Rearrange String k Distance Apart
We can use a greedy approach. Start by counting the frequency of each character in the string. Then, repeatedly choose the character with the highest frequency that hasn’t been used in the last k positions. If it’s not possible to rearrange the string, return an empty string. Otherwise, return the rearranged string.
Python solution:


This solution uses a maxheap to prioritize characters with the highest frequency. The queue
keeps track of characters that need to be readded to the heap after k positions. The result
list holds the final rearranged string.
10 Prerequisite LeetCode Problems
“Rearrange String k Distance Apart” involves string manipulation and queue data structure. Here are 10 problems to prepare for this:
 49. Group Anagrams
 76. Minimum Window Substring
 205. Isomorphic Strings
 242. Valid Anagram
 283. Move Zeroes
 387. First Unique Character in a String
 438. Find All Anagrams in a String
 451. Sort Characters By Frequency
 567. Permutation in String
 621. Task Scheduler
“358. Rearrange String k Distance Apart” involves string manipulation, greedy algorithms, and usage of priority queues (heap data structure).
Here are 8 problems as good preparation:
767. Reorganize String: The approach to this problem is similar to “358. Rearrange String k Distance Apart”, as you have to rearrange the string so that no two identical characters are adjacent.
215. Kth Largest Element in an Array: This problem can give you a good understanding of using priority queues.
347. Top K Frequent Elements: This problem involves finding the k most frequent elements, using priority queues.
295. Find Median from Data Stream: This problem provides practice for handling and manipulating data in a stream using heap data structures.
692. Top K Frequent Words: This problem is similar to “347. Top K Frequent Elements” but with words.
75. Sort Colors: This problem deals with the idea of Dutch National Flag problem, where you can learn about rearranging elements.
394. Decode String: This problem can help you understand the manipulation of strings using stacks.
239. Sliding Window Maximum: This problem provides practice for using a deque to maintain a sliding window, which is a helpful concept for string manipulation problems.
Problem Boundary
To establish the boundary of this problem, we need to clarify the limits within which the solution must operate. These boundaries come from both the problem statement and inherent properties of the problem.
String Length: The length of the string
s
can range from 1 to 3 * 10^5. This sets an upper boundary on the computational complexity of the algorithm.Character Set: The string consists only of lowercase English letters. This means a maximum of 26 unique characters, setting a limit on the variety of characters you need to account for.
Distance Constraint
k
: The value ofk
can range from 0 to the length of the string. A value of 0 means no constraint, while a value equal to the string length requires each character to be unique.Output: The output is either an empty string (when a solution is not possible) or a rearranged string that meets the distance constraint. This sets the range for possible outputs.
Feasibility: One inherent boundary is the feasibility of rearrangement. If there are too many occurrences of a single character, such that placing them at least
k
distance apart is impossible, then the problem has no solution.Uniqueness of Solution: The problem doesn’t demand a unique solution, which broadens the boundary by allowing multiple valid outputs.
Computational Time: While not explicitly stated, it’s implied that the solution should be computationally feasible given the constraints. For example, a solution with time complexity higher than O(n) may not be practical for large n (like 3 * 10^5).
By understanding these boundaries, we get a clear picture of what is permitted and what is not. This helps in formulating a solution that is both correct and efficient.
Problem Classification
This problem falls under the domain of “String Manipulation” and “Combinatorial Optimization”. It involves rearranging characters within a string while adhering to a specific distance constraint.
‘What’ Components
Input String
s
: A string composed of lowercase English letters. We need to rearrange its characters.Distance Constraint
k
: An integer specifying the minimum distance that must be maintained between the same characters.Output String: Another string where characters are rearranged to meet the distance constraint. If rearrangement is impossible, the output is an empty string.
Constraints:
 1 <= s.length <= 3 * 10^5
 0 <= k <= s.length
s
consists of only lowercase English letters.
Feasibility Check: Determine if the string
s
can be rearranged based on the value ofk
.Rearrangement Strategy: Find the specific sequence of characters that satisfies the distance constraint
k
.Optimization: Minimize the computational time while ensuring that the rearranged string meets the constraints.
By identifying these elements, we are classifying the problem as a combinatorial optimization task with constraints. We have to explore the possible arrangements of the string while obeying the distance rule specified by k
. The task includes aspects of both feasibility checking and optimization.
