**π‘ Problem Formulation:** The task at hand involves creating a data structure in Python capable of efficiently calculating the sum of a range of elements. This means that given an array or list, we need a structure that can return the sum of elements between two indices (inclusive) with minimal computation. For example, given the list `[1, 2, 3, 4, 5]`

, a query for the sum from index 1 to 3 should output `9`

, as `2 + 3 + 4 = 9`

.

## Method 1: Cumulative Sum Array

This method involves generating an extra array where each element is the sum of all the previous elements in the input array. It’s very efficient for multiple range sum queries, as the sum between two indices can be calculated by a simple subtraction operation.

Here’s an example:

def create_cumulative_sum_array(arr): cum_sum = [0] for num in arr: cum_sum.append(cum_sum[-1] + num) return cum_sum def range_sum_query(cum_sum, start, end): return cum_sum[end + 1] - cum_sum[start] # Example usage input_array = [1, 2, 3, 4, 5] cumulative_sum_array = create_cumulative_sum_array(input_array) print(range_sum_query(cumulative_sum_array, 1, 3))

Output:`9`

This code snippet introduces a function `create_cumulative_sum_array()`

which builds the cumulative sum array, and a function `range_sum_query()`

which utilizes the cumulative sum array to calculate the sum of a specific range with a constant time complexity.

## Method 2: Fenwick Tree (Binary Indexed Tree)

Fenwick Tree is a data structure that provides efficient methods for range sum queries and updates. It offers a great balance between the two operations, making it suitable for scenarios where the array is mutable.

Here’s an example:

class FenwickTree: def __init__(self, size): self.size = size self.tree = [0] * (size + 1) def update(self, index, value): while index 0: result += self.tree[index] index -= index & -index return result def range_sum(self, start, end): return self.query(end) - self.query(start - 1) # Example usage fenwick = FenwickTree(5) for idx, val in enumerate([1, 2, 3, 4, 5], start=1): fenwick.update(idx, val) print(fenwick.range_sum(2, 4))

Output:`9`

In this code snippet, class `FenwickTree`

is implemented with methods to update individual elements and calculate the prefix sum. `range_sum()`

method is provided to calculate the range sum by computing the difference between two prefix sums, ensuring an efficient query and update time.

## Method 3: Segment Tree

Segment Trees are powerful data structures used for a variety of range query problems, including range sum queries. They allow for both fast updates and queries, making them ideal for dynamic datasets with numerous changes.

Here’s an example:

class SegmentTree: def __init__(self, arr): self.n = len(arr) self.tree = [0] * (4 * self.n) self.build(arr, 0, 0, self.n - 1) def build(self, arr, node, start, end): if start == end: self.tree[node] = arr[start] else: mid = (start + end) // 2 self.build(arr, 2 * node + 1, start, mid) self.build(arr, 2 * node + 2, mid + 1, end) self.tree[node] = self.tree[2 * node + 1] + self.tree[2 * node + 2] def range_sum_query(self, node, start, end, l, r): if r end: return 0 if l <= start and end <= r: return self.tree[node] mid = (start + end) // 2 left_sum = self.range_sum_query(2 * node + 1, start, mid, l, r) right_sum = self.range_sum_query(2 * node + 2, mid + 1, end, l, r) return left_sum + right_sum # Example usage arr = [1, 2, 3, 4, 5] seg_tree = SegmentTree(arr) print(seg_tree.range_sum_query(0, 0, len(arr) - 1, 1, 3))

Output:`9`

The code here demonstrates the implementation of a Segment Tree which can be used for sum range queries. The `SegmentTree`

class initializes the tree and provides a `range_sum_query()`

method which is capable of efficiently calculating the sum in a given range.

## Method 4: Prefix Sum Function

Instead of storing the cumulative sum in an array, a prefix sum function can be written to calculate the sum on the fly. This method may be more intuitive for simple use cases and requires less memory.

Here’s an example:

def prefix_sum(arr, start, end): return sum(arr[start:end+1]) # Example usage input_array = [1, 2, 3, 4, 5] print(prefix_sum(input_array, 1, 3))

Output:`9`

This method defines a simple `prefix_sum()`

function that calculates the sum of a range via Python’s built-in `sum()`

function. It can be a straightforward choice for one-off or infrequent queries, though it has a higher time complexity for repeated range sum calculations.

## Bonus One-Liner Method 5: List Slicing with Built-In Sum

For the simplest use cases, a one-liner involving list slicing and the built-in sum function can quickly deliver the desired result with minimal effort.

Here’s an example:

# Example usage input_array = [1, 2, 3, 4, 5] print(sum(input_array[1:4]))

Output:`9`

The one-liner provided executes a list slicing operation `input_array[1:4]`

to extract the desired range and then calculates the sum using Python’s `sum()`

function. This method is very concise and perfect for ad hoc queries but not optimized for performance in scenarios with multiple range sum queries.

## Summary/Discussion

**Method 1:**Cumulative Sum Array. Efficient for multiple range queries. Extra space required for the cumulative sum array.**Method 2:**Fenwick Tree (Binary Indexed Tree). Balanced between query and update operations. Requires non-trivial understanding and implementation.**Method 3:**Segment Tree. Versatile and fast for dynamic datasets. More complex to understand and implement.**Method 4:**Prefix Sum Function. Straightforward and easy to write. Not as efficient for multiple queries due to higher time complexity.**Bonus Method 5:**List Slicing with Built-In Sum. Simple one-liner ideal for ad hoc use. Not suitable for optimal performance in repeated queries scenarios.