a. Illustration of iterative lengthening search algorithm is optimal for the general path costs: Uniform cost just iteratively inspects the unexplored node nearest to the start node. Iterative lengthening search is a uniform cost search that runs multiple times with a growing cost limit from start to finish. The iterative lengthening search has not that much advantage as the iterative deepening search possesses. The weakness of iterative lengthening search: The algorithm simply searches the same node each time and adds more at the ends than the nearest first node and the next closest node are invariant...

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Answer 1

Expert Solution & Answer

Chapter 3, Problem 17E

Explanation of Solution

a.

Illustration of iterative lengthening search algorithm is optimal for the general path costs:

Uniform cost just iteratively inspects the unexplored node nearest to the start node.

Iterative lengthening search is a uniform cost search that runs multiple times with a growing cost limit from start to finish.

The iterative lengthening search has not that much advantage as the iterative deepening search possesses.

The weakness of iterative lengthening search:

The algorithm simply searches the same node each time and adds more at the ends than the nearest first node and the next closest node are invariant...

Explanation of Solution

b.

Number of iterations required for iterative deepening:

The uniform cost search will run through the nodes in the same order as that of the first breadth search.

The search progression will go as follows,

Assume that the goal node is found halfway through the search operation of depth, on average.

Thus the goal iteration = “b+b2+...+bd/2”

Explanation of Solution

c.

Number of iterations required in the worst case:

In the worst case, the limit begins at “0” and increments by a minimum positive cost “ε”, until it reaches the value “d”

Explanation of Solution

d.

Performance of algorithm compared to uniform cost search algorithm:

The algorithm requires only “k” uniform cost searches where iteration simply expands the boundary for the closest nodes inspected...

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We know that when we have a graph with negative edge costs, Dijkstra’s algorithm is not guaranteed to work. (a) Does Dijkstra’s algorithm ever work when some of the edge costs are negative? Explain why or why not. (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 Dijkstra’s algorithm. To be clear, your algorithm takes as input (i) a directed graph, G, given in adjacency list form. (ii) a weight function f, which, given two adjacent nodes, v,w, returns the weight of the edge between them. For non-adjacent nodes v,w, you may assume f(v,w) returns +1. (iii) a pair of nodes, s, t. If the input contains a negative cycle, you should find one and output it. Otherwise, if the graph contains at least one…

Write the algorithm that finds and returns how many paths in k units of length between any given two nodes (source node, destination node; source and target nodes can also be the same) in a non-directional and unweighted line of N nodes represented as a neighborhood matrix. (Assume that each side in the unweighted diagram is one unit long.) Note: By using the problem reduction method of the Transform and Conquer strategy, you have to make the given problem into another problem. Algorithm howManyPath (M [0..N-1] [0..N-1], source, target, k)// Input: NxN neighborhood matrix, source, target nodes, k value.// Ouput: In the given line, there are how many different paths of k units length between the given source and target node.

Please answer the following question in detail and explain all the proofs and assumptions for all parts. The question has three parts, (a), (b) and (c). Iterative lengthening search is an iterative analogue of uniform-cost search. The basic idea is to use increasing limits on path cost. If a node is generated whose path cost exceeds the current limit, it is immediately discarded. For each new iteration, the limit is set to the lowest path cost of any node discarded in the previous iteration. (a) Show that this algorithm is optimal for general path costs. You may assume that all costs are integers (this is not a loss of generality if the search space is finite). You may wish to consider the minimal path cost C; what happens when we set the path cost to be some limit l < C? (b) Consider a uniform tree with branching factor b, solution depth d, and unit step costs (each action costs one unit). How many iterations will iterative lengthening require? (c) (7 points) Now consider the…

Answer 2

Expert Solution & Answer

Chapter 3, Problem 17E

Explanation of Solution

a.

Illustration of iterative lengthening search algorithm is optimal for the general path costs:

Uniform cost just iteratively inspects the unexplored node nearest to the start node.

Iterative lengthening search is a uniform cost search that runs multiple times with a growing cost limit from start to finish.

The iterative lengthening search has not that much advantage as the iterative deepening search possesses.

The weakness of iterative lengthening search:

The algorithm simply searches the same node each time and adds more at the ends than the nearest first node and the next closest node are invariant...

