python coding test course, bridge building

Problem Description

The ‘Bridge Laying’ problem is about calculating the number of ways to place N bridges on M pillars given the two numbers N and M.
The bridges must not cross each other, and the number of bridges placed on the pillars must be less than or equal to the number of pillars.
This problem can be efficiently solved using the concept of combinations and dynamic programming (DP).

Problem Definition

Input:
The first line contains two integers N (number of bridges) and M (number of pillars) separated by a space. (1 ≤ N ≤ 100, 1 ≤ M ≤ 100)
Output:
Print the number of ways to lay the bridges.

Approach to Solve the Problem

This problem is solved using the properties of combinations. The number of ways to place N bridges on M pillars can be represented by the following mathematical formula.

C(M, N) = M! / (N! * (M – N)!)

Here, C(M, N) refers to the number of combinations of choosing N from M.
In Python, this problem can be easily solved using the factorial function from the math library.

Algorithm Implementation

Now let’s write Python code to solve the problem. The code below calculates the number of ways to lay the bridges:


import math

def bridge_count(N, M):
    return math.factorial(M) // (math.factorial(N) * math.factorial(M - N))

# Get input
N, M = map(int, input("Enter the number of bridges (N) and the number of pillars (M): ").split())
result = bridge_count(N, M)
print(result)

Code Explanation

1. import math: Imports the math library in Python.
2. def bridge_count(N, M):: Defines a function that takes the number of bridges (N) and the number of pillars (M) as arguments.
3. math.factorial(M): Calculates the factorial of M.
4. return: Calculates and returns the number of combinations.
5. N, M = map(int, input().split()): Receives input from the user and converts it to integers.
6. Finally, the result is printed.

Test Cases

Let’s validate the algorithm through various test cases.

  • Test Case 1: N = 2, M = 4 -> Output: 6 (e.g., Bridge 1-Pillar 1, Bridge 2-Pillar 2; Bridge 1-Pillar 2, Bridge 2-Pillar 3, etc.)
  • Test Case 2: N = 3, M = 5 -> Output: 10 (placing 3 bridges on 5 pillars)
  • Test Case 3: N = 1, M = 1 -> Output: 1 (1 bridge can only be placed on 1 pillar)

Time Complexity Analysis

The time complexity of this algorithm is O(M). The overall performance varies according to the size of M,
so appropriate results can be derived within a suitable time considering the range of input values.

Conclusion

In this posting, we have implemented a simple algorithm to calculate the arrangement of bridges using the fundamental principles of combinations through the ‘Bridge Laying’ problem.
While solving this problem, we learned that complex formulas can be easily calculated using the concept of combinations and Python’s math module.
The bridge laying problem is a fundamental algorithm problem that is frequently asked in coding tests and competitions.
Therefore, one should be able to adapt to various scenarios through algorithm practice.
Additionally, improving algorithmic sense through solving diverse problems can enhance one’s skills further.