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In this article we will write a go language program to find the diameter of a graph. The diameter of a graph is the maximum distance between any two vertices in the graph. There are several Algorithms that can be used to find the diameter of a graph, including Dijkstra’s Algorithm, Floyd-Warshall Algorithm, and Breadth-First Search Algorithm. Since, dijkastra Algorithm finds the shortest distance between source vertex and other vertices. We can also use it to find the largest distance by comparing the length of vertices recieved.

Syntax func len(v Type) int

The len() function is used to get the length of a any parameter. It takes one parameter as the data type variable whose length we wish to find and returns the integer value which is the length of the variable.

Algorithm

Step 1 − First, we need to import the fmt and math packages. Then Define the edge struct which represents an edge in a graph with a to vertex and a weight.

Step 2 − Define the priorityQueue type as a slice of edge structs.

Step 3 − Define the dijkstra function which takes a graph represented as a 2D slice of edge structs and a starting vertex as arguments and returns the shortest distances from the starting vertex to all other vertices using Dijkstra’s Algorithm.

Step 4 − Initialize a dist slice to store the shortest distances, with each element set to the maximum integer value.

Step 5 − Set the distance of the starting vertex to 0. Create a priority queue pq and push the starting vertex with a weight of 0 onto the queue.

Step 6 − If the distance to the neighbor through the popped vertex is less than the current shortest distance to the neighbor, update the distance and push the neighbor onto the priority queue.

Step 7 − Return the dist slice with the shortest distances to all vertices.

Step 8 − Define the getDiameter function and Initialize the start variable to the first vertex in the graph.

Step 9 − Calculate the shortest distances from the end vertex to all other vertices using dijkstra. Initialize the diameter variable to 0.

Step 10 − If the distance is greater than the current diameter and less than the maximum integer value, update the diameter. Return the diameter.

Step 11 − Now, start the main() function and define the edges. Further call the getDiameter() function by passing the edges as arguments and store the result obtained in a variable.

Step 12 − Print the result obtained on the screen

Example

In this Example we will write a go language program to find the diameter of a graph by using the dijkastra Algorithm. Dijkastra Algorithm is used to find the shortest path between vertices in a graph.

package main import ( "container/heap" "fmt" "math" ) type edge struct { to int weight int } type priorityQueue []edge func (pq priorityQueue) Len() int { return len(pq) } func (pq priorityQueue) Less(i, j int) bool { return pq[i].weight < pq[j].weight } func (pq priorityQueue) Swap(i, j int) { pq[i], pq[j] = pq[j], pq[i] } func (pq *priorityQueue) Push(x interface{}) { *pq = append(*pq, x.(edge)) } func (pq *priorityQueue) Pop() interface{} { old := *pq n := len(old) x := old[n-1] *pq = old[0 : n-1] return x } func dijkstra(graph [][]edge, start int) []int { dist := make([]int, len(graph)) for i := range dist { dist[i] = math.MaxInt32 } dist[start] = 0 pq := make(priorityQueue, 0) heap.Push(&pq, edge{to: start, weight: 0}) node := heap.Pop(&pq).(edge) if dist[node.to] < node.weight { continue } for _, e := range graph[node.to] { if nd := node.weight + e.weight; nd < dist[e.to] { dist[e.to] = nd heap.Push(&pq, edge{to: chúng tôi weight: nd}) } } } return dist } func getDiameter(graph [][]edge) int { var start int for i := range graph { start = i break } dist := dijkstra(graph, start) var end int for i, d := range dist { if d < math.MaxInt32 && dist[end] < d { end = i } } dist = dijkstra(graph, end) var diameter int for _, d := range dist { diameter = d } } return diameter } func main() { graph := [][]edge{ {{1, 5}, {2, 3}}, {{1, 5}, {2, 1}, {3, 6}, {4, 8}}, {{1, 3}, {1, 1}, {4, 7}}, {{1, 6}, {4, 9}}, {{1, 8}, {2, 7}, {3, 9}}, } fmt.Println("The given vertices are:", graph) diameter := getDiameter(graph) fmt.Println("Diameter obtained is:", diameter) } Output The given vertices are: [[{1 5} {2 3}] [{1 5} {2 1} {3 6} {4 8}] [{1 3} {1 1} {4 7}] [{1 6} {4 9}] [{1 8} {2 7} {3 9}]] Diameter obtained is: 9 Conclusion

In this article, we discussed method to find the diameter of a graph using the Dijkstra Algorithm in Go. This method is used to obtain the smallest length of vertices from the graph. To calculate the diameter we are comparing the distance of each vertex with the distance of the current value of diameter. If the distance is greater than the diameter then we need to update that value and continue till we get the largest distance which is actually the diameter of the circle.

