C# Coding Test Course, Finding the Placement of Parentheses to Minimize Value

Problem Description

The placement of parentheses can change the calculation result of an expression. For example,
2 * 3 + 4 and (2 * 3) + 4 yield the same result, but
2 * (3 + 4) gives a different result.

The goal is to solve the problem of finding the possible minimum value based on the placement of parentheses in a given expression.
The expression consists of numbers and operators (+, -, *).

Problem Definition

Given an array of integers and operators, perform the task of appropriately placing parentheses to
produce the smallest result.
Specifically, when the expression takes the following form:

2 * 3 - 5 + 7

You need to find a way to minimize this expression using parentheses.

Problem Approach

A strategy to solve this problem is to use recursive exploration to consider all placements of parentheses.
Generate all combinations of placements and compute the result for each case to find the smallest value.

1. Simplifying the Problem

First, let’s express the following expression in an array format.
For example, 2 * 3 - 5 + 7 is transformed into the following structure:

[2, '*', 3, '-', 5, '+', 7]

2. Recursive Approach

To place parentheses controlling long expressions at each possible position,
we will use a recursive function to explore all cases.
The main steps are as follows:

Recursively divide the expression and add parentheses.
Calculate the result of each sub-expression.
Update the minimum value among the calculated results.

3. Code Implementation

Below is an example code implemented in C#.


using System;
using System.Collections.Generic;

class Program
{
    static void Main()
    {
        string expression = "2*3-5+7";
        int result = MinValue(expression);
        Console.WriteLine("Minimum Value: " + result);
    }

    static int MinValue(string expression)
    {
        var numbers = new List();
        var operators = new List();

        // Split the input string into numbers and operators.
        for (int i = 0; i < expression.Length; i++)
        {
            if (char.IsDigit(expression[i]))
            {
                int num = 0;
                while (i < expression.Length && char.IsDigit(expression[i]))
                {
                    num = num * 10 + (expression[i] - '0');
                    i++;
                }
                numbers.Add(num);
                i--; // Adjust i value
            }
            else
            {
                operators.Add(expression[i]);
            }
        }

        return CalculateMin(numbers, operators);
    }

    static int CalculateMin(List numbers, List operators)
    {
        // Base case: When only one number is left
        if (numbers.Count == 1)
            return numbers[0];

        int minValue = int.MaxValue;

        for (int i = 0; i < operators.Count; i++)
        {
            char op = operators[i];
            List leftNumbers = numbers.ToList();
            List rightNumbers = numbers.ToList();
            List leftOperators = operators.GetRange(0, i);
            List rightOperators = operators.GetRange(i + 1, operators.Count - i - 1);

            // Divide the left and right expressions based on the operator.
            int leftValue = CalculateMin(leftNumbers.GetRange(0, i + 1), leftOperators);
            int rightValue = CalculateMin(rightNumbers.GetRange(i + 1, rightNumbers.Count - i - 1), rightOperators);

            // Perform the operation.
            int result = PerformOperation(leftValue, rightValue, op);

            // Update the minimum value
            if (result < minValue)
                minValue = result;
        }

        return minValue;
    }

    static int PerformOperation(int left, int right, char op)
    {
        switch (op)
        {
            case '+':
                return left + right;
            case '-':
                return left - right;
            case '*':
                return left * right;
            default:
                throw new InvalidOperationException("Unsupported operator.");
        }
    }
}

4. Code Explanation

The functions used in the above code serve the following purposes:

MinValue: Splits the given expression string into numbers and operators and prepares initial data for minimum value calculation.
CalculateMin: Recursively calculates all possible sub-expressions and finds the minimum value.
PerformOperation: Performs calculations using two numbers and an operator.

Through this structure, all combinations of parentheses placements are explored, and the minimum value among the calculated results is derived.

Conclusion

I hope this example problem has helped you understand the placement of parentheses and algorithmic approaches.
Based on this method and code, you can tackle various problems to find the minimum values of different expressions.
By always simplifying the problem and practicing recursive approaches, you can maximize your algorithmic thinking.

Additional Learning Resources

For a deeper understanding of algorithms, please refer to the following materials: