Consider a general topology (that is, not the specific network shown above) and a synchronous version of the distance-vector algorithm. Suppose that at each iteration, a node exchanges its distance vectors with its neighbors and receives their distance vectors. Assuming that the algorithm begins with each node knowing only the costs to its immediate neighbors, what is the maximum number of iterations required before the distributed algorithm converges? Justify your answer

Let G = (V, E) be an undirected graph and each edge e ∈ E is associated with a positive weight ℓ(e).For simplicity we assume weights are distinct. Is the following statement true or false? Let P be the shortest path between two nodes s, t. Now, suppose we replace each edge weight ℓ(e) withℓ(e)^2, then P is still a shortest path between s and t.

One can manually count path lengths in a graph using adjacency matrices. Using the simple example below, produces the following adjacency matrix: A B A 1 1 B 1 0 This matrix means that given two vertices A and B in the graph above, there is a connection from A back to itself, and a two-way connection from A to B. To count the number of paths of length one, or direct connections in the graph, all one must do is count the number of 1s in the graph, three in this case, represented in letter notation as AA, AB, and BA. AA means that the connection starts and ends at A, AB means it starts at A and ends at B, and so on. However, counting the number of two-hop paths is a little more involved. The possibilities are AAA, ABA, and BAB, AAB, and BAA, making a total of five 2-hop paths. The 3-hop paths starting from A would be AAAA, AAAB, AABA, ABAA, and ABAB. Starting from B, the 3-hop paths are BAAA, BAAB, and BABA. Altogether, that would be eight 3-hop paths within this graph. Write a program…

4. For the graph below: a. A snow plow starts at vertex 0 and must return to vertex 0 using a route which visits each edge at least once and repeats a minimum number of edges. Find such a route and how many repeated edges does it have? Assume that the edges all have the length, say length 1.

Consider the case where a data point has more than one nearest center. Most implementations assign such a point to the nearest center with the smallest index. If, instead, you assign such a point to a nearest center selected uniformly at random, does the algorithm still converge? If not, why not

Implement Algorithm (highest label preflow push algorithm). Let N = (G,c,s,t) be a flow network, where G is a symmetric digraph given by incidence lists Av. Moreover, let Q be a priority queue with priority function d, and rel a Boolean variable.

Consider a dataset of N = 3 x 105 nodes and L = 2 × 105 links. Assuming that the data is well described by a random graph model would you expect to find a giant component in the network? Select one: O a. Yes O b. No

Lower bounds on parallel algorithms: You must solve a problem using a parallel algorithm on a given interconnection network. The complexity to solve the problem on a EREW PRAM is known. Can you solve the problem faster on your given interconnection network or any other interconnection network? Explain how and justify your answer

In the small world experiment, it was observed that anyone in the world could reach any other person in the world in about 5.2 intermediates (thus 6-degree of separation). This observation was made more than 50 years ago. The Internet should have shortened this average path distance. Suppose with the use of Internet and online social networks, every person in the world increases their number of friends by doubling it (2x). Please estimate the new degree of separation. Explain your estimation.

3. Suppose that we want to know which agents are connected by a walk of length two in a given network. That is, want to know the set of ordered pairs (i, j) such that there exists a walk of length 2 between any i and j. Define n as the number of nodes and m as the number of links. (a) One way to do this is to input the adjacency matrix A, calculate A2, then output all pairs (i, j) for which A(i) > 0. Defining scalar multiplication and addition as "basic operations," up to a constant c, how many "basic operations" are required to calculate A², in terms of n and m? (Hint: First think about how many operations are required to calculate each entry of A²2, then multiply by the number of entries.) (b) Suppose that we want to know the number of agents who are connected by a walk of length k in a given network, where k≥ 1. Up to a constant, how many basic operations are required to calculate Ak, in terms of n and m? (c) (Harder) We showed in class (Lecture 4) that a simple adaptation of…