Range Sum Making Queries Without Updates

Range Sum Making Queries Without Updates

Calculating the sum of elements within a range with the prefix sum technique in a list is common in many programming scenarios.

These include competitive coding and data analysis.

As an expert in algorithm optimization, I have extensive experience in leveraging efficient techniques like prefix sums to solve such problems with minimal computational overhead.

This article will introduce you to a Python implementation that efficiently computes range sums using the prefix sum approach, saving time and computational resources.

Understanding the Prefix Sum Approach

Let's dive in!

What is the Prefix Sum Technique?

The prefix sum technique is a powerful method to preprocess an array to allow for quick sum queries over any subarray.

We can efficiently retrieve the sum of any range in constant time by calculating cumulative sums once.

Implementing the Prefix Sum Function

Here is a Python function that demonstrates how to compute the sum of elements in a given range using the prefix sum technique:

def sor(arr: list, i: int, j: int) -> int:
    """
    Calculate the sum of elements in the given range [i, j] using a prefix sum approach.

    Args:
        arr (list): The input list of integers.
        i (int): The starting index of the range (inclusive).
        j (int): The ending index of the range (inclusive).

    Returns:
        int: The sum of the elements from index i to j.
    """
    n = len(arr)  # Length of the input array
    dp = {}  # Dictionary to store the prefix sums

    # Form the prefix sum array (dp)
    dp[0] = arr[0]
    for k in range(1, n):
        dp[k] = arr[k] + dp[k - 1]

    print(f"dp: {dp}")  # Debug statement to print the prefix sum array

    # Fetch the value for the range from i to j
    if i == 0:
        return dp[j]  # If starting index is 0, return the prefix sum up to j
    else:
        return dp[j] - dp[i - 1]  # Subtract the prefix sum up to i-1 from the sum up to j

# Example usage
print(sor([10, 14, 5, -10, 3], 2, 4))  # Output: -2
print(sor([10, 14, 5, -10, 3], 1, 3))  # Output: 9

Code Walkthrough

Initialization: The function begins by calculating the length of the input array. This is followed by initializing a dictionary dp to store the prefix sums.

Prefix Sum Calculation: The prefix sums are computed iteratively. In each entry dp[k] holds the sum of elements from the start of the array up to index k.

Range Sum Retrieval: To get the sum of a subarray from index i to j, the function checks if i is zero. If it is, the prefix sum at j is returned directly. Otherwise, the prefix sum up to i-1 is subtracted from the sum to j to get the desired result.

Debugging Output:  A debug print statement is included to display the computed prefix sums.

Harness the Power of Prefix Sums

By understanding and implementing the prefix sum technique, you can significantly improve the efficiency of range sum queries in your applications.

Prefix Sums are especially useful in frequent sum calculations over varying ranges of scenarios.

That is because it reduces the time complexity from linear to constant for each query.

Now, it’s your turn to apply this technique to optimize your code and enhance project performance.