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- One of the most important mathematical problems through all times has been to find the area of a polygon. For example, real estate areas often had the shape of polygons, and the tax was proportional to the area. Suppose we have some polygon with vertices ("corners") specified by the coordinates (x1,y1), (x2,y2), …, (xn,yn), numbered either in a clockwise or counter-clockwise fashion around the polygon. The area A of the polygon can amazingly be computed by just knowing the boundary coordinates: A=12|(x1y2+x2y3+⋯+xn−1yn+xny1)−(y1x2+y2x3+⋯+yn−1xn+ynx1)|.(17) Write a function polygon_area(x, y) that takes two coordinate lists with the vertices as arguments and returns the area. Test the function on a triangle, a quadrilateral, and a pentagon where you can calculate the area by alternative methods for comparison. Hint. Since Python lists and arrays has 0 as their first index, it is wise to rewrite the mathematical formula in terms of vertex coordinates numbered as x0,x1,…,xn−1 and…A.) Running Dijkstra's algorithm with vertex A as the starting point, list the order in which the vertices would be marked. (Don't forget to include the starting vertex.) B.)What is the shortest distance discovered to vertex E? C.In the course of running Dijkstra's algorithm on the graph shown above, how many different (non-infinite) distances were found for vertex E?improvements of the given algorthims.
- In Computer Science a Graph is represented using an adjacency matrix. Ismatrix is a square matrix whose dimension is the total number of vertices.The following example shows the graphical representation of a graph with 5 vertices, its matrixof adjacency, degree of entry and exit of each vertex, that is, the total number ofarrows that enter or leave each vertex (verify in the image) and the loops of the graph, that issay the vertices that connect with themselvesTo program it, use Object Oriented Programming concepts (Classes, objects, attributes, methods), it can be in Java or in Python.-Declare a constant V with value 5-Declare a variable called Graph that is a VxV matrix of integers-Define a MENU procedure with the following textGRAPHS1. Create Graph2.Show Graph3. Adjacency between pairs4.Input degree5.Output degree6.Loops0.exit-Validate MENU so that it receives only valid options (from 0 to 6), otherwise send an error message and repeat the reading-Make the MENU call in the main…[Introduction to the Design and Analysis of Algorithms, 3rd Edition] Maxima search. A point (xi, yi) in the Cartesian plane is said to be dominated by point (xj , yj ) if xi ≤ xj and yi ≤ yj with at least one of the two inequalities being strict. Given a set of n points, one of them is said to be a maximum of the set if it is not dominated by any other point in the set. For example, in the figure below, all the maximum points of the set are circled. Design an efficient algorithm for finding all the maximum points of a given set of n points in the Cartesian plane. What is the time efficiency class of your algorithm?(c) Assume you are having a graph in which there are two shortest paths from the source node u to a node v with different number of intermediate vertices. Modify Dijkstra's algorithm in such a way that it always prefers the path with fewest intermediate edges/vertices. What is the complexity of this modified algorithm?
- Determine the loop invariant of the code written then prove that the loop invariant holds on Initialization, Maintainance and Termination.Find Regular Expression of: “Even number of a’s and odd number of b’s and start with ‘aba’ and end on ‘bba’ define over ={a,b})Write a computer program for Gauss elimination method using C programming language. Decide the number of significant figures yourselves. While writing your program, consider the effects of the number of significant figures, pivoting, scaling and do not forget to check if the system is ill conditioned.2)Repeat the same procedures for Gauss-Jordan method.3)Solve an example system using your Gauss elimination and Gauss-Jordan method. Measure the time your computer solves the system for both programs.4)Write a report in which you discuss and compare your Gauss elimination and Gauss-Jordan programs.
- If n points are connected to fom a closed polygon as shown below, the area of the polygon can be compuled as n-2 Area = (%)E (*»1 + x ) (y»1 - y ) =0 Notice that although the ilustrated polygon has only 6 distinct comers, n for his polygon is 7 because the algorithmexpects that the last point (x.ya) will be repeat of the initial point, (Ko.yo). Define a structure for a point. Each point contains x coordinate and y coordinate. The represe ntation of a Polygon must be an array of structures in your program. Write a C program that takes the number of actual points (n-1) from the user. After that, user enters x and y coordinates of each point. (The last point will be repeat of the initial point). Writo a compute Are a function which returns the area of the Polygon. Print he area of the Polygon in main. Display the area with wo digts after the decimal point. Note: The absolute value can be computed with fabs function. Example: double x.50: fabs(x) is 5.0 double x 0.0: fabs(x) is 0.0 double…Find a closed form representation for the following recursively defined function. Give the run-time complexity of each recursively defined function.Theory of Computation Prove the following