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Swift Program To Find The Area Of The Rectangle

This tutorial will discuss how to write a Swift program to find the area of the rectangle.

Formula for Area of Rectangle

Following is the formula for the area of the rectangle −

Area = Length(L) * Width(W)

Below is a demonstration of the same −

Input

Suppose our given input is −

Length = 10 Width = 5

Output

The desired output would be −

Area = 10 * 5 = 50. Area = 50 Algorithm

Following is the algorithm −

Step 1 − Declare two variables of integer type to store the value of the length and width of the rectangle. Here the value of these variables can be pre-defined or user-defined.

Step 2 − Declare an Area variable to store the area of the rectangle.

var RectArea = RectLength * RectWidth

Step 3 − Print the output.

Example 1

The following program shows how to find the area of the rectangle.

import

Glibc

var

RectLength

:

Int

=

20

var

RectWidth

:

Int

=

10

var

RectArea

=

RectLength

*

RectWidth

print

(

“Length – “

,

RectLength

)

print

(

“Width – “

,

RectWidth

)

print

(

“So, Area of the rectangle- “

,

RectArea

)

Output Length - 20 Width - 10 So, Area of the rectangle- 200

Here, in the above code, we find the area of the rectangle by multiplying the given length and width of the rectangle.

var RectArea = RectLength * RectWidth

That is ReactArea = 20 * 10 = 200. And display the area that is 200

Example 2

The following program shows how to find the area of the rectangle with user-defined inputs.

import

Foundation

import

Glibc

print

(

"Please enter the length - "

)

var

RectLength

=

Int

(

(

)

!

)

!

print

(

"Please enter the width - "

)

var

RectWidth

=

Int

(

(

)

!

)

!

var

RectArea

=

RectLength

*

RectWidth

print

(

"Entered Length - "

,

RectLength

)

print

(

"Entered Width - "

,

RectWidth

)

print

(

"So, Area of the rectangle- "

,

RectArea

)

STDIN Input Please enter the length - 6 Please enter the width - 3 Output Entered Length - 6 Entered Width - 3 So, Area of the rectangle - 18

Here, in the above code, we find the area of the rectangle by multiplying the given length and width of the rectangle.

var RectArea = RectLength * RectW idth

Here we take the value of RectLength and RectWidth from the user using readLine() function and convert the entered value in the integer type using Int() function. So ReactArea = 6 * 3 = 18, And display the result that is 18.

Area of a rectangle in terms of diagonals

We can also calculate the area of the rectangle with the help of diagonal and width. A rectangle has two diagonals and both are equal.

Formula for Area of Rectangle using Diagonals

Following is the formula for the area of rectangle in terms of diagonals

Area = Width(√ (Diagonal)2 - (Width)2) Example

The following program shows how to find the area of the rectangle using diagonals.

import

Glibc

var

RectDiagonal

=

30.0

var

RectWidth

=

10.0

var

RectArea

=

RectWidth

*

(

sqrt

(

pow

(

RectDiagonal

,

2

)

pow

(

RectWidth

,

2

)

)

)

print

(

“Diagonal – “

,

RectDiagonal

)

print

(

“Width – “

,

RectWidth

)

print

(

“So, Area of the rectangle- “

,

RectArea

)

Output Diagonal - 30.0 Width - 10.0 So, Area of the rectangle- 282.842712474619

Here in the above code, we find the area of the rectangle with the help of diagonal and width using the formula −

var RectArea = RectWidth * (sqrt(pow(RectDiagonal, 2) - pow(RectWidth, 2)))

Here we find the square root using sqrt() function and pow() function is used to find the power. So the working of the above code is

ReactArea = 10.0 * (sqrt(pow(30.0, 2) – pow(10.0, 2))) = 10.0 * (sqrt(900 – 100)) = 10.0 * (sqrt(800)) = 10.0 * (28.2842712474619) = 282.842712474619

Java Program To Compute The Area Of A Triangle Using Determinants

Introduction

The Java program to compute the area of a triangle using determinants is a concise and efficient program that calculates the area of a triangle given the coordinates of its three vertices.

