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16410f03_smidsol (1)

# 16410f03_smidsol (1) - 16.410-13 Principles of Automated...

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16.410-13: Principles of Automated Reasoning and Decision Making Midterm Solutions October 20 th , 2003 Note: Budget your time wisely. Some parts of this quiz could take you much longer than others. Move on if you are stuck and return to the problem later. Name E-mail Problem Number Max Score Grader Problem 1 26 Problem 2 22 Problem 3 24 Problem 4 22 Total 94

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Problem 1 Uninformed Search (26 points) You are searching for a path from S to G in the following graph using depth-first, breadth first and iterative deepening search: S Make the following assumptions: ± Each search algorithm explores a node’s children in alphabetical order . ± Unless otherwise stated, the search algorithms use a visited list to prune nodes. ± The search stops as soon as the goal is expanded. Write the sequence of nodes expanded by the specified search methods. A node is expanded when the search algorithm takes it off the queue, and attempts to create its children. Part A Depth-first Search (4 points) Show the depth-first expansion sequence (we have started it for you): Part B Breadth-first Search (4 points) Show the breadth-first expansion sequence (we have started it for you): A F C E D J I H B G S-A-D-E-B-F-I-J-C-G S-A-B-C-D-E-F-G
Graph repeated for your convenience: S Part C Iterative Deepening (4 points) Show the iterative-deepening expansion sequence (we have started it for you): Part D Visited List (4 points) Now assume that the depth first search algorithm does not use a visited list . Show the search expansion, terminating either when the goal is reached or after the sequence reaches length 15. Show the depth-first expansion sequence (we have started it for you): A F C E D J I H B G S-S-A-B-C-S-A-D-E-B-C-F-C-G S-A-D-E-E-D-B-C-G 3

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Part E Analysis – Lower Bounds on Breadth First Search (10 points) Assume you have a tree with branching factor b and depth d. Furthermore, assume that the goal node appears on level m, where m < d. Specify the exact cost of the search algorithm as a function of m, d and b. What is the minimum number of nodes that might be generated by breadth-first search in order to reach the goal? (b^m+b-2)/(b-1) For full credit you must show your derivation below, including the solution to any recurrence: The minimum number of nodes generated occurs when the goal node is placed at the far left, on level m. In this case, all the nodes above level m are generated, in addition to the one goal node. There is 1 node at level 0, and the number of nodes generated at each successive level is multiplied by b. Level 0: 1 Level 1: b Level 2: b^2 Level m-1: b^(m-1) Level m: 1 The total number of nodes generated is: Nodes=2+b+b^2+…+b^(m-1) Solving the recurrence: b*nodes=2b+b^2+b^3…+b^m -nodes=2+ b+b^2+…+b^(m-1) hence, (b-1)*nodes=b^m+b-2 or nodes=(b^m+b-2)/(b-1) 4
Problem 2 Planning (22 points) The Smith residence is a small house consisting of two rooms, A and B, connected by a door, as shown below: R1 A B Box1 Dr. Smith is away, but he has left a robot, R1, in charge of the house.

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16410f03_smidsol (1) - 16.410-13 Principles of Automated...

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