#553

Medium

auto_awesomeState transition dynamic programming

LeetCode Problem Workspace

Optimal Division

Maximize the value of an expression by optimally placing parentheses for division operations.

Problem Statement

You are given an integer array nums. The adjacent integers will perform the float division. However, you are allowed to add parentheses at any position to change the order of operations. Your goal is to return the corresponding expression in string format that maximizes the evaluated value.

To solve this, you need to determine the optimal placement of parentheses so the expression evaluates to its maximum possible value. For example, in the array [1000, 100, 10, 2], the correct expression is "1000/(100/10/2)" which evaluates to 200, providing the maximum result.

Examples

Example 1

Input: nums = [1000,100,10,2]

Output: "1000/(100/10/2)"

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)"

(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.

Solution Approach

Dynamic Programming Approach

The key to solving this problem is dynamic programming (DP). We can use a DP table to track the maximum and minimum values achievable for each subarray of nums. The challenge lies in determining how to handle divisions and optimize the placement of parentheses. By iterating through all possible subarrays, we calculate the maximum result by adjusting the operation precedence using parentheses.

State Transition Optimization

State transition dynamic programming is employed by considering all possible ways to divide the array and recursively combining results from smaller subarrays. This technique ensures that we explore all possible parentheses placements and choose the one that results in the maximum value. Each DP entry stores the result of operations between two indices of the array.

Minimizing Division Impact

In division problems, the placement of parentheses impacts the outcome. To minimize division effects, you need to place parentheses strategically. The optimal division occurs when numbers following a division operator are minimized, thereby maximizing the overall result. Using a DP table helps achieve this by considering all possible ways to group the divisions.

Complexity Analysis

Metric	Value
Time	Depends on the final approach
Space	O(n^2)

The time complexity depends on the DP approach, with a time complexity of O(n^3) in the worst case. The space complexity is O(n^2) due to the storage required for the DP table to track maximum and minimum values for each subarray.

What Interviewers Usually Probe

Can the candidate explain how to apply dynamic programming to a problem with state transitions?
Does the candidate understand how to optimize the placement of parentheses for maximum value?
Is the candidate able to describe the time and space complexity of the solution?

Common Pitfalls or Variants

Common pitfalls

Misunderstanding the division operation and how parentheses affect the evaluation order.
Failing to recognize the need to track both the maximum and minimum values for subarrays.
Overcomplicating the solution by not using dynamic programming to efficiently explore subarray possibilities.

Follow-up variants

What if the array has only two elements?
How does the problem change if other operations, like addition or multiplication, are involved?
What if the array contains decimal values instead of integers?

FAQ

What is the optimal strategy for placing parentheses in the "Optimal Division" problem?

The optimal strategy involves using dynamic programming to track the maximum and minimum results of subarrays, then strategically placing parentheses to maximize the result.

How does dynamic programming apply to the "Optimal Division" problem?

Dynamic programming helps by storing the maximum and minimum results for subarrays, allowing you to combine smaller subproblems and find the best placement of parentheses.

What is the time complexity of the "Optimal Division" problem solution?

The time complexity is O(n^3) in the worst case due to the need to evaluate all possible subarrays and placements of parentheses.

What are some common pitfalls in solving "Optimal Division"?

Common pitfalls include not properly handling division precedence, failing to track both maximum and minimum results, and overcomplicating the solution by ignoring dynamic programming.

How does GhostInterview help with the "Optimal Division" problem?

GhostInterview provides detailed guidance on applying dynamic programming, optimizing parentheses placement, and understanding the time and space complexity involved in solving the problem.

terminal

Solution

Solution 1

#### Python3

1

2

3

4

5

6

7

8

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:]))})'

Continue Practicing

#486 predict the winner #396 rotate function #368 largest divisible subset