Deterministic PeArmutation (DP) AlgorithmInput: a positive integer n.Output: a permutation graph G[π] with ∆(G) = n 2 all of its vertices are at distance at most two apart to each other. write shortest algo....

Write a pseudocode to find all pairs shortest paths using the technique used in Bellman-Ford's algorithm so that it will produce the same matrices like Floyd-Warshall algorithm produces. Also provide the algorithm to print the paths for a source vertex and a destination vertex. For the pseudocode consider the following definition of the graph - Given a weighted directed graph, G = (V, E) with a weight function wthat maps edges to real-valued weights. w(u, v) denotes the weight of an edge (u, v). Assume vertices are labeled using numbers from1 to n if there are n vertices.

Dijkstra's shortest path algorithm is run on the graph, starting at vertex D. When a vertex is dequeued, 0 or more adjacent vertices' distances are updated. For each iteration of the while loop in Dijkstra's algorithm, find the vertex dequeued and the adjacent vertices updated. Enter updated vertices as A, B, C or "none" if no adjacent vertices are updated. 10 B 4 C 8 6 9 A LO 5 D E Iteration Vertex dequeued Adjacent vertices updated Ex: A, B, C or none 1 2 3 4 5 SHIRA Ex: C

The Floyd-Warshall algorithm is a dynamic algorithm for searching the shortest path in a graph. Each vertex pair has its assigned weight. You are asked to draw the initial directed graph and show the tables for each vertex from Mo to Ms by finding all the shortest paths. Below is the algorithm as a guide. Algorithm 1: Pseudocode of Floyd-Warshall Algorithm Data: A directed weighted graph G(V, E) Result: Shortest path between each pair of vertices in G for each de V do | distance|d][d] «= 0; end for each edge (s, p) € E do | distance[s][p] + weight(s, p); end n = cardinality(V); for k = 1 to n do for i = 1 to n do for j = 1 to n do if distancefi][j] > distance/i][k] + distance/k][j] then | distance i]lj] + distancefi|[k] + distance/k|[j]; end end end end Consider the relation R = {(1,4) =4, (2,1)=3, (2,5)=-3, (3,4)=2, (4,2)=1, (4,3)=1, (5,4)=2 } on A = (1,2,3,4,5) solve the Floyd-Warshall Algorithm.

A complete graph on n vertices Kn has 15 egdes. Find the following: (i) Number of vertices of Kn (ii) Sum of all vertices degree (iii) Degree of graph 7- 42 - 6 а. O b.5 - 20 - 5 10 - 90 - 9 О с. 9 - 72 - 8 d. 9 - 81 - 8 е. 6 - 30 - 5 O f. 8 - 72 - 9 g.

You are given a connected, undirected graph G. Devise an algorithm that produces a path that traverses each edge in G exactly once in each direction. A vertex may occur multiple times on the path. Provide a short justification about why your algorithm is correct, and analyze its efficiency.

give an example of a graph on n vertices where ford and fulkerson algorithm achieve its worst time complexity

Generate Random Graph • Contain 10 vertices • Contain 12 Edges • Every vertex has the random weight between 1 and 3 Define Related Vertex Cover Problem Solve it using Approximation Algorithm.

The minimum vertex cover problem is stated as follows: Given an undirected graph G = (V, E) with N vertices and M edges. Find a minimal size subset of vertices X from V such that every edge (u, v) in E is incident on at least one vertex in X. In other words you want to find a minimal subset of vertices that together touch all the edges. For example, the set of vertices X = {a,c} constitutes a minimum vertex cover for the following graph: a---b---c---g d e Formulate the minimum vertex cover problem as a Genetic Algorithm or another form of evolutionary optimization. You may use binary representation, OR any repre- sentation that you think is more appropriate. you should specify: • A fitness function. Give 3 examples of individuals and their fitness values if you are solving the above example. • A set of mutation and/or crossover and/or repair operators. Intelligent operators that are suitable for this particular domain will earn more credit. • A termination criterion for the…