LeetCode #689 — HARD

Maximum Sum of 3 Non-Overlapping Subarrays

Break down a hard problem into reliable checkpoints, edge-case handling, and complexity trade-offs.

The Problem

Problem Statement

Given an integer array nums and an integer k, find three non-overlapping subarrays of length k with maximum sum and return them.

Return the result as a list of indices representing the starting position of each interval (0-indexed). If there are multiple answers, return the lexicographically smallest one.

Example 1:

Input: nums = [1,2,1,2,6,7,5,1], k = 2
Output: [0,3,5]
Explanation: Subarrays [1, 2], [2, 6], [7, 5] correspond to the starting indices [0, 3, 5].
We could have also taken [2, 1], but an answer of [1, 3, 5] would be lexicographically larger.

Example 2:

Input: nums = [1,2,1,2,1,2,1,2,1], k = 2
Output: [0,2,4]

Constraints:

1 <= nums.length <= 2 * 10⁴
1 <= nums[i] < 2¹⁶
1 <= k <= floor(nums.length / 3)

Step 01

Brute Force Baseline

Problem summary: Given an integer array nums and an integer k, find three non-overlapping subarrays of length k with maximum sum and return them. Return the result as a list of indices representing the starting position of each interval (0-indexed). If there are multiple answers, return the lexicographically smallest one.

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

[1,2,1,2,6,7,5,1]
2

Example 2

[1,2,1,2,1,2,1,2,1]
2

Core Insight

What unlocks the optimal approach

No official hints in dataset. Start from constraints and look for a monotonic or reusable state.

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

Largest constraint values

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 #689: Maximum Sum of 3 Non-Overlapping Subarrays
class Solution {
    public int[] maxSumOfThreeSubarrays(int[] nums, int k) {
        int[] ans = new int[3];
        int s = 0, s1 = 0, s2 = 0, s3 = 0;
        int mx1 = 0, mx12 = 0;
        int idx1 = 0, idx121 = 0, idx122 = 0;
        for (int i = k * 2; i < nums.length; ++i) {
            s1 += nums[i - k * 2];
            s2 += nums[i - k];
            s3 += nums[i];
            if (i >= k * 3 - 1) {
                if (s1 > mx1) {
                    mx1 = s1;
                    idx1 = i - k * 3 + 1;
                }
                if (mx1 + s2 > mx12) {
                    mx12 = mx1 + s2;
                    idx121 = idx1;
                    idx122 = i - k * 2 + 1;
                }
                if (mx12 + s3 > s) {
                    s = mx12 + s3;
                    ans = new int[] {idx121, idx122, i - k + 1};
                }
                s1 -= nums[i - k * 3 + 1];
                s2 -= nums[i - k * 2 + 1];
                s3 -= nums[i - k + 1];
            }
        }
        return ans;
    }
}

// Accepted solution for LeetCode #689: Maximum Sum of 3 Non-Overlapping Subarrays
func maxSumOfThreeSubarrays(nums []int, k int) []int {
	ans := make([]int, 3)
	s, s1, s2, s3 := 0, 0, 0, 0
	mx1, mx12 := 0, 0
	idx1, idx121, idx122 := 0, 0, 0
	for i := k * 2; i < len(nums); i++ {
		s1 += nums[i-k*2]
		s2 += nums[i-k]
		s3 += nums[i]
		if i >= k*3-1 {
			if s1 > mx1 {
				mx1 = s1
				idx1 = i - k*3 + 1
			}
			if mx1+s2 > mx12 {
				mx12 = mx1 + s2
				idx121 = idx1
				idx122 = i - k*2 + 1
			}
			if mx12+s3 > s {
				s = mx12 + s3
				ans = []int{idx121, idx122, i - k + 1}
			}
			s1 -= nums[i-k*3+1]
			s2 -= nums[i-k*2+1]
			s3 -= nums[i-k+1]
		}
	}
	return ans
}

