**π‘ Problem Formulation:** This article tackles the challenge of computing the minimum number of operations needed to transform a given array into a strictly increasing sequence. Each operation can increment a selected element by 1. For instance, given an input array [1, 2, 1], the desired output after making the array strictly increasing is [1, 2, 3], hence 2 operations are required.

## Method 1: Iterative Approach

This method involves iterating through the array and comparing each element with the previous one. If an element is not greater than the previous, we perform operations (incrementing) until it is. We keep track of the number of operations needed.

Here’s an example:

def min_operations_to_increase(arr): count = 0 for i in range(1, len(arr)): if arr[i] <= arr[i - 1]: count += arr[i - 1] - arr[i] + 1 arr[i] = arr[i - 1] + 1 return count print(min_operations_to_increase([1, 2, 1]))

Output: 2

The provided code defines a function `min_operations_to_increase()`

that takes an array as an argument and iterates through it. If the current element isn’t greater than the previous one, it increments the current element to be one more than the previous and adds the difference to the operation count.

## Method 2: Using Max Function Within Loop

The second method improves on the first by using the max function to set the current element to either its original value or one more than the previous element, depending on which is greater. This reduces the need for an if-statement within the loop.

Here’s an example:

def min_operations_to_increase(arr): count = 0 for i in range(1, len(arr)): required_value = max(arr[i], arr[i - 1] + 1) count += required_value - arr[i] arr[i] = required_value return count print(min_operations_to_increase([1, 2, 1]))

Output: 2

The function `min_operations_to_increase()`

iterates through the array using a for-loop, on each iteration it calculates the `required_value`

, which is the higher of the current array value or one more than the previous. It then increments the count by the difference between the `required_value`

and the current element, setting the current element to the `required_value`

.

## Method 3: Functional Style with Accumulate

Using Python’s functional programming capabilities, we can utilize `itertools.accumulate`

to elegantly apply a custom function across the array, ensuring each element is at least one greater than the previous.

Here’s an example:

from itertools import accumulate def min_operations_to_increase(arr): def custom_acc(acc, x): return max(x, acc + 1) return sum(x - y for x, y in zip(accumulate(arr, custom_acc), arr)) print(min_operations_to_increase([1, 2, 1]))

Output: 2

This snippet uses the `accumulate()`

function with a custom accumulator `custom_acc()`

that ensures the current element is at least one more than the accumulated value. The difference between the resulting array and the original array gives the count of operations.

## Method 4: Using NumPy Library

For high-performance numerical computation, we can leverage the NumPy library. This method can be beneficial when dealing with large data sets due to optimized operations.

Here’s an example:

import numpy as np def min_operations_to_increase(arr): arr_np = np.array(arr) for i in range(1, len(arr_np)): if arr_np[i] <= arr_np[i - 1]: arr_np[i] = arr_np[i - 1] + 1 return sum(arr_np - arr) print(min_operations_to_increase([1, 2, 1]))

Output: 2

By converting the list to a NumPy array, we can use the efficient numpy operations for incrementing elements that are not greater than their predecessors. The sum of the differences between the updated NumPy array and the original array is the count of operations.

## Bonus One-Liner Method 5: Pythonic Expressiveness

Adopting Python’s expressiveness, we can reduce the solution to a one-liner using list comprehension and the `enumerate()`

function combining the logic of our previous methods.

Here’s an example:

def min_operations_to_increase(arr): return sum(max(0, prev + 1 - curr) for prev, curr in zip([float('-inf')] + arr, arr)) print(min_operations_to_increase([1, 2, 1]))

Output: 2

This succinct code uses a generator expression with `zip()`

to iterate through pairs of previous and current values. It calculates the required number of increments, if any, ensuring an elegant and compact solution to the problem.

## Summary/Discussion

**Method 1:**Iterative Approach. Strengths: Simple and straightforward. Weaknesses: May not be the most efficient with larger data sets due to multiple increment operations.**Method 2:**Using Max Function Within Loop. Strengths: More optimized than Method 1 with fewer update operations. Weaknesses: Still not as fast as functional or numpy-based approaches for large arrays.**Method 3:**Functional Style with Accumulate. Strengths: Clean and functional style. Weaknesses: Can be less intuitive for those unfamiliar with functional programming paradigms.**Method 4:**Using NumPy Library. Strengths: Great speed and efficiency for large datasets. Weaknesses: Requires an external library and understanding of NumPy.**Bonus One-Liner Method 5:**Pythonic Expressiveness. Strengths: Brief and elegant. Weaknesses: May sacrifice some readability for succinctness.