Category: Python Coding Test

This course covers the problem of Finding the Fastest Bus Route, which is frequently presented in actual job exam algorithm problems. This problem can be solved using graph theory and shortest path algorithms. Throughout the course, we will enhance our algorithm problem-solving skills and learn various techniques necessary for writing Python code.

Problem Description

There is a city with N vertices from 1 to N. There are several bus routes between each pair of vertices, and each bus route has a departure and arrival vertex along with a travel time. Based on the given information, find the fastest route from vertex 1 to vertex N and the time it takes.

Input Format

• The first line contains the number of vertices N (1 ≤ N ≤ 1000) and the number of bus routes M (1 ≤ M ≤ 10000).
• The next M lines provide the information for each bus route in the format u v w, where w is the travel time from u to v (1 ≤ w ≤ 10000).

Output Format

• Print the fastest travel time from vertex 1 to vertex N. If there is no route, print -1.

Problem Solving Process

1. Understanding the Problem

This problem is a representative type of graph problem in finding the shortest path. When bus routes are represented as a graph, each vertex corresponds to a bus stop, edges represent bus routes, and the weights of the edges can be thought of as the travel times. What we want to solve is the fastest way to move from vertex 1 to vertex N.

2. Selecting the Algorithm

There are various algorithms to find the shortest path, but here we will use the Dijkstra’s Algorithm. Dijkstra’s algorithm is efficient for finding the shortest paths in graphs with non-negative weights. Since the travel times for the given bus routes are all positive, this algorithm is appropriate.

3. Implementing the Algorithm

The basic idea of Dijkstra’s algorithm is to maintain an array that records the shortest path distances to each vertex while selecting the vertex with the currently shortest distance. The detailed implementation process is as follows.

# Import required libraries
import sys
import heapq

def dijkstra(N, edges):
    # Initialize distance array
    distance = [float('inf')] * (N + 1)
    distance[1] = 0  # Distance to starting vertex 1 is 0
    priority_queue = [(0, 1)]  # (distance, vertex)

    # Initialize graph
    graph = [[] for _ in range(N + 1)]
    for u, v, w in edges:
        graph[u].append((v, w))

    while priority_queue:
        current_distance, current_vertex = heapq.heappop(priority_queue)
        
        if current_distance > distance[current_vertex]:
            continue

        for neighbor, weight in graph[current_vertex]:
            distance_via_neighbor = current_distance + weight
            
            if distance_via_neighbor < distance[neighbor]:
                distance[neighbor] = distance_via_neighbor
                heapq.heappush(priority_queue, (distance_via_neighbor, neighbor))

    return distance[N] if distance[N] != float('inf') else -1

# Input
N, M = map(int, input().split())
edges = [tuple(map(int, input().split())) for _ in range(M)]
result = dijkstra(N, edges)

print(result)

4. Example and Test Cases

To verify the appropriateness of this algorithm, let's consider several input examples.

Example 1

Input:
4 5
1 2 1
1 3 4
2 3 2
3 4 1
2 4 5

Output:
3

Explanation: The shortest path from vertex 1 to vertex 4 is 1 → 2 → 3 → 4, with a total travel time of 3.

Example 2

Input:
3 3
1 2 1
1 3 4
3 2 2

Output:
-1

Explanation: There is no route from vertex 1 to vertex 2, so we output -1.

Conclusion

In this course, we learned how to solve the problem of finding the fastest bus route using Dijkstra's algorithm for the shortest path. I hope this has been helpful in understanding and implementing the algorithm. As you may encounter similar problems in future coding tests, I encourage you to continue practicing.

In this course, we will cover the problem of finding the “Longest Increasing Subsequence” (LIS), which is one of the important concepts in Python coding tests. This problem frequently appears in coding algorithm tests of various IT companies and requires efficient algorithm design and implementation skills. Therefore, it is important to understand and practice it properly.

1. Problem Description

The problem is to find the length of the longest increasing subsequence in a given sequence. A subsequence is formed by selecting some elements from the original sequence while maintaining their order. For example, given the sequence [10, 20, 10, 30, 20, 50], the longest increasing subsequence is [10, 20, 30, 50], which has a length of 4.

