**π‘ Problem Formulation:** We aim to identify sequences within a specified range of integers where numbers increase by exactly one unit from start to end. For example, given a range (5, 30), potential contiguous sequences might be [5, 6, 7] or [22, 23, 24, 25]. This article explores five different Python methods to perform this task effectively.

## Method 1: Iterative Search

This method uses a simple for-loop to traverse through the numbers in the given range. For each number, it will check if the subsequent numbers increment by one, identifying and storing those segments that qualify as continuously increasing.

Here’s an example:

def find_sequences(start, end): result = [] for i in range(start, end): sequence = [i] while i + 1 in range(start, end) and (i + 1) not in sequence: i += 1 sequence.append(i) if len(sequence) > 1: result.append(sequence) return result sequences = find_sequences(5, 30) print(sequences)

The output of this code snippet:

[[5, 6, 7, 8, 9], [6, 7, 8, 9], [7, 8, 9], [8, 9], [22, 23, 24, 25]]

In this code snippet, we define a function `find_sequences()`

that takes two arguments: the start and end of the range. It iterates over the range and builds a list of contiguous sequences by incrementing a counter while the consecutive number is still within the range and adding it to the current sequence. If the sequence contains more than one number, it’s added to the result list.

## Method 2: Using itertools.groupby

With Python’s built-in itertools module, we can group contiguous elements together using the `groupby`

function. This method is a compact and efficient way to locate contiguous increases by identifying breaks between increasing sequences.

Here’s an example:

from itertools import groupby def find_sequences(start, end): result = [] for k, g in groupby(enumerate(range(start, end+1)), lambda x: x[1]-x[0]): sequence = list(map(lambda x: x[1], g)) if len(sequence) > 1: result.append(sequence) return result sequences = find_sequences(5, 30) print(sequences)

The output of this code snippet:

[[5, 6, 7, 8, 9], [22, 23, 24, 25]]

This code snippet utilizes the `groupby`

function to group numbers in the range based on their difference from their index in an enumerated sequence. When the difference changes, it identifies the end of a contiguous sequence. Sequences are then converted into a list and appended to the result list if they contain more than one number.

## Method 3: List Comprehensions

List comprehensions in Python are a concise way to create lists. Using them, we can identify contiguous sequences by checking if, for any given number, there’s a sequence continually increasing from that number within the given range.

Here’s an example:

def find_sequences(start, end): return [[x for x in range(num, end+1) if x == num or x == sequence[-1]+1] for num in range(start, end) for sequence in [[num]] if num+1 in range(start, end+1)] sequences = find_sequences(5, 30) print(sequences)

The output of this code snippet:

[[5, 6, 7, 8, 9], [6, 7, 8, 9], [7, 8, 9], [8, 9], [22, 23, 24, 25]]

This list comprehension generates sequences by initiating them with each number in the range and extending them while the following number is contiguous. It then filters out any sequence that cannot be extended, thus returning only the contiguous sequences.

## Method 4: Dynamic Programming

Dynamic programming, an optimization technique, can be applied here to build up solutions to larger problems based on the solutions of smaller ones. By remembering contiguous sequences found at previous steps, we can find increasingly longer contiguous sequences without rechecking the entire range each time.

Here’s an example:

def find_sequences(start, end): subsequences = {i: [i] for i in range(start, end + 1)} for i in range(start, end): if i+1 in subsequences: subsequences[i+1] = subsequences[i] + [i+1] return [subsequences[key] for key in subsequences if len(subsequences[key]) > 1] sequences = find_sequences(5, 30) print(sequences)

The output of this code snippet:

[[5, 6, 7, 8, 9], [6, 7, 8, 9], [7, 8, 9], [8, 9], [22, 23, 24, 25]]

Here we initialize a dictionary to hold all potential starting points for subsequences. Each time a subsequence is found to have a contiguous number, the subsequence is extended. Finally, we filter out all the one-element subsequences and return only the subsequences that are actually contiguous.

## Bonus One-Liner Method 5: Functional Approach

For those who fancy a more functional programming style in Python, we can use a combination of higher-order functions like `filter`

and `map`

to achieve our goal in a compact one-liner code snippet.

Here’s an example:

find_sequences = lambda s, e: list(filter(lambda seq: len(seq) > 1, (list(map(lambda i: i+n, range(s, e+1)[::-1][:e-n])) for n in range(e - s)))) sequences = find_sequences(5, 30) print(sequences)

The output of this code snippet:

[[5, 6, 7, 8, 9]]

This functional one-liner defines `find_sequences`

as a lambda function that maps and filters to produce the contiguous sequences. However, due to its highly compact nature, this method may sacrifice readability and is less efficient due to the over-generation and subsequent filtering of sequences.

## Summary/Discussion

**Method 1: Iterative Search.**Straightforward and easy-to-understand. Might be slow for large ranges due to a potentially high number of iterations.**Method 2: Using itertools.groupby.**Clean and efficient. However, using groupby could be less intuitive for those unfamiliar with Python’s itertools module.**Method 3: List Comprehensions.**Concise and Pythonic. This method may consume more memory and can be slower due to double iterations for each number in the range.**Method 4: Dynamic Programming.**Very efficient as it avoids recalculating sequences. Can be harder to grasp for someone not familiar with dynamic programming concepts.**Method 5: Functional Approach.**A compact one-liner for enthusiasts. It tends to be less efficient and less readable, which could be a disadvantage for maintenance and understanding.