Artificial Intelligence: A Modern Approach
3rd Edition
ISBN: 9780136042594
Author: Stuart Russell, Peter Norvig
Publisher: Prentice Hall
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Chapter 5, Problem 12E
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Minimax and alpha-beta algorithms for two-player, non-zero games
- The minimax
algorithm for non-zero-sum games works exactly as for multiplayer games. - The evaluation function is a
vector of values, one for each player...
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In a lecture the professor said that for every minimum spanning tree T of G there is an execution of the algorithm of Kruskal which delivers T as a result. ( Input is G).
The algorithm he was supposedly talking about is:
Kruskal()
Precondition. N = (G, cost) is a connected network with n = |V| node and m = |E| ≥ n − 1 edges.All edges of E are uncolored.
postcondition: All edges are colored. The green-colored edges together with V form one MST by N.
Grand Step 1: Sort the edges of E in increasing weight: e1 , e2, . . . , em
Grand step 2: For t = 0.1, . . . , m − 1 execute: Apply Kruskal's coloring rule to the et+1 edge
i dont really understand this statement or how it is done. can someone explain me what he meant?
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.
Suppose a biking environment consists of n ≥ 3 landmarks,which are linked by bike route in a cyclical manner. That is, thereis a bike route between landmark 1 and 2, between landmark 2 and 3,and so on until we link landmark n back to landmark 1. In the centerof these is a mountain which has a bike route to every single landmark.Besides these, there are no other bike routes in the biking environment.You can think of the landmarks and the single mountain as nodes, andthe bike routes as edges, which altogether form a graph G. A path is asequence of bike routes.What is the number of paths of length 2 in the graph in termsof n?What is the number of cycles of length 3 in the graph in termsof n?What is the number of cycles in the graph in terms of n?
Chapter 5 Solutions
Artificial Intelligence: A Modern Approach
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- n = 9. sign and Analysis of Algorithms (3rd ed.) [Levitin 2011-10-09].pdf ✓ 1 2 3 zoom O 4 O 5. 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? 1 3 4 5 6 8.1 Three Basic Examples 2 X O 5 XXX 2 O Xo X O Search Fri May 12 291 6. Rod-cutting problem Design a dynamic programming algorithm for the fol- lowing problem. Find the maximum total sale price that can be obtained by cutting a rod of n units long into integer-length pieces if the sale price of a piece i units long is p; for i = 1, 2, ..., n. What are the time and space efficiencies of your algorithm? 7. Shortest-path counting A chess rook can move horizontally or vertically to any square in the same row or in the same column of a chessboard. Find the number of shortest paths by which a rook can move…arrow_forwardA weighted graph consists of 5 nodes and the connections are described by: i. ii. iii. iv. Node 1 is connected with nodes 2 and 3 using weights 2 and 6 respectively.Node 2 is connected with nodes 3, 4 and 5 using weights 7, 3 and 4 respectively.Node 3 is connected with nodes 4 and 5 using weights -2 and 2 respectively.Node 4 is connected with node 5 using weight -1. Solve the problem for finding the minimum path to reach each nodes from the first nodearrow_forwardYou are given a graph G = (V, E) with positive edge weights, and a minimum spanning tree T = (V, E') with respect to these weights; you may assume G and T are given as adjacency lists. Now suppose the weight of a particular edge e in E is modified from w(e) to a new value w̃(e). You wish to quickly update the minimum spanning tree T to reflect this change, without recomputing the entire tree from scratch. There are four cases. In each case give a linear-time algorithm for updating the tree. Note, you are given the tree T and the edge e = (y, z) whose weight is changed; you are told its old weight w(e) and its new weight w~(e) (which you type in latex by widetilde{w}(e) surrounded by double dollar signs). In each case specify if the tree might change. And if it might change then give an algorithm to find the weight of the possibly new MST (just return the weight or the MST, whatever's easier). You can use the algorithms DFS, Explore, BFS, Dijkstra's, SCC, Topological Sort as…arrow_forward
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