Introduction to Algorithms
3rd Edition
ISBN: 9780262033848
Author: Thomas H. Cormen, Ronald L. Rivest, Charles E. Leiserson, Clifford Stein
Publisher: MIT Press
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Chapter 35.2, Problem 5E
Program Plan Intro
To show that the optimal tour never crosses itself in a travelling salesmen problem of a given vertices with Euclidean distance.
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Using the image provided, please answer the following questions.
(a). Find a path from a to g in the graph G using the search strategy of depth-first search. Is the returned solution path an optimal one? Give your explanation and remarks on "why-optimal" or "why-non-optimal".
(b). Find a path from a to g in the graph G using the search strategy of breadth-first search. Is the returned solution path an optimal one? Give your explanation and remarks on "why-optimal" or "why-non-optimal".(c). Find a path from a to g in the graph G using the search strategy of least-cost first search. Is the returned solution path an optimal one? Give your explanation and remarks on "why-optimal" or "why-non-optimal".
(d). Find a path from a to g in the graph G using the search strategy of best-first search. The heuristics for these nodes are: h(a,25); h(b, 43); h(c,5); h(d, 64); h(g, 0). Is the returned solution path an optimal one? Give your explanation and remarks on "why-optimal or "why-non-optimal".…
Let G be this weighted undirected graph, containing 7 vertices and 11 edges.
A
C
7
8
В
5
5
9.
15
E
D
8.
6
F
11
G
For each of the 10 edges that do not appear
(AC, AE, AF, AG, BF, BG, CD,CF,CG, DG), assign a weight of 1000. It is easy to
see that the optimal TSP tour has total cost 51.
Generate an approximate TSP tour using the algorithm from Questlon 5, and calculate the total
cost of your solution. Explain why your solution is not a 2-approximation.
Suppose we have the following undirected graph, and we know that the two
bolded edges (B-E and G-E) constitute the global minimum cut of the graph.
1. If we run the Karger’s algorithm for just one time to find the global minimum cut, what is the probability for the algorithm to find the minimum cut correctly? Please show your reasoning process.
2. How many times do we need to run the Karger’s algorithm if we want to guarantee that the probability of success is greater than or equal to
0.95, by “success” we mean that there is at least one time the Karger’s algorithm correctly found the minimum cut. Please show your reasoning process. [You do not have to work out the exact value of a logarithm]
Chapter 35 Solutions
Introduction to Algorithms
Ch. 35.1 - Prob. 1ECh. 35.1 - Prob. 2ECh. 35.1 - Prob. 3ECh. 35.1 - Prob. 4ECh. 35.1 - Prob. 5ECh. 35.2 - Prob. 1ECh. 35.2 - Prob. 2ECh. 35.2 - Prob. 3ECh. 35.2 - Prob. 4ECh. 35.2 - Prob. 5E
Ch. 35.3 - Prob. 1ECh. 35.3 - Prob. 2ECh. 35.3 - Prob. 3ECh. 35.3 - Prob. 4ECh. 35.3 - Prob. 5ECh. 35.4 - Prob. 1ECh. 35.4 - Prob. 2ECh. 35.4 - Prob. 3ECh. 35.4 - Prob. 4ECh. 35.5 - Prob. 1ECh. 35.5 - Prob. 2ECh. 35.5 - Prob. 3ECh. 35.5 - Prob. 4ECh. 35.5 - Prob. 5ECh. 35 - Prob. 1PCh. 35 - Prob. 2PCh. 35 - Prob. 3PCh. 35 - Prob. 4PCh. 35 - Prob. 5PCh. 35 - Prob. 6PCh. 35 - Prob. 7P
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- Run the 2-approximation algorithm for the Traveling Salesman problem on the following graph. First, specify a minimal spanning tree in the graph. Then make it clear at which node you start the tour and in which order the edges are considered and skipped if necessary. Then give the cost of the computed tour. Also determine the optimal tour cost and state the factor by which the tour found by the approximation algorithm is more expensive.arrow_forwardLet G be this weighted undirected graph, containing 7 vertices and 11 edges. For each of the 10 edges that do not appear (AC, AE, AF, AG, BF, BG, CD, CF, CG, DG), assign a weight of 1000. It is easy to see that the optimal TSP tour has total cost 51. Generate an approximate TSP tour using the algorithm, and calculate the total cost of your solution. Explain why your solution is not a 2-approximation.arrow_forwardConsider the following two-person game on a general (not necessarily bipartite) graph G = (V, E). Players A and B alternate and each selects a (yet unchosen) edge e of the graph so that e together with the previously selected edges form a simple path. The first player unable to select such an edge loses. Show that if G has a perfect matching then player A has a winning strategy.arrow_forward
- Suppose that you want to get from vertex s to vertex t in an unweighted graph G = (V, E), but you would like to stop by vertex u if it is possible to do so without increasing the length of your path by more than a factor of α. Describe an efficient algorithm that would determine an optimal s-t path given your preference for stopping at u along the way if doing so is not prohibitively costly. (It should either return the shortest path from s to t or the shortest path from s to t containing u, depending on the situation)arrow_forwardComputer Science In a triangle graph, the spanning-tree-doubling approximation to the TSP (in the version described in class for finding a walk that touches every vertex at least once, without shortcuts) will always produce a four-edge walk, even for inputs where there is a shorter three-edge cycle. Find weights for the edges of this graph for which the ratio of (solution found by the approximation)/(optimal solution) is as large as possible. What is this ratio for your weights?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_forward
- Given a directed graph w/ nonnegative edge lengths & two distinct nodes, a and z. Let X denote a shortest path from a to z. If we add 10 to the length of every edge in the graph, then: i) X definitely remains a shortest a-z path. ii) X definitely does not remain a shortest a-z path. iii) X might or might not remain a shortest a-z path (depending on the graph). iv) If X has only one edge, then X definitely remains a shortest a-z path. O i and iv only O iv only O None of the choices O i only O i and iv only O iii only O ii onlyarrow_forwardou are given a directed graph G = (V, E) and two vertices s and t. Moreover, each edge of this graph is colored either blue or red. Your goal is to find whether there is at least one path from s to t such that all red edges in this path appear after all blue edges (the path may not contain any blue edges or any red edges, but if it has both types of edges, all red edges should appear after all blue edges). Design and analyze an algorithm for solving this problem in O(n + m) time.arrow_forwardShow that this version of the choice problem is NP-complete; Does the graph G have a spanning tree in which the biggest cost paid by any point is less than the goal cost c?arrow_forward
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