Suppose you are given an undirected, connected, weighted graph G. You want to find the heaviest...

Suppose you are given an undirected, connected, weighted graph G. You want to find the heaviest weight set of edges that you can remove from G so that G is still connected. Propose an optimal greedy algorithm for this problem and prove that it is optimal. (Hint: use matroid theory!).

Expert Solution

Colby Messinger answered 6 months ago

5. Suppose we are given both an undirected graph G with weighted edges and a minimum...

5. Suppose we are given both an undirected graph G with weighted edges and a minimum spanning tree T of G . (a) Describe an algorithm to update the minimum spanning tree when the weight of a single edge e is decreased. (b) Describe an algorithm to update the minimum spanning tree when the weight of a single edge e is increased. In both cases, the input to your algorithm is the edge e and its new weight; your algorithms...

You are given an undirected graph G = ( V, E ) in which the edge...

You are given an undirected graph G = ( V, E ) in which the edge weights are highly restricted. In particular, each edge has a positive integer weight from 1 to W, where W is a constant (independent of the number of edges or vertices). Show that it is possible to compute the single-source shortest paths in such a graph in O(E+V) time.

Using C++: Create a function to search an undirected weighted graph and find the highest weighted...

Using C++: Create a function to search an undirected weighted graph and find the highest weighted edge between two specific values. This should include a class declaration and the definition for all required members that are needed to support the search. NO NEED to code those members. You can assume any other data structures such as stack, heap, or linked list is defined (so you can use their standard functions without declaring or defining them).

Is it always possible to make an undirected graph with two connected components connected by adding...

Is it always possible to make an undirected graph with two connected components connected by adding a single edge? • Why or why not? (Proof or counter example) [if true, give a proof, if false, give counter example] • Does the same hold true for directed graphs (and strongly connected components rather than connected components)? (Proof or counter example) [if true, give a proof, if false, give counter example]

Construct a connected undirected graph with 25 or more nodes. - Draw your graph in a...

Construct a connected undirected graph with 25 or more nodes. - Draw your graph in a diagram. - Draw the adjacency matrix of your graph. - Draw the adjacency list of your graph. - Run the BFS algorithm on your graph and complete the chart discussed in class. Detailed steps, including the changes of node color, predecessor, etc., as discussed in class, are required.

# Problem Description Given a directed graph G = (V, E), find the number of connected...

# Problem Description Given a directed graph G = (V, E), find the number of connected components in G. # Input The graph has `n` vertices and `m` edges. There are m + 1 lines, the first line gives two numbers `n` and `m`, describing the number of vertices and edges. Each of the following lines contains two numbers `a` and `b` meaning there is an edge (a,b) belong to E. All the numbers in a line are separated by...

Given an undirected graph G=(V, E) with weights and a vertex , we ask for a...

Given an undirected graph G=(V, E) with weights and a vertex , we ask for a minimum weight spanning tree in G where is not a leaf (a leaf node has degree one). Can you solve this problem in polynomial time? Please write the proof.

Let G(V, E,w) be a weighted undirected graph, where V is the set of vertices, E...

Let G(V, E,w) be a weighted undirected graph, where V is the set of vertices, E is the set of edges, and w : E → R + is the weight of the edges (R + is the set of real positive numbers). Suppose T(G) is the set of all minimum spanning trees of G and is non-empty. If we know that the weight function w is a injection, i.e., no two edges in G have the same weight, then:...

Given a connected graph G with n vertices. We say an edge of G is a...

Given a connected graph G with n vertices. We say an edge of G is a bridge if the graph becomes a disconnected graph after removing the edge. Give an O(m + n) time algorithm that finds all the bridges. (Partial credits will be given for a polynomial time algorithm.) (Hint: Use DFS)

You are given an undirected graph A. The problem P asks, if there exists a cycle...

You are given an undirected graph A. The problem P asks, if there exists a cycle that contains more than half of the nodes in graph A? Is this problem in NP(nondeterministic polynomial time)? How is it related to the Hamiltonian cycle problem? Construct your example in python. Input: Graph A as either an edge list or adjacent list. Output: Yes or No. If the output is yes, display the cycle as a sequence of nodes.

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