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Python Coding Test Course, Exploring Dynamic Programming

1. What is Dynamic Programming?

Dynamic Programming (DP) is an algorithmic approach to solving computational problems by breaking down complex problems into simpler subproblems. Generally, it improves performance by remembering the results of subproblems through a recursive approach (memoization technique), preventing repeated calculations.

Dynamic programming is primarily used for solving optimization problems and is effective in finding the optimal solution for a given problem. Many problems can be solved using dynamic programming, with Fibonacci sequences, the longest common subsequence, and the minimum edit distance problem being representative examples.

2. Applied Problem: Maximum Subarray Sum

Problem Description: This problem involves finding the maximum sum of a subarray within a given integer array. A subarray is formed by selecting contiguous elements from the array. For example, in the array [−2,1,−3,4,−1,2,1,−5,4], the maximum sum of a subarray is 6. (This is the sum of [4,−1,2,1].)

2.1 Problem Approach

This problem can be solved using dynamic programming. By iterating through each element of the array, we calculate the maximum sum that includes the current element. We compare the case where the current element is included and where it is not, selecting the larger value. We determine the maximum subarray sum for the current element by utilizing the maximum subarray sum of the previous elements.

3. Problem Solving Process

3.1 Define Variables

First, we will define the following variables:

  • nums: Given integer array
  • max_sum: Maximum subarray sum so far
  • current_sum: Sum of the subarray up to the current position

3.2 Define the Recurrence Relation

The recurrence relation can be defined as follows:

current_sum = max(nums[i], current_sum + nums[i])

Where nums[i] is the current element. We will select the maximum value between the sum that includes the current element and the sum that does not. We then update max_sum each time.

3.3 Initialization and Loop

After initialization, we write a loop to iterate through each element and calculate the maximum sum of the subarray.


def max_sub_array(nums):
    max_sum = nums[0]
    current_sum = nums[0]

    for i in range(1, len(nums)):
        current_sum = max(nums[i], current_sum + nums[i])
        max_sum = max(max_sum, current_sum)

    return max_sum

In the code above, the first element of the array is set as the initial value, and the max_sub_array function is performed repeatedly starting from the second element.

3.4 Code Explanation

Let’s go through the code line by line:

  • max_sum = nums[0]: Initializes the maximum subarray sum to the first element.
  • current_sum = nums[0]: Initializes the current subarray sum to the first element.
  • for i in range(1, len(nums)):: Iterates over the elements starting from the second element of the array.
  • current_sum = max(nums[i], current_sum + nums[i]): Updates the current_sum.
  • max_sum = max(max_sum, current_sum): Updates the max_sum.

3.5 Execution Result

print(max_sub_array([-2,1,-3,4,-1,2,1,-5,4])) # 6

Running the above code will output the maximum subarray sum 6.

4. Techniques of Dynamic Programming

4.1 Memoization and Bottom-Up Approach

Dynamic programming is typically divided into two main techniques:

  • Memoization: A method that saves the results of subproblems to reduce unnecessary calculations. It uses recursive calls, checking for already computed results in each function call.
  • Bottom-Up: A method that systematically solves smaller subproblems before progressing to larger ones. It is generally implemented using loops, which can reduce memory usage.

These techniques can be used to solve a variety of problems.

5. Conclusion

Dynamic programming is a very useful algorithmic technique for solving various optimization problems. Through the maximum subarray sum problem discussed in this lecture, we have learned the fundamental concepts of dynamic programming and methods for problem-solving. This can be applied to various algorithmic problem-solving and is a frequently covered topic in coding tests.

Additionally, I encourage you to practice various problems to deepen your understanding of dynamic programming. This will enhance your algorithmic thinking and help you achieve good results in coding tests.

python coding test course, Dijkstra

Hello, everyone! Today, we will take a deep dive into the Dijkstra’s Algorithm and solve problems that are frequently asked in coding tests. Dijkstra’s Algorithm is an algorithm used to find the shortest path in a weighted graph, primarily used to solve network shortest path problems.

1. Overview of Dijkstra’s Algorithm

Dijkstra’s Algorithm, developed by Edsger Dijkstra in 1956, is a shortest path algorithm used to find the shortest path from a specific vertex to all other vertices. This algorithm does not allow negative weights and assumes that the graph is connected.

1.1 Algorithm Operating Principle

  • Select a starting vertex and set its distance to 0.
  • Initialize the distance of all other vertices to infinity.
  • Select the vertex with the shortest distance among the vertices whose minimum distance has not yet been determined.
  • Update the distances of adjacent vertices based on the selected vertex.
  • Repeat until the distances of all vertices are determined.

2. Problem Description

Now let’s apply Dijkstra’s Algorithm through a real problem.

Problem: Find the Shortest Path

Write a program to find the shortest path from a specific starting vertex to all other vertices given a graph provided as input.

Input Conditions

  • The first line contains the number of vertices V and the number of edges E. (1 ≤ V ≤ 20, 1 ≤ E ≤ 100)
  • From the second line onwards, information about each edge is provided over E lines. Edge information is given in the form of the starting vertex A, ending vertex B, and weight C.
  • The last line contains the starting vertex K.

Output Conditions

Output the shortest distance from the starting vertex K to all other vertices. If there is no path to any vertex, output INF.

