Artificial Intelligence: A Modern Approach
3rd Edition
ISBN: 9780136042594
Author: Stuart Russell, Peter Norvig
Publisher: Prentice Hall
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Chapter 5, Problem 13E
Explanation of Solution
Assertion
- Consider a MIN node whose children are terminal nodes.
- If MIN plays suboptimally, then the value of the node is greater than or equal to the value it would have if MIN played optimally...
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Students have asked these similar questions
* Simulate the node expansion from start state (s) to reach goal state (G)
- Using Geedy, A*
- Given heuristic function:
h(a) = 8.0
h(b) = 9.0
h(c) = 7.0
h(d) = 5.0
h(f) = 4.0
h(h) = 2.0
h(g) = 0
h(j) = 3.0
h(i) = 1.0
Develop the Branch and Bound tree for the following problem and find the optimal solution. For convenience, always select x1 as the branching variable at node 0.max ? = 2?1 + 3?2 s.t. 5?1 + 7?2 ≤ 354?1 + 9?2 ≤ 36 ?1, ?2 ≥ 0 and integer
we concretize the game tree algorithms with an iterator abstraction.In this exercise use the following state changing functions instead:move(v) Iterates all the possible moves from node v of playerlabel(v) and returns nil when they are exhausted.apply(v, m) Returns the successor node of v for move m.cancel(u, m) Yields the parent node of u when move m is taken back.You can assume that a move is reversible (i.e. v = cancel(apply(v, m), m))
Chapter 5 Solutions
Artificial Intelligence: A Modern Approach
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- Using Java Design an algorithm for the following operations for a binary tree BT, and show the worst-case running times for each implementation:preorderNext(x): return the node visited after node x in a pre-order traversal of BT.postorderNext(x): return the node visited after node x in a post-order traversal of BT.inorderNext(x): return the node visited after node x in an in-order traversal of BT.arrow_forwardWhat is the time complexity for these operation in average case (balanced tree), respectively? What is the time complexity for these operation in average case (balanced tree), respectively? In a BST with n nodes, what is the time complexity for searching, insertion, and deletion in worst case? What is the height of node 3, and node 9, respectively? What is the depth of node 3, and node 9, respectively? What is the level of node 17, and node 11, respectively? Insert integers 11, 22, 15, 44, 6, 9, 3, 33, 17, 5 into a BST in that order. Draw the final tree.arrow_forwardConsider the Fibonacci sequence F(0)=0, F(1)=1 and F(i)=F(i-1)+F(i-2) for i > 1. For the sake of this exercise we define the height of a tree as the maximum number of vertices of a root-to-leaf path. In particular, the height of the empty tree is zero, and the height of a tree with a single vertex is one. Prove that the number of nodes of an AVL tree of height h is at least F(h) and this inequality is tight only for two values of h.arrow_forward
- Q1) how many graphs are there on 20 nodes? (To make this question precise, we have to make sure we known what it mean that two graphs are the same . For the purpose of this exercise,we consider the nodes given and labeled,say,asAlice ,Bob,...... The graph consisting of a single edge connecting Alice and Bob is different from the graph consisting of a single edge connecting Eve and frank.) Q2) Formulate the following assertion as a theorem about graphs and prove it :At every party one can find two people who know the same number of other people (like Bob and Eve in our first example).arrow_forwardDraw the portion of the state space tree generated by LCBB for the following instances. n = 4, m = 15, (P₁, ..., P) = (10, 10, 12, 18) (w₁,..... W 4) = (2, 4, 6, 9).arrow_forwardTrue or False (If your answer to the question is "False", explain why, and provide correction when possible). (a) Let h(n) be the heuristics for the node n, h(m) be the heuristics for the node m, d(m,n) be the actual minimal cost from node m to n in a graph. A* satisfies the monotone restriction iff d(m,n) <= |h(n)-h(m)|. (b) If an A* heuristics is admissible then it satisfies the monotone restriction. (c) Best-first search guarantees optimality in its returned solution. (d) Least-cost-first search guarantees optimality in its returned solution. (e) If all edges are with unit cost, then Breadth-first search guarantees optimality in its returned solution.arrow_forward
- a) Draw the connected subgraph of the given graph above which contains only four nodes ACGB and is also a minimum spanning tree with these four nodes. What is its weighted sum? Draw the adjacency matrix representation of this subgraph (use boolean matrix with only 0 or 1, to show its adjacency in this case).b) Find the shortest path ONLY from source node D to destination node G of the given graph above, using Dijkstra’s algorithm. Show your steps with a table as in our course material, clearly indicating the node being selected for processing in each step.c) Draw ONLY the shortest path obtained above, and indicate the weight in each edge in the diagram. Also determine the weighted sumarrow_forwardWhat scenario is possible for a node-based situation in computer science. Please explainin set of 9 nodes, every node has 3 branches In a set of 9 nodes, one node has two branches, and the rest have exactly three.arrow_forward: Given a binary tree, your task is to determine its maximum value. Thefirst step is to decide how to create subinstances for your friends. As said, when theinput instance is a binary trees, the most natural subinstances are its left and rightsubtrees. Your friends must solve the same problem that you do. Hence, assume thatthey provide you with the maximum value within each of these trees. Luckily, themaximum value within a tree is either the maximum on the left, the maximum onthe right, or the value at the root. Our only job then is to determine which of thesethree is the maximum. As described in the beginning of Chapter 10, the maximum ofthe empty list is −∞.algorithm Max(tree) pre-cond: tree is a binary tree. post-cond: Returns the maximum of data fields of nodes.arrow_forward
- : Given a binary tree, your task is to determine its maximum value. Thefirst step is to decide how to create subinstances for your friends. As said, when theinput instance is a binary trees, the most natural subinstances are its left and rightsubtrees. Your friends must solve the same problem that you do. Hence, assume thatthey provide you with the maximum value within each of these trees. Luckily, themaximum value within a tree is either the maximum on the left, the maximum onthe right, or the value at the root. Our only job then is to determine which of thesethree is the maximum. As described in the beginning of Chapter 10, the maximum ofthe empty list is −∞.algorithm Max(tree) pre-cond: tree is a binary tree. post-cond: Returns the maximum of data fields of nodes.beginarrow_forwardApply the exhaustive parsing algorithm to determine whether or not bbaaaba ∈ L(G5), where G5 is this CFG: S ⟶ bSR | a (1) (2) R ⟶ aRb | a (3) (4) To illustrate the workings of the algorithm, show the breadth-first tree that it, in effect, traversed during its execution. (Each node of that tree is labeled by a sentential form.)arrow_forwardConsider the array t = [1, 2, 3, 4, 5, 8, 0 , 7, 6] of size n = 9, . a) Draw the complete tree representation for t. b) What is the index of the first leaf of the tree in Part a (in level order)? In general, give a formula for the index of the first leaf in the corresponding complete binary tree for an arbitrary array of size n. c) Redraw the tree from Part a after each call to fixheap, in Phase 1 of heapsort. Remember, the final tree obtained will be a maxheap. d) Now, starting with the final tree obtained in Part c, redraw the tree after each call to fixheap in Phase 2 of heap sort. For each tree, only include the elements from index 0 to index right (since the other elements are no longer considered part of the tree). e) For the given array t, how many calls to fixheap were made in Phase 1? How many calls to fixheap were made in Phase 2? f) In general , give a formula for the total number of calls to fixheap in Phase 1, when heapsort is given an arbitrary array of size n. Justify…arrow_forward
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