Prefix Sum Problems | Problems in Arrays - 3 | Lecture 16 | C++ and DSA Foundation Course

Key Concepts

• Prefix Sum (Running Sum): The sum of all elements in an array up to a given index.
• Two-Pointer Approach: Using two pointers to traverse an array, often from opposite ends.
• Subarray: A contiguous part of an array.
• Suffix Sum: The sum of all elements in an array from a given index to the end.
• One-Based Indexing: Array indexing that starts at 1 instead of 0.
• Query-Based Problems: Problems that involve processing multiple queries on a given dataset.

Prefix Sum Concept and Calculation

The video explains the prefix sum (or running sum) concept. Given an array, the prefix sum array contains, at each index i, the sum of all elements from the beginning of the array up to index i.

• Example: For the array [5, 4, 1, 2, 3], the prefix sum array is [5, 9, 10, 12, 15].
• Calculation: The prefix sum at index i can be calculated using the formula: prefix_sum[i] = prefix_sum[i-1] + array[i]. The first element of the prefix sum array is the same as the first element of the original array.
• Code Implementation: The code iterates through the array, updating each element with the sum of itself and the previous element. This modifies the original array to store the prefix sums.

Partitioning an Array into Two Subarrays with Equal Sums

The video presents a problem where you need to determine if an array can be partitioned into two subarrays with equal sums. This is equivalent to checking if there exists an index where the prefix sum up to that index is equal to the suffix sum from the next index to the end of the array.

• Approach: Instead of calculating prefix and suffix sums separately for each index, the video suggests calculating the total sum of the array first. Then, for each index, calculate the prefix sum and derive the suffix sum by subtracting the prefix sum from the total sum.
• Formula: suffix_sum = total_sum - prefix_sum.
• Code Implementation: The code calculates the total sum of the array. Then, it iterates through the array, calculating the prefix sum at each index. It compares the prefix sum with the calculated suffix sum. If they are equal, it returns true; otherwise, it continues iterating. If no such index is found, it returns false.

Query-Based Problem: Sum of Values in a Range

The video discusses a query-based problem where you are given an array and a series of queries. Each query consists of two indices, L and R, and you need to find the sum of the elements in the array between these indices (inclusive).

• Challenge: Calculating the sum for each query using a loop would be inefficient, especially with a large number of queries.
• Solution: Use the prefix sum concept to pre-calculate the prefix sums of the array. Then, for each query, the sum of the elements between L and R can be calculated using the formula: sum(L, R) = prefix_sum[R] - prefix_sum[L-1].
• One-Based Indexing: The video highlights that the problem uses one-based indexing, meaning the first element of the array is at index 1. To handle this, the code creates a vector of size n+1, where n is the size of the original array. The element at index 0 is initialized to 0, and the elements from index 1 to n store the actual array values.
• Code Implementation: The code first calculates the prefix sum array. Then, for each query, it calculates the sum of the elements between L and R using the formula prefix_sum[R] - prefix_sum[L-1] and prints the result.

Synthesis/Conclusion

The video effectively demonstrates the prefix sum concept and its applications in solving array-related problems. It covers calculating prefix sums, using them to efficiently partition an array into subarrays with equal sums, and applying them to solve query-based problems involving range sums. The video emphasizes the importance of understanding the problem constraints, such as one-based indexing, and adapting the solution accordingly. The key takeaway is that pre-calculating prefix sums can significantly improve the efficiency of algorithms that involve repeated calculations of sums over ranges of elements.