Explanation of Solution

b.

Number of iterations required for iterative deepening:

The uniform cost search will run through the nodes in the same order as that of the first breadth search.

The search progression will go as follows,

Assume that the goal node is found halfway through the search operation of depth, on average.

Thus the goal iteration = “b+b2+...+bd/2”

Explanation of Solution

c.

Number of iterations required in the worst case:

In the worst case, the limit begins at “0” and increments by a minimum positive cost “ε”, until it reaches the value “d”

Explanation of Solution

d.

Performance of algorithm compared to uniform cost search algorithm:

The algorithm requires only “k” uniform cost searches where iteration simply expands the boundary for the closest nodes inspected...

Expert Solution & Answer

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Check out a sample textbook solution

See solution

Answer 3

Expert Solution & Answer

Chapter 3, Problem 17E

Explanation of Solution

a.

Illustration of iterative lengthening search algorithm is optimal for the general path costs:

Uniform cost just iteratively inspects the unexplored node nearest to the start node.

Iterative lengthening search is a uniform cost search that runs multiple times with a growing cost limit from start to finish.

The iterative lengthening search has not that much advantage as the iterative deepening search possesses.

The weakness of iterative lengthening search:

The algorithm simply searches the same node each time and adds more at the ends than the nearest first node and the next closest node are invariant...

Explanation of Solution

b.

Number of iterations required for iterative deepening:

The uniform cost search will run through the nodes in the same order as that of the first breadth search.

The search progression will go as follows,

Assume that the goal node is found halfway through the search operation of depth, on average.

Thus the goal iteration = “b+b2+...+bd/2”

Explanation of Solution

c.

Number of iterations required in the worst case:

In the worst case, the limit begins at “0” and increments by a minimum positive cost “ε”, until it reaches the value “d”

Explanation of Solution

d.

Performance of algorithm compared to uniform cost search algorithm:

The algorithm requires only “k” uniform cost searches where iteration simply expands the boundary for the closest nodes inspected...

Expert Solution & Answer

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Check out a sample textbook solution

See solution

Answer 4

Expert Solution & Answer

Chapter 3, Problem 17E

Explanation of Solution

a.

Illustration of iterative lengthening search algorithm is optimal for the general path costs:

Uniform cost just iteratively inspects the unexplored node nearest to the start node.

Iterative lengthening search is a uniform cost search that runs multiple times with a growing cost limit from start to finish.

The iterative lengthening search has not that much advantage as the iterative deepening search possesses.

The weakness of iterative lengthening search:

The algorithm simply searches the same node each time and adds more at the ends than the nearest first node and the next closest node are invariant...

Explanation of Solution

b.

Number of iterations required for iterative deepening:

The uniform cost search will run through the nodes in the same order as that of the first breadth search.

The search progression will go as follows,

Assume that the goal node is found halfway through the search operation of depth, on average.

Thus the goal iteration = “b+b2+...+bd/2”

Explanation of Solution

c.

Number of iterations required in the worst case:

In the worst case, the limit begins at “0” and increments by a minimum positive cost “ε”, until it reaches the value “d”

Explanation of Solution

d.

Performance of algorithm compared to uniform cost search algorithm:

The algorithm requires only “k” uniform cost searches where iteration simply expands the boundary for the closest nodes inspected...

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Answer 5

Expert Solution & Answer

Answer 6

Expert Solution & Answer

Answer 7

Chapter 3, Problem 17E

Explanation of Solution

a.

Illustration of iterative lengthening search algorithm is optimal for the general path costs:

Uniform cost just iteratively inspects the unexplored node nearest to the start node.

Iterative lengthening search is a uniform cost search that runs multiple times with a growing cost limit from start to finish.

The iterative lengthening search has not that much advantage as the iterative deepening search possesses.

The weakness of iterative lengthening search:

The algorithm simply searches the same node each time and adds more at the ends than the nearest first node and the next closest node are invariant...

Explanation of Solution

b.

Number of iterations required for iterative deepening:

The uniform cost search will run through the nodes in the same order as that of the first breadth search.

The search progression will go as follows,

Assume that the goal node is found halfway through the search operation of depth, on average.