Distilling the Problem to Its Core Elements
Fundamental Concept: The fundamental concept this problem is based on is rearrangement and spacing constraints in an array or string. Specifically, it’s about how to reorder the elements (characters in this case) to satisfy a given conditionâ€”making sure that the same characters are at least
k
distance apart.Simplest Description: Imagine you have a basket of different colored balls. You need to line them up in such a way that the same colored balls are at least a certain distance apart from each other.
Core Problem: The core problem is rearranging the characters in a string so that identical characters are separated by at least
k
other characters. Simplified Statement: “Reorder the letters in a word so that the same letters are not too close together; they must be at least
k
letters apart.”
 Simplified Statement: “Reorder the letters in a word so that the same letters are not too close together; they must be at least
Key Components:
 Input string
s
 Distance
k
 Frequency of each character in
s
 Feasibility check based on
k
and character frequency  Rearrangement strategy to place characters at least
k
apart
 Input string
Minimal Set of Operations:
 Count the frequency of each character in the string.
 Check for feasibility: Ensure that it’s possible to place the most frequent character at least
k
distance apart.  Rearrange the characters based on their frequency and the
k
value constraint.  Return the rearranged string, or an empty string if rearrangement is not possible.
By addressing these points, we can better understand the problem’s nuances and constraints, thereby making it easier to work towards a solution.
Visual Model of the Problem
Visualizing this problem can be particularly helpful for understanding its constraints and working towards a solution. Here are some ways to visualize the problem:
Character Frequency Table: Start by drawing a table or bar graph showing the frequency of each character in the input string. This gives a quick overview of the elements you’ll be working with.
Array or String Representation: Use an array or string representation to simulate the arrangement process. Place characters in this array as you iterate through the original string, adhering to the ‘at least k distance apart’ rule.
 Empty spots can be visualized with placeholders (like dots or dashes).
 Fill the spots with characters from the input string, obeying the distance constraint.
For example, for s = “aabbcc” and k = 3:
Step 1: a _ _ _ Step 2: a _ _ b Step 3: a c _ b Step 4: a c a b Step 5: a c a b c Step 6: a c a b c b
Timeline or Distance Tracker: Create a timeline or distance tracker to keep tabs on the last occurrence of each character. This can be a secondary array or a mapping. Use it to decide where to place the next occurrence of the same character.
Constraint Boundaries: Explicitly mark or visualize the minimum required distance (k) between the same characters. For example, you might represent this using colored lines or brackets that span
k
positions on your array representation.Feasibility Check: Before starting the rearrangement, visualize a “worstcase” scenario to check for feasibility. Place the most frequent characters at kdistance apart and see if they fit within the length of the array. If they don’t, you can immediately see that the task is impossible.
Outcome Visualization: At the end, compare your original frequency table/bar graph with your final array or string representation to ensure that all characters have been included and the ‘kdistance’ condition has been met.
This multilevel visualization can help break down the problem into smaller, more manageable tasks, making it easier to think about the logic and constraints involved.
Problem Restatement
The problem asks us to rearrange a given string so that the same characters are spaced at least ‘k’ positions apart from each other. If we can’t do that, we return an empty string. The string contains only lowercase English letters, and ‘k’ can range from 0 up to the length of the string. The input string has a length between 1 and 3 * 10^5.
Abstract Representation of the Problem
In abstract terms, you’re given an array of elements and a parameter ‘k.’ The task is to reorder this array so that identical elements are at least ‘k’ distance away from each other. If it’s not feasible to rearrange the array under these conditions, an empty array should be returned. This abstract representation focuses on the inherent constraints and goals of the problem without relying on specific realworld context.
Terminology
There are some specialized terms and concepts that help in understanding this problem:
Array (or String) Rearrangement: Modifying the order of elements in an array to meet specific conditions. In this problem, the rearrangement is based on a minimum distance between identical elements.
Distance: In this context, the term refers to the number of positions between two identical elements in the array. The distance is determined by the parameter ‘k.’
Constraints: These are the rules or limitations placed on potential solutions. In this problem, you’re constrained by the length of the string and the value of ‘k.’
Feasibility: This is a check to determine if the problem has a valid solution based on the given constraints. In this case, if rearranging is not possible, an empty array must be returned.
InPlace: A term often used in array manipulation problems to specify whether you’re allowed to create a new array to store the result or need to rearrange elements within the existing array itself. This problem doesn’t specify, so both methods could be considered.
