Describe the state graph of the Towers of Hanoi puzzle for general n. Prove that the puzzle is always solvable, and find the number of moves in the shortest possible solution. (It will be useful to have a recursive definition of the state space.)
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Computer Science
Describe the state graph of the Towers of Hanoi puzzle for general n. Prove that the puzzle is always solvable, and find the number of moves in the shortest possible solution. (It will be useful to have a recursive definition of the state space.)
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- 5. Given an undirected graph with n vertices and m edges, find an O(n+m) time algorithm that determines whether it is possible to color all the vertices red and blue such that every edge is between a red vertex and blue vertex. If such a coloring exists, your algorithm should produce one.a. Given an undirected graph G = (V, E), develop a greedy algorithm to find a vertex cover of minimum size. b. What is the time complexity of your algorithm. c. Apply your algorithm on the graph below and state whether it correctly finds it or not. 2 6 3 7 1 4 8 5 94. Königsberg bridges The Königsberg bridge puzzle is universally accepted as the problem that gave birth to graph theory. It was solved by the great Swiss-born mathematician Leonhard Euler (1707–1783). The problem asked whether one could, in a single stroll, cross all seven bridges of the city of Königsberg exactly once and return to a starting point. Following is a sketch of the river with its two islands and seven bridges: a. State the problem as a graph problem. b. Does this problem have a solution? If you believe it does, draw such a stroll; if you believe it does not, explain why and indicate the smallest number of new bridges that would be required to make such a stroll possible.
- 3) The graph k-coloring problem is stated as follows: Given an undirected graph G= (V,E) with N vertices and M edges and an integer k. Assign to each vertex v in V a color c(v) such that 13) The graph k-coloring problem is stated as follows: Given an undirected graph G = (V,E) with N vertices and M edges and an integer k. Assign to each vertex v in Va color c(v) such that 1< c(v)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…Suppose are you given an undirected graph G = (V, E) along with three distinct designated vertices u, v, and w. Describe and analyze a polynomial time algorithm that determines whether or not there is a simple path from u to w that passes through v. [Hint: By definition, each vertex of G must appear in the path at most once.]Please solve with the computer Question 2: Draw a simple undirected graph G that has 11 vertices, 7 edges.6. Give an algorithm that determines whether or not a given undirected graph contains a cycle. Your algorithm should run in time O(n) and independent of m.A two dimensional array of red and green entries represents a maze. Green entries are passable and red entries are blocked (like a wall). Two special green entries en and ex denote the entrance and exit of the maze. (1) Abstract the problem as a graph; (2) Design an efficient algorithm (pseudo code) to find a shortest path from en to ex if it exists and then print out the path; (3) Analyze the complexity of the algorithm.The graph-coloring problem is usually stated as the vertex-coloring problem: assign the smallest number of colors to vertices of a given graph so that no two adjacent vertices are the same color. Consider the edge-coloring problem: assign the smallest number of colors possible to edges of a given graph so that no two edges with the same end point are the same color. Explain how the edge-coloring problem can be polynomial reduced to a vertex-coloring problem. Give an example.Based on the definition given in Chapter 14 about tractable and intractable problems in computer science, classify each of the following problems based on their complexity, whether they are tractable or intractable. Problem Sorting a list of n integers Searching a list of integers Finding a Clique in a graph The traveling salesman problem K-colorability problem Classification (Tractable Intractable)SEE MORE QUESTIONSRecommended textbooks for youDatabase System ConceptsComputer ScienceISBN:9780078022159Author:Abraham Silberschatz Professor, Henry F. Korth, S. SudarshanPublisher:McGraw-Hill EducationStarting Out with Python (4th Edition)Computer ScienceISBN:9780134444321Author:Tony GaddisPublisher:PEARSONDigital Fundamentals (11th Edition)Computer ScienceISBN:9780132737968Author:Thomas L. FloydPublisher:PEARSONC How to Program (8th Edition)Computer ScienceISBN:9780133976892Author:Paul J. Deitel, Harvey DeitelPublisher:PEARSONDatabase Systems: Design, Implementation, & Manag…Computer ScienceISBN:9781337627900Author:Carlos Coronel, Steven MorrisPublisher:Cengage LearningProgrammable Logic ControllersComputer ScienceISBN:9780073373843Author:Frank D. PetruzellaPublisher:McGraw-Hill EducationDatabase System ConceptsComputer ScienceISBN:9780078022159Author:Abraham Silberschatz Professor, Henry F. Korth, S. SudarshanPublisher:McGraw-Hill EducationStarting Out with Python (4th Edition)Computer ScienceISBN:9780134444321Author:Tony GaddisPublisher:PEARSONDigital Fundamentals (11th Edition)Computer ScienceISBN:9780132737968Author:Thomas L. FloydPublisher:PEARSONC How to Program (8th Edition)Computer ScienceISBN:9780133976892Author:Paul J. Deitel, Harvey DeitelPublisher:PEARSONDatabase Systems: Design, Implementation, & Manag…Computer ScienceISBN:9781337627900Author:Carlos Coronel, Steven MorrisPublisher:Cengage LearningProgrammable Logic ControllersComputer ScienceISBN:9780073373843Author:Frank D. PetruzellaPublisher:McGraw-Hill Education