Swift Coding Test Course, Bridge Building

Hello! Today, I would like to talk about how to prepare for coding tests using Swift. In this tutorial, we will cover the algorithmic problem called Bridge Building. This problem can be solved using the concepts of combination and dynamic programming.

Problem Description

The Bridge Building problem assumes the following situation: there is a piece of land that is N units wide. On both sides of the land, pillars numbered from 1 to N are erected. Given a fixed number of pillars and bridges, the goal is to calculate the number of ways to place bridges between two pillars.

Since the bridges cannot cross each other, the ways of placing the bridges must not overlap. In other words, to place a bridge between pillar A and pillar B, A must be before B. The problem is to take the number of pillars N as input and output the number of possible ways to place bridges.

Problem Input

• Input: Integer N (1 ≤ N ≤ 30)

Problem Output

• Output: The number of possible ways to place bridges

Problem Solving Process

To solve this problem, the following steps are taken:

1. Understanding Combinations

This problem can be reduced to a combination problem. When there are N pillars, the number of ways to place bridges among the N pillars is defined as the number of combinations of choosing 2 from N pillars. Mathematically, this can be expressed as:

C(N, 2) = N! / (2! * (N - 2)!)

However, this problem is not a simple combination problem; it also considers the order and overlap of the bridges placed between the pillars.

2. Dynamic Programming Approach

This problem can be approached using dynamic programming. We define dp[i] as the number of ways to place bridges when there are i pillars. The state transition equation is defined as follows:

dp[i] = dp[i-1] + dp[i-2] * (i - 1)

This equation is based on the following ideas:

• When not placing a bridge among i-1 pillars: This case is represented by dp[i-1].
• When placing one bridge: This case is expressed as dp[i-2] * (i - 1). After placing a bridge between two pillars, i-2 pillars remain.

3. Setting Initial Conditions

Now we need to establish the initial conditions. We can set dp[1] = 1 and dp[2] = 1. When there is only one pillar, no bridge can be placed, and when there are two pillars, only one bridge can be placed.

4. Implementing in Swift

import Foundation

func bridgeCombinations(n: Int) -> Int {
    var dp = [0] * (n + 1)
    dp[1] = 1
    if n > 1 {
        dp[2] = 1
    }
    
    for i in 3...n {
        dp[i] = dp[i - 1] + dp[i - 2] * (i - 1)
    }
    
    return dp[n]
}

// Testing the Bridge Building Problem
let n = 5 // Set the desired N value here.
let result = bridgeCombinations(n: n)
print("Number of ways to place bridges: \(result)")

The code above implements a solution for the Bridge Building problem. It outputs the result based on the n value. This is an example solved using dynamic programming with Swift arrays.

Conclusion

We learned how to solve the Bridge Building algorithm problem using Swift. Understanding how dynamic programming and combinations combine is essential, especially in similar problems. You may often encounter such problems in actual coding tests, so don’t forget to practice!

I hope this post has been helpful for your coding test preparation. I will return with more posts covering various algorithms and coding problems.