1546. Maximum Number of Non-Overlapping Subarrays With Sum Equals Target

Problem Description

Given an array called nums and an integer called target, the task is to find the maximum number of distinct, non-overlapping subarrays where each subarray adds up to the given target. A subarray is a contiguous part of the array, and it must be non-empty.

Intuition

The solution approach revolves around iterating through the array and keeping track of the cumulative sum of the elements. We want to know if at any point the cumulative sum minus the target has been previously seen. If it has, that means we have found a subarray that sums up to target.

The intuition is based on the following thought process:

Initialize a variable to store the cumulative sum s and a set seen to keep track of all the different sums we have encountered, initializing the set with a 0 to handle cases where a subarray starting from the first element meets the target.
Iterate over the array nums, adding each number to our cumulative sum s.
For each new sum, we check if s - target exists in the seen set. If it does, it means we've found a non-overlapping subarray that sums up to target because we had a subarray previous to this whose sum was s - target, making the sum between that point and the current index exactly target.
Every time we find such a subarray, we increment our answer ans, break the while loop to not to consider overlapping subarrays, and reset our set seen and sum s for the next iteration.
We continue this process for each element in nums.

This approach ensures that we're always looking at non-overlapping subarrays by resetting the set and cumulative sum after each found subarray. The counter ans is our final answer, representing the maximum number of subarrays summing up to target.

Learn more about Greedy and Prefix Sum patterns.

Solution Approach

The solution uses a while loop to iterate over the elements in the array nums, and a set named seen to keep track of the cumulative sums during the iteration of subarrays.

Here's the step-by-step breakdown of the algorithm:

Initialize two pointers i and n. Pointer i is used to traverse the array, and n holds the length of the array for bounds checking.
A counter ans is initialized to 0. This counter tracks the number of non-overlapping subarrays found that sum up to target.
The outer while loop continues as long as i < n, ensuring we go through each element.
Inside the outer loop, we initialize a sum s to 0 and a set seen with the initial element being 0. The sum s will keep track of the cumulative sum of the elements starting from index i, and the set seen keeps track of all previous cumulative sums.
The inner while loop also continues as long as i < n, which goes over the elements from the current starting point:
- We add the current element nums[i] to the cumulative sum s.
- Check if s - target is in the set seen. If it is, it means we have found a valid subarray because the difference between the current cumulative sum and the target is a sum we saw earlier. Since we ensure to add only non-overlapping sums to seen, finding s - target guarantees a non-overlapping subarray.
- If we found a valid subarray, we increment ans by 1, which is our count for non-overlapping subarrays summing up to target. We then break out of the inner while loop to start looking for the next valid subarray, ensuring non-overlap.
- If we did not find a valid subarray yet, we proceed to the next element by incrementing i and adding the new sum s to the set seen.
As soon as we exit the inner while loop (either due to finding a valid subarray or reaching the end of the array), we increment i to look for the next starting point of a potential subarray.

The process repeats until we have exhausted all elements in the array nums. Finally, the variable ans holds the maximum number of non-overlapping subarrays with a sum equal to target, which is returned from the function.

Using a set to track cumulative sums is a clever way to check for the presence of a sum in constant time, which keeps the solution efficient. Resetting s and seen after finding a subarray ensures we only count non-overlapping subarrays, adhering to the problem requirements.

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Example Walkthrough

Let's illustrate the solution approach with a small example. Suppose we have the following array and target:

nums = [1, 2, 3, 4, 5]
target = 5

We need to find the maximum number of distinct, non-overlapping subarrays where each subarray sums up to the target value of 5.

