Category: Python Coding Test

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.

Calculating the area of a polygon is an important problem that requires a fundamental understanding of programming and mathematical thinking.
This problem is commonly presented in actual coding tests, and the process relies on various logical foundations.
In this course, we will first define the problem, explain the corresponding algorithm, and finally write the code using the Python language.

Problem Definition

Let’s assume we form a polygon using multiple points given as input.
The given points are represented as a series of coordinates (x, y), and these points are connected to form the polygon.
At this point, we need to write an algorithm to calculate the area of the polygon.
An example of input is as follows:

        4
        0 0
        4 0
        4 3
        0 3
    

The input above indicates that we can connect four points (0, 0), (4, 0), (4, 3), and (0, 3) to form a rectangle.
The formula for calculating the area is as follows:
Area = 1/2 × |Σ (x_i * y_{i+1} - x_{i+1} * y_i)|
Note that the last point (x_n, y_n) must be connected to the first point (x_0, y_0).

Approach to Problem Solving

The approach to calculating the area of a polygon can be broadly divided into two methods.
First, directly implementing a formula using the coordinates of the vertices of the polygon.
Second, utilizing a library to easily calculate the area.
In this course, we will implement the algorithm using the first method.

1. Data Input

We will input the number of vertices of the polygon and the coordinates of each vertex.
The data received from the user will be stored in a list for management.
We can use Python’s basic input functions to handle multiple points.

2. Implementing the Area Calculation Algorithm

Let’s implement the area calculation formula in Python.
One of the advantages of polygons is that their area can be easily calculated, and our algorithm will work based on the ‘Shoelace formula’.

Example Code

        def polygon_area(coords):
            n = len(coords)
            area = 0
            for i in range(n):
                x1, y1 = coords[i]
                x2, y2 = coords[(i + 1) % n]
                area += x1 * y2 - x2 * y1
            return abs(area) / 2
    

3. Complete Coding Process

Let’s wrap the entire algorithm into a single function.
The code below includes the part that receives input and the function for calculating the area.
This code requires the user to input integers and expects ideal data.

Final Code

        def polygon_area(coords):
            n = len(coords)
            area = 0
            for i in range(n):
                x1, y1 = coords[i]
                x2, y2 = coords[(i + 1) % n]
                area += x1 * y2 - x2 * y1
            return abs(area) / 2

        def main():
            n = int(input("Please enter the number of vertices of the polygon: "))
            coords = []
            for _ in range(n):
                x, y = map(int, input("Please enter the coordinates of the vertex (x y): ").split())
                coords.append((x, y))
            print("The area of the polygon is: ", polygon_area(coords))

        if __name__ == "__main__":
            main()
    

Conclusion and Additional Tips

Now we have implemented an algorithm to calculate the area of a polygon using Python.
In actual coding interviews, it is highly likely that you will encounter such basic problems.
Therefore, it is recommended to practice with various types of polygons and input data.
Additionally, if necessary, it is important to analyze and optimize the time complexity of the algorithm to improve efficiency.

References

• Mathematical Foundations for Calculating the Area of a Polygon
• Understanding Python’s Basic Functions and I/O Processing
• Understanding and Utilizing the Shoelace Formula
• Preparing for Coding Tests – Solving Various Algorithm Problems

Breadth-First Search (BFS) is a method for exploring graphs or trees, starting from a vertex and visiting all its adjacent vertices first, followed by the unvisited adjacent vertices. It can also be used as a shortest path search algorithm in certain cases. Today, we will proceed with a deep problem-solving approach along with the basic concept of BFS.

Problem: Numbering Complexes

Problem Description

In a given n x n array consisting of 0s and 1s, 1 represents the location of a house and 0 represents the absence of a house. Write a program that assigns each complex a unique number to distinguish all the houses. The size of a complex is defined as the number of houses belonging to it.

Input

n (1 ≤ n ≤ 25)
An n x n array composed of 0s and 1s

Output

Print the size of each complex in ascending order

Example Input

7
0 0 1 0 0 1 1
0 0 1 0 1 1 1
0 1 1 0 0 0 0
0 0 0 0 1 0 0
1 1 1 0 1 1 0
0 0 0 0 0 0 0
0 0 1 0 0 0 0

Example Output

1
7

Problem Approach

To solve this problem, we will use the BFS algorithm to explore the graph. By traversing the provided 2D array, whenever we encounter a house (1), we will count all the connected houses using BFS to calculate the size of the complex. In the next step, we will explain the concept of BFS in detail.

Breadth-First Search (BFS) Concept

BFS is implemented using a queue. It follows the following process:

1. Add the starting node to the queue and mark it as visited.
2. Remove a node from the queue and add all its adjacent nodes (that have not yet been visited) to the queue, marking them as visited.
3. If the queue is not empty, return to step 2.

