Artificial Intelligence: A Modern Approach
3rd Edition
ISBN: 9780136042594
Author: Stuart Russell, Peter Norvig
Publisher: Prentice Hall
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Expert Solution & Answer
Chapter 4, Problem 5E
Explanation of Solution
Determining the information and using it by the
- See figure S4.1 for the adapted algorithm. For states that OR-SEARCH finds a solution for it records the solution found.
- If it later visits state and the solution is immediately returns by that state.
- The user has to be careful when the OR-SEARCH algorithm fails to find the solution.
- As the user do not allow the cycles, the state can be solved by depending on the path to that solution...
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The Graph Data Structure is made up of nodes and edges. (A Tree Data Structure is a special kind of a
Graph Data Structure). A Graph may be represented by an Adjacency Matrix or an Adjacency List. Through
this exercise, you should be able to have a better grasp the Adjacency Matrix concept. You are expected to
read about the Adjacency Matrix concept as well as the Adjacency List concept.
Suppose the vertices A, B, C, D, E, F, G and H of a Graph are mapped to row and column
indices(0,1,2,3,4,5,6,and 7) of a matrix (i.e. 2-dimensional array) as shown in the following table.
Vertex of Graph
Index in the 2-D Array Adjacency Matrix
Representation of Graph
A
B
2
F
6.
H
7
Suppose further, that the following is an Adjacency Matrix representing the Graph.
3
4
5.
6.
7
0.
1
1
1
1
01
1
01
1.
3
14
1
1
1
6.
1
Exercise:
Show/Draw the Graph that is represented by the above Adjacency matrix. Upload the document that contains
your result. (Filename: AdjacencyMatrixExercise.pdf)
Notes:
-The nodes of the…
3. Kleinberg, Jon. Algorithm Design (p. 519, q. 28) Consider this version of the Independent Set Problem. You are given an undirected graph G and an integer k. We will call a set of nodes I "strongly independent" if, for any two nodes v, u € I, the edge (v, u) is not present in G, and neither is there a path of two edges from u to v. That is, there is no node w such that both (v, w) and (u, w) are present. The Strongly Independent Set problem is to decide whether G has a strongly independent set of size at least k.
Show that the Strongly Independent Set Problem is NP-Complete.
Warshall's algorithm is applied on the following graph. Show the values of the table element "c to a"
for each step of the algorithm. Use -1 for infinity. Initially (at step 0) the table element is -1. (Table
elements are listed in alphabetical order from a to g, and this order is used for the steps of the
algorithm. Accordingly in the first step the paths going through a are used.)
a
...
1
2
2
2
(f
1
2
4
Chapter 4 Solutions
Artificial Intelligence: A Modern Approach
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- For problem 1, you should provide a list of numbers of states for each of the search methods. Problem 1 In the state space shown below ( refer to image ), write the order in which states are expanded if the initial state is 4 and there are two goal states: 5 and 8. For example, for breadth first search the answer would be: 4 2 7 1 3 6 8 using depth-first search using iterative deepening.arrow_forwardThe following graph is connected. True or False True Falsearrow_forward5. (This question goes slightly beyond what was covered in the lectures, but you can solve it by combining algorithms that we have described.) A directed graph is said to be strongly connected if every vertex is reachable from every other vertex; i.e., for every pair of vertices u, v, there is a directed path from u to v and a directed path from v to u. A strong component of a graph is then a maximal subgraph that is strongly connected. That is all vertices in a strong component can reach each other, and any other vertex in the directed graph either cannot reach the strong component or cannot be reached from the component. (Note that we are considering directed graphs, so for a pair of vertices u and v there could be a path from u to v, but no path path from v back to u; in that case, u and v are not in the same strong component, even though they are connected by a path in one direction.) Given a vertex v in a directed graph D, design an algorithm for com- puting the strong connected…arrow_forward
- a) Given a depth-first search tree T, the set of edges in T are referred to as "tree edges" while those not in T are referred to as "back edges". Modify the implementation of the Depth-First Search algorithm to print out the set of tree edges and the set of back edges for the following graph. 1(0 1 1 0 0 1 o) 210 10 0 0 31 10 1 0 0 1 40 0 10 0 0 0 50 0 0 0 0 1 1 6 10 001 70 0 10 1 1 0 0 1arrow_forwardGiven a graph that is a tree (connected and acyclic). (I) Pick any vertex v.(II) Compute the shortest path from v to every other vertex. Let w be the vertex with the largest shortest path distance.(III) Compute the shortest path from w to every other vertex. Let x be the vertex with the largest shortest path distance. Consider the path p from w to x. Which of the following are truea. p is the longest path in the graphb. p is the shortest path in the graphc. p can be calculated in time linear in the number of edges/verticesarrow_forwardLet G be an undirected graph whose vertices are the integers 1 through 8, and let the adjacent vertices of each vertex be given by the table below: look at the picture sent Assume that, in a traversal of G, the adjacent vertices of a given vertex are returned in the same order as they are listed in the table above. Which statement of the following is correct? group of answer choices a) The sequence of vertices visited using a DFS traversal starting at vertex 1: 1, 2, 3, 4, 6, 5, 7, 8. b) The sequence of vertices visited using a BFS traversal starting at vertex 1: 1, 2, 3, 4, 6, 5, 7, 8. c) Both sequences are wrong. d) Both sequences are correct.arrow_forward
- 2. An undirected graph G can be partitioned into connected components, where two nodes are in the same connected component if and only if there is a path connecting them. Design and analyze an efficient algorithm that computes the connected components of a graph G given in adjacency list format. Be sure to give a correctness argument and detailed time analysis. You can use algorithms from class as a sub-procedure, but be sure to use the claims proven about them carefully. A good algorithm has time approximately 0(n + m) where the graph has n nodes and m edges.arrow_forwardGiven a graph that is a tree (connected and acyclic). (1) Pick any vertex v. (II) Compute the shortest path from v to every other vertex. Let w be the vertex with the largest shortest path distance. (III) Compute the shortest path from w to every other vertex. Let x be the vertex with the largest shortest path distance. Consider the path p from w to x. Which of the following are true a. p is the longest path in the graph b. p is the shortest path in the graph c. p can be calculated in time linear in the number of edges/vertices a,c a,b a,b,c b.carrow_forward2. (Error-correcting codes x planarity.) Let n be a positive integer. Define the graph H, as follows: • Vertices: the set of all length-n strings over the alphabet {0, 1, 2}. • Edges: draw an edge from string s to string t if the Hamming distance dy(s, t) is 1. (a) Draw H2 (no justification needed.) (b) Prove that for every n 2 3, H, is nonplanar.arrow_forward
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