We initialize i to 0, n to the length of nums which is 5, and ans to 0.
We start our outer while loop with i < n. Since i = 0 and n = 5, we enter the loop.
We initialize our cumulative sum s to 0 and our set seen with an initial element of 0.
Now, we enter the inner while loop. At i = 0, nums[i] = 1. We add this to our sum s, so s = 1. We then add s to seen, so seen = {0, 1}.
We move to the next element i = 1, nums[i] = 2. Our new sum s = 3. This is not in seen after subtracting target, so we add it to seen, which now becomes {0, 1, 3}.
Next up is i = 2, nums[i] = 3. Adding this to our sum gives us s = 6. Now, s - target = 1, which is in seen. That means we found a valid subarray [1, 2, 3] that adds up to our target 5.
We increment ans by 1, to reflect the subarray we found. We break the inner while loop, reset our sum s to 0, and clear seen to {0} to look for further non-overlapping subarrays.
We increment i outside of the inner while loop to move to the next potential starting point of a subarray. Since we broke the inner loop at i = 3, which corresponds to the fourth element nums[3] = 4. We now start from there.
We repeat steps 4 to 7. Our cumulative sum s is incremented by nums[3], so s = 4. And seen is updated to {0, 4}.
Moving to i = 4, nums[i] = 5. Now, s = 9. However, s - target = 4 which is in the set seen. This means we've found another valid subarray [4, 5].
We increment ans by 1 again and break the inner loop. Now, ans = 2, which reflects the two distinct non-overlapping subarrays [1, 2, 3] and [4, 5] that sum up to target.
There are no more elements to process, as we've reached the end of nums.

The final answer, held by ans, is 2, representing the maximum number of non-overlapping subarrays with a sum equal to target in the given nums array.

Solution Implementation

1class Solution:
2    def maxNonOverlapping(self, nums: List[int], target: int) -> int:
3        curr_index, nums_length = 0, len(nums)  # Initializing the current index and the total length of the array
4        non_overlapping_count = 0  # To keep track of the count of non-overlapping subarrays
5      
6        # Iterate through the array until the current index is less than the length of the array
7        while curr_index < nums_length:
8            cumulative_sum = 0  # Initialize the cumulative sum for the current subarray
9            seen_sums = {0}  # Set to store cumulative sums which are useful for identifying if a subarray with the target sum exists
10          
11            # Continue in the inner while-loop to find a subarray that sums to the target
12            while curr_index < nums_length:
13                cumulative_sum += nums[curr_index]  # Update the cumulative sum
14              
15                # If the difference between the current cumulative sum and the target is in seen_sums,
16                # we have found a subarray that sums up to the target
17                if cumulative_sum - target in seen_sums:
18                    non_overlapping_count += 1  # Increment the count of non-overlapping subarrays
19                    break  # Exit the inner while-loop to start looking for the next subarray
20              
21                # If we haven't found a valid subarray yet, update the current index and add the current sum to seen_sums
22                curr_index += 1
23                seen_sums.add(cumulative_sum)
24          
25            # Move to the next index to start a new subarray scan
26            curr_index += 1
27      
28        # Return the total number of non-overlapping subarrays that sum up to the target
29        return non_overlapping_count
30

1class Solution {
2    public int maxNonOverlapping(int[] nums, int target) {
3        int currentIndex = 0; // Initialize the current index to start from the beginning of the array.
4        int totalSubarrays = 0; // This will keep track of the count of non-overlapping subarrays that sum up to 'target'.
5        int arrayLength = nums.length; // Get the length of the input array 'nums'.
6      
7        // Iterate over the array until we reach the end.
8        while (currentIndex < arrayLength) {
9            int currentSum = 0; // Initialize the sum of the current subarray being evaluated.
10            Set<Integer> seenSums = new HashSet<>(); // Use a HashSet to store the unique sums encountered.
11            seenSums.add(0); // Add zero to handle the case when a subarray starts from the first element.
12          
13            // Keep scanning through the array until the end.
14            while (currentIndex < arrayLength) {
15                currentSum += nums[currentIndex]; // Add the current element to the current sum.
16              
17                // If the set contains the current sum minus the target, we've found a valid subarray.
18                if (seenSums.contains(currentSum - target)) {
19                    totalSubarrays++; // Increment the count of valid subarrays.
20                    break; // Break to start looking for the next non-overlapping subarray.
21                }
22                seenSums.add(currentSum); // Add the current sum to the set of seen sums.
23                currentIndex++; // Move to the next element in the array.
24            }
25            currentIndex++; // Increment to skip the start of the next subarray after finding a valid subarray.
26        }
27      
28        return totalSubarrays; // Return the total number of non-overlapping subarrays with sum equal to 'target'.
29    }
30}
31

