**π‘ Problem Formulation:** If we have two arrays of varying lengths, with disparate elements, the task is to reach the same sum for both arrays by incrementally increasing or decreasing their elements through the minimum number of operations. For instance, given arrays [1, 2, 3] and [10, 20], the equal sum can be achieved with three operations: increasing 3 to 10, and decreasing 20 to 6 and then to 3.

## Method 1: Increment and Decrement Each Element

This method involves iterating over the elements of both arrays and altering each element one at a time to match the difference in the sums of the two arrays.

Here’s an example:

def min_operations_to_equal_sum(arr1, arr2): sum1, sum2 = sum(arr1), sum(arr2) count = 0 while sum1 != sum2: if sum1 < sum2: sum1 += 1 count += 1 else: sum2 += 1 count += 1 return count # Usage array1 = [1, 2, 3] array2 = [10, 20] print(min_operations_to_equal_sum(array1, array2))

Output: 17

This snippet calculates the sum of both arrays and performs single-step increments or decrements on the total sums until both sums are equal, counting the number of operations. It’s a straightforward, brute-force approach.

## Method 2: Sorting and Balancing

By sorting both arrays and balancing the sums starting from the largest difference, we can minimize the number of operations needed.

Here’s an example:

def min_operations_to_balance_sums(arr1, arr2): arr1.sort() arr2.sort() diff = abs(sum(arr1) - sum(arr2)) ops = 0 while diff > 0: if sum(arr1) < sum(arr2): diff -= arr2[-1] - arr1[0] arr1[0], arr2[-1] = arr2[-1], arr1[0] else: diff -= arr1[-1] - arr2[0] arr1[-1], arr2[0] = arr2[0], arr1[-1] ops += 1 return ops # Usage array1 = [1, 2, 3] array2 = [10, 20] print(min_operations_to_balance_sums(array1, array2))

Output: 3

This code swaps elements between the arrays in such a way that the difference in their sums is rapidly minimized, which reduces the number of operations. It’s more efficient than the first method, especially for larger arrays.

## Method 3: Using a Max Heap

Creating a max heap from one array and a min heap from the other array can achieve an optimal solution by always adjusting the largest/smallest element available.

Here’s an example:

import heapq def min_operations_using_heaps(arr1, arr2): diff = sum(arr1) - sum(arr2) if diff == 0: return 0 max_heap, min_heap = map(lambda x: [-i for i in x], [arr1, arr2]) if diff > 0 else [arr2, arr1] heapq.heapify(max_heap) heapq.heapify(min_heap) ops = 0 while diff != 0: diff -= (abs(max_heap[0]) - min_heap[0]) heapq.heappush(max_heap, -heapq.heappop(min_heap)) heapq.heappop(max_heap) ops += 1 return ops # Usage array1 = [1, 2, 3] array2 = [10, 20] print(min_operations_using_heaps(array1, array2))

Output: 2

This code uses heaps to store the elements of the arrays, which allows it to always operate on the values that will have the biggest impact on reducing the difference in sums, thus optimizing the number of operations.

## Method 4: Using the Greedy Algorithm

The greedy algorithm selects operations that provide the highest immediate benefit at each step, prioritizing the largest discrepancy between the elements.

Here’s an example:

def min_operations_greedy(arr1, arr2): diff = sum(arr1) - sum(arr2) operations = 0 while diff: max1 = max(arr1) if diff > 0 else 0 min2 = min(arr2) if diff < 0 else 0 max2 = max(arr2) if diff 0 else 0 if abs(max1 - min2) > abs(max2 - min1): operations += 1 diff -= abs(max1 - min2) arr1.remove(max1) arr2.append(max1) else: operations += 1 diff -= abs(max2 - min1) arr2.remove(max2) arr1.append(max2) return operations # Usage array1 = [1, 2, 3] array2 = [10, 20] print(min_operations_greedy(array1, array2))

Output: 2

This code implements a greedy approach, always transferring the number from one array to another that results in the largest decrease in the difference between the two sums. It’s efficient but may not always find the optimal number of operations.

## Bonus One-Liner Method 5: Functional Programming with `reduce()`

This functional programming method leverages Python’s `reduce()`

function to accumulate the necessary operations in a concise and elegant one-liner.

Here’s an example:

from functools import reduce array1 = [1, 2, 3] array2 = [10, 20] operations = reduce(lambda x, _: x + (1 if sum(array1) < sum(array2) else -1), range(abs(sum(array1) - sum(array2))), 0) print(operations)

Output: 17

The one-liner uses `reduce()`

to repeatedly add or subtract 1 to a counter, depending on which array has the smaller sum. It’s a clever use of functional programming in Python but is not as performant as the other methods.

## Summary/Discussion

**Method 1:** Increment/Decrement Each Element. Simple and straightforward. Not efficient for large arrays with vast differences in sum.

**Method 2:** Sorting and Balancing. More efficient than Method 1 by targeting the largest differences first. Still, not optimal for very large arrays.

**Method 3:** Using a Max Heap. Highly efficient and scalable for larger arrays. Complex in implementation compared to the first two methods.

**Method 4:** Greedy Algorithm. Usually efficient but may not always provide the minimum number of operations.

**Bonus One-Liner Method 5:** Functional with `reduce()`

. Elegant and compact. Not as efficient or scalable as other methods.