Unique Summations within Each Sub-Array Aggregated Together
In the realm of computational problem-solving, hashing plays a significant role, particularly when it comes to problems related to sub-array sums. This article introduces the concept of hashing and demonstrates how it can be used to efficiently find the sum of all unique sub-array sums in an array of positive elements.
The key idea behind this approach is the use of prefix sums and a hash set (unordered_set) to track distinct sums, ensuring no duplicates are counted. Since all elements are positive, the prefix sums array is strictly non-decreasing, allowing for efficient sub-array sum calculations.
Here's an outline of the algorithm:
1. Compute prefix sums: Calculate the sum of each sub-array by adding up the elements from the start to the current index. 2. Iterate over all sub-arrays: Generate all possible sub-arrays using two loops (start and end indices). 3. For each sub-array, calculate the sum: Subtract the prefix sum of the start index from the prefix sum of the end index plus one to get the sum of the sub-array. 4. Insert this sum into the unordered_set to keep track of unique sums. 5. After processing all sub-arrays, sum all values in the unordered_set to get the total sum of all unique sub-array sums.
The time complexity of this approach is O(n^2) due to the nested loops, which is often acceptable for moderate-sized arrays. The space complexity can be O(n^2) in the worst case if all sub-array sums are unique, needed for the hash set and prefix sums.
Below is a sample C++ implementation of this algorithm:
```cpp #include
long long sumOfUniqueSubarraySums(const vector
// Compute prefix sums for (int i = 0; i < n; ++i) { prefixSum[i + 1] = prefixSum[i] + arr[i]; }
unordered_set
// Generate all sub-array sums for (int start = 0; start < n; ++start) { for (int end = start; end < n; ++end) { long long subarraySum = prefixSum[end + 1] - prefixSum[start]; uniqueSums.insert(subarraySum); } }
// Sum all unique sums long long totalSum = 0; for (auto val : uniqueSums) { totalSum += val; }
return totalSum; }
int main() { vector
In this implementation, the prefix sums enable constant-time sub-array sum computation. The unordered_set ensures that each unique sub-array sum is stored once (hashing provides average O(1) insertion and lookup). Finally, the sum of these unique sums is computed.
By leveraging hashing and prefix sums, this approach efficiently finds the sum of all unique sub-array sums in an array of positive integers, making it a valuable tool for solving such problems in C++. Additionally, in the Efficient Approach, the count of each sum in the hash table can be incremented, providing more information about the frequency of each unique sum.
In this C++ implementation, the hashing provided by the ensures that each unique sub-array sum is stored once, allowing for efficient tracking of distinct sums without duplicates. Furthermore, data-and-cloud-computing technology facilitates the efficient computation of prefix sums, enabling constant-time sub-array sum computation in this algorithm.