Each of these terms and concepts plays a role in defining the problem’s requirements and the kinds of solutions that are considered valid.
Problem Simplification and Explanation
Let’s break it down:
Key Concepts:
 String: Think of it as a necklace with beads, where each bead is a letter.
 Integer k: This is the minimum distance that the same type of beads need to be from each other.
 Rearrange: You can rethread the necklace, changing the order of the beads.
Interaction of Concepts:
 You have to rearrange the beads (letters) on the necklace (string) so that the same type of beads are at least ‘k’ beads apart from each other.
 If it’s not possible to do so given the number and types of beads you have, then you have to indicate that (return an empty string).
Metaphor or Analogy:
Imagine you’re arranging guests at a dinner party and some of the guests aren’t on speaking terms. You have a rule: such guests must be seated at least ‘k’ seats apart from each other. Your task is to create a seating chart following this rule.
You’ve got a string full of letters, and you need to rearrange these letters so that the same letters are at least ‘k’ positions apart. If you can’t do it with the letters you’ve got, you have to make it clear that it’s not possible (return an empty string).
Constraints
Certainly. Here are some aspects that could be exploited for an efficient solution:
Character Count: The problem is about rearranging characters. Counting the frequency of each character can help prioritize which characters need to be placed first.
String Length: The constraints mention that the string length is between 1 and 3 * 10^5. This suggests that our solution should aim for linear time complexity, or it won’t be efficient enough.
Value of k: The value of ‘k’ can be as low as 0 and as high as the string length. When ‘k’ is 0, we don’t have to rearrange anything. When ‘k’ is greater than or equal to the string length, rearranging is impossible unless all characters are unique. Knowing ‘k’ gives us important bounds.
Lowercase English Letters: The problem states that the string only contains lowercase English letters, which limits the character set to 26 possible characters. This can make counting and storing character frequencies more manageable.
Empty String for Impossible Scenarios: The problem allows us to return an empty string if rearranging is not possible, which simplifies edge cases. We don’t need to look for alternative ways to indicate an impossibility.
Specific Ranges for k: If ‘k’ is exactly half of the string length, then the most frequent character should appear no more than twice. Similar patterns can be found for other specific ‘k’ values.
By identifying these specific conditions or patterns, you can start thinking of more targeted solutions that exploit these characteristics for efficiency.
The key insights from analyzing the constraints are:
Linear Time Complexity: Given the string length can go up to 3 * 10^5, the solution should aim for linear time complexity to be efficient.
Limited Character Set: The string consists only of lowercase English letters, giving us a maximum of 26 unique characters. This constraint simplifies counting and storing character frequencies.
Value of k: The range for ‘k’ varies from 0 to the length of the string. This helps in setting boundaries for our logic. For example, when ‘k’ is 0, no rearrangement is necessary. When ‘k’ is greater than or equal to the string length, the task becomes impossible unless all characters are unique.
Early Exit: If rearrangement is impossible, the problem allows us to return an empty string. This simplifies edge case handling.
Understanding these constraints can guide the design of an efficient algorithm for solving the problem.
Case Analysis
Certainly. Here are some test cases categorized to cover different aspects of the problem:
Case 1: Minimal Inputs
 Input:
s = "a", k = 0
 Output: “a”
 Analysis: This is the smallest possible input. The value of ‘k’ is zero, so no rearrangement is needed.
Case 2: Impossible Rearrangement
 Input:
s = "aa", k = 2
 Output: ""
 Analysis: Here, it’s impossible to rearrange the string to satisfy the condition, as ‘k’ equals the string length.
Case 3: Single Character Frequency Exceeds Possible Positions
 Input:
s = "aaaabc", k = 3
 Output: ""
 Analysis: ‘a’ appears 4 times but we have only 3 positions to place ‘a’ such that another ‘a’ is k distance apart.
Case 4: Multiple Characters with Same Highest Frequency
 Input:
s = "aabcc", k = 2
 Output: “acbac”
 Analysis: Both ‘a’ and ‘c’ appear twice. The key insight is that these most frequent characters should be spaced first.
Case 5: All Unique Characters
 Input:
s = "abcdef", k = 3
 Output: “abcdef”
 Analysis: If all characters are unique, the original string already satisfies the condition.
Case 6: k = 0
 Input:
s = "aabbcc", k = 0
 Output: “aabbcc”
 Analysis: When k is zero, any arrangement satisfies the condition. Original string can be returned.
