LeetCode #3640 — HARD

Trionic Array II

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

The Problem

Problem Statement

You are given an integer array nums of length n.

A trionic subarray is a contiguous subarray nums[l...r] (with 0 <= l < r < n) for which there exist indices l < p < q < r such that:

nums[l...p] is strictly increasing,
nums[p...q] is strictly decreasing,
nums[q...r] is strictly increasing.

Return the maximum sum of any trionic subarray in nums.

Example 1:

Input: nums = [0,-2,-1,-3,0,2,-1]

Output: -4

Explanation:

Pick l = 1, p = 2, q = 3, r = 5:

nums[l...p] = nums[1...2] = [-2, -1] is strictly increasing (-2 < -1).
nums[p...q] = nums[2...3] = [-1, -3] is strictly decreasing (-1 > -3)
nums[q...r] = nums[3...5] = [-3, 0, 2] is strictly increasing (-3 < 0 < 2).
Sum = (-2) + (-1) + (-3) + 0 + 2 = -4.

Example 2:

Input: nums = [1,4,2,7]

Output: 14

Explanation:

Pick l = 0, p = 1, q = 2, r = 3:

nums[l...p] = nums[0...1] = [1, 4] is strictly increasing (1 < 4).
nums[p...q] = nums[1...2] = [4, 2] is strictly decreasing (4 > 2).
nums[q...r] = nums[2...3] = [2, 7] is strictly increasing (2 < 7).
Sum = 1 + 4 + 2 + 7 = 14.

Constraints:

4 <= n = nums.length <= 10⁵
-10⁹ <= nums[i] <= 10⁹
It is guaranteed that at least one trionic subarray exists.

Step 01

Brute Force Baseline

Problem summary: You are given an integer array nums of length n. A trionic subarray is a contiguous subarray nums[l...r] (with 0 <= l < r < n) for which there exist indices l < p < q < r such that: nums[l...p] is strictly increasing, nums[p...q] is strictly decreasing, nums[q...r] is strictly increasing. Return the maximum sum of any trionic subarray in nums.

Baseline thinking

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

Pattern signal: Array · Dynamic Programming

Example 1

[0,-2,-1,-3,0,2,-1]

Example 2

[1,4,2,7]

Step 02

Core Insight

What unlocks the optimal approach

Use dynamic programming
Let four arrays <code>dp0...dp3</code> where <code>dpk[i]</code> is the max sum of a subarray ending at <code>i</code> after finishing <code>k</code> of the four phases (start -> inc -> dec -> inc)
Process each <code>i>0</code>
If <code>nums[i] > nums[i‑1]</code>, set <code>dp1[i]=max(dp1[i‑1]+nums[i], dp0[i‑1]+nums[i])</code>, <code>dp3[i]=max(dp3[i‑1]+nums[i], dp2[i‑1]+nums[i])</code>
If <code>nums[i] < nums[i‑1]</code>, set <code>dp2[i]=max(dp2[i‑1]+nums[i], dp1[i‑1]+nums[i])</code>
Always carry over <code>dp0[i]=dp0[i‑1]+nums[i]</code> when <code>nums[i]>nums[i‑1]</code>

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 #3640: Trionic Array II
class Solution {
    public long maxSumTrionic(int[] nums) {
        int n = nums.length;
        int i = 0;
        long ans = Long.MIN_VALUE;
        while (i < n) {
            int l = i;
            i += 1;
            while (i < n && nums[i - 1] < nums[i]) {
                i += 1;
            }
            if (i == l + 1) {
                continue;
            }

            int p = i - 1;
            long s = nums[p - 1] + nums[p];
            while (i < n && nums[i - 1] > nums[i]) {
                s += nums[i];
                i += 1;
            }
            if (i == p + 1 || i == n || nums[i - 1] == nums[i]) {
                continue;
            }

            int q = i - 1;
            s += nums[i];
            i += 1;
            long mx = 0, t = 0;
            while (i < n && nums[i - 1] < nums[i]) {
                t += nums[i];
                i += 1;
                mx = Math.max(mx, t);
            }
            s += mx;

            mx = 0;
            t = 0;
            for (int j = p - 2; j >= l; j--) {
                t += nums[j];
                mx = Math.max(mx, t);
            }
            s += mx;

