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 3, Problem 8E
a.
Explanation of Solution
Reason:
- Any path that may appear to be bad, but might lead to an arbitrarily large reward (negative cost)...
b.
Explanation of Solution
Effects of insisting that step costs should be greater than or equal to some negative constant:
- If the greatest possible reward is assumed to be ācā.
- Then if the maximum depth of the state space (e.g. when the state space is a tree) is also known, then any path with d levels r...
c.
Explanation of Solution
Justification:
- In the given case, a set of actions is given that forms a loop in the state space such that executing the set in different order results in no net change to the state...
d.
Explanation of Solution
State-space search:
- Value of a scenic loop is decreased each time it is revisited; a novel scenic sight is a great reward.
- But seeing the same one for the tenth time in an hour is tedious and not rewarding...
e.
Explanation of Solution
Real domain example:
- There are many real domain examples that include steps that may cause looping...
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Students have asked these similar questions
Consider the following graph. We are finding the lengths of the shortest paths from vertex a to all
vertices by the Relaxation algorithm from the lecture:
At the beginning, every vertex v is set as unmarked and h(v) = +ā.
Then a is set as open and h(a) = 0.
How many ways there are to select five (at that moment) open vertices so that after their relaxation
(and closing) the value of h(v) will represent the shortest path length from a to every v in the graph?
Specify the number of ways and one such way separated by commas and spaces, for example
25, b, Š°, Ń, Š°, d.
d
3
1
a
1
-4
-4
Question 9. Present an algorithm for the following problem. The input is a weighted graph G, two vertices s
and t, and a positive number k. The goal is to find a path from s to t such that all edges along the path have weight
ā¤ k (if there is such a path), or to print "no good path", if there is no such path.
Let G = (V, E) be an undirected graph and each edge e ā E is associated with a positive weight ā(e).For simplicity we assume weights are distinct. Is the following statement true or false?Ā Let P be the shortest path between two nodes s, t. Now, suppose we replace each edge weight ā(e) withā(e)^2, then P is still a shortest path between s and t.
Chapter 3 Solutions
Artificial Intelligence: A Modern Approach
Ch. 3 - Explain why problem formulation must follow goal...Ch. 3 - Prob. 2ECh. 3 - Prob. 3ECh. 3 - Prob. 4ECh. 3 - Prob. 5ECh. 3 - Prob. 6ECh. 3 - Prob. 8ECh. 3 - Prob. 9ECh. 3 - Prob. 10ECh. 3 - Prob. 11E
Ch. 3 - Prob. 12ECh. 3 - Prob. 13ECh. 3 - Prob. 14ECh. 3 - Prob. 15ECh. 3 - Prob. 16ECh. 3 - Prob. 17ECh. 3 - Prob. 18ECh. 3 - Prob. 20ECh. 3 - Prob. 21ECh. 3 - Prob. 22ECh. 3 - Trace the operation of A search applied to the...Ch. 3 - Prob. 24ECh. 3 - Prob. 25ECh. 3 - Prob. 26ECh. 3 - Prob. 27ECh. 3 - Prob. 28ECh. 3 - Prob. 29ECh. 3 - Prob. 31ECh. 3 - Prob. 32E
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- Suppose we want to use UCS and the A* algorithm on the graph below to find the shortest path from node S to node G. Each node is labeled by a capital letter and the value of a heuristic function. Each edge is labeled by the cost to traverse that edge. Perform A*, UCS, and BFS on this graph. Indicate the f, g, and h values of each node for the A*. e.g., S = 0 + 6 = 6 (i.e. S = g(S) + h(S) = f(S)). Additionally, show how the priority queue changes with time. Show the order in which the nodes are visited for BFS and UCS. Show the path found by the A*, UCS, and BFS algorithms on the graph above. Make this example inadmissible by changing the heuristic value at one of the nodes. What node do you choose and what heuristic value do you assign? What would be the A* algorithm solution then.arrow_forwardTwo vertices, s and t, are provided together with a directed graph G = (V, E). Additionally, the graph's edges are all either blue or red in color. Finding a road from s to t such that all red edges appear after all blue edges is your objective. The path need not have any red or blue edges, but if it does, all red edges should appear after all blue edges. Create and evaluate an algorithm that will solve this issue in O(n + m) time.arrow_forwardGiven a graph that is a tree (connected and acyclic). Ā (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.Ā Consider the path p from w to x.Ā Ā 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_forward
- TRY YOUR BEST PLEASE Let A and B each be sets of N labeled vertices, and consider bipartite graphs between A and B. 1. Starting with no edges between A and B, if |E| many edges are added between A and B uniformly at random, what is the expected number of perfect matchings in the resulting graph? Hint: if S is a set of edges in a potential perfect matching, let X sub S = 1 if all the edges in S are added to the graph, and X sub S= 0 if any of them are missing. What isE[X sub S]?arrow_forwardThe given inputs consist of two nodes (s, t) and a directed graph G = (V, E). In addition, each edge of the graph is either blue or red. The goal is to find a path from point s to point t such that red edges always follow blue edges. There need not be any red or blue borders on the route, but if there are, the red ones should follow the blue ones. Develop an algorithm that does the task in O(n + m) time and analyze its performance.arrow_forward3) 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)arrow_forwardWe know that when we have a graph with negative edge costs, Dijkstraās algorithm is not guaranteed to work. Ā (a) Does Dijkstraās algorithm ever work when some of the edge costs are negative? Explain why or why not. Ā (b) Find an algorithm that will always find a shortest path between two nodes, under the assumption that at most one edge in the input has a negative weight. Your algorithm should run in time O(m log n), where m is the number of edges and n is the number of nodes. That is, the runnning time should be at most a constant factor slower than Dijkstraās algorithm. To be clear, your algorithm takes as input Ā (i) a directed graph, G, given in adjacency list form. 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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_forwardin a graph G = (N,E,C), where N are nodes, E edges between nodes, and the weight of an edge e ā E is given by C(e), where C(e) > 1, for all e ā E. the heuristic h that counts the least amount of edges from an initial state to a goal state. now removing edges from the graph, while keeping the heuristic values unchanged. Is the heuristic still consistent?arrow_forwardConsider the Minimum-Weight-Cycle Problem: Input: A directed weighted graph G (V, E) (where the weight of edge e is w(e)) and an integer k. Output: TRUE if there is a cycle with total weight at most k and FALSE if there is no cycle with total weight at most k. 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