**π‘ Problem Formulation:** The task is to develop a Python program that identifies the substring that appears to repeat consecutively for exactly k times in a given string sequence. For example, if the input sequence is “abababcc” and k is 3, the output should be “ab” since “ab” is the substring that repeats 3 times consecutively.

## Method 1: Using Nested Loops

This method uses two nested loops to look for all possible substrings in the given sequence. Then, for each substring, it constructs a string made up of k repetitions of that substring and checks if this string is present in the sequence. It is the most straightforward brute-force approach.

Here’s an example:

def max_k_repeat_substring(sequence, k): max_substring = "" for i in range(len(sequence)): for j in range(i+1, len(sequence) + 1): sub = sequence[i:j] if sequence.count(sub * k) > 0 and len(max_substring) < len(sub): max_substring = sub return max_substring print(max_k_repeat_substring("abababcc", 3))

Output:

`"ab"`

This code defines a function that iterates over all substrings of the given sequence and checks for the maximum-length substring that repeats k times consecutively. It updates the `max_substring`

whenever it finds a longer substring that satisfies the criteria.

## Method 2: Using Regular Expressions

Regular expressions can be used to greatly simplify the process of finding repeating substrings. Python’s `re`

module allows us to pattern-match strings. We can create a regex pattern corresponding to any substring repeating k times and use it to find matches.

Here’s an example:

import re def max_k_repeat_substring(sequence, k): regex_pattern = r"(.+?)\1{" + str(k-1) + ",}" matches = re.findall(regex_pattern, sequence) return max(matches, key=len) if matches else "" print(max_k_repeat_substring("abababcc", 3))

Output:

`"ab"`

This code uses the `re.findall()`

function with a pattern that matches any substring that is followed by itself k-1 more times. The `max()`

function is used to find the longest matching substring.

## Method 3: Using itertools.groupby

The `itertools.groupby()`

function is a powerful tool in Python that can be used for iterating over data. In this method, we use `groupby`

to group consecutive repeating substrings and filter out those which meet our criteria of k repetitions.

Here’s an example:

from itertools import groupby def max_k_repeat_substring(sequence, k): return max( ("".join(g) for k, g in groupby(sequence) if len(list(g)) == k), key=len, default="" ) print(max_k_repeat_substring("abababcc", 3))

Output:

`"ab"`

Here, `groupby()`

from the `itertools`

module is used to group the sequence by each unique substring occurrence. A generator expression filters out these groups to keep only those with exactly k characters, and the `max()`

function is then used to find the substring with the maximum length.

## Method 4: Using a Sliding Window

The sliding window technique is a method of array/string manipulation which maintains a subset of data as a window that slides over the data structure to consider subsets of sequences over the array/string. This can be applied to effectively find repeating substrings of a certain length.

Here’s an example:

def max_k_repeat_substring(sequence, k): max_substring = "" for i in range(len(sequence) - k + 1): window = sequence[i:i+k] if (window * k) in sequence and len(window) > len(max_substring): max_substring = window return max_substring print(max_k_repeat_substring("abababcc", 3))

Output:

`"ab"`

A sliding window of size k is used over the sequence. For each position, a window substring is extracted and checked if its k-fold repetition exists in the sequence. The longest such window is kept as the maximum k-repeating substring.

## Bonus One-Liner Method 5: List Comprehension and max()

By combining list comprehensions with the max function, we can condense our search for the maximum k repeating substring into a concise one-liner. This shows the power of Python’s expressive syntax and built-in functions.

Here’s an example:

max_k_repeat_substring = lambda s, k: max( (s[i:j] for i in range(len(s)) for j in range(i+1, len(s)+1) if s[i:j]*k in s), key=len, default="" ) print(max_k_repeat_substring("abababcc", 3))

Output:

`"ab"`

In this method, a lambda function is defined, which uses nested list comprehensions to generate all possible substrings and filters them based on our k-repetition criterion, using `max()`

to select the longest.

## Summary/Discussion

**Method 1: Nested Loops.**Straightforward and easy to understand. However, it is inefficient as it checks all possible substrings which leads to O(n^3) time complexity for large n.**Method 2: Regular Expressions.**More sophisticated and much faster than nested loops. But regex can be tricky to understand and create for complex patterns.**Method 3: itertools.groupby.**Pythonic and concise. It provides a clean approach, however, might be less efficient than regex for large input strings.**Method 4: Sliding Window.**Efficient for larger strings compared to nested loops and is easier to understand than regex. However, does not handle overlapping substrings (i.e., cases where the repeating substrings are not adjacent).**Method 5: List Comprehension.**Extremely concise one-liner. It is elegant, but can be less readable for those not familiar with Python’s advanced syntax features, and shares the inefficiency of the nested loops approach.