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Given a list of positive integers nums, you need to return a string that represents the optimal division of these numbers such that the value of the expression is maximized. In other words, you want to maximize the result of nums[0] / nums[1] / nums[2] / ... / nums[n-1].

Example 1:

• Input: nums = [1000, 100, 10, 2]
• Output: "1000/(100/10/2)"
• Explanation: 1000/(100/10/2) = 1000/5 = 200, but 1000/100/10/2 = 0.5.

Example 2:

• Input: nums = [2, 3, 4]
• Output: "2/(3/4)"
• Explanation: 2/(3/4) = 2 * 4/3 = 8/3, but 2/3/4 = 1/6.

Example 3:

• Input: nums = [2]
• Output: "2"

Your task is to return the optimal division as a string.

Clarifying Questions:

1. Can the list have only one element?
   • Yes, the list can have only one element, in which case the output is simply that element.
2. Can the list have exactly two elements?
   • Yes, and in this situation, the optimal division is simply represented as nums[0] / nums[1].
3. Is there always a valid solution?
   • Yes, because the list will have at least one positive integer.
4. Are there any constraints on the size of nums or the value of its elements?
   • The problem doesn’t specify constraints, so we assume typical constraints as per LeetCode problems.

Strategy:

The key to maximize the result of division is to minimize the denominator resulting from the division.

• For a list with more than two elements, the strategy is to enclose all numbers after the first in parenthesis. This ensures that we divide nums[0] by a value that is as small as possible.
• For nums = [a], output just a.
• For nums = [a, b], output a/b.
• For nums = [a, b, c, ...], output a/(b/c/...).

Code:

def optimalDivision(nums):
    if not nums:
        return ""
        
    # If there's only one number, just return that number
    if len(nums) == 1:
        return str(nums[0])
    
    # If there are two numbers, just return the division of the two
    if len(nums) == 2:
        return f"{nums[0]}/{nums[1]}"
    
    # For more than two numbers, encloses the rest of the numbers in parenthesis
    result = '{}/({})'.format(nums[0], '/'.join(map(str, nums[1:])))
    return result

# Test cases
print(optimalDivision([1000, 100, 10, 2]))  # "1000/(100/10/2)"
print(optimalDivision([2, 3, 4]))           # "2/(3/4)"
print(optimalDivision([2]))                 # "2"
print(optimalDivision([2, 3]))              # "2/3"

Time Complexity:

The time complexity for this problem is O(n), where n is the length of the nums list. The most time-consuming operation is joining the strings with slashes, which is linear in the size of the list.

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