This program is useful for anyone studying or working with geometry, as it demonstrates how to use basic arithmetic and algebraic calculations in Java, and how to read in user input using the Scanner class. The program prompts the user to enter the coordinates of three points of the triangle, which are then read in and used to calculate the determinant of the matrix of coordinates. The absolute value of the determinant is used to ensure that the area is always positive, and the area of the triangle is then calculated using the formula and displayed to the user. This program can be easily modified to accept input in different formats or to perform additional calculations, making it a versatile tool for geometry calculations.

Determinants

Determinants are a mathematical concept that are used to determine certain properties of a matrix. In linear algebra, a determinant is a scalar value that can be calculated from the elements of a square matrix. The determinant can be used to determine whether a matrix has an inverse, whether a system of linear equations has a unique solution, and the area or volume of a parallelogram or parallelepiped.

Syntax Algorithm

Import the Scanner class.

Define a public class named TriangleArea.

Inside the class, define a main method.

Create a Scanner object to read user input.

Prompt the user to enter the coordinates of three points separated by a space.

Read the user input for the coordinates and store them in six double variables (x1, y1, x2, y2, x3, y3).

Calculate the determinant of the matrix of coordinates using the formula −

| x2 y2 1 | = x1*y2 + x2*y3 + x3*y1 – y1*x2 – y2*x3 – y3*x1 | x3 y3 1 |

Then we calculate the area of the triangle using the formula −

Example 1 Approach

First, we prompt the user to enter the coordinates of three points of the triangle.

We use the Scanner class to read in the user input for the coordinates and store them in six double variables (x1, y1, x2, y2, x3, y3).

Next, we calculate the determinant of the matrix of coordinates using the formula −

| x2 y2 1 | = x1*y2 + x2*y3 + x3*y1 – y1*x2 – y2*x3 – y3*x1 | x3 y3 1 |

Then we calculate the area of the triangle using the formula −

Here is a Java program to compute the area of a triangle using determinants −

import java.util.Scanner; public class TriangleArea { public static void main(String[] args) { Scanner scanner = new Scanner(System.in); System.out.println("Enter the coordinates of three points separated by a space:"); double x1 = scanner.nextDouble(); double y1 = scanner.nextDouble(); double x2 = scanner.nextDouble(); double y2 = scanner.nextDouble(); double x3 = scanner.nextDouble(); double y3 = scanner.nextDouble(); double determinant = x1 * y2 + x2 * y3 + x3 * y1 - y1 * x2 - y2 * x3 - y3 * x1; double area = Math.abs(determinant / 2); System.out.println("The area of the triangle is " + area); } } Explanation

Note that the Math.abs() function is used to ensure that the area is always positive, since the determinant can be negative if the vertices are listed in a counterclockwise order.

Output Enter the coordinates of three points separated by a space: 4 3 2 6 7 4 The area of the triangle is 5.5 Example 2

This approach works for any triangle, regardless of its orientation or size. The program assumes that the user enters valid numeric coordinates for the three points, otherwise, it may throw an exception if the input is not valid.

Here is a Java program to compute the area of a triangle using determinants −

import java.util.Scanner; public class TriangleArea { public static void main(String[] args) { Scanner sc = new Scanner(System.in); System.out.print("Enter the coordinates of the first point: "); double x1 = sc.nextDouble(); double y1 = sc.nextDouble(); System.out.print("Enter the coordinates of the second point: "); double x2 = sc.nextDouble(); double y2 = sc.nextDouble(); System.out.print("Enter the coordinates of the third point: "); double x3 = sc.nextDouble(); double y3 = sc.nextDouble(); double area = calculateTriangleArea(x1, y1, x2, y2, x3, y3); System.out.println("The area of the triangle is " + area); } public static double calculateTriangleArea(double x1, double y1, double x2, double y2, double x3, double y3) { double determinant = x1 * (y2 - y3) + x2 * (y3 - y1) + x3 * (y1 - y2); return Math.abs(determinant) / 2.0; } } Explanation

This program prompts the user to enter the coordinates of three points that form a triangle, and then uses the calculateTriangleArea() method to compute the area of the triangle using the determinant formula. Finally, it prints the computed area to the console.

