Artificial Intelligence: A Modern Approach
3rd Edition
ISBN: 9780136042594
Author: Stuart Russell, Peter Norvig
Publisher: Prentice Hall
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Chapter 4, Problem 3E
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Hill-climbing
Step 1: Connect every city into an arbitrary path.
Step 2: Pick two points along the path at random.
Step 3: Split the path at those points and produce three pieces...
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How would you modify the dynamic programming algorithm for the coin collecting problem if some cells on the board are inaccessible for the robot? Apply your algorithm to the board below, where the inaccessible cells are shown by X’s. How many optimal paths are there for this board? You need to provide 1) a modified recurrence relation, 2) a pseudo code description of the algorithm, and 3) a table that stores solutions to the subproblems.
Consider the following problem for path finding where S is the source, G is the Goal and O are obstacles. We can only move horizontally and vertically (not diagonally). We will not re-visit an already visited cell.
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- Simulate the application of Depth-first search tree to find all paths from S to G. Provide the order of visit for each node.
S
O
G
O
Write a short essay (1 short paragraph for each) to answer the questions in the space provided below.
a) Explain Backtracking and how it is used in the 8-Queens puzzle solution.
b) Explain Branch and Bound Search and its purpose.
c) Define Hill Climbing and why is it used in search algorithms to get from a start to a goal node?
d) Explain how Hill climbing can be used in the underestimate of a path from start to goal to obtain the optimal search path? (
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Artificial Intelligence: A Modern Approach
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- Consider the search problem represented in Figure, where a is the start node and e is the goal node. The pair [f, h] at each node indicates the value ofthe f and h functions for the path ending at that node. Given this information, what is the cost ofeach path?1. The cost < a, c >= 2 is given as a hint.2. Is the heuristic function h admissible? Explainarrow_forwardGive the sequence of S and G boundary sets computed by the CANDIDATE-ELIMINATION algorithm if it is given the sequence of training examples from Table 2.1 in reverse order. Although the final version space will be the same regardless of the sequence of examples (why?), the sets S and G computed at intermediate stages will, of course, depend on this sequence. Can you come up with ideas for ordering the training examples to minimize the sum of the sizes of these intermediate S and G sets for the H used in the EnjoySport example?arrow_forwardPlease answer the following question in detail. With all the proofs and assumptions explained. We have seen various search strategies in class, and analyzed their worst-case running time. Prove that any deterministic search algorithm will, in the worst case, search the entire state space. More formally, we define a search problem by a finite set of states S, a set of actions A, and a cost function c : S ×A×S → {1,∞} (i.e. the cost is uniform, but some states cannot be reached from some others, or equivalently have a cost of ∞), and a starting state s0 ∈ S. We assume that every state s ∈ S is reachable from s0, i.e. that there is a sequence of actions one can take from state s0 such that one reaches the state s after performing this sequence of actions, and such that the cost of reaching s is finite. Prove the following theorem. Theorem1. LetAlgbesomecomplete, deterministic uninformed search algorithm. Then for any search problem defined as above, there exists some choice of a state…arrow_forward
- For a double knapsack problem, assume follwoing 2 algorithms used. (1) Use the regular subset sum knapsack algorithm to pick a maximum-value solution S1 that fits in the first knapsack, and then use it again on the remaining items to pick a maximum-value solution S2 that fits in the second knapsack. (2) Use the Knapsack algorithm to pick a maximum-value solution S that would fit in a knapsack with capacity C1+C2, then partition S arbitrarily into two sets S1 and S2 with total sizes at most C1 and C2, respectively. Which of the following statements are true? (Choose all that apply.) a) Algorithm (1) is guaranteed to produce an optimal solution to the double-knapsack problem but algorithm (2) is not. b) Algorithm (2) is guaranteed to produce an optimal solution to the double-knapsack problem but algorithm (1) is not. c) Algorithm (1) is guaranteed to produce an optimal solution to the double-knapsack problem when C1=C2. d) Neither algorithm is guaranteed to produce an…arrow_forwardImplement the algorithm for an optimal parenthesization of a matrix chain product as dis-cussed in the class.Use the following recursive function as part of your program to print the outcome, assumethe matrixes are namedA1, A2, ..., An.PRINT-OPTIMAL-PARENS(s, i, j){if (i=j) thenprint “A”i else{print “(”PRINT-OPTIMAL-PARENS(s,i,s[i, j])PRINT-OPTIMAL-PARENS(s, s[i, j] + 1, j)print “)”} }a- Test your algorithm for the following cases:1. Find and print an optimal parenthesization of a matrix-chain product whose sequenceof dimensions is<5,10,3, X,12,5,50, Y,6>.2. Find and print an optimal parenthesization of a matrix-chain product whose sequenceof dimensions is<5,10,50,6, X,15,40,18, Y,30,15, Z,3,12,5>. 3. Find and print an optimal parenthesization of a matrix-chain product whose sequenceof dimensions is<50,6, X,15,40,18, Y,5,10,3,12,5, Z,40,10,30,5>. X=10 Y=20 Z=30arrow_forwardConsider the use of a genetic algorithm on this 0-1 Knapsack Problem W = 19 P1 = 20, w1 = 2 P2 = 30, w2 = 6 P3 = 36, w2 = 9 P4 = 16, w3 = 8 If an individual in the population is given the string: "0,1,0,1" then the measure of fitness would be?arrow_forward
- We are given the following training examples in 2D ((-3,5), +), ((-4, –2), +), ((2,1), -), ((4,3),–) Use +1 to map positive (+) examples and -1 to map negative (-) examples. We want to apply the learning algorithm for training a perceptron using the above data with starting weights wo = w1 = w2 = 0 and learning rate 7 = 0.1. In each iteration process the training examples in the order given above. Complete at most 3 iterations over the above training examples. What are the weights at the end of each iteration? Are these weights finalarrow_forwardConsider the Trisection method, which is analogous to the Bisection method except that at each step, the interval is subdivided into 3 equal subintervals, instead of 2; then takes a subinterval where the function values at the endpoints are of opposite signs. Is the Trisection method guaranteed to converge if for the initial interval [a, b], we havef(a)f(b) < 0? Why or why not? Considering computational cost (e.g. function evaluations, oating point operations), would you prefer Bisection or Trisection method? Explain.arrow_forwardI need a solution to this question quickly -Given the following search tree, apply the Expected Minmax algorithm to it and show the search tree that would be built by this algorithm. -Who is the winner player and what is his final utility value?arrow_forward
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