Case 7: k Exceeds Length
 Input:
s = "abc", k = 5
 Output: ""
 Analysis: If ‘k’ is greater than or equal to the length, then the task becomes impossible unless all characters are unique.
Case 8: Equal Frequency Characters
 Input:
s = "aabb", k = 2
 Output: “abab”
 Analysis: When all characters have the same frequency and ‘k’ allows for proper spacing, any alternating sequence is valid.
Case 9: Exact Fit
 Input:
s = "aaab", k = 2
 Output: “aba”
 Analysis: The characters can be rearranged exactly to meet the ‘k’ distance condition.
Case 10: Random Complexity
 Input:
s = "abcabcabcabc", k = 4
 Output: “aabbccccaaaa”
 Analysis: This case tests the solution’s ability to handle a larger string with multiple repeating characters.
By considering these test cases, you can ensure your solution is robust and handles edge and boundary conditions.
Analyzing different test cases yields several key insights:
Single Character Frequency: If the frequency of a single character exceeds the possible positions where it could be placed while satisfying the ‘k’ distance, the rearrangement is impossible.
Equal Frequencies: If all characters have equal frequencies and ‘k’ allows for proper spacing, multiple valid rearrangements can exist.
k Value: The value of ‘k’ significantly impacts the solution. When ‘k’ is zero or smaller than the frequency of each character, the task becomes trivial. When ‘k’ equals or exceeds the string length, rearrangement is often impossible unless all characters are unique.
Unique Characters: If all characters in the string are unique, the string already satisfies the condition, making the problem trivial.
Highest Frequency Characters: Placing the most frequent characters first can be a useful strategy, as they impose the most constraints on the arrangement.
Edge Cases: Small inputs like singlecharacter strings or small values of ‘k’ can have unique conditions that might not be captured when considering only the general case.
Exact Fit: Some cases will have just one correct answer that exactly satisfies all conditions, making it important for the solution to be precise.
Random Complexity: Larger test cases with multiple repeating characters are important to test the algorithm’s efficiency.
These insights help in formulating a robust solution and considering all possible scenarios the algorithm might encounter.
Identification of Applicable Theoretical Concepts
Here are some mathematical and algorithmic concepts that can be applied:
Frequency Count: Utilize hashing to calculate the frequency of each character in the string. This is foundational for understanding the constraints of possible rearrangements.
Greedy Approach: Place the most frequent characters first, as they have the most constraints. This approach can lead to a more straightforward solution.
Queue/Heap Structure: Use a priority queue or maxheap to manage characters based on their frequency. This ensures that you always pick the most limiting character first.
Sliding Window: When ‘k’ is specified, you could use a sliding window approach to keep track of the characters that should not be placed within a distance ‘k’ from the current position.
Modular Arithmetic: To place characters at a distance ‘k’ from each other, modular arithmetic can be used to find the correct index for each character.
Combinatorial Theory: If the frequency of characters and the value of ‘k’ are known, combinatorial formulas might be used to quickly determine if a solution is possible or not, avoiding the need for complete enumeration.
Dynamic Programming: Although likely overkill for this specific problem, DP could potentially be used for more complex variations of the problem, where different characters have different distance constraints.
Graph Theory: If you treat each character as a node and edges as possible rearrangements, this problem can be reframed as finding a valid path in the graph, more specifically an Eulerian path if such a path exists.
Complexity Analysis: Understanding the time and space complexity of different approaches can help in optimizing the algorithm for scalability.
Using these concepts, the problem becomes more manageable and can be solved more efficiently.
Simple Explanation
Imagine you have a box of different colored balls, and you need to arrange them in a row. The rule is that the same color of balls must be at least ‘k’ spaces apart from each other. If you can’t do that, you can’t make a row.
Think of a string as a necklace with differently colored beads. You need to arrange the beads in such a way that beads of the same color should be ‘k’ beads apart. If it’s not possible, the necklace can’t be made.
You have a bunch of letters, like ABCs, and you want to make a line with them. But if you have two ‘A’s, they can’t be right next to each other; they have to have at least ‘k’ other letters between them. If you can’t do that, then you can’t make the line.
It’s like arranging people in a queue but making sure that people wearing the same color tshirts are spaced ‘k’ persons apart. If you can’t do it, the queue can’t be formed.