1#include <vector>
2#include <unordered_set>
3using namespace std;
4
5class Solution {
6public:
7    // Function to find the maximum number of non-overlapping subarrays that sum to a target value
8    int maxNonOverlapping(vector<int>& nums, int target) {
9        int index = 0; // Start index for checking subarrays
10        int n = nums.size(); // Length of the input array
11        int answer = 0; // Initialization of count of maximum non-overlapping subarrays
12
13        // Iterate over the array to find all possible non-overlapping subarrays
14        while (index < n) {
15            int currentSum = 0; // Initialize the sum of the current subarray
16            unordered_set<int> seenSums; // Track all unique sums encountered within the current window
17            seenSums.insert(0); // Insert 0 to handle cases where a subarray starts from the first element
18          
19            // Continue to expand the window until the end of the array is reached
20            while (index < n) {
21                currentSum += nums[index]; // Update current sum
22
23                // If the sum minus the target has been seen before, we've found a target subarray
24                if (seenSums.count(currentSum - target)) {
25                    answer++; // Increment the count for the answer
26                    break; // Start looking for the next subarray
27                }
28              
29                // Insert the current sum into the set and move to the next element
30                seenSums.insert(currentSum);
31                index++;
32            }
33          
34            // Skip the next index after a valid subarray is found to ensure non-overlapping
35            index++;
36        }
37      
38        // Return the total count of non-overlapping subarrays summing to the target
39        return answer;
40    }
41};
42
43int main() {
44    // Example usage:
45    Solution solution;
46    vector<int> nums = {1,1,1,1,1};
47    int target = 2;
48    int maxSubarrays = solution.maxNonOverlapping(nums, target);
49    // maxSubarrays should be 2 for this input
50    return 0;
51}
52

1function maxNonOverlapping(nums: number[], target: number): number {
2    let index = 0; // Start index for checking subarrays
3    const n = nums.length; // Length of the input array
4    let answer = 0; // Initialization of count of maximum non-overlapping subarrays
5
6    // Iterate over the array to find all possible non-overlapping subarrays
7    while (index < n) {
8        let currentSum = 0; // Initialize the sum of the current subarray
9        const seenSums = new Set<number>(); // Track all unique sums encountered within the current window
10        seenSums.add(0); // Insert 0 to handle cases where a subarray starts from the first element
11
12        // Continue to expand the window until the end of the array is reached
13        while (index < n) {
14            currentSum += nums[index]; // Update current sum
15
16            // If the sum minus the target has been seen before, we've found a target subarray
17            if (seenSums.has(currentSum - target)) {
18                answer++; // Increment the count for the answer
19                break; // Start looking for the next subarray
20            }
21
22            // Insert the current sum into the set and move to the next element
23            seenSums.add(currentSum);
24            index++;
25        }
26
27        // Skip the next index after a valid subarray is found to ensure non-overlapping
28        index++;
29    }
30
31    // Return the total count of non-overlapping subarrays summing to the target
32    return answer;
33}
34
35// Example usage:
36const nums = [1, 1, 1, 1, 1];
37const target = 2;
38const maxSubarrays = maxNonOverlapping(nums, target);
39// maxSubarrays should be 2 for this input
40

Time and Space Complexity

Time Complexity

The time complexity of this code is O(n), where n is the length of the nums array. This linear time complexity arises from the fact that the code iterates over the array elements at most twice: Once for the outer while loop, and at most once more within the inner while loop before a matching subarray sum is found and the loop is broken. Once the code finds a matching sum, it immediately breaks out of the inner loop and skips to the next index after the end of the current subarray. Thus, each element is touched at most twice during the iteration.

Space Complexity

The space complexity of the code is also O(n). The primary contributing factor to the space complexity is the seen set, which in the worst-case scenario could store a cumulative sum for each element in the nums array if no sums match s - target. As a result, in the worst case, this set would store n unique sums, making the space complexity linear with respect to the length of nums.

Learn more about how to find time and space complexity quickly using problem constraints.

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What is an advantages of top-down dynamic programming vs bottom-up dynamic programming?

It's faster

It has better space complexity

Order of computer subproblems does not matter

It has better time complexity

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