Category: Python Coding Test

One of the common problems presented in coding tests is finding a way to reduce a given integer to 1. In this tutorial, we will explain the algorithms, approaches, and provide actual coding examples needed to solve this problem in detail. Our goal is to perform the minimum number of operations until the given integer becomes 1.

Problem Description

Given an integer X. The problem is to reduce this X to 1 using the following three operations and find the minimum number of operations required:

• 1. X - 1
• 2. X / 2 (only possible if X is divisible by 2)
• 3. X / 3 (only possible if X is divisible by 3)

For example, when X = 10, it can be calculated as follows:

• 10 → 9 (1 operation)
• 9 → 3 (2 operations)
• 3 → 1 (3 operations)

Therefore, the total number of operations is 3.

Approach to Problem Solving

To solve this problem efficiently, we can use Dynamic Programming (DP). DP involves breaking the problem into smaller subproblems and storing the solutions to those subproblems to reduce redundant calculations. This approach will be explained in detail in the following steps.

Step 1: Initialize DP Table

Create a DP table to store the minimum number of operations required to reduce each index i to 1. The size of the table is set to X + 1.

X = int(input("Enter an integer: "))
dp = [0] * (X + 1)
    

Step 2: Set Base Case

By default, the value of dp[1] is 0 because 1 is already the target, so no additional operations are needed.

dp[1] = 0  # 1 can be represented with 0 operations.
    

Step 3: Set Recurrence Relation

For each integer i, perform all possible operations to update the value of dp[i].

• First, for the case of subtracting 1 from i: dp[i] = dp[i-1] + 1
• Then, if i is divisible by 2: dp[i] = min(dp[i], dp[i // 2] + 1)
• Also, if i is divisible by 3: dp[i] = min(dp[i], dp[i // 3] + 1)

In other words, we update the value of dp[i] with the minimum operations.

Step 4: Derive Final Result

After filling the DP table for all integers, dp[X] will be the desired result, that is, the minimum number of operations needed to reduce X to 1.

for i in range(2, X + 1):
    dp[i] = dp[i - 1] + 1
    if i % 2 == 0:
        dp[i] = min(dp[i], dp[i // 2] + 1)
    if i % 3 == 0:
        dp[i] = min(dp[i], dp[i // 3] + 1)

print("Minimum number of operations:", dp[X])
    

Full Code

Below is the complete code based on the aforementioned explanations:

def min_operations_to_one(X):
    dp = [0] * (X + 1)
    dp[1] = 0  # Base case: 1 can be represented with 0 operations.

    for i in range(2, X + 1):
        dp[i] = dp[i - 1] + 1  # Case of subtracting 1
        if i % 2 == 0:
            dp[i] = min(dp[i], dp[i // 2] + 1)  # Case of dividing by 2
        if i % 3 == 0:
            dp[i] = min(dp[i], dp[i // 3] + 1)  # Case of dividing by 3

    return dp[X]

# User input
X = int(input("Enter an integer: "))
result = min_operations_to_one(X)
print("Minimum number of operations:", result)
    

Complexity Analysis

The above algorithm has a time complexity of O(N). This is because the DP table is filled using a loop iterating over X. The space complexity is also O(N) due to the space required to store the DP array.

Conclusion

In this article, we explained the process of solving the problem of reducing a given integer to 1 using dynamic programming techniques. It is essential to learn various applications of DP in preparation for coding tests. We hope you enhance your skills through more practice problems and achieve great results in coding tests!

In today’s lecture, we will take a detailed look at how to implement an absolute value heap. An absolute value heap is a data structure that finds the minimum or maximum based on the absolute value of the given numbers. It operates similarly to a regular heap but works based on absolute values. This lecture will cover everything from problem definition to algorithm design and implementation.

Problem Definition

To sort the given integers based on their absolute values, we need to write a program that performs the following tasks:

• An absolute value heap can insert integer values.
• It can return and remove the minimum value.

The input values are provided as follows:

• The first line of input contains the number of operations N.
• Then, N lines follow, each containing an integer X. If X is 0, it returns and removes the minimum value from the heap.