2. Problem Approach and Understanding

The longest increasing subsequence problem has the following characteristics:

• The elements of the subsequence must maintain their order in the original sequence.
• We need to find the length of the subsequence, not the subsequence itself.

To solve this problem, two approaches can be used.

1. Dynamic Programming approach
2. Binary Search approach

2.1 Dynamic Programming Approach

Using dynamic programming, we can maintain the length of the longest increasing subsequence based on each element of the sequence. The time complexity of this approach is O(n^2).

The algorithm using dynamic programming can be described as follows:

1. Initialize dp[i] as the length of the increasing subsequence, setting all values to 1.
2. For each element i, traverse the previous elements j (j < i), and if arr[j] < arr[i], update dp[i] to dp[j] + 1.
3. The final result is the maximum value in the dp array.

2.2 Binary Search Approach

The method using binary search is more efficient, with a time complexity of O(n log n). This approach uses an array called ‘tails’ to store the last elements of the longest increasing subsequences found so far.

The algorithm for the binary search approach can be described as follows:

1. Initialize an empty array.
2. Traverse the sequence and perform a binary search to find the position to insert the current element in the tails array.
3. If the found position is equal to the length of the tails array, add the current element; otherwise, update that position with the current element.
4. The final result is the length of the tails array.

3. Algorithm Implementation

3.1 Dynamic Programming Implementation

def longest_increasing_subsequence(arr):
    n = len(arr)
    dp = [1] * n  # Initialization
    for i in range(n):
        for j in range(i):
            if arr[j] < arr[i]:
                dp[i] = max(dp[i], dp[j] + 1)
    return max(dp)

3.2 Binary Search Implementation

from bisect import bisect_left

def longest_increasing_subsequence(arr):
    tails = []
    for x in arr:
        i = bisect_left(tails, x)  # Find index of the first element greater than x
        if i == len(tails):
            tails.append(x)  # Add new element
        else:
            tails[i] = x  # Update element
    return len(tails)

4. Examples and Results

Now we will run some examples using the algorithms implemented above.

arr = [10, 20, 10, 30, 20, 50]
print(longest_increasing_subsequence(arr))  # Output: 4

arr = [3, 2, 5, 6, 3, 7, 1]
print(longest_increasing_subsequence(arr))  # Output: 5

arr = [1, 2, 3, 4, 5]
print(longest_increasing_subsequence(arr))  # Output: 5

5. Conclusion

The problem of finding the longest increasing subsequence frequently appears in coding interviews, and can be solved using both dynamic programming and binary search approaches. Through this problem, you can learn both the concept of dynamic programming and the application of binary search, laying the foundation for solving complex algorithm problems.

This concludes the Python coding test course on finding the longest increasing subsequence. I hope this has been helpful in solving algorithm problems. Keep practicing to improve your algorithm skills and prepare for the next coding test!

Author: [Author Name] | Date: [Date]

Problem Description

This is a problem to find a number n that is a “good number.”
The conditions for a “good number” are as follows.

• A “good number” is a natural number with two or more digits.
• A number is defined as a “good number” if the sum of its digits is odd.
• Find all “good numbers” less than or equal to the given n and return them as a list.

For example, if n is 30, the “good numbers” are 11, 13, 15, 17, 19, 21, 23, 25, 27, 29.
We will design an algorithm to solve this problem and implement the code.

Problem-Solving Approach

Let’s approach the problem step by step.
First, since we only consider cases where n is a natural number with two or more digits, we will use a loop from 10 to n to calculate the sum of the digits for each number.

To calculate the sum of the digits of a number, we can convert the given number to a string,
then convert each digit back to an integer and add it to the total sum.
Next, we will check if that sum is odd, and if so, add the number to the list of good numbers.

