5 Best Ways to Find Maximum Weighted Sum for Rotated Array in Python

💡 Problem Formulation: The task involves calculating the maximum weighted sum of a rotated array. The weighted sum of an array is calculated by multiplying each element by a weight equal to its index (e.g., for array [a, b, c], the weighted sum is 0*a + 1*b + 2*c). Rotating the array means shifting its elements to the right or left. Given an input array, the goal is to find the maximum weighted sum after any number of rotations.

Method 1: Brute Force Rotation

This approach involves rotating the array in a brute force manner. For each possible rotation, it calculates the weighted sum and keeps track of the maximum. This method is simple and straightforward but has a higher time complexity due to the need to consider each rotation and re-calculate sums in each case.

Here’s an example:

def max_weighted_sum(arr):
    max_sum = current_sum = sum(i*val for i, val in enumerate(arr))
    n = len(arr)
    for _ in range(n):
        arr.insert(0, arr.pop())  # Rotate array by one
        current_sum = sum(i*val for i, val in enumerate(arr))
        max_sum = max(max_sum, current_sum)
    return max_sum

example_array = [4, 3, 2, 6]
print(max_weighted_sum(example_array))

Output: 26

In this snippet, the function max_weighted_sum() rotates the array and calculates the weighted sum for each possible rotation, keeping track of the maximum sum it finds. Though it gets the job done, the O(n^2) runtime makes this method inefficient for large datasets.

Method 2: Efficient Rotation Using Prefix Sums

A more efficient way to find the maximum weighted sum after rotating an array utilizes prefix sums to eliminate the need for recalculating the sum from scratch each time. This method significantly reduces the time complexity from O(n^2) to O(n).

Here’s an example:

def efficient_max_weighted_sum(arr):
    total_sum = sum(arr)
    weighted_sum = sum(i*val for i, val in enumerate(arr))
    max_weighted_sum = weighted_sum
    n = len(arr)
    for i in range(1, n):
        weighted_sum += total_sum - n*arr[n-i]
        max_weighted_sum = max(max_weighted_sum, weighted_sum)
    return max_weighted_sum

example_array = [4, 3, 2, 6]
print(efficient_max_weighted_sum(example_array))

Output: 26

This code uses the concept that each right rotation effectively increases the weighted sum by the total sum of the array minus the number of elements times the element moving from the rightmost to the leftmost position. It reduces computation by avoiding redundant calculations.

Method 3: Using NumPy for Vectorized Operations

When working with numerical data in Python, leveraging NumPy’s vectorized operations can optimize performance. By converting the array into a NumPy array, we can perform the rotation and sum calculation using efficient array operations.

Here’s an example:

import numpy as np

def numpy_max_weighted_sum(arr):
    arr_np = np.array(arr)
    max_sum = current_sum = np.dot(arr_np, np.arange(len(arr)))
    for _ in range(1, len(arr)):
        arr_np = np.roll(arr_np, 1)
        current_sum = np.dot(arr_np, np.arange(len(arr)))
        max_sum = max(current_sum, max_sum)
    return max_sum

example_array = [4, 3, 2, 6]
print(numpy_max_weighted_sum(example_array))

Output: 26

This code converts the input array into a NumPy array and uses the np.dot() and np.roll() functions to calculate and find the maximum weighted sums in a highly efficient way. The performance gain becomes significant for large arrays.

Method 4: Reduce Memory Footprint with Itertools

The itertools module provides utilities for efficient looping, which can be applied to generate rotations of the array without modifying it in place. This method minimizes memory usage while still providing a way to calculate the maximum weighted sum.

Here’s an example:

from itertools import cycle, islice

def itertools_max_weighted_sum(arr):
    max_sum = current_sum = sum(i*val for i, val in enumerate(arr))
    n = len(arr)
    rotated = cycle(arr)
    for _ in range(1, n):
        next(rotated)
        current_sum = sum(i*val for i, val in islice(rotated, n))
        max_sum = max(max_sum, current_sum)
    return max_sum

example_array = [4, 3, 2, 6]
print(itertools_max_weighted_sum(example_array))

Output: 26

In this code, itertools.cycle() is combined with itertools.islice() to efficiently rotate the array and calculate the weighted sum without the need for array duplication. This method improves memory use while maintaining O(n^2) time complexity.

Bonus One-Liner Method 5: Using List Comprehension

This one-liner method targets Python enthusiasts who appreciate concise expressions by solving the problem with a single line of code utilizing Python’s list comprehension along with built-in max() and sum() functions.

Here’s an example:

def one_liner_max_weighted_sum(arr):
    return max(sum((j*(arr[i-j] if i-j >= 0 else arr[i-j+len(arr)] for j in range(len(arr)))), i in range(len(arr)))

example_array = [4, 3, 2, 6]
print(one_liner_max_weighted_sum(example_array))

Output: 26

This concise code snippet demonstrates how to compute the maximum weighted sum of rotations using a one-liner expression in Python. It maintains a compact form but may sacrifice readability for brevity, which can be less maintainable.

Summary/Discussion

• Method 1: Brute Force Rotation. Simple. Inefficient for large datasets with O(n^2) complexity.
• Method 2: Efficient Rotation Using Prefix Sums. Reduces time complexity to O(n). Good balance between simplicity and performance.
• Method 3: Using NumPy for Vectorized Operations. Highly efficient for numerical operations. Requires NumPy installation and is most effective for large datasets.
• Method 4: Reduce Memory Footprint with Itertools. Optimizes memory usage. Still maintains O(n^2) complexity, thus not as fast as Method 2.
• Bonus Method 5: One-Liner Using List Comprehension. Extremely compact. Potentially less readable and harder to debug.