Output Enter the coordinates of the first point: 0 0 Enter the coordinates of the second point: 4 0 Enter the coordinates of the third point: 0 3 The area of the triangle is 6.0 Conclusion

The Java program to compute the area of a triangle using determinants is a simple and efficient way to calculate the area of a triangle given its coordinates. The program uses basic arithmetic and algebraic calculations to determine the determinant of a matrix of coordinates, and then uses this determinant to calculate the area of the triangle using a simple formula. The program demonstrates the use of the Scanner class for user input, the Math class for mathematical operations, and the use of methods for code organization and modularity.

The time complexity of the program is constant time, which means that it performs a fixed number of operations regardless of the size of the input. This makes it a fast and efficient program for calculating the area of a triangle. The space complexity of the program is also constant, as it only uses a fixed amount of memory to store its variables and does not require any additional memory allocation.

C++ Program To Construct Transitive Closure Using Warshall’s Algorithm

If a directed graph is given, determine if a vertex j is reachable from another vertex i for all vertex pairs (i, j) in the given graph. Reachable mean that there is a path from vertex i to j. This reach-ability matrix is called transitive closure of a graph. Warshall algorithm is commonly used to find the Transitive Closure of a given graph G. Here is a C++ program to implement this algorithm.

Algorithm Begin    1.Take maximum number of nodes as input.    2.For Label the nodes as a, b, c …..    3.To check if there any edge present between the nodes make a for loop:       for i = 97 to less than 97 + number of nodes          for j = 97 to less than 97 + number of nodes             if edge is present do,                adj[i - 97][j - 97] = 1             else                adj[i - 97][j - 97] = 0             end loop       end loop.    4.To print the transitive closure of graph:       for i = 0 to number of nodes          c = 97 + i       end loop.       for i = 0 to number of nodes          c = 97 + i;       for j = 0 to n_nodes          print adj[I][j]       end loop    end loop using namespace std; const int n_nodes = 20; int main() {    int n_nodes, k, n;    char i, j, res, c;    int adj[10][10], path[10][10];    cout << "ntMaximum number of nodes in the graph :";    n_nodes = n;    cout << "nEnter 'y' for 'YES' and 'n' for 'NO' n";    for (i = 97; i < 97 + n_nodes; i++)       for (j = 97; j < 97 + n_nodes; j++) {          cout << "ntIs there an edge from " << i << " to " << j << " ? ";          if (res == 'y')             adj[i - 97][j - 97] = 1;          else             adj[i - 97][j - 97] = 0;       }       cout << "nTransitive Closure of the Graph:n";       cout << "nttt ";       for (i = 0; i < n_nodes; i++) {          c = 97 + i;          cout << c << " ";       }       cout << "nn";       for (int i = 0; i < n_nodes; i++) {          c = 97 + i;          cout << "ttt" << c << " ";          for (int j = 0; j < n_nodes; j++)             cout << adj[i][j] << " ";             cout << "n";       }       return 0; } Output Maximum number of nodes in the graph :4 Enter 'y' for 'YES' and 'n' for 'NO' Is there an edge from a to a ? y Is there an edge from a to b ?y Is there an edge from a to c ? n Is there an edge from a to d ? n Is there an edge from b to a ? y Is there an edge from b to b ? n Is there an edge from b to c ? y Is there an edge from b to d ? n Is there an edge from c to a ? y Is there an edge from c to b ? n Is there an edge from c to c ? n Is there an edge from c to d ? n Is there an edge from d to a ? y Is there an edge from d to b ? n Is there an edge from d to c ? y Is there an edge from d to d ? n Transitive Closure of the Graph: a b c d a 1 1 0 0 b 1 0 1 0 c 1 0 0 0 d 1 0 1 0

Golang Program To Cyclically Permutes The Elements Of The Array

Cyclically means that in every cycle we wish to move the array elements by one position till we get the original array again. Cyclic permutation can be useful in various matrix manipulation and linear algebra operations. In this article, we are going to see different examples to permute the elements of the array cyclically using go programming language.

