**π‘ Problem Formulation:** We need to write a program that can efficiently calculate K-prefix (KPR) sums for a given list of numbers, for multiple queries. Each query specifies a value of K, and the program should return the sum of the first K numbers in the list. For example, given a list `[1, 3, 5, 7]`

and a query K = 2, the KPR sum would be `1 + 3 = 4`

.

## Method 1: Brute Force Approach

This method involves a direct approach to solving the problem by iterating through the list up to K elements for each query and calculating the sum. This is the most straightforward method but can be inefficient for large lists with many queries, as it has a time complexity of O(N*K) where N is the number of queries.

Here’s an example:

def kpr_sum(numbers, queries): results = [] for k in queries: results.append(sum(numbers[:k])) return results nums = [1, 3, 5, 7] queries = [2, 3] print(kpr_sum(nums, queries))

Output: `[4, 9]`

This code defines a function `kpr_sum()`

which iterates through each query value, slices the list up to the Kth element, and computes the sum. Although simple, the repeated slicing can lead to high time complexity with larger lists and many queries.

## Method 2: Cumulative Sum

To optimize our code for multiple queries, we can preprocess the list by creating a cumulative sum array. Once this array is in place, we can calculate the KPR sum for any query in O(1) time by subtracting the cumulative sum of elements just before K from the cumulative sum at K.

Here’s an example:

def kpr_sum_cumulative(numbers, queries): cum_sum = [0] for num in numbers: cum_sum.append(cum_sum[-1] + num) results = [] for k in queries: results.append(cum_sum[k]) return results nums = [1, 3, 5, 7] queries = [2, 3] print(kpr_sum_cumulative(nums, queries))

Output: `[4, 9]`

In the above code, we’ve reduced the complexity of multiple queries by avoiding repeated computation. The cumulative sum array, `cum_sum`

, is calculated first, which is then used to find the KPR sum for each query in constant time.

## Method 3: Using Numpy Library

For numerical computations, the Numpy library offers an efficient and fast approach. With its optimized operations, Numpy can compute cumulative sums efficiently, which we can then use for our KPR sum queries.

Here’s an example:

import numpy as np def kpr_sum_numpy(numbers, queries): cum_sum = np.cumsum([0] + numbers) results = [cum_sum[k] for k in queries] return results nums = [1, 3, 5, 7] queries = [2, 3] print(kpr_sum_numpy(nums, queries))

Output: `[4, 9]`

Using Numpy, we prepend a 0 to the list of numbers to align the indexes with our K-values. The function `np.cumsum()`

efficiently calculates the cumulative sum, and we retrieve KPR sums for all queries with a list comprehension.

## Method 4: itertools.accumulate()

Python’s built-in library `itertools`

provides a function `accumulate()`

, which can be used to create the cumulative sum array in a memory-efficient and readable manner.

Here’s an example:

from itertools import accumulate def kpr_sum_itertools(numbers, queries): cum_sum = list(accumulate(numbers, initial=0)) results = [cum_sum[k] for k in queries] return results nums = [1, 3, 5, 7] queries = [2, 3] print(kpr_sum_itertools(nums, queries))

Output: `[4, 9]`

The above snippet uses `accumulate()`

to generate the cumulative sum list with an initial value of 0, allowing for easy access to KPR sums in constant time. This function is part of the Python standard library, making it widely available for use.

## Bonus One-Liner Method 5: Lambda and map

A compact and functional approach can be utilized by combining `lambda`

functions and the `map()`

function to calculate KPR sums in a one-liner. This method emphasizes readability and conciseness over performance.

Here’s an example:

nums = [1, 3, 5, 7] queries = [2, 3] kpr_sums = map(lambda k: sum(nums[:k]), queries) print(list(kpr_sums))

Output: `[4, 9]`

The above code succinctly expresses the KPR sum calculation as a `map()`

application of a `lambda`

function across the queries. It is less efficient for large datasets but provides a quick and expressive solution for smaller instances.

## Summary/Discussion

**Method 1: Brute Force Approach.**It’s simple and easy to implement. However, it’s inefficient for large inputs and multiple queries due to its high time complexity.**Method 2: Cumulative Sum.**Significantly more efficient for multiple queries. Precomputing the cumulative sum array reduces the per-query time complexity to constant time, but requires additional space to store the cumulative sums.**Method 3: Using Numpy Library.**It takes advantage of Numpy’s performance optimizations for numerical computations. Provides a fast and concise approach but introduces a dependency on the external Numpy library.**Method 4: itertools.accumulate().**Uses a Python standard library function for a clean and memory-efficient solution. It’s both readable and efficient for any size of input data.**Method 5: Lambda and map.**A concise one-liner that is aesthetically pleasing but not as performance-optimized as the other methods for large data sets.