3. Problem Solving Process

3.1 Graph Representation

First, we represent the graph as an adjacency list based on the given edge information. In Python, we can use a dictionary to store each vertex and the other vertices connected to it along with their weights.

from collections import defaultdict
import heapq

def create_graph(edges):
    graph = defaultdict(list)
    for A, B, C in edges:
        graph[A].append((B, C))
    return graph

3.2 Implementing Dijkstra’s Algorithm

Now we will implement Dijkstra’s Algorithm. This algorithm efficiently updates the shortest path found so far using a priority queue.

def dijkstra(graph, start, V):
    distances = {vertex: float('inf') for vertex in range(1, V + 1)}
    distances[start] = 0
    priority_queue = [(0, start)]

    while priority_queue:
        current_distance, current_vertex = heapq.heappop(priority_queue)

        if current_distance > distances[current_vertex]:
            continue

        for neighbor, weight in graph[current_vertex]:
            distance = current_distance + weight

            if distance < distances[neighbor]:
                distances[neighbor] = distance
                heapq.heappush(priority_queue, (distance, neighbor))

    return distances

3.3 Final Code

We compile all elements to write the final code, which also includes the input handling and result processing parts.

def main():
    V, E = map(int, input().split())
    edges = [tuple(map(int, input().split())) for _ in range(E)]
    K = int(input())

    graph = create_graph(edges)
    distances = dijkstra(graph, K, V)

    for vertex in range(1, V + 1):
        if distances[vertex] == float('inf'):
            print("INF")
        else:
            print(distances[vertex])

if __name__ == "__main__":
    main()

4. Time Complexity

The time complexity of Dijkstra’s Algorithm varies depending on the data structures used. Typically, when using a priority queue, the time complexity is O((V + E) log V), where V is the number of vertices and E is the number of edges.

5. Conclusion

In this lecture, we learned how to solve the shortest path problem using Dijkstra’s Algorithm and handled real input. This algorithm can be utilized in various fields such as networking and game development, so it’s good to have a firm grasp of it.

Practice various scenarios through additional problems. In the next lecture, we will look at another algorithm. Thank you!

Python coding test course, building bridges

Hello everyone! In this post, we will explore how to solve algorithm problems frequently encountered in coding tests using Python. The topic is ‘Building Bridges’. The ‘Building Bridges’ problem is a classic problem that actually has various variations, including important concepts related to graph theory.

Problem Description

There are several given islands. Each island is located in different positions, and we want to build bridges between the islands. At this time, bridges can only be constructed vertically or horizontally, and the bridges must stretch out in a straight line. Additionally, we need to minimize the total length of the bridges so that as many islands as possible can be connected.

Write a function that outputs the minimum total length of the bridges when the coordinates of the given islands are provided as (x, y).

def minimum_bridge_length(islands: List[Tuple[int, int]]) -> int:
    pass

Input

The number of islands N (2 ≤ N ≤ 10,000) and a list of coordinates (x, y) for each island will be provided.

Output

Return the minimum total length of the bridges as an integer.

Problem Solving Process

1. Understanding the Problem

The goal of the problem is to find the minimum total length of bridges connecting the given islands. The key to this problem is understanding how the bridge length between each pair of islands is calculated. If there are coordinates (x1, y1) and (x2, y2) for two islands, the bridge length between them can be calculated as |x1 – x2| + |y1 – y2|. In other words, it is the sum of the differences in the x-coordinates and y-coordinates of the islands.

2. Algorithm Design

We can solve the problem by considering all combinations of the islands to build bridges. However, since N can be as large as 10,000, considering all combinations would lead to an O(N^2) time complexity, which is inefficient. Instead, we can use a Minimum Spanning Tree (MST) algorithm to connect the bridges minimally.

To construct the MST, we can use either Kruskal’s or Prim’s algorithm. Here, we will use Prim’s algorithm. This algorithm starts from one vertex of the graph and builds the MST by adding the shortest edge one at a time.

3. Implementing Prim’s Algorithm

First, we prepare a list to store the coordinates of each island and a priority queue to store the connection costs. Then, starting from one island, we calculate the lengths of the bridges to all connected islands and repeat the process of adding the island with the shortest length.


import heapq
from typing import List, Tuple

def minimum_bridge_length(islands: List[Tuple[int, int]]) -> int:
    n = len(islands)
    visited = [False] * n
    min_heap = []
    
    # Start from the first island
    for i in range(1, n):
        dist = abs(islands[0][0] - islands[i][0]) + abs(islands[0][1] - islands[i][1])
        heapq.heappush(min_heap, (dist, i))
    
    visited[0] = True
    total_length = 0
    edges_used = 0
    
    while min_heap and edges_used < n - 1:
        dist, island_index = heapq.heappop(min_heap)
        
        if visited[island_index]:
            continue
        
        visited[island_index] = True
        total_length += dist
        edges_used += 1
        
        for i in range(n):
            if not visited[i]:
                new_dist = abs(islands[island_index][0] - islands[i][0]) + abs(islands[island_index][1] - islands[i][1])
                heapq.heappush(min_heap, (new_dist, i))
    
    return total_length

4. Complexity Analysis

The time complexity of this algorithm is O(E log V), where E is the number of edges and V is the number of vertices. However, since we do not consider all combinations and generate bridges based on each island, the actual performance is better.

5. Example

Now, let's solve an example using this algorithm.


islands = [(0, 0), (1, 1), (3, 1), (4, 0)]
print(minimum_bridge_length(islands))  # Output: 4

In the above example, we can connect the bridge from (0, 0) to (1, 1), from (1, 1) to (3, 1), and from (3, 1) to (4, 0). The total length of the bridges becomes 4.

Conclusion

In this tutorial, we examined the 'Building Bridges' problem using Python. There are various ways to solve algorithmic problems, and these are very useful skills for preparing for coding tests. I hope you continue to practice based on a deep understanding of the data structures and algorithms used in solving problems. If you need more problems, I will continue to provide updates.

Thank you!

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.

python coding test course, calculating the area of a polygon

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