Syntax func append(slice, element_1, element_2…, element_N) []T

The append function is used to add values to an array slice. It takes number of arguments. The first argument is the array to which we wish to add the values followed by the values to add. The function then returns the final slice of an array containing all the values.

func make ([] type, size, capacity)

The make function in go language is used to create an array/map it accepts the type of variable to be created, its size, and capacity as arguments and returns the slice that we can store in the variable.

Algorithm

Step 1 − First, we need to import the fmt package.

Step 2 − Then start the main() function. Here in this function initialize an array of integers using make() function and append values to it using append() function.

Step 3 − Declare a new variable called n and store the length of array to it. Further, print the array on the screen.

Step 4 − Use a for loop to iterate over the array and shift the elements of the array to left by one position and print the new array thus formed.

Step 5 − Print the new array in each Step until whole of the array is shifted cyclically.

Example 1

In this example, we will use a for loop to permute the elements of the array in cyclic manner. We will use the for loop to iterate over the array and shift the element of the array to left by one position in each Step.

package main import "fmt" func main() { arr := make([]int, 0, 5) arr = append(arr, 1, 2, 3, 4, 5) n := len(arr) fmt.Println("The given array is: ", arr) for i := 0; i < n; i++ { last := arr[n-1] arr[j] = arr[j-1] } arr[0] = last fmt.Println("Permuted Array: ", arr) } } Output The given array is: [1 2 3 4 5] Permuted Array: [5 1 2 3 4] Permuted Array: [4 5 1 2 3] Permuted Array: [3 4 5 1 2] Permuted Array: [2 3 4 5 1] Permuted Array: [1 2 3 4 5] Example 2

In this example, we will use the internal append() function of the go programming language to cyclically permute the elements of the array till we get the original array again.

package main import "fmt" func main() { arr := make([]int, 0, 5) arr = append(arr, 1, 2, 3, 4, 5) n := len(arr) fmt.Println("Original Array: ", arr) for i := 0; i < n; i++ { arr = append(arr[1:], arr[0]) fmt.Println("Permuted Array: ", arr) } } Output Original Array: [1 2 3 4 5] Permuted Array: [2 3 4 5 1] Permuted Array: [3 4 5 1 2] Permuted Array: [4 5 1 2 3] Permuted Array: [5 1 2 3 4] Permuted Array: [1 2 3 4 5] Conclusion

We have successfully compiled and executed a go language program to cyclically permute the elements of the array. Here we have used two programs. In the first program we are using a for loop to shift the elements of the array by one position till we get the original array again while in the second program we are using an internal function append() to perform the logic and print the array after each Step.

Program To Check The Similarity Of Given Two Triangles

In this problem, we will learn to check the similarity of two given triangles, which have many real-world use cases from the viewpoint of a programmer.

To construct and manage 2D and 3D models of things, CAD systems are utilized, and one key function is the capability to compare two triangles.

For instance, engineers working in design and construction may need to make that the foundational measurements of a building match the blueprint. Engineers can rapidly evaluate whether the angles and sides of the foundation game the layout by utilizing a CAD tool that has a built-in feature to check the resemblance of two triangles. This can aid in ensuring the building’s structural stability and safety.

In addition, 3D models of objects are produced using CAD software by 3D printing technology. To make sure that the model is printed precisely and to the desired scale in this situation, similarity checking can be helpful. This is crucial for complicated models because manually verifying similarity can be tedious and error-prone.

The precision of robot motions can be ensured by programmers in the field of robotics by using similarity-checking tools. Checking the similarity of two triangles can be useful in ensuring that complex movements made by robot arms, which frequently have numerous joints, are precise and constant.

Explanation

Now let us understand some mathematics involved in calculating the similarity of triangles.

Two triangles are similar if they share the following characteristics −

The angles of both triangles are equal.

Corresponding sides of the triangle share the same ratio.

There are three ways to determine whether two triangles are similar: SSS, SAS, and AA. Let’s discuss each of the theorems in brief.

SSS (Side-Side-Side) Criteria

In two given triangles, if the three pairs of sides are in the same ratio then the two triangles are similar.