Imagine you have a tray of cookies to arrange for a party. You have different types of cookies, like chocolate chip, sugar, and oatmeal. You need to place them on the tray so that no two chocolate chip cookies, for example, are closer than ‘k’ cookies apart. If you can’t manage to do that, then you have to leave the tray empty.
Problem Breakdown and Solution Methodology
To solve this problem, you can imagine a puzzle where each piece is a letter, and you’re trying to fit them into a line. The rule is that the same letters can’t be too close to each other.
Steps to Solve the Problem
Count Letters: First, count the frequency of each letter in the string. Imagine you’re putting each letter into separate piles based on their type, like separating red, blue, and yellow balls.
Sort by Frequency: Next, sort the letters by how often they appear, most frequent first. This is like prioritizing the biggest piles of balls to be placed first.
Initialize Result: Create an empty list that will hold the final string’s characters. This list will act as a ’tray’ where you will place your ‘balls’ following the rules.
Placement: Starting with the most frequent letter, place each letter ‘k’ distance apart in the result list. If the letter frequency and ‘k’ distance requirement makes it impossible to fit a letter, return an empty string.
Fill Gaps: Once the most frequent letters are placed, fill in the gaps with the less frequent letters, making sure to maintain the ‘k’ distance rule.
Construct String: Finally, join the list back into a string to get the rearranged string.
Effects of Parameters
 If ‘k’ is zero or one, the problem is straightforward, as the initial string already satisfies the conditions.
 The larger the ‘k’, the harder it becomes to rearrange the string. For very large ‘k’, you might not find a valid arrangement.
 If a letter appears too many times, it might not be possible to keep a ‘k’ distance, and you’ll return an empty string.
Example
Let’s take the example of s = “aaadbbcc” and k = 2:
 The frequency count gives us: a:3, b:2, c:2, d:1
 Sorted by frequency: a, b, c, d
 Initialize Result: [_, _, _, _, _, _, _, _] (8 slots)
 Place ‘a’ (most frequent) at positions 0, 3, 6: [a, _, _, a, _, _, a, _]
 Place ‘b’ and ‘c’ next, while maintaining the distance: [a, b, c, a, b, c, a, d]
 Join and return the string “abcaabcd”
With this approach, you can rearrange the letters so that the same characters are at least ‘k’ distance apart.
Inference of ProblemSolving Approach from the Problem Statement
There are several key terms and concepts that help frame the approach to solving this problem:
String: The data type of the main input; a sequence of characters. Knowing this leads to string manipulation techniques like splitting and joining.
Integer k: The minimum distance between the same characters in the rearranged string. This constraint informs us that a loop or an array indexbased manipulation will likely be involved.
Rearrange: The need to reorder the elements of the string. Sorting or shuffling techniques could be employed, but with custom rules.
Same Characters: Indicates that we must account for characters that appear multiple times. This suggests frequency counting as a first step.
At least distance k: The rule for rearranging. This implies that we can’t just sort the string; we must place elements in a way that they are ‘k’ distance apart if they are the same.
Empty String: The output when rearrangement isn’t possible. This is a fallback and indicates that our solution needs to have a way to recognize when the task is impossible under the given conditions.
Constraints: These guide the optimization of the solution. Knowing the upper bound of string length can inform us about acceptable time complexity.
Each keyword and its associated concept guide the approach and methods used to solve the problem. For example, “Rearrange” and “Integer k” together suggest that we need a sorting algorithm that respects custom distance rules. “Same Characters” and “Frequency” make it clear that the first step should likely involve counting the occurrences of each character in the string. “Empty String” shows us that the function should have a defined exit condition when the task is impossible.
How did you infer from the problem statement that this problem can be solved using ?
Stepwise Refinement
Stepwise Refinement of Approach
Frequency Count: The first step is to count the occurrences of each character in the string ’s’. This gives us an idea of how many of each character we have to rearrange.
Sort by Frequency: Sort the counted characters in descending order based on their frequency. The character that appears the most often should be processed first to ensure it can be placed at least ‘k’ distance apart.
Initialize Result Array: Create an empty array of the same length as the input string ’s’. We will use this array to place characters while ensuring they are ‘k’ distance apart.
Character Placement: Starting with the character that has the highest frequency, place it in the result array such that all instances of that character are ‘k’ distance apart. Proceed to the next most frequent character and do the same. If you can’t place a character meeting the ‘k’ distance requirement, return an empty string.
Convert to String: Once all characters have been placed in the result array, join it to form a string, which will be the output.
