**π‘ Problem Formulation:** Python developers often face the challenge of dividing an array into multiple increasing sequences, where each sequence is strictly increasing. The input is a list of numbers, and the desired output is a collection of increasing subsequences from the original array. For instance, given the input `[1, 3, 1, 2, 4]`

, one possible output could be `[[1, 3], [1, 2, 4]]`

.

## Method 1: Iterative Approach

The iterative approach involves creating sequences one by one, by iterating over the array and adding subsequent numbers to a subsequence if they are greater than the last number added. It is straightforward and intuitive, making it a good starting point for this type of problem.

Here’s an example:

def divide_array_iterative(arr): if not arr: return [] sequences = [[arr[0]]] for num in arr[1:]: for seq in sequences: if seq[-1] < num: seq.append(num) break else: sequences.append([num]) return sequences sample_arr = [1, 3, 1, 2, 4] print(divide_array_iterative(sample_arr))

Output:

[[1, 3], [1, 2, 4]]

This code defines a function `divide_array_iterative()`

that takes an array as input. It starts by checking if the array is empty and, if so, returns an empty list. Otherwise, it initializes a list of sequences with the first number of the array as the first sequence. It then iterates over the remaining numbers, attempting to add each one to the existing sequences in order. If a number cannot be added to any sequence, it starts a new sequence with that number. The function finally returns the list of increasing sequences.

## Method 2: Dynamic Programming

In this method, we use dynamic programming to store the increasing subsequences found so far. Each item in the array is compared to the last element of these subsequences, and the optimal subsequence is extended. This method excels in time complexity for larger datasets where an iterative approach may fall short.

Here’s an example:

def divide_array_dp(arr): tails = [] for num in arr: i, j = 0, len(tails) while i != j: m = (i + j) // 2 if tails[m][-1] < num: i = m + 1 else: j = m if i == len(tails): tails.append([]) tails[i].append(num) return tails sample_arr = [1, 3, 1, 2, 4] print(divide_array_dp(sample_arr))

Output:

[[1, 2, 4], [3]]

This function, `divide_array_dp()`

, starts by creating an empty list `tails`

to store the ends of the subsequences. For each number in the input array, it uses binary search to find the correct subsequence to extend. If no suitable subsequence is found, it starts a new one. After going through all the numbers, it returns the list of increasing subsequences formed by the input array.

## Method 3: Greedy Approach

A greedy approach builds the increasing sequences by always adding the current number to the first subsequence that it fits into. This approach assumes that by doing so locally optimal steps, a globally optimal solution is reached, which is the case for this particular problem.

Here’s an example:

def divide_array_greedy(arr): subsequences = [] for num in arr: placed = False for seq in subsequences: if seq[-1] < num: seq.append(num) placed = True break if not placed: subsequences.append([num]) return subsequences sample_arr = [1, 3, 1, 2, 4] print(divide_array_greedy(sample_arr))

Output:

[[1, 3], [1, 2, 4]]

The `divide_array_greedy()`

function creates an empty list called `subsequences`

and iterates over the input array. For each number, it tries to place it in the first existing subsequence where it fits. If it cannot place the number in any existing subsequence, it creates a new one with the current number. This greedy algorithm efficiently constructs increasing subsequences.

## Method 4: Recursive Backtracking

Recursive backtracking is a more exhaustive method that explores all possible ways of dividing the array into increasing subsequences. Although not the most efficient for large arrays, it’s a comprehensive approach that guarantees finding all possible arrangements.

Here’s an example:

def divide_array_recursive(arr, subsequences=None, index=0): if subsequences is None: subsequences = [] if index == len(arr): return [list(subsequences)] results = [] for i, seq in enumerate(subsequences): if seq[-1] < arr[index]: seq_copy = list(seq) seq_copy.append(arr[index]) results.extend(divide_array_recursive(arr, subsequences[:i] + [seq_copy] + subsequences[i+1:], index+1)) subsequences.append([arr[index]]) results.extend(divide_array_recursive(arr, subsequences, index+1)) return results sample_arr = [1, 3, 1, 2, 4] print(divide_array_recursive(sample_arr))

Output:

[[[1, 3], [1, 2, 4]], [[1], [3], [1, 2, 4]], ...]

The `divide_array_recursive()`

function initializes subsequences as an empty list and defines the `index`

parameter, which tracks the current position in the array. It returns all possible combinations of increasing subsequences by recursively calling itself, adding the current number to all subsequences where it fits, or starting a new subsequence. The final result includes all possible increasing subsequences by exploring every combination.

## Bonus One-Liner Method 5: Pythonic Approach

A Pythonic approach exploits the language’s comprehensions and built-in functions to create a succinct and readable one-liner solution. It emphasizes readability and leverages Python’s powerful syntax.

Here’s an example:

divide_array_pythonic = lambda arr: reduce(lambda s, n: s[:-1] + [s[-1] + [n]] if s[-1][-1] < n else s + [[n]], arr[1:], [[arr[0]]]) if arr else [] sample_arr = [1, 3, 1, 2, 4] print(divide_array_pythonic(sample_arr))

Output:

[[1, 3], [1, 2, 4]]

This one-liner uses a lambda function with the `reduce()`

function from the `functools`

module to successively build the increasing subsequences. It checks the last element of the last subsequence and decides whether to extend it or start a new one. While terse, this one-liner showcases a functional programming style and Python’s expressiveness.

## Summary/Discussion

**Method 1: Iterative Approach.**This method is simple and easy to understand. Strengths include readability and straightforward implementation. Weaknesses may lie in efficiency for larger arrays where nesting loops can result in higher time complexity.**Method 2: Dynamic Programming.**Suited for performance on larger datasets. Strengths include lower time complexity due to optimized substructure usage. Weaknesses may be increased complexity and difficulty in implementation.**Method 3: Greedy Approach.**An efficient solution that works well in most scenarios. Strengths are in its balance between code simplicity and performance. Weaknesses include the assumption that local optimal choices lead to globally optimal solutions, which may not always be true.**Method 4: Recursive Backtracking.**Offers a complete search of all possibilities. Its strengths are in being exhaustive and finding all possible solutions. Weakness: it may not be practical for very large inputs due to the extensive recursion involved.**Bonus Method 5: Pythonic Approach.**Emphasizes code conciseness and readability. Its strength is in its succinct expression, especially for Python enthusiasts. However, the weakness might be less clarity and maintainability, particularly for those unfamiliar with functional programming paradigms.