# Accepted solution for LeetCode #689: Maximum Sum of 3 Non-Overlapping Subarrays
class Solution:
    def maxSumOfThreeSubarrays(self, nums: List[int], k: int) -> List[int]:
        s = s1 = s2 = s3 = 0
        mx1 = mx12 = 0
        idx1, idx12 = 0, ()
        ans = []
        for i in range(k * 2, len(nums)):
            s1 += nums[i - k * 2]
            s2 += nums[i - k]
            s3 += nums[i]
            if i >= k * 3 - 1:
                if s1 > mx1:
                    mx1 = s1
                    idx1 = i - k * 3 + 1
                if mx1 + s2 > mx12:
                    mx12 = mx1 + s2
                    idx12 = (idx1, i - k * 2 + 1)
                if mx12 + s3 > s:
                    s = mx12 + s3
                    ans = [*idx12, i - k + 1]
                s1 -= nums[i - k * 3 + 1]
                s2 -= nums[i - k * 2 + 1]
                s3 -= nums[i - k + 1]
        return ans

// Accepted solution for LeetCode #689: Maximum Sum of 3 Non-Overlapping Subarrays
struct Solution;

impl Solution {
    fn max_sum_of_three_subarrays(nums: Vec<i32>, k: i32) -> Vec<i32> {
        let k = k as usize;
        let sums: Vec<i32> = nums.windows(k).map(|w| w.iter().sum()).collect();
        let n = sums.len();
        let mut left = vec![];
        let mut left_max = 0;
        let mut left_index = 0;
        for i in 0..n {
            if sums[i] > left_max {
                left_max = sums[i];
                left_index = i;
            }
            left.push((left_max, left_index));
        }

        let mut right = vec![];
        let mut right_max = 0;
        let mut right_index = n;
        for i in (0..n).rev() {
            if sums[i] >= right_max {
                right_max = sums[i];
                right_index = i;
            }
            right.push((right_max, right_index));
        }
        right.reverse();
        let mut mid_max = 0;
        let mut res = vec![0, 0, 0];
        for i in k..n - k {
            let sum_3 = sums[i] + left[i - k].0 + right[i + k].0;
            if sum_3 > mid_max {
                mid_max = sum_3;
                res = vec![left[i - k].1, i, right[i + k].1];
            }
        }
        res.into_iter().map(|x| x as i32).collect()
    }
}

#[test]
fn test() {
    let nums = vec![1, 2, 1, 2, 6, 7, 5, 1];
    let k = 2;
    let res = vec![0, 3, 5];
    assert_eq!(Solution::max_sum_of_three_subarrays(nums, k), res);
}

// Accepted solution for LeetCode #689: Maximum Sum of 3 Non-Overlapping Subarrays
function maxSumOfThreeSubarrays(nums: number[], k: number): number[] {
    const n: number = nums.length;
    const s: number[] = Array(n + 1).fill(0);

    for (let i = 0; i < n; ++i) {
        s[i + 1] = s[i] + nums[i];
    }

    const pre: number[][] = Array(n)
        .fill([])
        .map(() => new Array(2).fill(0));
    const suf: number[][] = Array(n)
        .fill([])
        .map(() => new Array(2).fill(0));

    for (let i = 0, t = 0, idx = 0; i < n - k + 1; ++i) {
        const cur: number = s[i + k] - s[i];
        if (cur > t) {
            pre[i + k - 1] = [cur, i];
            t = cur;
            idx = i;
        } else {
            pre[i + k - 1] = [t, idx];
        }
    }

    for (let i = n - k, t = 0, idx = 0; i >= 0; --i) {
        const cur: number = s[i + k] - s[i];
        if (cur >= t) {
            suf[i] = [cur, i];
            t = cur;
            idx = i;
        } else {
            suf[i] = [t, idx];
        }
    }

    let ans: number[] = [];
    for (let i = k, t = 0; i < n - 2 * k + 1; ++i) {
        const cur: number = s[i + k] - s[i] + pre[i - 1][0] + suf[i + k][0];
        if (cur > t) {
            ans = [pre[i - 1][1], i, suf[i + k][1]];
            t = cur;
        }
    }

    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)

Space

O(n)

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.