters are provided.
Binary output lets us short-circuit on first invalid string detected.
Upper bound on string size enables simpler solutions without complex optimizations.
Incremental construction is possible by appending “abc” without full generation.
Overlapping insertions simplify logic by avoiding duplicate checks.

These constraints allow us to focus optimization efforts on efficient string operations and cut out unnecessary work around validation, ordering, duplication, etc. Insights like incremental construction and leveraging the fixed substring pattern are key. The constraints provide clarity.

Case Analysis

Here are some additional test cases covering different aspects:

Empty string

Input: "" Output: True

Analysis: Base case, valid by definition.

Simple valid

Input: “abcabc” Output: True

Analysis: Basic case with repeated insertions.

Invalid characters

Input: “abd” Output: False

Analysis: Strings with invalid chars should be rejected.

Out of order

Input: “cbabaac” Output: True

Analysis: Checks commutative property is handled.

Overlapping insertions

Input: “aaabcabcabccc” Output: True

Analysis: Allows overlapping inserted substrings.

Edge cases:

Empty string
Repeated characters
Max length string
Strings with all possible substrings

Other cases help cover commute times, duplicate characters, and strings constructed in non-intuitive ways.

Here are some ideas for visualizing these test cases:

Show start and end strings with insertion operation transforming one to other.
For invalid cases, indicate where getting to end fails.
Animate the incremental insertions building up the final string.
Use diagrams mapping start to end strings as states with insertion operations on edges.
Color code strings to highlight character positions enabled by insertions.
Split strings into chunks that can/cannot be created by insertions.
Leverage regex visualizers to highlight pattern matches.
Compare valid and invalid examples side-by-side.
Show how overlapping insertions work through character labeling.
Keep animations and diagrams simple and focused for clarity.
Provide counter examples demonstrating incorrect logic.
Use video to show animations and walkthroughs of cases.

The goal is to vividly illustrate how the insertions incrementally construct the strings across different cases. Animation helps demonstrate the time-based building process. diagrams summarize states and transitions.

Here are some key insights gained from analyzing these test cases:

Empty string edge case reveals assumptions about minimum valid length.
Simple examples are crucial to validating core logic and assumptions.
Invalid character cases show input validation needs.
Out of order cases emphasize the need to support any insertion order.
Overlapping cases highlight that duplicate checks are unnecessary.
Max length strings test optimizations and assumptions.
Varied examples validate handling of diverse construction sequences.
Visualization and animation make the incremental process more intuitive.
Counter examples are useful for conveying negative logic and pitfalls.
Fundamentally, these cases validate the core idea of reachability through repeated insertions from diverse angles.
They help refine assumptions, cover edge cases, identify validation needs, stress test limits, and check correctness across the valid input space.

Identification of Applicable Theoretical Concepts

Here are some mathematical and algorithmic concepts that could help solve this problem:

String matching algorithms like KMP can quickly find occurrences of the “abc” pattern within the input string. This helps validate reachability.
Dynamic programming can store intermediate results of substring checks, avoiding recomputing them. Useful for incremental validation.
Bitmasking and bitwise operations can optimize encoding and manipulating strings as sequence of bits rather than characters.
Finite state automata provide a framework for modeling the incremental construction of the string across valid state transitions.
Regular expressions give a tool for matching and validating the presence of the required “abc” insertion pattern within the final string.
Set theory offers ways to model the state space of valid strings, and operations transforming between states.
Commutativity of the insertion operation allows reordering steps without changing the end result.

Concepts from combinatorics, algorithms, automata theory, and abstract algebra provide useful techniques for optimizing substring operations, state modeling, pattern matching, and sequencing. The problem also lends itself to dynamic programming solutions.

Simple Explanation

Here’s how I would explain this problem in simple non-technical terms:

Imagine you have a machine that can add the sequence of letters “abc” into any existing string of letters. For example, if your starting string is:

“hi”

I can use the machine to insert “abc” and make it:

“hiabc”

Now imagine I gave you a final string with only a’s, b’s and c’s. Your task is to figure out if you could generate that final string starting with an empty initial string, only using this “abc” inserting machine.

For example, say the target string is:

“abcabc”

To generate this, you could:

Start with empty ""
Insert “abc” to get “abc”
Insert “abc” again to get “abcabc”

Since you can get the final string using the machine, you would return true.

But if the target was “ac”, you couldn’t generate it, so you’d return false.

It’s like figuring out if you can spell a word only by inserting a fixed sequence of letters repeatedly. The specific steps don’t matter as long as the end result is achievable.