Return Result: Return the rearranged string, or an empty string if the rearrangement is not possible.
Granular, Actionable Steps
 Use a hash map to count the frequency of each character in ’s’.
 Sort the hash map based on character frequency, in descending order.
 Initialize an empty array of length equal to ’s’.
 Iterate through the sorted hash map. For each character:
 Find suitable positions in the result array.
 Place the character in those positions.
 If not possible, return an empty string.
 Join the result array into a string and return it.
Independent Parts
 Counting character frequencies can be done independently as the initial step.
 Sorting the frequency count is also a separate task.
 Initialization of the result array is independent and doesn’t depend on other steps.
Repeatable Patterns
 The act of placing each character in the result array based on ‘k’ distance is a repeatable pattern, iterated for each unique character in the string.
 Checking whether a position in the result array is suitable for a character is also a repeatable action.
By breaking down the problem into these steps and identifying patterns and independent parts, we can tackle the problem in a structured and organized manner.
Solution Approach and Analysis
Step 1: Frequency Count
The first thing to do is count how many times each letter appears in the string s
. Imagine you have a bag of Scrabble tiles, each representing a letter from s
. Sorting them into different piles based on the letter will show you how many of each you have.
Step 2: Sort by Frequency
Sort these counts in descending order. You want to start placing the letter that appears most frequently first. This is because if you can place the most frequent letter successfully, you are more likely to place others.
Step 3: Initialize Result Array
Create an empty ‘result’ array of the same size as the string s
. Think of this as your game board where you’re going to place your Scrabble tiles (letters) in a way that follows the rules (at least ‘k’ distance apart).
Step 4: Character Placement
Now, start placing the most frequent letter at intervals of ‘k’ in the ‘result’ array. If it’s not possible to do so, then you know right away it’s impossible to solve the problem; hence, return an empty string.
Step 5: Convert to String
Once the ‘result’ array is populated correctly, convert it back to a string to get the final answer.
Step 6: Return Result
Return the string if successfully generated; otherwise, return an empty string.
Changes in Parameters
 Length of String
s
: The larger the string, the more computational time it will require.  Value of
k
: Ifk
is closer to zero, the problem becomes easier because you don’t need to space characters as much. A largerk
will require more strategic placement.
Example Cases
Example 1: s = “aabbcc”, k = 3
Sorted Frequencies: a: 2, b: 2, c: 2
Result Array: a _ _ a _ _ b _ _ b _ _ c _ _ c
Output: “abcabc”
Example 2: s = “aaabc”, k = 3
Sorted Frequencies: a: 3, b: 1, c: 1
Here, you can’t place three ‘a’s at least 3 spaces apart in a 5length string.
Output: ""
Example 3: s = “aaadbbcc”, k = 2
Sorted Frequencies: a: 3, b: 2, c: 2, d: 1
Result Array: a _ a _ a b _ b c _ c d
Output: “abacabcd”
By following these steps, we can tackle this problem in a systematic way. It’s like solving a puzzle: you must place each piece carefully, considering the constraints and the pieces you have left to place.
Identify Invariant
The invariant in this problem is the minimum distance k
between the same characters after rearrangement. No matter how you rearrange the string, if a solution exists, the same characters must be spaced at least k
positions apart from each other in the final string. This constraint remains constant throughout the problemsolving process and serves as a rule that any solution must adhere to.
Identify Loop Invariant
The loop invariant in this problem is that at any point during the iteration, the result
array should maintain the property that the same characters are at least k
distance apart up to the current index.
This invariant ensures that as you continue to populate the array or string, you maintain the core constraint defined by the problem. Whenever you place a new character in the array, it should adhere to this invariant; otherwise, you know that the problem has no feasible solution, and you can break out of the loop to return an empty string.
Ensuring this loop invariant helps you maintain the constraints and conditions laid out in the problem statement, allowing for a correct and efficient implementation.
In this particular problem, the invariant and the loop invariant can be considered different.
Invariant: The overarching invariant for the entire problem is that the characters in the final rearranged string must be at least
k
distance apart from each other. This is a global condition that the final output string needs to satisfy.Loop Invariant: This is a more localized condition that applies during each iteration of your loop, ensuring that the string built up to that point meets the distance requirement. The loop invariant is a condition that you ensure holds true at the beginning and end of each iteration of your loop.
So while the loop invariant serves to maintain the global invariant, they serve different purposes and occur at different levels of granularity in your solution.
