Swift Coding Test Course, Finding Binomial Coefficient 2

This course covers the problem of finding binomial coefficients. The binomial coefficient is an important concept in combinatorics, representing the number of ways to choose k items from n items. This type of problem is frequently asked in algorithm coding tests.

Problem Description

Write a program to calculate the binomial coefficient C(n, k) for the given two integers n (0 ≤ n ≤ 30) and k (0 ≤ k ≤ n). The binomial coefficient C(n, k) is defined as follows:

C(n, k) = n! / (k! * (n - k)!)

An example of the problem is as follows:

Example

• Input: 5 2
• Output: 10

Problem Solving Process

Recursive Property of Binomial Coefficient

The binomial coefficient has the following recursive property:

C(n, k) = C(n - 1, k - 1) + C(n - 1, k)

Here, C(n, 0) = 1 and C(n, n) = 1. By utilizing this property, we can use a recursive function to calculate the binomial coefficient. However, this method can be inefficient due to deep recursive calls.

Solution via Dynamic Programming

This problem can be solved using dynamic programming. Dynamic programming improves algorithm performance by avoiding redundant calculations. The following table can be used to derive the value of C(n, k).

Dynamic Programming Approach

To calculate the binomial coefficient using dynamic programming, we declare the following 2D array. dp[i][j] will store the value for C(i, j).

var dp = [[Int]](repeating: [Int](repeating: 0, count: n + 1), count: n + 1)

for i in 0...n {
    for j in 0...i {
        if j == 0 || j == i {
            dp[i][j] = 1
        } else {
            dp[i][j] = dp[i - 1][j - 1] + dp[i - 1][j]
        }
    }
}

Swift Code Implementation

Based on the above dynamic programming approach, let’s write a Swift program to calculate the binomial coefficient.

import Foundation

func binomialCoefficient(n: Int, k: Int) -> Int {
    var dp = [[Int]](repeating: [Int](repeating: 0, count: k + 1), count: n + 1)

    for i in 0...n {
        for j in 0...min(i, k) {
            if j == 0 || j == i {
                dp[i][j] = 1
            } else {
                dp[i][j] = dp[i - 1][j - 1] + dp[i - 1][j]
            }
        }
    }
    return dp[n][k]
}

// Example Input
let n = 5
let k = 2

// Print Result
let result = binomialCoefficient(n: n, k: k)
print("C(\(n), \(k)) = \(result)")

Result Verification

Running the above code will output 10 for C(5, 2). This accurately calculates the number of ways to choose 2 items from 5 items.

Time Complexity Analysis

The time complexity of this algorithm is O(n*k). In this case, n is 30, and k is proportional to n, so it will be at most 30. The number of binomial coefficients calculated in this manner is efficient, making it highly suitable for solving the problem.

Conclusion

In this course, we addressed the problem of calculating binomial coefficients and learned how to implement efficient algorithms using dynamic programming. By using a 2D array to store each binomial coefficient, we avoided recursive calls and improved performance. This problem helps establish foundational concepts in combinatorics and dynamic programming.

In the next course, we will cover another algorithm problem. I hope you continue to enhance your algorithm-solving skills through ongoing learning!