Problem Breakdown and Solution Methodology

Here is a step-by-step explanation of how I would approach solving this problem:

Check for empty string edge case

If input s is empty, return true, as empty string is valid by definition.

Iterate through string s

Check if each substring of length 3 equals “abc”.
If yes, remove that substring.
Repeat until no more removals possible.

If fully reduced to empty string, return true

If s was reduced to empty, it is reachable by insertions.

If not empty, return false.

If removals did not reduce to empty, s is not valid.

This incrementally tries removing “abc” substrings as valid steps. If the string fully reduces to empty, it is valid.

For example, input “abcabc”:

s = “abcabc”
Remove “abc” -> s = “abc”
Remove “abc” -> s = ""
s is empty, return true

If the input was “ac”:

s = “ac”
No “abc” to remove
s != “”, return false

The key is iteratively trying to deconstruct s by reversing the allowed insertions. Visualizing the incremental removal helps.

If duplicate removals were allowed, we’d need to prevent infinite loops. We could also optimize matching “abc” with KMP algorithm.

Inference of Problem-Solving Approach from the Problem Statement

Here are some key terms and how they guide my approach:

String - The input/output being strings informs using string algorithms and data structures.
Insertion - The defined insertion operation of a fixed substring tells me to leverage substring search and matching.
Reachability - Checking if target string is reachable from start means using search algorithms and incrementally validating constructions.
Validity - Need to determine validity based on rules suggests pattern matching the required substring.
Commutativity - Order not matering allows flexibility in construction sequencing.
States - Concept of start and end strings as states means modeling as a state graph and transitions.
Incremental - Building up string incrementally points towards iterative, dynamic programming approaches.
Transformation - Insertions transform strings, implying use of automata theory modeling.

The core strings, transformations, reachability and validity concepts lead towards usingsearch algorithms, graph representations, automata theory, dynamic programming for incrementally validating constructions. The terms guide both the modeling and implementation.

We can visualize some key properties and aspects of this problem using tables and diagrams:

State Space:

"" “a” “b” “c” “aa” …

List out some examples strings as valid states

State Transitions:

 ""
  |
  v

"" -> “abc” | v “abc”

Show state changes from insertions as a graph

Incremental Construction:

"" -> “abc” -> “abcabc” -> “abcabcabc”

Visualize insertions building up string

Matching:

“abcabcabc” | | | v v v “abc” detected

Highlight pattern matches

These diagrams help illustrate the state space, transitions between states via insertions, incremental string construction, and pattern matching - all core aspects of the problem.

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?

Here is one way to break this problem down into more granular steps:

High-level approach:

Check if input string is constructible by insertions
Return validity result

Refined steps:

Check base case of empty string
Loop through string
- Try removing “abc” substring
- If removed, repeat loop
- If cannot remove, break
After loop, check if fully reduced to empty
Return true if empty, false otherwise

Independent parts:

Base case checking
Matching and removing “abc”

Repeatable patterns:

Substring matching
String manipulation after removal
Updating loop condition and variables

This breaks the problem down into base case handling, a loop trying removals and checking the result, and determining final validity. Substring matching and updating state are reusable patterns within the loop.

Solution Approach and Analysis

Here is a step-by-step explanation of how I would approach solving this problem:

Check base case

If input string s is empty, return true. Empty string is always valid.

Iterate through string s

Use a for loop through indices i in s:

Extract substring s[i:i+3]
Check if it equals “abc”
If so, remove it from s by splicing strings before and after
Repeat loop on modified s

After loop, check if s is empty

If s is empty, it was fully reduced and is valid.

If s not empty, it had characters that couldn’t be removed so invalid.
Return final validity result

This tries removing “abc” substrings incrementally, tracking if fully reducible to empty.

For example on input “abcabc”:

s = “abcabc”
s[0:3] = “abc”, remove it
s = “abc”
s[0:3] = “abc”, remove it
s = ""
s is empty, return true

Visualizing the incremental removals helps illustrate the process.

If duplicate removals are allowed, we’d need to prevent infinite loops. We could also optimize finding “abc” using KMP algorithm.

Identify Invariant

An invariant in this problem is a condition or statement that remains true throughout the execution of the solution. Some potential invariants are:

The input string s remains unchanged. The original string is not modified.
The allowed insertion operation of adding “abc” stays the same. This defines what transformations are valid.
Any substring removed during the loop once removed, stays removed for the remainder.
The rules for validity remain fixed - i.e. s is valid if reducible to empty string by removals.
The output boolean indicating validity starts out undefined and is only set once after full processing.
Any substring reducible before the loop remains reducible in the same way after the loop. Reducibility is invariant.