Thought Process
Understanding Constraints: The first cue is the constraint that the same characters should be at least
k
distance apart. This tells us we can’t just randomly rearrange the characters; we need a specific strategy.Character Frequency: We need to know how many times each character appears in the input string
s
. This could guide us in placing each character at appropriate intervals.Sort by Frequency: Sorting characters by their frequency can help us distribute the most frequent characters first, ensuring they are
k
distance apart.Initialize Result Array: Create an empty array or string of the same length as the input. We’ll populate this with characters, maintaining the
k
distance constraint.Place Characters: Start placing the most frequent characters first at intervals of
k
, then move on to less frequent characters.Failure Condition: At any point, if you can’t place a character while adhering to the
k
distance rule, return an empty string.Convert to String: Once the array is fully populated without breaking the rule, convert it back to a string format, if you used an array.
Python3 Code


WRONG ANSWER
Insights
 Character frequency is crucial; we can’t solve the problem without it.
 The problem naturally leads to sorting characters by their frequency.
 We need to be cautious while placing characters; the highest frequency characters should be dealt with first to make space for others.
By understanding these cues and constraints, we can systematically approach the problem and solve it efficiently.
Establishing Preconditions and Postconditions
Problem Name:
 Rearrange String k Distance Apart
Method Name:
rearrangeString
Parameters:
s: str
: The input string to be rearranged.k: int
: The minimum distance between identical characters in the rearranged string. Both parameters represent constraints and input data for the problem at hand.
Preconditions:
s
must only consist of lowercase English letters.1 <= s.length <= 3 * 10^5
0 <= k <= s.length
 No other specific state is required for the program.
Method Functionality:
 The method rearranges the string
s
such that the same characters are at leastk
distance apart.  If it’s not possible to rearrange the string as per the given conditions, it returns an empty string.
 The method rearranges the string
Postconditions:
 The method returns a new string that meets the criteria of having identical characters at least
k
distance apart, or an empty string if it’s not possible.  The return value indicates the successful rearrangement or impossibility of such a rearrangement.
 No side effects.
 The method returns a new string that meets the criteria of having identical characters at least
Error Handling:
 If the preconditions are not met, the method should throw a ValueError exception with a corresponding error message.
 If it’s not possible to rearrange the string according to the problem statement, it returns an empty string.
Problem Decomposition
Problem Name:
 Rearranging Characters in a String to Be K Distance Apart
Problem Understanding:
 The task is to rearrange the letters in a given string such that the same characters are at least ‘k’ distance apart. If it’s impossible, return an empty string. Key components are the input string and the distance ‘k’.
Initial Breakdown:
 Count the frequency of each character in the string.
 Determine if the rearrangement is possible.
 Perform the rearrangement.
Subproblem Refinement:
 For frequency counting: Loop through the string and populate a frequency map.
 For determining possibility: Calculate the maximum number of occurrences of any character and check if it can fit in the rearranged string.
 For rearrangement: Insert characters at appropriate positions, respecting the ‘k’ distance.
Task Identification:
 Counting frequency can be a separate reusable task.
 Checking for possibility based on frequency and ‘k’ can also be a reusable task.
Task Abstraction:
 Counting frequency: Takes a string, returns a frequency map. Clear and reusable.
 Checking possibility: Takes a frequency map and ‘k’, returns a boolean. Clear and contextual.
 Rearrangement: Takes a frequency map and ‘k’, returns a rearranged string.
Method Naming:
countFrequency
isRearrangementPossible
performRearrangement
Subproblem Interactions:
 First,
countFrequency
should run to get the frequency map.  Second,
isRearrangementPossible
should run to determine if we should proceed. It depends on the output ofcountFrequency
.  Finally, if possible,
performRearrangement
should run to get the final output. It also depends on the output ofcountFrequency
.
 First,
From Brute Force to Optimal Solution
Brute Force Solution
The brute force way to solve this problem would involve generating all possible permutations of the string and then checking each one to see if it meets the ‘k’ distance requirement.
Pseudocode:
 Generate all permutations of the string.
 For each permutation, check if any two identical characters are less than ‘k’ distance apart.
 If you find a valid string, return it.
Inefficiencies:
 Time Complexity: Generating all permutations would take O(n!), which is extremely inefficient for large strings.
 Space Complexity: Storing all permutations would also require a lot of memory.
