JavaScript Coding Test Course, Finding the Largest Palindrome

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.