So in essence, the key invariants are:

Original input string s remains unmodified
Valid insertion operation definition
Monotonically reduced string in loop
Consistent validity rules
Single final assignment of output value

The loop cannot alter these core aspects defining the problem’s input space and validity constraints. The final output also remains unset until full traversal.

Identify Loop Invariant

The loop invariant in this problem is a condition that remains true before and after each iteration of the main string processing loop. Some possibilities for loop invariants are:

The original input string s remains unmodified by the loop.
The substring insertion operation definition is constant within the loop.
Any substrings already removed before the loop remain removed after each iteration.
The rules for valid string construction do not change within the loop body.
The output variable indicating overall validity is not updated within the loop.
Any substring removable before an iteration remains removable after. Removability is invariant.
The string s only gets smaller or remains the same, never grows in size.

So in summary, the key loop invariants center around the unchanged original string, consistent insertion rules, monotonically shrinking state, and preserved removability of substrings. The loop cannot alter these core aspects that determine correctness.

The invariant refers to high-level conditions that are true for the entire problem, regardless of implementation details.

The loop invariant focuses specifically on what must remain true before and after each iteration of the main processing loop.

There is overlap - the invariants describing the unchanged input string, fixed insertion rules, and substring removability also qualify as loop invariants.

However, some invariants like the single assignment of the output variable are important for the problem but not relevant inside the loop body.

And some loop invariants like the monotonically reduced string length don’t necessarily apply globally, only locally within the loop.

So while there is a strong overlap between the invariant and loop invariant properties in this problem, they are not identical sets. The loop invariants are focused on preserving key attributes directly impacted by the loop logic itself.

The global invariants provide wider constraints that the full solution must respect, even outside the core processing loop.

Thought Process

Here is how I would think through solving this problem step-by-step:

The problem statement cues:

Transforming an initial empty string by insertions indicates an incremental construction approach
Repeated fixed substring insertion points to a loop trying to match and remove that pattern
Only requirement is final string validity, not full sequence

So the steps would be:

Check if input string s is empty
If so return true, empty string is valid
Otherwise iterate through s:
- Try to remove “abc” substring
- If removed, repeat loop on modified s
- If not removed, break loop
After loop, check if s is fully reduced to empty
If so, return true, else return false

This tries reversing the insertions by removing “abc” substrings incrementally, then checks if fully reduced to validate.

In Python:

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def isValid(s):

  if s == '':
    return True
  
  for i in range(len(s)):
  
    if s[i:i+3] == 'abc':
      s = s[:i] + s[i+3:]
      i -= 3

  return s == ''  

We incrementally try removing “abc” from s in a loop, then check if fully reduced to empty.

The key is the incremental construction perspective, rather than trying to recreate the steps.

Establishing Preconditions and Postconditions

Here are the details for the isValid method:

Parameters:

s: String - The input string to validate

Preconditions:

s only contains characters ‘a’, ‘b’, ‘c’
s is not null

Method Functionality:

Iteratively tries removing “abc” substrings from s
Checks if s can be fully reduced to empty string by removals
Returns a boolean indicating if s is valid

Postconditions:

s remains unmodified
Returns true if s is valid, false otherwise

Error Handling:

If s is null, throw IllegalArgumentException
If s contains invalid characters, throw IllegalArgumentException
Otherwise return false for invalid string

So in summary, this takes an input string, tries incrementally reducing it by “abc” removals, and returns if the string is valid based on reducibility rules. The input string remains unmodified.

Problem Decomposition

Here’s how I would break down this problem:

Problem Understanding: Given a string, determine if it can be constructed by repeatedly inserting “abc” into an initially empty string.
Initial Breakdown:

Check empty string base case
Iteratively try removing “abc”
Check if completely reducible
Return result

Subproblem Refinement:

Implement removal logic
- Extract substring
- Check if “abc”
- Remove substring
- Update string
Validate final string
- Check if empty after removals

Task Identification:

Base case handling
Substring extraction
Equality check
String manipulation
Validity checking

Task Abstraction:

Base case
Substring matching
String mutation
Reduction validation

Method Naming:

checkBaseCase()
matchSubstring()
removeSubstring()
validateReduced()

Subproblem Interactions:

Handle base case
Use matching and removal in loop
After loop, validate reducibility
Return result

The key stages are the base case, a removal loop, final validation, and result return. Subtasks are abstracted into reusable string operations.

