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, Problem 6P
a.
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To give an example of a graph with at least4 vertices for which set of maximum weight edges
b.
Program Plan Intro
To give an example of a graph with at least4 vertices for which set of maximum weight edges
c.
Program Plan Intro
To prove that
d.
Program Plan Intro
To prove that
e.
Program Plan Intro
To give an
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Let G = (V, E) be an undirected graph and each edge e ∈ E is associated with a positive weight ℓ(e).For simplicity we assume weights are distinct. Is the following statement true or false?
Let T be a minimum spanning tree for the graph with the original weight. Suppose we replace eachedge weight ℓ(e) with ℓ(e)^2, then T is still a minimum spanning tree.
Let the graph G be a cycle of n nodes in which x edges have the weight 100 and y edges have weight 200. How many minimum spanning trees does G have?
Let G = (V, E) be an undirected graph with vertices V and edges E. Let w(e) denote the weight of e E E. Let T C E be a
spanning tree of G.
Select all of the following that imply that T is not a minimum spanning tree (MST) for G. Incorrect choices will be penalized.
There exists e'
(u, v) g T, u, v E V such that w(e') w(e').
O There exists e' g T such that w(e') w(e) for all e E E.
O There exists e'
(u, v) É T, u, v E V such that w(e') < w(e) for all e on the shortest path from u to v in T.
O There exists e E T, e' ¢ T with w(e) < w(e').
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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- Be G=(V, E)a connected graph and u, vEV. The distance Come in u and v, denoted by d(u, v), is the length of the shortest path between u'and v, Meanwhile he width from G, denoted as A(G), is the greatest distance between two of its vertices. a) Show that if A(G) 24 then A(G) <2. b) Show that if G has a cut vertex and A(G) = 2, then Ġhas a vertex with no neighbors.arrow_forwardLet G = (V, E) be a connected graph that has two distinct spanning trees. Prove that |E| > |V] – 1.arrow_forwardGiven an undirected graph G = <V,E>, a vertex cover is a subset of vertices S V such that for each edge (u,v) belongs to E, either u S or v S or both. The Vertex Cover Problem is to find minimum size of the set S. Consider the following algorithm to Vertex Cover Problem: (1) Initialize the result as {} (2) Consider a set of all edges in given graph. Let the set be E’. (3) Do following while E’ is not empty ...a) Pick an arbitrary edge (u,v) from set E’ and add u and v to result ...b) Remove all edges from E which are either incident on u or v. (4) Return result. It claim that this algorithm is exact for undirected connected graphs. Is this claim True or False? Justify the answer.arrow_forward
- 4. Let G (V, E) be a directed graph. Suppose we have performed a DFS traversal of G, and for each vertex v, we know its pre and post numbers. Show the following: (a) If for a pair of vertices u, v € V, pre(u) < pre(v) < post(v) < post(u), then there is a directed path from u to v in G. (b) If for a pair of vertices u, v € V, pre(u) < post(u) < pre(v) < post(v), then there is no directed path from u to v in G.arrow_forwardLet G = (V, E) be an undirected and connected graph, where each edge (u, v) E E has a weight wt(u, v) > 0. Moreover, assume that all weights are equal. (a) G. The running time of your algorithm must be faster than the running time of Kruskal's algorithm. Design an algorithm to compute a minimum spanning tree (MST) of You must describe your algorithm in plain English (no pseudocode). You must write the running time of your algorithm and explain why you get this running time. (b) In at most 50 words, explain why your algorithm is correct.arrow_forwardG = (V,E,W) is a weighted connected (undirected) graph where all edges have distinct weights except two edges e and e′ which have the same weight. Suppose there is a Minimum Spanning Tree of G containing both e and e′. Prove that G has a unique Minimum Spanning Tree.arrow_forward
- 1.6 If G is a simple graph of with number of vertices n > 0, show that deg(v) id(v) = od(v) = q. VEV VEVarrow_forwardGiven a graph that is a tree (connected and acyclic). (1) Pick any vertex v. (II) Compute the shortest path from v to every other vertex. Let w be the vertex with the largest shortest path distance. (III) Compute the shortest path from w to every other vertex. Let x be the vertex with the largest shortest path distance. Consider the path p from w to x. Which of the following are true a. p is the longest path in the graph b. p is the shortest path in the graph c. p can be calculated in time linear in the number of edges/vertices a,c a,b a,b,c b.carrow_forwardA complete graph G with 5 nodes has positive weight edges, each node has a distinct weight with an integer value and maximum weight is equal to number of edges in G. What can be the maximum weight of minimum spanning tree for graph G?arrow_forward
- Let G = (V, E) be a directed graph, and let wv be the weight of vertex v for every v ∈ V . We say that a directed edgee = (u, v) is d-covered by a multi-set (a set that can contain elements more than one time) of vertices S if either u isin S at least once, or v is in S at least twice. The weight of a multi-set of vertices S is the sum of the weights of thevertices (where vertices that appear more than once, appear in the sum more than once).1. Write an IP that finds the multi-set S that d-cover all edges, and minimizes the weight.2. Write an LP that relaxes the IP.3. Describe a rounding scheme that guarantees a 2-approximation to the best multi-setarrow_forwardLet G be a graph with n vertices. If the maximum size of an independent set in G is k, clearly explain why the minimum size of a vertex cover in G is n - k.arrow_forwardGiven a graph that is a tree (connected and acyclic). (I) Pick any vertex v.(II) Compute the shortest path from v to every other vertex. Let w be the vertex with the largest shortest path distance.(III) Compute the shortest path from w to every other vertex. Let x be the vertex with the largest shortest path distance. Consider the path p from w to x. Which of the following are truea. p is the longest path in the graphb. p is the shortest path in the graphc. p can be calculated in time linear in the number of edges/verticesarrow_forward
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