Author: root

Problem Description

There are N islands given, and plans are in place to build bridges between these islands to connect them. However, bridges can only be built once between each pair of islands, and the length of the bridge is calculated based on the shortest distance between the two islands.
Additionally, in order to build a bridge, the height of each island must be considered according to their terrain.
It is necessary to consider how the height difference between the two islands affects the bridge length.

The problem is to calculate the minimum bridge length required to connect all the islands based on the given height information of the islands.
The length of the bridge is proportional to the height difference between the two islands.
Write a program that finds the minimum bridge length required to connect all islands by building bridges for all possible pairs based on the given heights of the islands.

Input Format

The first line contains the number of islands N (2 ≤ N ≤ 100).
The second line provides N integers separated by spaces, representing the height H (0 ≤ H ≤ 100) of each island.

Output Format

Output the minimum bridge length required to connect all the islands.

Problem Solving Process

1. Understanding the Problem and Designing an Approach

To solve the bridge construction problem, we will approach it through the Minimum Spanning Tree (MST) of a graph. This problem requires calculating the bridge lengths between each pair of islands to connect all islands and minimizing this total,
hence we can use either Kruskal’s algorithm or Prim’s algorithm.
As negative weights are not allowed, when building a bridge between A and B, we define the bridge length as the height difference between A and B, resulting in separate edges in the graph based on the height of the islands.

2. Graph Creation

The edges between each pair of islands correspond to the height difference of the two islands, and the length of the bridge is defined as |H[i] – H[j]|.
We can now calculate the bridge lengths between all pairs of islands to construct the graph.
To achieve this, we will use a nested loop to calculate the bridge lengths for all pairs of islands.

3. Applying the MST Algorithm

Now we will use Kruskal’s algorithm to calculate the minimum bridge length required to connect all islands.
The Kruskal algorithm constructs the MST by first sorting the edges by weight and then adding the edges one by one to avoid forming cycles.

4. Implementation and Validation

Below is the code implemented in Kotlin as described above.
This code takes input to calculate bridge lengths and ultimately outputs the minimum bridge length required to connect all the islands.

        
        import java.util.*

        fun main() {
            val scanner = Scanner(System.`in`)
            val n = scanner.nextInt()
            val heights = IntArray(n)
            for (i in 0 until n) {
                heights[i] = scanner.nextInt()
            }

            val edges = mutableListOf()
            for (i in 0 until n) {
                for (j in i + 1 until n) {
                    val weight = Math.abs(heights[i] - heights[j])
                    edges.add(Edge(i, j, weight))
                }
            }

            edges.sortBy { it.weight }

            val parent = IntArray(n) { it }

            fun find(x: Int): Int {
                if (parent[x] != x) {
                    parent[x] = find(parent[x])
                }
                return parent[x]
            }

            fun union(x: Int, y: Int) {
                val rootX = find(x)
                val rootY = find(y)
                if (rootX != rootY) {
                    parent[rootY] = rootX
                }
            }

            var totalLength = 0
            for (edge in edges) {
                if (find(edge.start) != find(edge.end)) {
                    union(edge.start, edge.end)
                    totalLength += edge.weight
                }
            }

            println("The minimum bridge length required to connect all the islands is: $totalLength")
        }

        data class Edge(val start: Int, val end: Int, val weight: Int)
        
    

Optimization and Considerations

The bridge construction problem can be solved using simple graph and MST algorithms.
However, this algorithm may require relatively high memory usage because it stores all edges.
Therefore, if the height information of each island and the connection information are stored in a more optimized data structure,
better performance can be achieved.

Moreover, due to the nature of the graph, this problem can also work well with more islands or more complex shapes.
In actual coding tests, having a fundamental understanding and grasping the algorithms needed to solve such problems is crucial.
It is also beneficial to practice and familiarize oneself with the patterns of similar problems.