From Brute Force to Optimal Solution

Here is a brute force approach and how we can optimize it:

Brute force:

Generate all possible strings with ‘a’, ‘b’ and ‘c’ up to length n
Check if input string s exists in generated list
Return true if found, false otherwise

This enumerates all possible strings which is inefficient.

Complexity: Time - O(3^n) Space - O(3^n)

Optimizations:

Incrementally construct and validate string

Start with empty string and iteratively insert “abc”, checking if we construct s.

Complexity: Time - O(n^2) Space - O(n)

Match and remove “abc” substrings

Rather than constructing s from empty, remove “abc” from s and check if reducible to empty.

Complexity: Time - O(n^2)
Space - O(n)

Use KMP algorithm to optimize matching

Improves substring matching complexity.

Complexity: Time - O(n) Space - O(m) where m is pattern size

Track validity during construction

Avoid fully building s, return early if invalid prefix detected.

Complexity: Time - O(n) Space - O(n)

In summary, we optimized from brute force generation to incremental construction and validation, simplified logic with reversibility, efficient sub-string matching, and early return. This reduces exponential time to linear.

Code Explanation and Design Decisions

Here’s how I would explain the reasoning behind this code:

The initial parameter s is the input string to validate.
The loop iterates over s trying to remove “abc” substrings. This represents transforming s incrementally.
The condition checks for a full “abc” match, ensuring only valid removals occur. This respects the insertion rules.
The loop updates s after removals. This mutation models the incremental transformation.
An invariant is that “abc” maintains its meaning as the substring to remove. This consistent rule is key.
The final output indicates if s can be fully reduced by the transformations. This answers the core reachability question.

In summary, the code incrementally tries to deconstruct s according to fixed rules, while preserving those rules invariantly. The end result satisfies checking if s can be generated from the defined transformations.

Coding Constructs

Here are high-level explanations for this code:

It uses incremental construction and validation, trying to deconstruct the input string through valid reversable operations.
The code checks if a given string can be created by repeatedly inserting a specific substring pattern.
The logical constructs are loops, substring matching, string manipulation, and conditional checks on validity.
It tries removing a fixed substring in all possible ways from the input, checking if the string can be reduced to empty.
The key steps are matching and removing substrings, mutating the string, and validating the final reduced string. This incrementally tries reversing the construction rules.
The core algorithmic strategies are incremental validation through reversible operations and exhaustive substring search. This allows efficiently checking validity through transformation rules.

In summary, the code employs incremental string transformations and validation by trying to reverse valid construction operations. The explanations focus on the high-level reasoning rather than language-specific details.

Language Agnostic Coding Drills

Here is how I would break down this code into learning units:

Concepts in increasing difficulty:

Looping - Iterating through a string character-by-character. Basic looping construct.
Substring extraction - Getting a substring from a string. Core string manipulation.
Equality check - Comparing two strings. Fundamental conditional check.
String mutation - Removing part of a string. More advanced string skill.
Validity check - Checking final string against condition. Key validation concept.
Iterative construction - Building string incrementally. Intermediate technique.
Reversible operations - Trying to reverse/undo operations. More complex logic.
Exhaustive search - Trying all possible substrings. Important optimization.

The problem solving approach would involve:

Looping through string
Extracting substrings
Checking for equality with target substring
Removing matched substrings
Repeating loop on mutated string
Validating final result
Trying all substrings exhaustively

Each coding drill builds up essential skills in looping, strings, conditional logic, validity checking, and optimizations like exhaustive search. Combining these solves the problem.

Targeted Drills in Python

Here are Python coding drills for each concept:

Looping

1
2
for i in range(len(s)):
  print(s[i]) 

Substring extraction

1
sub = s[i:i+3] 

Equality check

1
2
if sub == "abc":
  print("Matched")

String mutation

1
s = s[:i] + s[i+3:]

Validity check

1
2
if s == "":
  print("Valid")

Iterative construction

1
2
result = ""
result += "abc"

Reversible operations

1
2
if substring in s:
  s.remove(substring)

Exhaustive search

1
2
3
for i in range(len(s)-2):
  sub = s[i:i+3]
  # Check substring

We can integrate these by:

Looping through string
Extracting substrings
Checking for match
Removing matched substrings
Constructing string iteratively
Checking final validity
Exhaustive substring search

Each drill builds up key concepts needed for the full solution. Combining them in this order solves the problem.