LeetCode #1031 — MEDIUM

Maximum Sum of Two Non-Overlapping Subarrays

Move from brute-force thinking to an efficient approach using array strategy.

The Problem

Problem Statement

Given an integer array nums and two integers firstLen and secondLen, return the maximum sum of elements in two non-overlapping subarrays with lengths firstLen and secondLen.

The array with length firstLen could occur before or after the array with length secondLen, but they have to be non-overlapping.

A subarray is a contiguous part of an array.

Example 1:

Input: nums = [0,6,5,2,2,5,1,9,4], firstLen = 1, secondLen = 2
Output: 20
Explanation: One choice of subarrays is [9] with length 1, and [6,5] with length 2.

Example 2:

Input: nums = [3,8,1,3,2,1,8,9,0], firstLen = 3, secondLen = 2
Output: 29
Explanation: One choice of subarrays is [3,8,1] with length 3, and [8,9] with length 2.

Example 3:

Input: nums = [2,1,5,6,0,9,5,0,3,8], firstLen = 4, secondLen = 3
Output: 31
Explanation: One choice of subarrays is [5,6,0,9] with length 4, and [0,3,8] with length 3.

Constraints:

1 <= firstLen, secondLen <= 1000
2 <= firstLen + secondLen <= 1000
firstLen + secondLen <= nums.length <= 1000
0 <= nums[i] <= 1000

Step 01

Brute Force Baseline

Problem summary: Given an integer array nums and two integers firstLen and secondLen, return the maximum sum of elements in two non-overlapping subarrays with lengths firstLen and secondLen. The array with length firstLen could occur before or after the array with length secondLen, but they have to be non-overlapping. A subarray is a contiguous part of an array.

Baseline thinking

Start with the most direct exhaustive search. That gives a correctness anchor before optimizing.

Pattern signal: Array · Dynamic Programming · Sliding Window

Example 1

[0,6,5,2,2,5,1,9,4]
1
2

Example 2

[3,8,1,3,2,1,8,9,0]
3
2

Example 3

[2,1,5,6,0,9,5,0,3,8]
4
3

Step 02

Core Insight

What unlocks the optimal approach

We can use prefix sums to calculate any subarray sum quickly. For each L length subarray, find the best possible M length subarray that occurs before and after it.

Interview move: turn each hint into an invariant you can check after every iteration/recursion step.

Step 03

Algorithm Walkthrough

Iteration Checklist

Define state (indices, window, stack, map, DP cell, or recursion frame).
Apply one transition step and update the invariant.
Record answer candidate when condition is met.
Continue until all input is consumed.

Use the first example testcase as your mental trace to verify each transition.

Step 04

Edge Cases

Minimum Input

Single element / shortest valid input

Validate boundary behavior before entering the main loop or recursion.

Duplicates & Repeats

Repeated values / repeated states

Decide whether duplicates should be merged, skipped, or counted explicitly.

Extreme Constraints

Upper-end input sizes

Re-check complexity target against constraints to avoid time-limit issues.

Invalid / Corner Shape

Empty collections, zeros, or disconnected structures

Handle special-case structure before the core algorithm path.

Step 05

Full Annotated Code

Source-backed implementations are provided below for direct study and interview prep.

// Accepted solution for LeetCode #1031: Maximum Sum of Two Non-Overlapping Subarrays
class Solution {
    public int maxSumTwoNoOverlap(int[] nums, int firstLen, int secondLen) {
        int n = nums.length;
        int[] s = new int[n + 1];
        for (int i = 0; i < n; ++i) {
            s[i + 1] = s[i] + nums[i];
        }
        int ans = 0;
        for (int i = firstLen, t = 0; i + secondLen - 1 < n; ++i) {
            t = Math.max(t, s[i] - s[i - firstLen]);
            ans = Math.max(ans, t + s[i + secondLen] - s[i]);
        }
        for (int i = secondLen, t = 0; i + firstLen - 1 < n; ++i) {
            t = Math.max(t, s[i] - s[i - secondLen]);
            ans = Math.max(ans, t + s[i + firstLen] - s[i]);
        }
        return ans;
    }
}

// Accepted solution for LeetCode #1031: Maximum Sum of Two Non-Overlapping Subarrays
func maxSumTwoNoOverlap(nums []int, firstLen int, secondLen int) (ans int) {
	n := len(nums)
	s := make([]int, n+1)
	for i, x := range nums {
		s[i+1] = s[i] + x
	}
	for i, t := firstLen, 0; i+secondLen-1 < n; i++ {
		t = max(t, s[i]-s[i-firstLen])
		ans = max(ans, t+s[i+secondLen]-s[i])
	}
	for i, t := secondLen, 0; i+firstLen-1 < n; i++ {
		t = max(t, s[i]-s[i-secondLen])
		ans = max(ans, t+s[i+firstLen]-s[i])
	}
	return
}