Conclusion

Through this problem, I was able to learn about the use of data structures and algorithms in Kotlin.
The Minimum Spanning Tree problem is one of the topics in coding tests, and it’s important to repeatedly study various modified problems.
A deep understanding of graph theory will be a great help in coding tests.

In today’s lecture, we will explain the problem of calculating the area of a polygon. This problem is one of the frequently asked questions in coding tests, and we will learn how to implement an algorithm to calculate the area when given various shapes of polygons.

Problem Description

Write an algorithm to calculate the area of a polygon formed by given multiple points. The points are represented as coordinates on a 2D plane, and it is assumed that the points are given in either clockwise or counterclockwise order.

Example Input

    4
    0 0
    4 0
    4 3
    0 4
    

Example Output

    14.0
    

Here, the area is determined by the coordinates of the vertices that make up the polygon. In the above example, the given points are the vertices of a polygon consisting of (0,0), (4,0), (4,3), and (0,4). In this case, the area of the polygon is 14.0.

Algorithm Explanation

There are various methods to calculate the area of a polygon, but one commonly used method is the ‘Shoelace Formula’. Using this formula, the area can be calculated with a time complexity of O(n).

Shoelace Formula

The Shoelace Formula is expressed as follows:

A = 0.5 * | ∑(xiyi+1 - xi+1yi) |

Here, x_i and y_i are the x and y coordinates of the i-th vertex of the polygon. When the polygon has n vertices, i increments from 0 to n-1. The last vertex closes back to the first vertex (i.e., when increasing i+1, it wraps around to n).

Problem Solving Process

1. Processing Input Data

First, we need to read the number of vertices and the vertex coordinates of the polygon provided as input. In Kotlin, we can use a list to store the coordinates.

In coding tests, a deep understanding of algorithms and data structures is required. In this course, we will explore the features and useful functions of Kotlin while solving an algorithm problem called “Sorting Digits in Descending Order”.

Problem Description

Reconstruct the number by sorting the digits of a given integer N in descending order. Return the number after sorting the digits. The integer N is between 0 and 2³¹-1.

For example:

• Input: 118372 → Output: 873211
• Input: 2143 → Output: 4321
• Input: 0 → Output: 0

Solution Process

Step 1: Understanding the Problem

The first step in solving the problem is to accurately understand the given problem. The input is in the form of an integer, and what we need to do is separate this integer into its digits and sort them in descending order. After sorting the digits, we need to combine them back into an integer and return it.

Step 2: Extracting Each Digit

We can convert the integer into a string to easily access each digit. Each character in the string represents a digit of the original integer, and we can use this to extract the digits.

Step 3: Sorting the Digits

To sort the digits, we can use Kotlin’s built-in function. We can use the sortedDescending() function to sort the digits in descending order.

Step 4: Combining the Sorted Digits

We need to convert the sorted digits back into an integer as our final result. We concatenate each digit to form a single string and then convert it back into an integer to return it.

Step 5: Implementing the Final Code

fun solution(N: Int): Int {
    // Convert integer N to string
    val strN = N.toString()
    
    // Sort the digits in descending order
    val sortedDigits = strN.toCharArray().sortedDescending()
    
    // Combine the sorted digits back into a single string
    val resultStr = sortedDigits.joinToString("")
    
    // Convert string to integer and return
    return resultStr.toInt()
}
            

Example Test

To test the solution we have written, we will execute each test case.

fun main() {
    println(solution(118372)) // 873211
    println(solution(2143))   // 4321
    println(solution(0))      // 0
}
            

When the above code is executed, each integer is expected to be printed in descending order as intended.

Time Complexity and Space Complexity

The time complexity of the algorithm used here is O(d log d), where d is the number of digits. There is a logarithmic time complexity due to the sorting of the digits. The space complexity is O(d), as an array is used to store the digits.

Conclusion

Through this course, we learned how to leverage Kotlin’s powerful features and concise syntax for solving algorithm problems. The digit sorting problem serves as a good starting point that can be extended to other algorithm problems. It is important to practice solving more problems to familiarize oneself with various algorithms and data structures.