Optimizing the Solution
Step 1: Frequency Count
Start by counting the frequency of each character in the string.
Step 2: Quick Check for Impossibility
Check if the rearrangement is even possible given the constraint of ‘k’. If the most frequent character appears more times than what can be accommodated by inserting ‘k’ distance apart, return an empty string.
Step 3: Priority Queue
Instead of generating permutations, use a priority queue to keep track of characters by their frequency. This way, you can place the most frequent characters first.
Step 4: Place Characters
Place each character from the priority queue in the first available position that is ‘k’ distance from its last occurrence.
Pseudocode for Optimized Version:


Time Complexity:
 Counting frequency takes O(n)
 Priority queue operations take O(n log n)
 Placing characters takes O(n)
Overall: O(n log n)
Space Complexity:
 Frequency counter: O(n)
 Priority queue: O(n)
Overall: O(n)
This optimized solution is far more efficient than the bruteforce approach in both time and space complexity.
Code Explanation and Design Decisions
1. Initial Parameters
s
: The input string we need to rearrange. It is the core data we are working on.k
: The minimum distance between any two identical characters in the rearranged string. It sets the constraints for the problem.
2. Primary Loop or Iteration
The primary loop is the one iterating over the max_heap
. Each iteration deals with a character that is most frequent among the remaining characters.
Significance:
 Places a character as many times as its frequency dictates.
 Ensures that the most frequent characters are spread apart before less frequent characters are considered.
3. Conditions or Branches
The condition
if (max_freq  1) * k >= len(s):
checks if the problem has no solution. It signifies whether rearrangement is possible given the value ofk
.The condition
if result[i] == '' and (i < k or result[i  k] != char):
ensures that the character is placed where there is an empty slot and where it is at leastk
distance apart from any of its duplicates.
4. Updates or Modifications to Parameters
freq = 1
: This reflects that we have successfully placed an occurrence of the character, reducing the number of remaining placements needed.result[i] = char
: This modification updates the resulting string, progressively building towards the final answer.
5. Invariant
The invariant is that for any character we place, the minimum distance between its adjacent occurrences should be k
. This invariant assures that the constraints of the problem are continually met.
6. Significance of the Final Output
The final output string represents the rearranged version of the input string such that no two identical characters are less than k
distance apart. It satisfies the problem’s requirements by either providing a valid rearranged string or confirming that it’s impossible by returning an empty string.
By maintaining these conditions, constraints, and invariants, the code effectively solves the problem within its defined parameters.
Coding Constructs
What are the highlevel problemsolving strategies or techniques being used by this code?
If you had to explain the purpose of this code to a nonprogrammer, 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 languageagnostic 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 problemsolving 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 stepbystep 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 Pythonbased 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 problemspecific 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.
Q&A
What are the reasons for making these mistakes in the given code?
Similar Problems
Here are 10 problems that involve similar underlying concepts:
Top K Frequent Elements (#347)
 Why: Utilizes a max heap to deal with frequencybased problems. Similar to our problem where we needed to find and place frequent characters.
Reorganize String (#767)
 Why: Involves rearranging a string based on character frequencies to satisfy certain conditions, much like our problem.
Task Scheduler (#621)
 Why: Requires tasks to be scheduled with a minimum cooling period between the same tasks, similar to the minimum distance required between characters in our problem.
Longest Substring Without Repeating Characters (#3)
 Why: The need to keep track of distances between identical characters has parallels in our problem.
Permutations II (#47)
 Why: It also deals with rearranging elements (in an array), although with different constraints.
Group Anagrams (#49)
 Why: Requires manipulation and reorganization of strings based on their characters, much like our problem.
Find All Anagrams in a String (#438)
 Why: Requires recognizing patterns in the distribution of characters within strings, which is a core part of our problem.
Maximum Frequency Stack (#895)
 Why: Relates to frequencybased data manipulation. Itâ€™s also another application where heapbased solutions are beneficial.
Wiggle Sort II (#324)
 Why: Involves rearranging elements in an array to satisfy specific positional constraints, a parallel concept to our problem.
Minimum Window Substring (#76)
 Why: The problem requires dealing with the frequency of characters in substrings. Frequency mapping is a common element with our original problem.
These problems are related to the original problem either by their utilization of similar data structures like heaps, by the need to arrange or rearrange elements based on frequencies or positional constraints, or by requiring pattern recognition within strings or arrays.