Code Implementation

Now, let’s implement the algorithm described above in Python.

def is_good_number(num):
    # Calculate the sum of the digits and check if it is odd
    digit_sum = sum(int(digit) for digit in str(num))
    return digit_sum % 2 == 1

def find_good_numbers(n):
    good_numbers = []
    for i in range(10, n + 1):  # digits with two or more
        if is_good_number(i):
            good_numbers.append(i)
    return good_numbers

# Test
n = 30
print(find_good_numbers(n))
            

The above code uses a function called is_good_number to calculate the sum of the digits of a given number and check
whether that sum is odd.

find_good_numbers function loops through all numbers from 10 to n,
finding “good numbers” and adding them to a list to return.

Execution Result

The result of executing the function for n = 30 is as follows:

[11, 13, 15, 17, 19, 21, 23, 25, 27, 29]
            

As shown above, the “good numbers” 11, 13, 15, 17, 19, 21, 23, 25, 27, 29 have been returned.
This result satisfies the condition of being “natural numbers with two or more digits whose sum of digits is odd.”

Complexity Analysis

Looking at the time complexity of the above algorithm,
since we have to iterate through all the numbers up to the given n, the time complexity is O(n).
For each number, we need additional constant time to calculate the sum of its digits, so overall it can be confirmed as O(n).

The space complexity is O(k)
(where k is the number of “good numbers”) due to the list used to store the “good numbers.”
In practice, since the smaller n is, the fewer “good numbers” will be stored, thus allowing for efficient space usage.

Conclusion

Through this problem, we have learned a method to effectively find “good numbers” by considering counterexamples and calculating the sum of the digits.
We were able to explore various ways to improve efficiency through a step-by-step approach to problem-solving and
implementation using Python code, as well as complexity analysis.

In the future, practice various ways of thinking and solving through more algorithm problems.
Understanding and approaching problems is very important in coding tests.

Understanding and practicing algorithm problem solving is essential for software developers, especially for students aiming for employment. By solving various types of problems, one must develop problem-solving skills and learn how to practically apply algorithms and data structures. In this course, we will cover the topic of “Finding the Kth Shortest Path” and explain the process of solving this problem in Python in detail.

Problem Description

Given a graph, the problem is to find the Kth shortest path between two specific nodes. The Kth shortest path means the Kth shortest path among the shortest paths between the two nodes. This problem can be approached through variations of shortest path algorithms like Dijkstra’s algorithm.

Problem Summary

• The graph can be a directed graph with or without weights.
• A graph is given (number of nodes N, number of edges M).
• The starting node S and the ending node E are given.
• The Kth shortest path needs to be found.

Input Format

N M K
a1 b1 c1
a2 b2 c2
...

In the input format above, N is the number of nodes, M is the number of edges, and K is the Kth shortest path you want to find. Each edge is given by three numbers: a is the starting node, b is the ending node, and c is the weight.

Output Format

The length of the Kth shortest path, or -1 if the Kth shortest path does not exist

Examples

Input Example

4 5 2
1 2 4
1 3 2
2 3 5
2 4 1
3 4 8

Output Example

6

Solution Process

To solve this problem, we need to implement an algorithm to find the Kth shortest path. We plan to modify a typical shortest path algorithm, Dijkstra’s algorithm, and devise a way to manage the costs of each path. Here, we will solve it by exploring paths using a Priority Queue.

Step-by-step Approach

Step 1: Representing the Graph

We will represent the graph in the form of an adjacency list. We will use a dictionary or list to hold information about each node and edge.

from collections import defaultdict
import heapq

def build_graph(edges):
    graph = defaultdict(list)
    for (u, v, w) in edges:
        graph[u].append((v, w))
    return graph

Step 2: Path Exploration Using a Priority Queue

We will use a priority queue to explore the shortest paths. The costs of each path and nodes can be saved and managed in tuple form. At this point, we should prepare a list to store K paths.

def kth_shortest_path(graph, start, end, k):
    # Initialization
    pq = []
    heapq.heappush(pq, (0, start))  # (cost, node)
    paths = defaultdict(list)

    while pq:
        cost, node = heapq.heappop(pq)
        