Example Input

    9
    1
    -1
    0
    -2
    0
    2
    0
    -3
    0
    

Example Output

    1
    -1
    2
    -2
    -3
    

Problem-Solving Strategy

The most important aspect of this problem is that we have to sort the integer values based on their absolute values when inserting. Therefore, when inserting integer values, we can initially use Python’s heapq module to implement a min-heap. However, we need to manage the heap based on absolute values ourselves.

The following strategy can be employed:

1. Transform the input values into a tuple with two elements and insert them into the heap: (absolute value, original value).
2. When retrieving the minimum value from the heap, return it according to the order of the original value if the absolute values are the same.
3. When the input is 0, remove and return the minimum value from the heap.

Implementation Steps

Now, let’s implement the absolute value heap based on the above strategy. We can solve the problem using the following code:

    import heapq
    import sys

    input = sys.stdin.readline

    def absolute_heap():
        heap = []
        N = int(input())
        
        for _ in range(N):
            X = int(input().strip())

            if X != 0:
                # Insert the absolute value and original value as a pair into the heap
                heapq.heappush(heap, (abs(X), X))
            else:
                # When 0 is input, remove and return the minimum value
                if heap:
                    print(heapq.heappop(heap)[1])
                else:
                    print(0)

    if __name__ == "__main__":
        absolute_heap()
    

Code Explanation

1. heapq: This is Python’s built-in heap module. It allows us to easily use a min-heap.

2. input: It allows for quick input using sys.stdin.readline.

3. heap: This is a list used to store pairs of absolute values and original values.

4. For each integer X read, if X is not 0, it inserts its absolute value and original value as a pair into the heap.

5. If X is 0, it removes the minimum value from the heap and outputs the original value.

Complexity Analysis

The time complexity of this algorithm is as follows:

• The time complexity for inserting an element into the heap is O(log N).
• The time complexity for removing an element from the heap is also O(log N).

Therefore, the overall time complexity of the algorithm is O(N log N).

Conclusion

In this lecture, we understood the concept of an absolute value heap and learned how to implement it in Python. The important aspect of implementing an absolute value heap is managing the original and absolute values properly. Based on what we learned today, we hope you will tackle various problems.

Additional Problems

Try solving the problem below to deepen your understanding of absolute value heaps:

Problem: Write a program to calculate the sum of given numbers using the absolute value heap. Find the minimum value through the absolute value heap and repeat until all inputs are processed.

To solve the above problem, you need to accumulate the values removed from the heap to keep track of the sum. Write the code yourself and apply the principles learned in this lecture.

Reference Materials

• heapq – The Python Standard Library
• Programmers – Dual Priority Queue Problem Page

Thank you. See you in the next lecture!

In this course, we will learn about algorithms to solve the critical path problem. The Critical Path shows the start and completion times of each task in project management, representing the longest path to complete the entire project. It is a very important concept in engineering and software projects and is one of the frequently tested topics in actual coding interviews.

Problem Definition

We aim to address the problem of finding the critical path when given a directed graph with the following structure. Each node in the graph represents a task, and edges indicate the precedence relationships between tasks. What we need to calculate is the minimum time required to complete all tasks.

Problem Description

Input:
- n (1 ≤ n ≤ 100): Number of tasks
- edges: A list representing the dependency relationships between tasks, where each element is a tuple (u, v, w), 
  where u is the starting node of the task, v is the ending node of the task, and w is the time required to complete the task.

Output:
- The longest time required to complete all tasks (length of the critical path)

Example Input

n = 5
edges = [
    (1, 2, 3),
    (1, 3, 2),
    (2, 4, 4),
    (3, 4, 5),
    (4, 5, 1)
]

Example Output

8

Problem Solving and Approach

To solve this problem, we can implement it by following these steps.

1. Representing the Directed Graph

First, we need to structure the graph based on the input edge information. To do this, we will construct the graph in the form of an adjacency list. We will also prepare an array to store the completion times of each task.

from collections import defaultdict
import sys

def critical_path(n, edges):
    graph = defaultdict(list)
    in_degree = [0] * (n + 1)
    completion_time = [0] * (n + 1)

    # Construct the graph and calculate in-degrees
    for u, v, w in edges:
        graph[u].append((v, w))
        in_degree[v] += 1

2. Topological Sorting

The next step is to perform a topological sort of the graph to safely process all tasks. Through topological sorting, we can determine the order in which each task should be completed while considering task dependencies.

