In this lecture, we will solve an algorithm problem titled “Finding the Parenthesis Arrangement that Produces the Minimum Value.”
The goal of this problem is to find a way to place parentheses appropriately around given numbers and operators to minimize the calculation result.
Problem Definition
From the given string of numbers and operators, find the minimum value by considering all possible arrangements using parentheses. For example, given input such as “3+5-2+6*3”,
we need to find the correct positions for the parentheses to minimize the calculation result.
Example Problem
Input: "3+5-2+6*3"
Output: 9 (e.g., (3+5-2)+(6*3) => 9 is the minimum value)
Problem Analysis
This problem has the property that the order of operations changes depending on the placement of parentheses.
Thus, it can be solved using dynamic programming techniques.
Several steps can be considered to solve the problem.
Step 1: Understanding the Input Format
As seen in the problem, parentheses can only be inserted after operators.
Therefore, it’s necessary to separate the input into numbers and operators.
Step 2: Finding Possible Combinations of Parentheses Placement
We need to find all possible combinations from the given numbers and operators.
This can be resolved through a recursive method, and for each combination,
we compare the outcomes and store the minimum value.
Step 3: Implementing the Calculation Function
We need to implement a function that actually performs the calculations for each combination.
Care must be taken to ensure that different operations are performed depending on the operator.
Code Implementation
The following code is the final implementation example for finding the optimal parenthesis arrangement that produces the minimum value.
def calculate(expression):
return eval(expression)
def min_parentheses(arr, ops):
min_value = float('inf')
def find_min(l, r):
nonlocal min_value
if l == r:
return arr[l]
for i in range(l, r):
left = find_min(l, i)
right = find_min(i + 1, r)
expr = f"({left}{ops[i]}{right})"
min_value = min(min_value, calculate(expr))
return min_value
find_min(0, len(arr) - 1)
return min_value
def min_parentheses_solution(s):
arr = list(map(int, s.replace('+', ' ').replace('-', ' ').replace('*', ' ').split()))
ops = [char for char in s if not char.isdigit()]
return min_parentheses(arr, ops)
# Example execution
print(min_parentheses_solution("3+5-2+6*3"))
Code Explanation
Let’s take a look at the functions used in the code one by one.
1. calculate Function
Evaluates the given expression string and returns the result.
It uses the eval function to compute the string as a formula.
However, it is generally advisable to avoid using eval,
and it can be modified later to implement mathematical operations safely.
2. min_parentheses Function
A function that implements the dynamic programming part, recursively dividing the passed array
to calculate the minimum value.
It performs all possible operations for each interval to update the minimum result.
3. min_parentheses_solution Function
This function separates the input string into numbers and operators,
and then calls the min_parentheses function to find the minimum value.
Result Verification
Through the code above, we can confirm that the minimum value for “3+5-2+6*3” is 9.
This example illustrates the basic structure of the algorithm, and it is a good problem to practice with custom data structures or syntax.
Conclusion
In this lecture, we learned how to tackle various cases of parenthesis arrangements to solve the problem.
Such problems frequently appear in coding tests, so it is important to establish your own algorithm and deepen your understanding of it.
Furthermore, it is recommended to experiment with different approaches to solving the problem.
Algorithm Reflection
Finally, the problem of arranging parentheses to create a minimum value requires exploring all possible cases,
making it effective to combine a brute force algorithm with dynamic programming.
Overcoming the challenges that arise during the problem-solving process greatly aids in developing algorithmic thinking skills.
Additional Practice Problems
Solve similar problems to apply the algorithm.
Example: Find the minimum value for input such as “1+2*3-4/2+5*6”.