            ans = Math.max(ans, s);
            i = q;
        }
        return ans;
    }
}

// Accepted solution for LeetCode #3640: Trionic Array II
func maxSumTrionic(nums []int) int64 {
	n := len(nums)
	i := 0
	ans := int64(math.MinInt64)
	for i < n {
		l := i
		for i++; i < n && nums[i-1] < nums[i]; {
			i++
		}
		if i == l+1 {
			continue
		}

		p := i - 1
		s := int64(nums[p-1]) + int64(nums[p])
		for i < n && nums[i-1] > nums[i] {
			s += int64(nums[i])
			i++
		}
		if i == p+1 || i == n || nums[i-1] == nums[i] {
			continue
		}

		q := i - 1
		s += int64(nums[i])
		i++
		var mx, t int64
		for i < n && nums[i-1] < nums[i] {
			t += int64(nums[i])
			i++
			mx = max(mx, t)
		}
		s += mx

		mx, t = 0, 0
		for j := p - 2; j >= l; j-- {
			t += int64(nums[j])
			mx = max(mx, t)
		}
		s += mx

		ans = max(ans, s)
		i = q
	}
	return ans
}

# Accepted solution for LeetCode #3640: Trionic Array II
class Solution:
    def maxSumTrionic(self, nums: List[int]) -> int:
        n = len(nums)
        i = 0
        ans = -inf
        while i < n:
            l = i
            i += 1
            while i < n and nums[i - 1] < nums[i]:
                i += 1
            if i == l + 1:
                continue

            p = i - 1
            s = nums[p - 1] + nums[p]
            while i < n and nums[i - 1] > nums[i]:
                s += nums[i]
                i += 1
            if i == p + 1 or i == n or nums[i - 1] == nums[i]:
                continue

            q = i - 1
            s += nums[i]
            i += 1
            mx = t = 0
            while i < n and nums[i - 1] < nums[i]:
                t += nums[i]
                i += 1
                mx = max(mx, t)
            s += mx

            mx = t = 0
            for j in range(p - 2, l - 1, -1):
                t += nums[j]
                mx = max(mx, t)
            s += mx

            ans = max(ans, s)
            i = q
        return ans

// Accepted solution for LeetCode #3640: Trionic Array II
impl Solution {
    pub fn max_sum_trionic(nums: Vec<i32>) -> i64 {
        let n = nums.len();
        let mut i = 0;
        let mut ans = i64::MIN;
        while i < n {
            let l = i;
            i += 1;
            while i < n && nums[i - 1] < nums[i] {
                i += 1;
            }
            if i == l + 1 {
                continue;
            }

            let p = i - 1;
            let mut s = nums[p - 1] as i64 + nums[p] as i64;
            while i < n && nums[i - 1] > nums[i] {
                s += nums[i] as i64;
                i += 1;
            }
            if i == p + 1 || i == n || nums[i - 1] == nums[i] {
                continue;
            }

            let q = i - 1;
            s += nums[i] as i64;
            i += 1;
            let mut mx = 0i64;
            let mut t = 0i64;
            while i < n && nums[i - 1] < nums[i] {
                t += nums[i] as i64;
                i += 1;
                mx = mx.max(t);
            }
            s += mx;

            mx = 0;
            t = 0;
            let mut j = p as isize - 2;
            while j >= l as isize {
                t += nums[j as usize] as i64;
                mx = mx.max(t);
                j -= 1;
            }
            s += mx;

            ans = ans.max(s);
            i = q;
        }
        ans
    }
}

// Accepted solution for LeetCode #3640: Trionic Array II
function maxSumTrionic(nums: number[]): number {
    const n = nums.length;
    let i = 0;
    let ans = -Infinity;

    while (i < n) {
        const l = i;
        i += 1;

        while (i < n && nums[i - 1] < nums[i]) i += 1;
        if (i === l + 1) continue;

        const p = i - 1;
        let s = nums[p - 1] + nums[p];

        while (i < n && nums[i - 1] > nums[i]) {
            s += nums[i];
            i += 1;
        }
        if (i === p + 1 || i === n || nums[i - 1] === nums[i]) continue;

        const q = i - 1;
        s += nums[i];
        i += 1;

        let mx = 0,
            t = 0;
        while (i < n && nums[i - 1] < nums[i]) {
            t += nums[i];
            i += 1;
            mx = Math.max(mx, t);
        }
        s += mx;

        mx = 0;
        t = 0;
        for (let j = p - 2; j >= l; j--) {
            t += nums[j];
            mx = Math.max(mx, t);
        }
        s += mx;

        ans = Math.max(ans, s);
        i = q;
    }

    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.