
Problem Statement
Given an integer array nums
, each element is divided by its subsequent neighbor forming a float division chain expression of the format "nums[0]/nums[1]/.../nums[n-1]". You can manipulate this expression by strategically placing parentheses to affect the computation order and maximize the final result. The goal is to determine where these parentheses should be placed to achieve the maximum possible division result, and then return the maximized expression in string format without any redundant parentheses.
Examples
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.
Approach and Intuition
To solve this problem, we need to understand how division operations are affected by the order of execution, particularly when parentheses alter this order.
Priority of Division: In floating-point divisions, dividing by a smaller number increases the quotient, hence to maximize the result, we sometimes need to ensure that divisions happen in a manner where smaller results are not prematurely further divided.
Strategic Parentheses Placement:
- When the array has more than two numbers, the maximum result can generally be achieved by dividing the first number by the result of the subsequent chain of divisions. This is because dividing by a bigger number results from the chain would decrease the overall value.
- Expressions like
A/(B/C/D)
ensure thatB
,C
, andD
are first evaluated to a smaller number which when divided intoA
yields a larger value.
Constructing the Expression:
- If the input list only contains one or two numbers, the maximum result is simply the straightforward division of these numbers.
- For more than two numbers, to maximize the division, you would ideally want to divide the first number by the division of all subsequent numbers. Therefore, the general form would be:
- For
nums = [a, b, c, d]
, the optimal division string would be"a/(b/c/d)"
. This setup reduces the impact of the division chain on reducinga
by first reducing the impact of allb, c, d
on each other.
- For
Generating the Optimal Expression:
- If
nums
length is 1, simply return the number. - If
nums
length is 2, return the expression"nums[0]/nums[1]"
. - For lengths greater than 2: Build the string such that the first number is divided by the enclosed expression of the remaining numbers. This is done by joining all numbers from the second to the last with "/" and wrapping them in a single parenthesis, prefixed by the division operation with the first number.
- If
With this approach, we ensure that the parentheses we add are purposeful and not redundant, strictly influencing the computation order to maximize the result.
Solutions
- Java
public class Solution {
public String divisionMaximization(int[] array) {
if (array.length == 1)
return String.valueOf(array[0]);
if (array.length == 2)
return array[0] + "/" + array[1];
StringBuilder result = new StringBuilder(array[0] + "/(" + array[1]);
for (int index = 2; index < array.length; index++) {
result.append("/" + array[index]);
}
result.append(")");
return result.toString();
}
}
This Java solution provides an efficient approach to solving the "Optimal Division" problem where the goal is to divide a list of numbers to achieve the maximum result. The solution uses a conditional structure based on the size of the input array to determine the output:
- If the input array contains only one element, the output is the string representation of that single element.
- For an array containing exactly two elements, the output is a direct division of the first element by the second.
- For arrays with more than two elements, the first element is divided by the subsequent elements treated as a group, enclosed in parentheses. This ensures that division operations within the parentheses are evaluated first, aiming to maximize the division result by dividing the first number by the result of a chain of divisions starting from the second element.
This method adopts straightforward approach utilizing StringBuilder
to dynamically construct the output based on these rules, which ensures efficient string manipulation and concatenation. This arrangement safeguards the calculation flow essential for achieving the optimal division result, particularly when dealing with more than two numbers where the division sequence significantly impacts the final outcome.
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