As we know, the time complexity of the BFS traversal on a graph with n vertices and m edges that is represented with the adjacency list structure is O(n + m). Calculate the time complexity in the case of a graph represented with adjacency matrix structure. Demonstrate the steps in your calculation.

Please help me with this practice problem in python : Implement two-level iterative method B = B_{TL} for graph Laplacian matrices. We want the symmetric B. Components: Given a graph, construct its graph Laplacian matrix. Then using Luby's algorithm, construct the P matrix that ensures a prescribed coarsening factor, e.g., 2, 4, or 8 times smaller number of coarse vertices. Since the graph Laplacian matrix is singular (it has the constants in its nullspace), to make it invertible, make its last row and columns zero, but keep the diagonal as it were (nonzero). The resulting modified graph Laplacian matrix A is invertible and s.p.d.. Form the coarse matrix A_c = P^TAP. To implement symmetric two-level cycle use one of the following M and M^T: (i) M is forward Gauss-Seidel, M^T - backward Gauss-Seidel (both corresponding to A) (ii) M = M^T - the ell_1 smoother. Compare the performance (convergence properties in terms of number of iterations) of B w.r.t. just using the smoother M…

Recall the Floyd-Warshall algorithm. For this problem, we are interested in the number of paths between each pair of vertices i and j in a directed acyclic graph. Suppose we know the number of paths between each pair of vertices where we restrict the intermediate vertices to be chosen from 1, 2, . . . , k − 1, show how we can extend the result to allow vertex k as an intermediate vertex as well. To conclude what would its complexity be?

1. Run DFS-with-timing on this graph G: give the pre and post number of each vertex. Whenever there is a choice of vertices to explore, always pick the one that is alphabetically first. 2. Draw the meta-graph of G. 3. What is the minimum number of edges you must add to G to make it strongly connected (i.e., it consists of a single connected component after adding these edges)? Give such a set of edges. b.

3. Kleinberg, Jon. Algorithm Design (p. 519, q. 28) Consider this version of the Independent Set Problem. You are given an undirected graph G and an integer k. We will call a set of nodes I "strongly independent" if, for any two nodes v, u € I, the edge (v, u) is not present in G, and neither is there a path of two edges from u to v. That is, there is no node w such that both (v, w) and (u, w) are present. The Strongly Independent Set problem is to decide whether G has a strongly independent set of size at least k.   Show that the Strongly Independent Set Problem is NP-Complete.

Hello, i have this Dijkstra’s exercise. could you help me to solve it?

Design an algorithm to find the maximum spanning tree  of any graph with a run time of O(m log m).

Think about the disadvantages of an adjacency list representation for a weighted graph.

Suppose you have a graph with 100 nodes and 500 edges and you want to find the shortest path between two nodes using Dijkstra's algorithm. What is the time complexity of this operation?

Write to code describes how to create the links dataset that represents the input graph. In this particular case, the links represent a directed network. The links dataset has only the nodes identification, which means, the from and to variables. The link weights in the transitive closure problem are irrelevant. In other words, it doesn’t matter the cost or the weight of the links, the algorithm searches for the possible paths to connect the nodes within the input graph. If there is a link or a set of links, no matter the weights, that connects node i to node j, that is the matter.