# Accepted solution for LeetCode #1031: Maximum Sum of Two Non-Overlapping Subarrays
class Solution:
    def maxSumTwoNoOverlap(self, nums: List[int], firstLen: int, secondLen: int) -> int:
        n = len(nums)
        s = list(accumulate(nums, initial=0))
        ans = t = 0
        i = firstLen
        while i + secondLen - 1 < n:
            t = max(t, s[i] - s[i - firstLen])
            ans = max(ans, t + s[i + secondLen] - s[i])
            i += 1
        t = 0
        i = secondLen
        while i + firstLen - 1 < n:
            t = max(t, s[i] - s[i - secondLen])
            ans = max(ans, t + s[i + firstLen] - s[i])
            i += 1
        return ans

// Accepted solution for LeetCode #1031: Maximum Sum of Two Non-Overlapping Subarrays
struct Solution;

use std::i32;

impl Solution {
    fn max_sum_two_no_overlap(mut a: Vec<i32>, l: i32, m: i32) -> i32 {
        let n = a.len();
        let l = l as usize;
        let m = m as usize;
        for i in 1..n {
            a[i] += a[i - 1];
        }
        let mut res = a[l + m - 1];
        let mut max_l = a[l - 1];
        let mut max_m = a[m - 1];
        for i in l + m..n {
            max_l = i32::max(a[i - m] - a[i - m - l], max_l);
            max_m = i32::max(a[i - l] - a[i - l - m], max_m);
            let last_l = a[i] - a[i - l];
            let last_m = a[i] - a[i - m];
            res = i32::max(i32::max(max_m + last_l, max_l + last_m), res);
        }
        res
    }
}

#[test]
fn test() {
    let a = vec![0, 6, 5, 2, 2, 5, 1, 9, 4];
    let l = 1;
    let m = 2;
    let res = 20;
    assert_eq!(Solution::max_sum_two_no_overlap(a, l, m), res);
    let a = vec![3, 8, 1, 3, 2, 1, 8, 9, 0];
    let l = 3;
    let m = 2;
    let res = 29;
    assert_eq!(Solution::max_sum_two_no_overlap(a, l, m), res);
    let a = vec![2, 1, 5, 6, 0, 9, 5, 0, 3, 8];
    let l = 4;
    let m = 3;
    let res = 31;
    assert_eq!(Solution::max_sum_two_no_overlap(a, l, m), res);
}

// Accepted solution for LeetCode #1031: Maximum Sum of Two Non-Overlapping Subarrays
// Auto-generated TypeScript example from java.
function exampleSolution(): void {
}
// Reference (java):
// // Accepted solution for LeetCode #1031: Maximum Sum of Two Non-Overlapping Subarrays
// class Solution {
//     public int maxSumTwoNoOverlap(int[] nums, int firstLen, int secondLen) {
//         int n = nums.length;
//         int[] s = new int[n + 1];
//         for (int i = 0; i < n; ++i) {
//             s[i + 1] = s[i] + nums[i];
//         }
//         int ans = 0;
//         for (int i = firstLen, t = 0; i + secondLen - 1 < n; ++i) {
//             t = Math.max(t, s[i] - s[i - firstLen]);
//             ans = Math.max(ans, t + s[i + secondLen] - s[i]);
//         }
//         for (int i = secondLen, t = 0; i + firstLen - 1 < n; ++i) {
//             t = Math.max(t, s[i] - s[i - secondLen]);
//             ans = Math.max(ans, t + s[i + firstLen] - s[i]);
//         }
//         return ans;
//     }
// }

Step 06

Interactive Study Demo

Use this to step through a reusable interview workflow for this problem.

Custom checkpoints (one per line)

Press Step or Run All to begin.

Step 07

Complexity Analysis

Time

O(n × m)

Space

O(n × m)

Approach Breakdown

RECURSIVE

O(2ⁿ) time

O(n) space

Pure recursion explores every possible choice at each step. With two choices per state (take or skip), the decision tree has 2ⁿ leaves. The recursion stack uses O(n) space. Many subproblems are recomputed exponentially many times.

DYNAMIC PROGRAMMING

O(n × m) time

O(n × m) space

Each cell in the DP table is computed exactly once from previously solved subproblems. The table dimensions determine both time and space. Look for the state variables — each unique combination of state values is one cell. Often a rolling array can reduce space by one dimension.

Shortcut: Count your DP state dimensions → that’s your time. Can you drop one? That’s your space optimization.

Coach Notes

Common Mistakes

Review these before coding to avoid predictable interview regressions.

Off-by-one on range boundaries

Wrong move: Loop endpoints miss first/last candidate.

Usually fails on: Fails on minimal arrays and exact-boundary answers.

Fix: Re-derive loops from inclusive/exclusive ranges before coding.

State misses one required dimension

Wrong move: An incomplete state merges distinct subproblems and caches incorrect answers.

Usually fails on: Correctness breaks on cases that differ only in hidden state.

Fix: Define state so each unique subproblem maps to one DP cell.

Shrinking the window only once

Wrong move: Using `if` instead of `while` leaves the window invalid for multiple iterations.

Usually fails on: Over-limit windows stay invalid and produce wrong lengths/counts.

Fix: Shrink in a `while` loop until the invariant is valid again.