    # Perform topological sort
    zero_degree_queue = []
    for i in range(1, n + 1):
        if in_degree[i] == 0:
            zero_degree_queue.append(i)
    
    topo_order = []
    while zero_degree_queue:
        node = zero_degree_queue.pop(0)
        topo_order.append(node)
        for neighbor, duration in graph[node]:
            in_degree[neighbor] -= 1
            if in_degree[neighbor] == 0:
                zero_degree_queue.append(neighbor)

3. Calculating Minimum Duration

Now, we calculate the completion times for each task in the order determined by the topological sort. Each time we perform a task, we update the completion times of all nodes connected to that task. This allows us to store the time taken to reach each task.

    for node in topo_order:
        for neighbor, duration in graph[node]:
            completion_time[neighbor] = max(completion_time[neighbor], completion_time[node] + duration)

    # The maximum time required to complete all tasks
    return max(completion_time)

4. Writing the Complete Code

We will write the final code by integrating all the steps together.

def critical_path(n, edges):
    graph = defaultdict(list)
    in_degree = [0] * (n + 1)
    completion_time = [0] * (n + 1)

    # Construct the graph and calculate in-degrees
    for u, v, w in edges:
        graph[u].append((v, w))
        in_degree[v] += 1

    # Perform topological sort
    zero_degree_queue = []
    for i in range(1, n + 1):
        if in_degree[i] == 0:
            zero_degree_queue.append(i)
    
    topo_order = []
    while zero_degree_queue:
        node = zero_degree_queue.pop(0)
        topo_order.append(node)
        for neighbor, duration in graph[node]:
            in_degree[neighbor] -= 1
            if in_degree[neighbor] == 0:
                zero_degree_queue.append(neighbor)

    # Calculate the completion time for each task
    for node in topo_order:
        for neighbor, duration in graph[node]:
            completion_time[neighbor] = max(completion_time[neighbor], completion_time[node] + duration)

    # The maximum time required to complete all tasks
    return max(completion_time)

# Example execution
if __name__ == "__main__":
    n = 5
    edges = [
        (1, 2, 3),
        (1, 3, 2),
        (2, 4, 4),
        (3, 4, 5),
        (4, 5, 1)
    ]
    result = critical_path(n, edges)
    print("Length of the critical path:", result)

Conclusion

In this lecture, we learned how to find the critical path. By using graph theory and topological sorting, we safely processed each task and calculated the time taken for each task to derive the critical path. This approach can ultimately be applied to various project management and scheduling problems.

Moreover, in practical applications, such algorithms form the basis of complex project management and will significantly aid in enhancing efficiency. I hope you continue to improve your programming skills by tackling various algorithm problems in the future.

In this blog post, we will take a closer look at how to calculate the binomial coefficient using Python.
The binomial coefficient is an important concept in combinatorics that represents the number of ways to select r elements from a given n elements.
The binomial coefficient is defined by the following formula.

Definition of Binomial Coefficient

The binomial coefficient C(n, r) is defined as follows:

C(n, r) = n! / (r! * (n - r)!)

Here, n! denotes the factorial of n, which is the product of all natural numbers from n to 1.
For example, 5! is 5 * 4 * 3 * 2 * 1 = 120. The binomial coefficient is very useful for calculating the number of combinations.

Problem Description

Write a function that calculates the binomial coefficient C(n, r) for given integers n and r.
n is a non-negative integer, and r is a non-negative integer less than or equal to n.

Input Format

The first line contains two integers n and r, separated by a space.

Output Format

Print the binomial coefficient C(n, r).

Example Input/Output

Input:
5 2
Output:
10

Solution Approach

There are several approaches to solve this problem. The most intuitive method is to directly calculate factorials using the mathematical definition.
However, calculating factorials directly can be inefficient for large n.
Therefore, we will use a dynamic programming (DP) approach to solve this problem more efficiently.

Calculating Binomial Coefficient Using Dynamic Programming

One of the properties of binomial coefficients is as follows:

C(n, r) = C(n - 1, r - 1) + C(n - 1, r)

Using the above formula, we can decompose C(n, r) into previous binomial coefficients.
This can be implemented in the form of a DP table as shown below.

