The core challenge of the problem is to compute the partial sums of an array efficiently. Partial sums are useful in various applications such as range sum queries, prefix sums, and cumulative frequency tables. A common pitfall is to recompute sums for each query, leading to inefficient solutions.

To solve this problem, we can use the prefix sum technique. The idea is to preprocess the array to create a prefix sum array where each element at index `i`

contains the sum of elements from the start of the array up to index `i`

. This allows us to answer range sum queries in constant time.

A naive approach would involve iterating through the array to compute the sum for each query. This approach has a time complexity of `O(n^2)`

for multiple queries, which is inefficient.

The optimized solution involves creating a prefix sum array in `O(n)`

time. Once the prefix sum array is built, any range sum query can be answered in `O(1)`

time.

Here is a step-by-step breakdown of the optimized algorithm:

- Initialize a prefix sum array of the same length as the input array.
- Set the first element of the prefix sum array to be the same as the input array.
- Iterate through the input array starting from the second element, and for each element, set the corresponding element in the prefix sum array to be the sum of the current element and the previous element in the prefix sum array.
- To get the sum of elements between indices
`i`

and`j`

, subtract the prefix sum at`i-1`

from the prefix sum at`j`

.

```
public class PartialSums {
// Function to compute the prefix sums
public static int[] computePrefixSums(int[] arr) {
int n = arr.length;
int[] prefixSums = new int[n];
prefixSums[0] = arr[0];
// Compute the prefix sums
for (int i = 1; i < n; i++) {
prefixSums[i] = prefixSums[i - 1] + arr[i];
}
return prefixSums;
}
// Function to get the sum of elements between indices i and j
public static int rangeSum(int[] prefixSums, int i, int j) {
if (i == 0) {
return prefixSums[j];
} else {
return prefixSums[j] - prefixSums[i - 1];
}
}
public static void main(String[] args) {
int[] arr = {1, 2, 3, 4, 5};
int[] prefixSums = computePrefixSums(arr);
// Example queries
System.out.println("Sum of elements from index 1 to 3: " + rangeSum(prefixSums, 1, 3)); // Output: 9
System.out.println("Sum of elements from index 0 to 4: " + rangeSum(prefixSums, 0, 4)); // Output: 15
}
}
```

The time complexity for computing the prefix sums is `O(n)`

, where `n`

is the length of the input array. The space complexity is also `O(n)`

due to the additional prefix sum array. Each range sum query can be answered in `O(1)`

time.

Potential edge cases include:

- Empty array: The function should handle an empty input array gracefully.
- Single element array: The prefix sum array should be the same as the input array.
- Negative numbers: The algorithm should correctly handle arrays with negative numbers.

To test the solution comprehensively, consider the following test cases:

- Simple cases with small arrays.
- Arrays with all positive numbers, all negative numbers, and a mix of both.
- Edge cases such as empty arrays and single element arrays.

When approaching such problems, it is crucial to think about preprocessing techniques that can optimize query times. Practice problems involving prefix sums, range queries, and cumulative sums to develop a deeper understanding.

Understanding and implementing the prefix sum technique is essential for efficiently solving range sum queries. This problem highlights the importance of preprocessing in algorithm design to achieve optimal performance.

For further reading and practice, consider the following resources: