Floyd-Warshall Algorithm.
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- Write down the tensor expression for the following matrix operations. Where A,B are 3 × 3, C is 3 × 4, and D is 4 × 7 matrices. And for a matrix M, its i-th row j-th column component will be denoted as M_ij . (a) det(A · B^T ).(b) Tr[A · B].(c) B · C · D, and explicitly spell out all the index summation.(d)δ_ii =?(e) ϵ_ijkϵ_ijk =?The highway distance between 6 cities named (A ... G) are illustrated in the following adjacency matrix:A B C D E FA 0 7 19 28B 7 0 10 18 40 C 19 10 0 16 17D 18 16 0 14 10E 40 14 0 12F 28 17 10 12 a) Draw the graph that represent such adjacency matrix b) List the right sequence of nodes traversed by the DFS and BFS algorithm starting from node A. c) Does this graph possess a Euler circuit/path? , why? If any of them does not exist, how the graph can be modified to have one? d) Draw the minimum spanning tree of this graph . e) Use the Dijkstra's algorithm to determine the shortest paths from city (A) to all other cities . Determine the shortest path and cost from node A to node E .[Hint: implement the algorithm step by step to show which node will be added in sequence] f) Determine the shortest paths between all pairs of nodes using Floyd-Warshall algorithm.Given and nxn matrix filled with either 0 or 1 your task is to turn as few 0 as possible into 1 such that every index meets the requirement that its neighbour (i+1,i-1,j+1,j-1) sum is even. return minimum no of zero you need to convert into 1, return -1 if no soltuion possible constraints : 1<=tc<=10 1<=n<=10 input: tc = 2 n = 3 1 1 1 1 1 1 0 0 0 n = 3 0 0 0 1 0 0 0 0 0 output : for first tc : -1 for second : 3
- (4) For the following Directed Graph, create an adjacency matrix. Then, via matrix multiplication, give me a COMPLETE list of all pairs of nodes that can reach one another in THREE moves. (Keep in mind when listing an ordered pair, the node you start from goes first, and the node you end on goes last.)Assume 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.Write a Java program that checks whether the relation of a matrix is reflexive, irreflexive, symmetric, anti-symmetric, asymmetric and transitive. (and get the square matrix)
- Create the matrix of the graph above and answer the following questions (If there isno arrow at the end of both sides of line, it is bidirectional).Answer the following questions by examining the matrix and its powers, show yourresults step by step clearly:(a) How many walks of length 2 are there from 2 to 4?(b) How many walks of length 2 are there from 3 to 5?(c) How many walks of length 3 are there from 0 to 5?(d) How many walks of length 3 are there from 2 to 4?What are the advantages of the Matrix Chain Multiplication algorithm? Why this algorithm is much effective for dynamic programming?Type in Latex **Problem**. Let $$A = \begin{bmatrix} .5 & .2 & .3 \\ .3 & .8 & .3 \\ .2 & 0 & .4 \end{bmatrix}.$$ This matrix is an example of a **stochastic matrix**: its column sums are all equal to 1. The vectors $$\mathbf{v}_1 = \begin{bmatrix} .3 \\ .6 \\ .1 \end{bmatrix}, \mathbf{v}_2 = \begin{bmatrix} 1 \\ -3 \\ 2 \end{bmatrix}, \mathbf{v}_3 = \begin{bmatrix} -1 \\ 0 \\ 1\end{bmatrix}$$ are all eigenvectors of $A$. * Compute $\left[\begin{array}{rrr} 1 & 1 & 1 \end{array}\right]\cdot\mathbf{x}_0$ and deduce that $c_1 = 1$.* Finally, let $\mathbf{x}_k = A^k \mathbf{x}_0$. Show that $\mathbf{x}_k \longrightarrow \mathbf{v}_1$ as $k$ goes to infinity. (The vector $\mathbf{v}_1$ is called a **steady-state vector** for $A.$) **Solution**. To prove that $c_1 = 1$, we first left-multiply both sides of the above equation by $[1 \, 1\, 1]$ and then simplify both sides:$$\begin{aligned}[1 \, 1\, 1]\mathbf{x}_0 &= [1 \, 1\, 1](c_1\mathbf{v}_1 +…
- In which of the following cases, linked list implementation of sparse matrices consumes the same memory space as the conventional way of storing the entire array? A. 5x6 matrix with 9 non-zero entries B. 5x6 matrix with 10 non-zero entries C. Efficient in accesing an entry D. Efficient if the sparse matrix is a band matrixThe Algebraic AlgorithmThe algebraic BFS algorithm uses the adjacency matrix A of a graph G = (V, E).We first initialize a vector x which consists of all zeros except the index of the sourcevertex s that we want to run the algorithm from; we then form the matrix A = A+ Iand then AT · x selects all nodes which are at distance 1 (level 1) from the sourcevertex. Multiplying the vector x with the matrix A2 gives the vertices that are at most2 hops. In general, the product Ak · x will give neighbors that are at most k hopsaway and the multiplication should be performed boolean as in Algorithm 7.2.Algorithm 7.2 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: levels[i] ← column i of N9: visited ← visited∪ {visited nodes…Computer Networks The book demonstrated that a poisoned reverse will prevent the count-to-infinity problem caused when there is a loop involving three directly connected nodes. However, other loops are possible. Will the poisoned reverse solve the general case count-to-infinity problem encountered by Bellman-Ford? Yes, the poisoned reverse will prevent a node from offering a path that includes preceding nodes in the loop. It will not, preceding nodes may still be used in the computation of the distance vector offered by a given node.