Let us consider the two triangles given above. The above two triangles can be similar by SSS criteria if the ratio of the three pairs of sides are equal, i.e. AC/PR = AB/PQ = CB/RQ

SAS (Side-Angle-Side) Criteria

In two given triangles, if the two pairs of sides are in the same ratio, and the angle between the two sides are same in both triangles, then the two triangles are similar.

If we take the above triangle as an example, then if AB/PQ = BC/QR and <B = <Q then both the triangles are similar by SAS criteria.

AA (Angle-Angle) Criteria

In two given triangles, if any two angles of the two triangles are equal, then the two triangles are similar.

If we take the above triangle as an example, then if <A =<P and <B=<Q or <A=<P and <C=<R or any such combinations exist then the two triangles are similar by AA criteria.

Normally, we are given the coordinates of the three points of the triangle and then we need to check similarity. In such cases, we will use this formula to calculate the distance.

Program to check the similarity of given two triangles when the coordinates are provided.

Approach

Let’s decode the entire program into step by step algorithm

Take the coordinates of the three points of both triangles as input.

Calculate the length in between the coordinates using the formula discussed above i.e. distance= Math. sqrt(Math. pow(y2-y1,2)+Math.pow(x2-x1,2))

After calculating the length of all the sides of both triangles, calculate the ratio of all the pairs.

Next, check if all three ratios are the same, if they are the same, then print the triangles are similar otherwise print the triangles are not similar.

Now, we will write the code implementing the above algorithm

Example

C++ Program to check the similarity of given two triangles when the coordinates are provided.

using namespace std;

int main() { double x1 = 0, y1 = 0, x2 = 3, y2 = 0, x3 = 0, y3 = 4; double p1 = 0, q1 = 0, p2 = 6, q2 = 0, p3 = 0, q3 = 8;

double dist1 = sqrt(pow((x2 – x1), 2) + pow((y2 – y1), 2)); double dist2 = sqrt(pow((x3 – x2), 2) + pow((y3 – y2), 2)); double dist3 = sqrt(pow((x1 – x3), 2) + pow((y1 – y3), 2));

double dist4 = sqrt(pow((p2 – p1), 2) + pow((q2 – q1), 2)); double dist5 = sqrt(pow((p3 – p2), 2) + pow((q3 – q2), 2)); double dist6 = sqrt(pow((p1 – p3), 2) + pow((q1 – q3), 2));

double ratio1 = dist1/dist4; double ratio2 = dist2/dist5; double ratio3 = dist3/dist6;

if ((ratio1 == ratio2) && (ratio2 == ratio3)) { cout << “The two triangles are similar.” << endl; } else { cout << “The two triangles are not similar.” << endl; }

return 0; }

Output The two triangles are similar. Complexities

Time complexity: O(1), As this code performs a fixed number of calculations, regardless of the size of the input.

Space complexity: O(1), As the code uses a fixed number of variables to store input values and results, regardless of the size of the input.

Program to check the similarity of given two triangles when the coordinates are provided.

Approach

Let’s decode the entire program into step by step algorithm

Take the angles of the triangles as input.

Compare the angles to check if any two angles of the triangle are the same, here we are using AA criteria.

If any two angles are the same then print the triangles are similar else print the triangles are not similar.

Now, we will write the code implementing the above algorithm.

Example

C++ Program to check the similarity of given two triangles when the angles are provided.

using namespace std;

bool check_aa(int a1,int a2,int a3,int a4,int a5,int a6){ return true; else return false;

} int main(){

double a1 = 30, a2 = 60, a3 = 90; double a4 = 60, a5 = 90, a6 = 30;

bool similar= check_aa(a1,a2,a3,a4,a5,a6);

if (similar) cout << “The two triangles are similar.” << endl; else cout << “The two triangles are not similar.” << endl; }

Output The two triangles are similar. Complexities

Time complexity: O(1), As this code performs a fixed number of calculations, regardless of the size of the input.

Space complexity: O(1), As the code uses a fixed number of variables to store input values and results, regardless of the size of the input.

Conclusion

In this article, we have tried to explain the approach to check the similarity of two triangles based on two situations, one where the sides are provided as input and another where the angles are provided as input. I hope this article helps you to learn the concept in a better way.

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