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- Given the following directed graph (binary relation) R: 10 11 (a) Determine R*. (b) If vertices on R are considered as tasks, determine a sequence that accomplishes all tasks sequentially by finding a topological sort of Rt. (c) Assume that all tasks can be completed in parallel (with as many parallelism as required), how many steps are needed to complete all tasks? Show how the tasks are partitioned into blocks consisting of parallel tasks.P R O B L E M D E S C R I P T I O N :1. As described in the reading and lecture, an adjacency matrix for a graph with n verticesnumber 0, 1, …, n – 1 is an n by n array matrix such that matrix[i][j] is 1 (or true) ifthere is an edge from vertex i to vertex j, and 0 (or false) otherwise.2. In this assignment, you will implement the methods identified in the stub code below insupport of the Graph ADT that uses an adjacency matrix to represent an undirected,unweighted graph with no self-loops.import java.util.*; // for all needed JCF classespublic class Graph { private int[][] matrix;// the adjacency matrix of the graph. // Creates an n x n array with all values initialized to 0. public Graph(int n) {// your code here } // end constructor // This method returns the number of nodes in the graph. public int getNumVertices() {// your code here } // end getNumVertices // This method returns the number of edges in the graph. public int getNumEdges() {// your code here } // end getNumEdges //…The Algorithm of AlgebraThe adjacency matrix A of a graph G = is used by the algebraic BFS algorithm (V, E).First, we create a vector x with all zeros except for the index of the source vertex s that we wish to use as the starting point for the algorithm; next, we create the matrix A = A+ I; last, AT x chooses all nodes that are at a distance of 1 (level 1) from the source vertex. The vertices with the fewest number of hops are obtained by multiplying the vector x by the matrix A2. As a rule, the product Ak x will produce neighbours that are at most k hops distant, and the multiplication should be done in a boolean manner as in the algorithm shown below. Algorithm for Algebraic BFS1: Input : Adjacency matrix An,n of a graph G = (V, E) connected, unweighted graph G and asource vertex s2: Output : N, visited, levels a matrix that shows level i vertices at its column i, the visitedvertices in sequence and their levels3: x[n] ← 04: x[s] ← 15: A ← A + I6: for i = 1 to n do7: N ← AT · x8:…
- 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.Consider the graph in following. Suppose the nodes are stored in memory in a linear array DATA as follows: A, B, C, D, E, F, G, H, I, J, K, L, M Find the path matrix P of graph using powers of the adjacency matrix AAssume for an undirected graph with 6 vertices and 6 edges that a vertex index requires 3 bytes, a pointer requires 5 bytes, and that edge weights require 3 bytes. Since the graph is undirected, each undirected edge is represented by two directed edges. Calculate the byte requirements for an adjacency matrix.
- PROBLEM 2. Let G be an undirected graph whose vertices are the integers 1 through 8, and let the adjacent vertices of each vertex be given by the table below: Assume that, in a traversal of G, the adjacent vertices of a given vertex are returned in the same order as they are listed in the table above.a. Draw G.b. Give the sequence of vertices of G visited using a DFS traversal starting at vertex 1.c. Give the sequence of vertices visited using a BFS traversal starting at vertex 1.For the undirected graph G shown in Example 9.1 and reproduced here for convenience, undertake Breadth first search for the key 9, by refining the Breadth first traversal algorithm (10) (a) Graph G 1 2 ني | 4 5 6 7 8 9 10 5 4 7 2 1 1 3 2 6 6 9 8 5 4 10 3 10 5 8 (b) Adjacency list of Graph GDiscrete mathematics. Let G = (V, E) be a simple graph4 with n = |V| vertices, and let A be its adjacency matrix of dimension n × n. We want to count the L-cycles : such a cycle, denoted by C = u0u1 · · · uL with uL = u0 contains L distinct vertices u0, . . . , uL-1 et L edges E(C) = {uiui+1 | 0 ≤ i ≤ L − 1} ⊆ E. Two cycles are distinct if the edge sets are different : C = C' if and only if E(C) = E(C'). We define the matrices D, T, Q, the powers of A by matrix multiplication : D = A · A = A2, T = A · D = A3, Q = A · T = A4. Consider the values on the diagonals. Prove that for any vertex u ∈ V with degree d(u), d(u) = Du,u.
- Discrete mathematics. Let G = (V, E) be a simple graph4 with n = |V| vertices, and let A be its adjacency matrix of dimension n × n. We want to count the L-cycles : such a cycle, denoted by C = u0u1 · · · uL with uL = u0 contains L distinct vertices u0, . . . , uL-1 et L edges E(C) = {uiui+1 | 0 ≤ i ≤ L − 1} ⊆ E. Two cycles are distinct if the edge sets are different : C = C' if and only if E(C) = E(C'). We define the matrices D, T, Q, the powers of A by matrix multiplication : D = A · A = A2, T = A · D = A3, Q = A · T = A4. Consider the values on the diagonals. Prove that the number of 3-cycles N3 in the whole graph is N3 = 1/6 ∑ u∈V Tu,uQuestion 5 Refer to the following graph and choose the option that reflects the row for node 3 of the D(2) matrix generated by the Floyd-Warshall Algorithm. If you are solving this programmatically, be careful about zero-based indices in most languages. The graph below has 1-based indexing. 1 O 32042-8 3 O-42 327620 -5 Infinity O 87 Infinity 9 0 Infinity O 0204-1-8 10 5 -8In Computer Science a Graph is represented using an adjacency matrix. Ismatrix is a square matrix whose dimension is the total number of vertices.The following example shows the graphical representation of a graph with 5 vertices, its matrixof adjacency, degree of entry and exit of each vertex, that is, the total number ofarrows that enter or leave each vertex (verify in the image) and the loops of the graph, that issay the vertices that connect with themselvesTo program it, use Object Oriented Programming concepts (Classes, objects, attributes, methods), it can be in Java or in Python.-Declare a constant V with value 5-Declare a variable called Graph that is a VxV matrix of integers-Define a MENU procedure with the following textGRAPHS1. Create Graph2.Show Graph3. Adjacency between pairs4.Input degree5.Output degree6.Loops0.exit-Validate MENU so that it receives only valid options (from 0 to 6), otherwise send an error message and repeat the reading-Make the MENU call in the main…