LeetCode #553 — MEDIUM

Optimal Division

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

The Problem

Problem Statement

You are given an integer array nums. The adjacent integers in nums will perform the float division.

For example, for nums = [2,3,4], we will evaluate the expression "2/3/4".

However, you can add any number of parenthesis at any position to change the priority of operations. You want to add these parentheses such the value of the expression after the evaluation is maximum.

Return the corresponding expression that has the maximum value in string format.

Note: your expression should not contain redundant parenthesis.

Example 1:

Input: nums = [1000,100,10,2]
Output: "1000/(100/10/2)"
Explanation: 1000/(100/10/2) = 1000/((100/10)/2) = 200
However, the bold parenthesis in "1000/((100/10)/2)" are redundant since they do not influence the operation priority.
So you should return "1000/(100/10/2)".
Other cases:
1000/(100/10)/2 = 50
1000/(100/(10/2)) = 50
1000/100/10/2 = 0.5
1000/100/(10/2) = 2

Example 2:

Input: nums = [2,3,4]
Output: "2/(3/4)"
Explanation: (2/(3/4)) = 8/3 = 2.667
It can be shown that after trying all possibilities, we cannot get an expression with evaluation greater than 2.667

Constraints:

1 <= nums.length <= 10
2 <= nums[i] <= 1000
There is only one optimal division for the given input.

Step 01

Brute Force Baseline

Problem summary: You are given an integer array nums. The adjacent integers in nums will perform the float division. For example, for nums = [2,3,4], we will evaluate the expression "2/3/4". However, you can add any number of parenthesis at any position to change the priority of operations. You want to add these parentheses such the value of the expression after the evaluation is maximum. Return the corresponding expression that has the maximum value in string format. Note: your expression should not contain redundant parenthesis.

Baseline thinking

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

Pattern signal: Array · Math · Dynamic Programming

Example 1

[1000,100,10,2]

Example 2

[2,3,4]

Step 02

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

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 #553: Optimal Division
class Solution {
    public String optimalDivision(int[] nums) {
        int n = nums.length;
        if (n == 1) {
            return nums[0] + "";
        }
        if (n == 2) {
            return nums[0] + "/" + nums[1];
        }
        StringBuilder ans = new StringBuilder(nums[0] + "/(");
        for (int i = 1; i < n - 1; ++i) {
            ans.append(nums[i] + "/");
        }
        ans.append(nums[n - 1] + ")");
        return ans.toString();
    }
}

// Accepted solution for LeetCode #553: Optimal Division
func optimalDivision(nums []int) string {
	n := len(nums)
	if n == 1 {
		return strconv.Itoa(nums[0])
	}
	if n == 2 {
		return fmt.Sprintf("%d/%d", nums[0], nums[1])
	}
	ans := &strings.Builder{}
	ans.WriteString(fmt.Sprintf("%d/(", nums[0]))
	for _, num := range nums[1 : n-1] {
		ans.WriteString(strconv.Itoa(num))
		ans.WriteByte('/')
	}
	ans.WriteString(fmt.Sprintf("%d)", nums[n-1]))
	return ans.String()
}

# Accepted solution for LeetCode #553: Optimal Division
class Solution:
    def optimalDivision(self, nums: List[int]) -> str:
        n = len(nums)
        if n == 1:
            return str(nums[0])
        if n == 2:
            return f'{nums[0]}/{nums[1]}'
        return f'{nums[0]}/({"/".join(map(str, nums[1:]))})'

// Accepted solution for LeetCode #553: Optimal Division
impl Solution {
    pub fn optimal_division(nums: Vec<i32>) -> String {
        let n = nums.len();
        match n {
            1 => nums[0].to_string(),
            2 => nums[0].to_string() + "/" + &nums[1].to_string(),
            _ => {
                let mut res = nums[0].to_string();
                res.push_str("/(");
                for i in 1..n {
                    res.push_str(&nums[i].to_string());
                    res.push('/');
                }
                res.pop();
                res.push(')');
                res
            }
        }
    }
}

// Accepted solution for LeetCode #553: Optimal Division
function optimalDivision(nums: number[]): string {
    const n = nums.length;
    const res = nums.join('/');
    if (n > 2) {
        const index = res.indexOf('/') + 1;
        return `${res.slice(0, index)}(${res.slice(index)})`;
    }
    return res;
}

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.

Overflow in intermediate arithmetic

Wrong move: Temporary multiplications exceed integer bounds.

Usually fails on: Large inputs wrap around unexpectedly.

Fix: Use wider types, modular arithmetic, or rearranged operations.

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.