        # If the Kth path is found
        if node == end:
            paths[end].append(cost)
            if len(paths[end]) >= k:
                break
        
        for neighbor, weight in graph[node]:
            new_cost = cost + weight
            heapq.heappush(pq, (new_cost, neighbor))
    
    return paths[end][k - 1] if len(paths[end]) >= k else -1

Step 3: Integrating the Overall Algorithm

We will combine the above two functionalities to complete the overall algorithm. We will receive input, create the graph, and implement the logic for finding the Kth shortest path.

def main():
    N, M, K = map(int, input().split())
    edges = [tuple(map(int, input().split())) for _ in range(M)]
    graph = build_graph(edges)
    result = kth_shortest_path(graph, 1, N, K)
    print(result)

if __name__ == "__main__":
    main()

Conclusion and Wrap-up

In this course, we have covered the Kth shortest path finding problem. We understood the structure of the graph and the variations of Dijkstra’s algorithm, and learned how to explore paths through a priority queue. As algorithm problems can be variously modified, it’s important to practice solving many different problems.

Now, you have laid the foundation to solve the given problem. The methods learned will be applicable not only to the Kth shortest path problem but also to various other graph-related problems, so developing your application skills is important. I hope you continue to practice more algorithms and problem-solving methods.

Happy Coding!

In this course, we will learn about preparing for coding tests using Python through the algorithm problem “Finding the Kth Number.” This problem requires basic sorting and list manipulation, making it useful for practicing relevant basic syntax and algorithm techniques.

Problem Description

The problem is as follows:

n: Number of integers
k: The Kth number to find
arr: A list of n integers

1. Find the k-th number in the list.
2. Note that the numbers are positive integers, and the conditions are 1 ≤ n ≤ 1000, 1 ≤ k ≤ n.
3. The k-th number refers to the number at the k-th position in ascending order.

Example

Assuming the following input values are given:

n = 5
k = 2
arr = [5, 2, 3, 1, 4]

For the above input, the output should be as follows:

The 2nd number is 2.

Problem-Solving Strategy

To solve this problem, you can approach it in the following steps:

1. Input: Obtain the values of n, k, and arr from the user.
2. Sort: Sort the list arr in ascending order.
3. Find Kth Number: Since the indexing of the list starts from 0, extract the value at the index k-1 and output it.

Code Implementation

Now, let’s implement the actual code based on the above-solving strategy.

def find_kth_number(n, k, arr):
    # 1. Sort the list
    arr.sort()
    
    # 2. Find the Kth number
    return arr[k - 1]

# Input
n = int(input("Enter the number of integers: "))
k = int(input("Enter the Kth number to find: "))
arr = list(map(int, input("Enter the list of integers (separated by spaces): ").split()))

# Find the Kth number
kth_number = find_kth_number(n, k, arr)
print(f"The {k}th number is {kth_number}.")

Code Explanation

The above code can be explained in three parts:

1. Function Definition: The find_kth_number function is defined to take n, k, and arr as parameters. This function returns the k-th number.
2. Sorting: The list is sorted in ascending order using arr.sort().
3. Return Result: The k-th number is returned through return arr[k - 1]. Since k is taken as user input, k-1 is used to match the 0-based indexing.

Design Considerations

When solving the problem, there are several factors to consider. It is advisable to think about these points before writing the code:

• Input Validation: It is essential to check if the values of n and k are given within the specified range.
• Handling Duplicates: When there are duplicate values, it is good to clarify which number to select among the multiple k-th numbers.
• Time Complexity: Sorting the arr takes O(n log n) time, so choosing an efficient algorithm is necessary.

Additional Practice Problems

If you have learned the basic algorithm for finding the Kth number through this problem, you can further develop your design skills by solving similar problems. Here are additional practice problems:

1. Find the k-th smallest number among the numbers from 1 to n in a given list.
2. Given two sorted lists, find the Kth number in the merged list of the two lists.
3. Solve the problem of searching for the k-th smallest number in a 2D array.

Conclusion

In this course, we have explored the algorithm problem of Finding the Kth Number using Python. When solving algorithm problems, it is essential to clearly understand the problem, devise a solution strategy, and then implement it in code. Through practice, I hope you encounter various types of problems and improve your skills in solving them.

Thank you.