How to Create a DP Table

Define dp[i][j] as C(i, j). The following rules can be applied to compute all binomial coefficients:

• dp[i][0] = 1 (for all i)
• dp[i][i] = 1 (for all i)
• Else dp[i][j] = dp[i - 1][j - 1] + dp[i - 1][j] (0 < j < i)

Python Implementation

Below is the Python code for calculating the binomial coefficient:

def binomial_coefficient(n, r):
    # Initialize 2D DP table
    dp = [[0] * (r + 1) for _ in range(n + 1)]

    # Fill the DP table
    for i in range(n + 1):
        for j in range(min(i, r) + 1):
            if j == 0 or j == i:
                dp[i][j] = 1
            else:
                dp[i][j] = dp[i - 1][j - 1] + dp[i - 1][j]

    return dp[n][r]

# Example call
n, r = map(int, input().split())
print(binomial_coefficient(n, r))

Time Complexity Analysis

The above algorithm has a time complexity of O(n * r).
Since there are two nested for loops, in the worst case, it will handle all values of n and r.
However, it can operate very quickly for small ranges of n and r.
This algorithm is a good method to effectively calculate binomial coefficients.

Conclusion

In this post, we explored various methods to compute the binomial coefficient C(n, r).
From traditional factorial-based methods to dynamic programming approaches,
we examined the advantages and disadvantages of each approach.
Since binomial coefficient problems frequently appear in coding tests,
it is important to have a good understanding of this content.

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

1. What is a Binomial Coefficient?

A binomial coefficient is defined in combinatorics for two integers n and k. It represents the number of ways to choose k items from n items, and is denoted as C(n, k) or (n choose k). The binomial coefficient is calculated as follows:

• C(n, k) = n! / (k! * (n-k)!)

Here, n! is the factorial of n, where n! = n × (n-1) × (n-2) × … × 1.
Binomial coefficients are very useful in solving combinatorial problems.

2. Problem Description

Problem: Write a function to calculate the binomial coefficient C(n, k) for given n and k.
n is an integer from 0 to 30, and k is an integer from 0 to n.

3. Problem Solving Approach

There are several methods to calculate the binomial coefficient.
We can use recursion, dynamic programming, or mathematical formulas to solve it.
Here, we will solve the problem using the Dynamic Programming approach.

3.1 Dynamic Programming

Dynamic programming is a method that divides the problem into smaller subproblems and saves the results of these subproblems for reuse in subsequent calculations. The binomial coefficient can be calculated using the following recurrence relation:

• C(n, k) = C(n-1, k) + C(n-1, k-1)
• Base cases: C(n, 0) = C(n, n) = 1

In the above recurrence relation, we can calculate the binomial coefficient recursively for each case, but this leads to redundant calculations. To avoid this, we will use a DP table for our lecture.

4. Implementation

Now, let’s write the code for a Python function to calculate the binomial coefficient.
We will implement the method using a DP table.

def binomial_coefficient(n, k):
    # Initialize DP table
    dp = [[0] * (k + 1) for _ in range(n + 1)]

    # Set base cases
    for i in range(n + 1):
        dp[i][0] = 1  # C(i, 0) = 1
        dp[i][i] = 1  # C(i, i) = 1

    # Fill the DP table according to the recurrence relation
    for i in range(1, n + 1):
        for j in range(1, min(i, k) + 1):
            dp[i][j] = dp[i - 1][j] + dp[i - 1][j - 1]

    return dp[n][k]

            

In the above code, we initialize the DP table for the given n and k, set the base cases, and then fill the DP table using the recurrence relation.
As a result, the binomial coefficient is stored in dp[n][k].

5. Testing

It’s time to test the function we implemented above. We will use a simple example like C(5, 2) to verify the accuracy of the function.

# Test
print(binomial_coefficient(5, 2))  # Output: 10
print(binomial_coefficient(30, 15))  # Output: 155117520

            

By calling the function this way, we can calculate the binomial coefficient and print the results.
Check if the results for the above input values are correct.

6. Conclusion

In this lecture, we learned about the definition and calculation methods of binomial coefficients.
We learned how to efficiently calculate binomial coefficients using dynamic programming.
We’ve also looked closely at the implementation process using Python.
This algorithm is frequently used in actual coding tests and algorithm problem solving, so
make sure to master it.

Additionally, try solving advanced problems related to binomial coefficients to further improve your skills.
Thank you.