Java Coding Test Course, Bridge Building

Problem Description

The Bridge Building Problem arises in the following context. You want to build N bridges and M piers. The goal of this problem is to calculate the number of possible combinations of bridge placements.
The bridges must not overlap, and each pier serves to connect two bridges. To calculate the number of ways to place the bridges, knowledge of combinations and dynamic programming is required.

Problem Definition

Input

  • Size N (1 ≤ N ≤ 30): Number of bridges
  • Size M (1 ≤ M ≤ N): Number of piers

Output

  • Print the total number of ways to place the bridges as an integer.

Problem Approach

The bridge building problem is essentially a combinatorial problem. We need to find the number of ways to select M bridges from N bridges. The number of combinations is calculated using the following formula: 

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

However, directly calculating the factorial can be time-consuming, so we can use dynamic programming to calculate it efficiently. We will use two arrays.
The first array will keep track of the number of piers chosen to place the bridges, and the second array will record the number of ways to place the bridges.

Problem Solution

Step 1: Initialize the Dynamic Programming Array

First, we initialize the DP array. dp[i][j] represents the number of ways to place i bridges and j piers. After initializing the array to 0,
we set dp[0][0] = 1. This means there is 1 way when there are no bridges and no piers.

Step 2: Find the Recurrence Relation

When adding one bridge, there is an option to place one pier. We can fill the dp array using the following recurrence relation. 

dp[i][j] = dp[i-1][j-1] + dp[i-1][j]

Here, dp[i-1][j-1] refers to the case of placing i-1 bridges and j-1 piers, adding a new bridge, while
dp[i-1][j] refers to the case of not adding a new bridge and using i-1 bridges to place j piers.

Step 3: Print the Final Result

After filling all the ranges, we can print dp[N][M] to get the final number of ways to place the bridges.

Java Code Implementation

The Java code to solve the bridge building problem is as follows:


public class BridgeBuilding {
    public static void main(String[] args) {
        int N = 5;  // Number of bridges
        int M = 3;  // Number of piers
        
        System.out.println(countWays(N, M));
    }
    
    public static int countWays(int n, int m) {
        int[][] dp = new int[n + 1][m + 1];

        for (int i = 0; i <= n; i++) {
            for (int j = 0; j <= Math.min(i, m); j++) {
                if (j == 0) {
                    dp[i][j] = 1; // The case of placing no piers
                } else if (i > 0) {
                    dp[i][j] = dp[i - 1][j - 1] + dp[i - 1][j];
                }
            }
        }
        
        return dp[n][m];
    }
}

Time Complexity

The time complexity of this algorithm is O(N*M). This time arises from filling the DP table using two nested loops. The space complexity is O(N*M), as we need space to store the DP array.

Conclusion

In this lecture, we covered the ‘Bridge Building’ problem. To solve this problem, an understanding of combinatorial theory and dynamic programming is essential.
The provided Java code allows us to solve the problem, and similar approaches can be used when encountering similar types of problems.
In the next lecture, we will discuss another algorithm problem.