maximum sum of 3 non overlapping subarrays leetcode 689

the problem "maximum sum of 3 non-overlapping subarrays" (leetcode 689) is an interesting challenge that requires a strategic approach to efficiently find the maximum sum of three non-overlapping subarrays of a given size. here’s a detailed tutorial on how to approach this problem.

problem statement

you are given an integer array `nums` and three integers `len1`, `len2`, and `len3`. the task is to find three non-overlapping subarrays of lengths `len1`, `len2`, and `len3`, respectively, such that their combined sum is maximized.

constraints
- the lengths of the subarrays (`len1`, `len2`, `len3`) must be non-negative integers.
- the sum of `len1`, `len2`, and `len3` must be less than or equal to the length of `nums`.

approach

1. **prefix sum array**:
- compute a prefix sum array to facilitate quick calculation of the sum of any subarray. the prefix sum at index `i` will contain the sum of all elements from the start of the array up to index `i`.

2. **calculate subarray sums**:
- precompute the sums of all possible subarrays of lengths `len1`, `len2`, and `len3` using the prefix sum array.

3. **dynamic programming**:
- use dynamic programming to keep track of the maximum sums of the first two non-overlapping subarrays up to each index.

4. **iterate and compute maximum**:
- for the third subarray, iterate through the possible starting positions, and for each position, check the maximum combined sum of the first two subarrays.

code example

here's a python implementation of the above approach:

explanation of the code

1. **prefix sum calculation**: the `prefix_sum` array is computed to allow o(1) time complexity for calculating any subarray sum.

2. **max sum arrays**:
- `max_sum1` keeps track of the maximum sum of the first subarray that can end at each index.
- `max_sum2` maintains the maximum sum of the first two non-overlapping subarrays.

3. **final calculation**: finally, we iterate over possible starting positions for the third s ...

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maximum sum
3 non overlapping subarrays
leetcode
sliding window
prefix sum
array partitioning
optimization
dynamic programming
greedy approach
contiguous subarrays
maximum sum problem
interval selection
constraints
algorithm design
problem solving