Thus the goal iteration = “b+b2+...+bd/2”

Explanation of Solution

c.

Number of iterations required in the worst case:

In the worst case, the limit begins at “0” and increments by a minimum positive cost “ε”, until it reaches the value “d”

Explanation of Solution

d.

Performance of algorithm compared to uniform cost search algorithm:

The algorithm requires only “k” uniform cost searches where iteration simply expands the boundary for the closest nodes inspected...

Answer 8

Chapter 3, Problem 17E

Explanation of Solution

a.

Illustration of iterative lengthening search algorithm is optimal for the general path costs:

Uniform cost just iteratively inspects the unexplored node nearest to the start node.

Iterative lengthening search is a uniform cost search that runs multiple times with a growing cost limit from start to finish.

The iterative lengthening search has not that much advantage as the iterative deepening search possesses.

The weakness of iterative lengthening search:

The algorithm simply searches the same node each time and adds more at the ends than the nearest first node and the next closest node are invariant...

Answer 9

Chapter 3, Problem 17E

Explanation of Solution

a.

Illustration of iterative lengthening search algorithm is optimal for the general path costs:

Uniform cost just iteratively inspects the unexplored node nearest to the start node.

Iterative lengthening search is a uniform cost search that runs multiple times with a growing cost limit from start to finish.

The iterative lengthening search has not that much advantage as the iterative deepening search possesses.

The weakness of iterative lengthening search:

The algorithm simply searches the same node each time and adds more at the ends than the nearest first node and the next closest node are invariant...

Answer 10

Explanation of Solution

b.

Number of iterations required for iterative deepening:

The uniform cost search will run through the nodes in the same order as that of the first breadth search.

The search progression will go as follows,

Assume that the goal node is found halfway through the search operation of depth, on average.

Thus the goal iteration = “b+b2+...+bd/2”

Answer 11

Explanation of Solution

b.

Number of iterations required for iterative deepening:

The uniform cost search will run through the nodes in the same order as that of the first breadth search.

The search progression will go as follows,

Assume that the goal node is found halfway through the search operation of depth, on average.

Thus the goal iteration = “b+b2+...+bd/2”

Answer 12

Explanation of Solution

c.

Number of iterations required in the worst case:

In the worst case, the limit begins at “0” and increments by a minimum positive cost “ε”, until it reaches the value “d”

Answer 13

Explanation of Solution

c.

Number of iterations required in the worst case:

In the worst case, the limit begins at “0” and increments by a minimum positive cost “ε”, until it reaches the value “d”

Answer 14

Explanation of Solution

d.

Performance of algorithm compared to uniform cost search algorithm:

The algorithm requires only “k” uniform cost searches where iteration simply expands the boundary for the closest nodes inspected...

Answer 15

Explanation of Solution

d.

Performance of algorithm compared to uniform cost search algorithm:

The algorithm requires only “k” uniform cost searches where iteration simply expands the boundary for the closest nodes inspected...

Answer 16

Expert Solution & Answer

Answer 17

Expert Solution & Answer

Answer 18

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Answer 19

Ch. 3 - Explain why problem formulation must follow goal...Ch. 3 - Prob. 2E Ch. 3 - Prob. 3E Ch. 3 - Prob. 4E Ch. 3 - Prob. 5E Ch. 3 - Prob. 6E Ch. 3 - Prob. 8E Ch. 3 - Prob. 9E Ch. 3 - Prob. 10E Ch. 3 - Prob. 11E

Explanation of Solution

Explanation of Solution

Explanation of Solution

c. Number of iterations required in the worst case: In the worst case, the limit begins at “0” and increments by a minimum positive cost “ε”, until it reaches the value “d”

Explanation of Solution

d. Performance of algorithm compared to uniform cost search algorithm: The algorithm requires only “k” uniform cost searches where iteration simply expands the boundary for the closest nodes inspected...

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Chapter 3 Solutions

c.

Number of iterations required in the worst case:

In the worst case, the limit begins at “0” and increments by a minimum positive cost “ε”, until it reaches the value “d”

d.

Performance of algorithm compared to uniform cost search algorithm:

The algorithm requires only “k” uniform cost searches where iteration simply expands the boundary for the closest nodes inspected...