Author: root

In this article, we will address the algorithm problem of finding the Catalan numbers using JavaScript. Catalan numbers are related to mathematical concepts such as structural concepts like nested parentheses, and this problem often appears in programming interviews. Therefore, we will explore a way to thoroughly understand and solve this problem.

What are Catalan numbers?

Catalan numbers are strings of length N composed of 0s and 1s that follow the following rules:

• The first and last characters of the string must be 0.
• The number of 1s in the string must always be at least 1, there cannot be consecutive 1s, and each 1 must be surrounded by 0s.

For example, the Catalan numbers of length 5 include 00000, 01000, 00100, 00010, 00001,
01010, 01100, 10000, etc. Catalan numbers are similar to the Fibonacci sequence, and
they can take different counts depending on the length N.

Problem Description

Given a natural number N, the problem is to output the count of Catalan numbers of length N.
For example, when N = 5, we need to find the number of Catalan numbers,
and the result should be 8.

Solution

To find the Catalan numbers, we can use recursive calls or dynamic programming (DP) methods.
Below is the formula for computing the Catalan numbers.

• p(n) = p(n-1) + p(n-2) (n ≥ 2, p(0) = 1, p(1) = 1)

This formula recursively computes the Catalan numbers.
It is important to note that it should return 1 when n is 0.
Moreover, the relationship between p(n-1) and p(n-2) refers to the counts of previous Catalan numbers.

Algorithm Implementation

        
            // Function to calculate Catalan numbers in JavaScript.
            function countCatalanNumbers(n) {
                // Array to store Catalan numbers
                const dp = new Array(n + 1).fill(0);
                dp[0] = 1; // Case for length 0
                dp[1] = 1; // Case for length 1

                // Using dynamic programming to calculate the Catalan numbers.
                for (let i = 2; i <= n; i++) {
                    dp[i] = dp[i - 1] + dp[i - 2];
                }

                return dp[n]; // Return the count of Catalan numbers of length n.
            }

            // Example of function call
            const n = 5; // Value of N
            console.log(`The number of Catalan numbers of length ${n} is: ${countCatalanNumbers(n)}`);
        
        

Process Explanation

1. Understanding the Problem: This is a problem to find the Catalan numbers for a natural number N.
2. Dynamic Programming Approach: We define the Catalan numbers structurally.
3. Array Initialization: Create the dp array and set the base values.
4. Executing the Loop: Calculate and store the count of Catalan numbers in the array.
5. Verifying the Result: Validate that the output of the function is correct.

Execution Result

        
            // Input: N = 5
            // Output: The number of Catalan numbers of length 5 is: 8
        
    

Conclusion

We have learned how to find the Catalan numbers and have been able to efficiently solve the problem through dynamic programming.
This problem frequently appears in programming interviews, so it is important to develop the ability to understand and implement the principles of the algorithm.

Coding tests have become a mandatory process for many companies these days. In particular, JavaScript is a programming language widely used in web development, and solving geometric problems greatly aids in understanding algorithms. In this article, we will introduce a geometric algorithm problem that can be solved with JavaScript and explain the solution process in detail.

Problem Description

The following is a problem of determining whether two points are inside a given rectangle.

Problem: Determine if Points are Inside a Rectangle

The coordinates of the bottom-left corner of the given rectangle are (x1, y1), and the coordinates of the top-right corner are (x2, y2). Additionally, there are two points A(xA, yA) and B(xB, yB). Write a function to determine whether these points are inside the rectangle.

Input

• Bottom-left corner of the rectangle: (x1, y1)
• Top-right corner of the rectangle: (x2, y2)
• Point A: (xA, yA)
• Point B: (xB, yB)

Output

Write a function that returns true if each point is inside the rectangle, and false otherwise. If both points are inside, return true; otherwise, return false.

Solution Process

Now let’s go through the steps to solve this problem.

Step 1: Understanding the Problem

We have the coordinates of the rectangle’s corners and need to verify the coordinates of each point. The conditions for being inside the rectangle are as follows:

• x1 < xA < x2
• y1 < yA < y2
• x1 < xB < x2
• y1 < yB < y2

Each point must satisfy all the above conditions to be considered inside the rectangle.

Step 2: Writing the Function

Here is the JavaScript code based on this logic.

function isPointInsideRectangle(x1, y1, x2, y2, xA, yA, xB, yB) {
    const isAInside = (x1 < xA && xA < x2) && (y1 < yA && yA < y2);
    const isBInside = (x1 < xB && xB < x2) && (y1 < yB && yB < y2);
    
    return isAInside && isBInside;
}

// Test
const result = isPointInsideRectangle(0, 0, 10, 10, 5, 5, 5, 5); // true
console.log(result);
    

Step 3: Testing and Verification

Let’s test the function. We confirmed that when the bottom-left corner of the rectangle is (0, 0) and the top-right corner is (10, 10), the point (5, 5) is inside.

In the example above, both points are the same (5, 5), so the function returns true. However, it should return false if one point is outside the rectangle.

Step 4: Checking Various Cases

We can validate the function through various cases. Here are additional test examples.

// Case 1: Both points are inside
console.log(isPointInsideRectangle(0, 0, 10, 10, 1, 1, 2, 2)); // true

// Case 2: Only one point is inside
console.log(isPointInsideRectangle(0, 0, 10, 10, 1, 1, 11, 11)); // false

// Case 3: Both points are outside
console.log(isPointInsideRectangle(0, 0, 10, 10, -1, -1, 11, 11)); // false

// Case 4: Located on the edge of the rectangle
console.log(isPointInsideRectangle(0, 0, 10, 10, 0, 5, 5, 10)); // false
    

Conclusion

Now we have completed the function to determine whether the points are inside a given rectangle using JavaScript. Problems like these are great for developing algorithmic thinking and are frequently presented in coding tests.

I hope this article helped you understand the basics of geometric problems and how to apply them. Continue to challenge yourself with various algorithmic problems!