LeetCode #1458 — HARD

Max Dot Product of Two Subsequences

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

The Problem

Problem Statement

Given two arrays nums1 and nums2.

Return the maximum dot product between non-empty subsequences of nums1 and nums2 with the same length.

A subsequence of an array is a new array which is formed from the original array by deleting some (can be none) of the characters without disturbing the relative positions of the remaining characters. (ie, [2,3,5] is a subsequence of [1,2,3,4,5] while [1,5,3] is not).

Example 1:

Input: nums1 = [2,1,-2,5], nums2 = [3,0,-6]
Output: 18
Explanation: Take subsequence [2,-2] from nums1 and subsequence [3,-6] from nums2.
Their dot product is (2*3 + (-2)*(-6)) = 18.

Example 2:

Input: nums1 = [3,-2], nums2 = [2,-6,7]
Output: 21
Explanation: Take subsequence [3] from nums1 and subsequence [7] from nums2.
Their dot product is (3*7) = 21.

Example 3:

Input: nums1 = [-1,-1], nums2 = [1,1]
Output: -1
Explanation: Take subsequence [-1] from nums1 and subsequence [1] from nums2.
Their dot product is -1.

Constraints:

1 <= nums1.length, nums2.length <= 500
-1000 <= nums1[i], nums2[i] <= 1000

Step 01

Brute Force Baseline

Problem summary: Given two arrays nums1 and nums2. Return the maximum dot product between non-empty subsequences of nums1 and nums2 with the same length. A subsequence of an array is a new array which is formed from the original array by deleting some (can be none) of the characters without disturbing the relative positions of the remaining characters. (ie, [2,3,5] is a subsequence of [1,2,3,4,5] while [1,5,3] is not).

Baseline thinking

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

Pattern signal: Array · Dynamic Programming

Example 1

[2,1,-2,5]
[3,0,-6]

Example 2

[3,-2]
[2,-6,7]

Example 3

[-1,-1]
[1,1]

Step 02

Core Insight

What unlocks the optimal approach

Use dynamic programming, define DP[i][j] as the maximum dot product of two subsequences starting in the position i of nums1 and position j of nums2.

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 #1458: Max Dot Product of Two Subsequences
class Solution {
    public int maxDotProduct(int[] nums1, int[] nums2) {
        int m = nums1.length, n = nums2.length;
        int[][] f = new int[m + 1][n + 1];
        for (var g : f) {
            Arrays.fill(g, Integer.MIN_VALUE);
        }
        for (int i = 1; i <= m; ++i) {
            for (int j = 1; j <= n; ++j) {
                int v = nums1[i - 1] * nums2[j - 1];
                f[i][j] = Math.max(f[i - 1][j], f[i][j - 1]);
                f[i][j] = Math.max(f[i][j], Math.max(f[i - 1][j - 1], 0) + v);
            }
        }
        return f[m][n];
    }
}

// Accepted solution for LeetCode #1458: Max Dot Product of Two Subsequences
func maxDotProduct(nums1 []int, nums2 []int) int {
	m, n := len(nums1), len(nums2)
	f := make([][]int, m+1)
	for i := range f {
		f[i] = make([]int, n+1)
		for j := range f[i] {
			f[i][j] = math.MinInt32
		}
	}
	for i := 1; i <= m; i++ {
		for j := 1; j <= n; j++ {
			v := nums1[i-1] * nums2[j-1]
			f[i][j] = max(f[i-1][j], f[i][j-1])
			f[i][j] = max(f[i][j], max(0, f[i-1][j-1])+v)
		}
	}
	return f[m][n]
}

# Accepted solution for LeetCode #1458: Max Dot Product of Two Subsequences
class Solution:
    def maxDotProduct(self, nums1: List[int], nums2: List[int]) -> int:
        m, n = len(nums1), len(nums2)
        f = [[-inf] * (n + 1) for _ in range(m + 1)]
        for i, x in enumerate(nums1, 1):
            for j, y in enumerate(nums2, 1):
                v = x * y
                f[i][j] = max(f[i - 1][j], f[i][j - 1], max(0, f[i - 1][j - 1]) + v)
        return f[m][n]

// Accepted solution for LeetCode #1458: Max Dot Product of Two Subsequences
impl Solution {
    pub fn max_dot_product(nums1: Vec<i32>, nums2: Vec<i32>) -> i32 {
        let m = nums1.len();
        let n = nums2.len();
        let mut f = vec![vec![i32::MIN; n + 1]; m + 1];

        for i in 1..=m {
            for j in 1..=n {
                let v = nums1[i - 1] * nums2[j - 1];
                f[i][j] = f[i][j].max(f[i - 1][j]).max(f[i][j - 1]);
                f[i][j] = f[i][j].max(f[i - 1][j - 1].max(0) + v);
            }
        }

        f[m][n]
    }
}

// Accepted solution for LeetCode #1458: Max Dot Product of Two Subsequences
function maxDotProduct(nums1: number[], nums2: number[]): number {
    const m = nums1.length;
    const n = nums2.length;
    const f = Array.from({ length: m + 1 }, () => Array.from({ length: n + 1 }, () => -Infinity));
    for (let i = 1; i <= m; ++i) {
        for (let j = 1; j <= n; ++j) {
            const v = nums1[i - 1] * nums2[j - 1];
            f[i][j] = Math.max(f[i - 1][j], f[i][j - 1]);
            f[i][j] = Math.max(f[i][j], Math.max(0, f[i - 1][j - 1]) + v);
        }
    }
    return f[m][n];
}

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(m × n)

Space

O(m × 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.