Find the largest subarray with 0 sum.

Subarrays are contiguous parts of an array. Finding maximum length subarrays with specific properties like having a 0 sum has various applications in computer science and mathematics. For instance, finding maximum length 0-sum subarrays aids in financial analysis to detect fraud transactions that nullify each other. Similarly, it helps detect data inconsistencies in domains like bioinformatics and physics.

In this article, we solve the problem of finding the length of the largest subarray having 0-sum in a given integer array. For example, in the collection [10, 2, -5, 1, 6], the longest 0-sum subarray is [2, -5, 1], having length 3. An efficient solution to this problem involves the usage of hashmap and tracking cumulative sum as we traverse the array from left to right.

The key idea relies on the fact that if two prefix sums upto two indices i and j are the same, the sum of elements between indices i and j is 0. We use this property to efficiently find the length of the longest 0-sum subarray by storing prefix sums and their indices in a hashmap.

Here is the outline for this article -

1. Formal problem statement
2. The algorithm logic explained
3. Python code implementation
4. Time and space complexity analysis
5. Example run with sample array as input.

Approach 1: Brute Force Method

The most straightforward approach to finding the longest subarray with 0 sum is to generate all possible subarrays of the given array and check their sums. This is essentially an exhaustive search exploring all combinations of start and end points marking subsequences in the array. More formally, we have two nested loops - the outer loop picks a starting element, and the inner loop considers all possible ending elements to construct subarrays beginning from the start element selected by the outer loop. We maintain a running sum in the inner loop, and if the sum becomes 0, we update the maximum length found so far. The time complexity of this brute force technique is O(n^2) since we have nested loops iterating upto n elements. Although simple to implement, this quadratic time complexity makes it infeasible for large input arrays. We will soon discuss a more efficient approach with better time complexity. But first, let's walk through the brute force method via some sample code for clarity, followed by an analysis of running time.

Output:

Explanation:

1. Define the function largest_zero_subarray that takes the input array (arr) as the parameter.
2. Initialize a variable max_len to store the result (length of longest 0-sum subarray). Initialize it to 0.
3. Start the outer loop from index 0 to len(arr). This loop picks the starting element of all possible subarrays.
4. Initialize curr_sum to 0 before starting the inner loop. curr_sum will store the sum of the current subarray.
5. Start the inner loop from the outer loop's current index to len(arr). This loop considers all possible ending elements for the recent start picked by the outer loop and finds the sum.
6. Keep adding a current element to curr_sum in the inner loop.
7. If at any point curr_sum becomes 0, update max_len to a maximum of current length, i.e. j-i+1 and existing max_len.
8. After the inner loop, return the final max_len, which contains the maximum length of the subarray with 0 sum.
9. Example usage: Define array and call function largest_zero_subarray by passing this array. Print the returned result.

Approach 2: Prefix Sum Technique

The brute force method to find the longest 0-sum subarray runs in quadratic time complexity, which can be infeasible for significant inputs. We can optimize it using the prefix sum technique combined with hashing. The prefix sum concept relies on the fact that if the prefix sum up to two indices i and j are equal, the sum of elements between indices i and j is 0. We can leverage this by traversing the array once while tracking the current prefix sum. If we reencounter any prefix sum value, we can deduce that elements between the previously occurring index and the current instance will sum to 0. We store indices of unique prefix sums in a hash table. Comparing the current index with the previously held index gives us the length of the current 0-sum subarray. This technique reduces time to O(n) as the array is traversed only once. Next, we will walk through efficient Python code implementing this prefix sum hashing logic followed by run-time analysis. But first, understanding the crux - linking the same prefix sums to deduce 0-sum subarrays is vital before proceeding further.

Output:

Explanation:

1. Define the function largest_zero_subarray that takes the input array (arr)
2. Initialize max_len to 0 to store the length of the result subarray.
3. Create an empty dictionary prefix_sum to store prefix sums as keys and their indices as values.
4. Initialize curr_sum to 0 to store the current prefix sum.
5. Start loop from 0 to the length of the array.
6. Keep adding a current element to curr_sum
7. If the current element is 0 itself, update max_len to 1. If it was 0
8. If current curr_sum becomes 0, update max_len as current index + 1
9. Check if the current curr_sum already exists in the prefix_sum dictionary.
10. If yes, then the current curr_sum had occurred earlier as well. So, by taking their difference, find the length between the previous occurrence and the current index. Update max_len if this difference is more significant.
11. If curr_sum does not exist in the dictionary, insert it with a current index.
12. Finally, return the computed max_len
13. Example usage - Pass input array to function and print returned result

Approach 3: Using Sets

In the previous prefix sum technique, we used a hash table to store the prefix sums encountered while traversing the array. This allowed us to efficiently find a 0-sum subarray in linear time by linking exact prefix sums. However, searching in a hash table can be optimized using a Set instead, which gives constant time-containing checks.

The key idea is to insert all prefix sums encountered into a Set. Checking the presence of any sum in the Set will convey whether it had occurred previously. Hence, exact sums indicate a 0-sum subarray from the index of the previous occurrence to the current instance. Sets provide average case O(1) search time, leading to linear time complexity.

The advantage of using Set over the hash table is that it eliminates unnecessary indices storage. We only need to check the existence of the current prefix sum. If it exists in Set, we have found a 0-sum subarray; otherwise, we simply insert the current sum. This space optimization is beneficial for large arrays. Now, let's walk through the Python code, applying this logic and analyzing its run time complexity.

Output:

Explanation:

1. Define the function largest_zero_subarray that takes the input array.
2. Initialize a variable max_len to store the result (length of longest 0 sum subarray).
3. Initialize curr_sum to 0 to store the current running sum.
4. Create a set sum_set and insert 0 into it. This Set will store the unique sums seen so far.
5. Start traversing the array from index 0 to the length of the array.
6. Keep updating curr_sum by adding a current element to it.
7. Check if the current curr_sum already exists in the set sum_set.
8. If yes, we have found a 0 sum subarray spanning from the index where this curr_sum had occurred previously to the current index. Update max_len.
9. If curr_sum does not exist, simply insert it into the Set.
10. Finally, return the maximum length stored in max_len.
11. To use - Pass the input array to the function and print the returned result.