Coding Implementation

I will write code to solve the problem using the BFS algorithm.

from collections import deque

def bfs(x, y, n, graph, visited):
    queue = deque()
    queue.append((x, y))
    visited[x][y] = True

    directions = [(-1, 0), (1, 0), (0, -1), (0, 1)]
    size = 1  # store the size of the complex

    while queue:
        cx, cy = queue.popleft()  # current position
        for dx, dy in directions:
            nx, ny = cx + dx, cy + dy
            if 0 <= nx < n and 0 <= ny < n and not visited[nx][ny] and graph[nx][ny] == 1:
                visited[nx][ny] = True
                queue.append((nx, ny))
                size += 1  # increase the count of connected houses

    return size  # return the size of the complex

def find_complexes(n, graph):
    visited = [[False] * n for _ in range(n)]
    complexes = []

    for i in range(n):
        for j in range(n):
            if graph[i][j] == 1 and not visited[i][j]:
                size = bfs(i, j, n, graph, visited)
                complexes.append(size)

    return sorted(complexes)  # return the sizes in ascending order

# Example input
n = 7
graph = [
    [0, 0, 1, 0, 0, 1, 1],
    [0, 0, 1, 0, 1, 1, 1],
    [0, 1, 1, 0, 0, 0, 0],
    [0, 0, 0, 0, 1, 0, 0],
    [1, 1, 1, 0, 1, 1, 0],
    [0, 0, 0, 0, 0, 0, 0],
    [0, 0, 1, 0, 0, 0, 0]
]

complex_sizes = find_complexes(n, graph)
print(len(complex_sizes))  # print the number of complexes
for size in complex_sizes:
    print(size)  # print the size of each complex

Code Explanation

The above code demonstrates how to calculate the size of complexes using BFS. The main functions are find_complexes and bfs.

• bfs(x, y, n, graph, visited): Executes BFS starting from the point (x, y) to calculate the size of the corresponding complex. It records each visited house using a queue and aggregates all connected houses.
• find_complexes(n, graph): Iterates through the given graph, finds houses (1), invokes BFS, and stores the calculated complex sizes in a list.

Conclusion

Breadth-First Search (BFS) is a useful search method that can be utilized in various fields. I hope you learned how to solve problems using BFS through this lecture. Understanding and applying the code during the problem-solving process will greatly assist you in your coding tests.

Additional Practice Problems

Try solving the following problems:

• Finding the shortest path between specific nodes in the given graph
• Solving a maze problem using BFS

References

• Breadth-first search - Wikipedia
• BFS for Graph - GeeksforGeeks

1. Problem Definition

The goal of this problem is to sort the given digits in descending order to form the largest possible number. The input number is an integer, and our task is to return the largest number possible using the digits of this number.

Example Input/Output

• Input: 2183
• Output: 8321

2. Approach to the Problem

This problem involves extracting the individual digits of the number and then sorting them in descending order. In Python, we can easily perform such sorting operations using the sort() method of a list. The specific steps are as follows:

1. Convert the integer to a string.
2. Split the string into individual digits (characters) to create a list.
3. Sort the list in descending order.
4. Merge the sorted list back into a string, convert it to an integer, and return it.

3. Algorithm Implementation

Now, let’s solve the problem using Python code based on the above approach.

def sort_digits_descending(n):
    # Step 1: Convert integer to string
    str_n = str(n)
    
    # Step 2: Convert string to list
    digits = list(str_n)
    
    # Step 3: Sort the list in descending order
    digits.sort(reverse=True)
    
    # Step 4: Merge the sorted list back into a string and convert to integer
    sorted_n = int(''.join(digits))
    
    return sorted_n

4. Code Explanation

The function sort_digits_descending takes the parameter n and performs the entire process. Each step works as follows:

• String Conversion: Converts the integer to a string using str(n).
• List Conversion: Converts each character (digit) of the string into a list using list(str_n).
• Sorting: Sorts the list in descending order using sort(reverse=True).
• Merging: Merges the list back into a string using "".join(digits), and converts it to an integer using int() before returning it.

5. Test Cases

Let’s use several test cases to verify that our function works correctly.

print(sort_digits_descending(2183)) # 8321
print(sort_digits_descending(1450)) # 5410
print(sort_digits_descending(9876543210)) # 9876543210
print(sort_digits_descending(0)) # 0
print(sort_digits_descending(1001)) # 1100

6. Result Analysis

After checking the results of each test case, we found that the expected output was returned in all cases. The function was implemented very simply and efficiently using Python’s built-in features.

7. Optimization and Considerations

The code written above has a time complexity of O(m log m) for sorting the digits, where m is the number of digits in the input integer. Since the maximum number of digits for integers is limited to 10, there are no performance issues, but it may be necessary to consider making it valid for larger numbers or reducing complexity for efficiency.

8. Conclusion

We have implemented an algorithm to sort the given integer in descending order to create the largest possible value. In this process, we were able to solve the problem simply with the use of Python’s list methods and string manipulation features. Future considerations for additional optimization and alternative approaches are encouraged. This will help improve problem-solving skills in coding tests.