I hope this helps you in preparing for coding tests. Stay tuned for the next course!

1. Problem Description

This problem is to find the shortest path between two nodes in a given graph. The graph is presented in the form of an adjacency list, providing the number of nodes and the information of nodes connected to each node. Your goal is to output the shortest path from a specific start node to a specific target node.

Problem

Input:
- First line: Number of vertices N (1 ≤ N ≤ 1000)
- Second line: Number of edges M (0 ≤ M ≤ 10000)
- Next M lines: Edge information (A B) - A and B represent the two connected nodes

- The last line gives the start node S and the target node T.

Output:
- Print the list of nodes in the path from S to T. If there is no path, print "PATH NOT FOUND!".

2. Problem Approach

This problem can be solved using a typical BFS algorithm. BFS is a breadth-first search method that explores nodes close to the start node first. This allows for finding the shortest path. The main characteristics of the BFS algorithm are as follows:

• It explores all nodes at the same depth, guaranteeing the shortest path.
• Implemented using a queue, it theoretically has a time complexity of O(V + E), where V is the number of nodes and E is the number of edges.

BFS Algorithm Steps

• Initialization: Mark the start node as visited and initialize distances. Insert the start node into the queue.
• Exploration Process: Repeat while the queue is not empty. Remove a node from the queue and add unvisited adjacent nodes to the queue.
• Path Tracking: Record the parent node of each node to allow for path tracking later.

3. Kotlin Code Implementation

Now, let’s implement BFS in Kotlin according to the approach above. Refer to the code below to see how to solve the given problem.

import java.util.*

data class Edge(val to: Int, val weight: Int)

fun bfs(graph: List>, start: Int, target: Int): List? {
    val queue: Queue = LinkedList()
    val visited = BooleanArray(graph.size)
    val parent = IntArray(graph.size) { -1 }  // Track parent nodes
    queue.add(start)
    visited[start] = true

    while (queue.isNotEmpty()) {
        val current = queue.poll()

        // Trace back the path if the target node is found
        if (current == target) {
            return tracePath(parent, start, target)
        }

        for (edge in graph[current]) {
            if (!visited[edge.to]) {
                visited[edge.to] = true
                parent[edge.to] = current // Set parent node
                queue.add(edge.to)
            }
        }
    }
    return null // No path found
}

fun tracePath(parent: IntArray, start: Int, target: Int): List {
    val path = mutableListOf()
    var current = target

    while (current != start) {
        path.add(current)
        current = parent[current]
    }
    path.add(start)
    path.reverse()  // Reverse the list as it was added in reverse order
    return path
}

fun main() {
    val scanner = Scanner(System.`in`)
    val n = scanner.nextInt()
    val m = scanner.nextInt()
    val graph = List(n) { mutableListOf() }

    // Create the graph
    for (i in 0 until m) {
        val a = scanner.nextInt() - 1
        val b = scanner.nextInt() - 1
        graph[a].add(Edge(b, 1)) // Undirected graph, so add both ways
        graph[b].add(Edge(a, 1))
    }

    val start = scanner.nextInt() - 1
    val target = scanner.nextInt() - 1
    val result = bfs(graph, start, target)

    if (result != null) {
        println(result.joinToString(" "))
    } else {
        println("PATH NOT FOUND!")
    }
}

4. Example Input and Output

Example 1

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

Output:
1 3 5

Example 2

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

Output:
PATH NOT FOUND!

5. Conclusion

In this tutorial, we addressed the problem of finding the shortest path between two nodes in a graph using breadth-first search (BFS). Due to the ease of exploration provided by BFS, it is widely useful in many algorithmic problems. It is essential to develop the ability to effectively utilize BFS to solve problems in actual exams or coding tests.

Now, we encourage you to tackle various problems using BFS. Enhance your understanding of algorithms through implementation in Kotlin, and develop the skills that can also be applied in real-world scenarios!