(1) Show the 3-CNF formula for which R builds the following graph. P -P T S -P 0 (P^Q ^T) V (P^Q ^S) V (T ^Q ^S) O (P V QVT) A (PVQVS) A (TV Q V S) O (PAQAT) V (P AQ AS) V (TA QAS) O (PV QVT) A (PV QV S) A (TV QV S) -S S T -T S
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Let R be the reduction from 3-SAT to VERTEX-COVER as discussed in class.
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- 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.5 Consider the degree-4 LFSR given by x to the power of 4 space plus space x squared space plus space x space plus space 1. Assume that the LFSR is initialized with the string left parenthesis s subscript 0 s subscript 1 s subscript 2 s subscript 3 right parenthesis equals left parenthesis 0110 right parenthesis. How long is the cycle generated?What is the time and space complexity of this function? It is a dfs function which goes through every possible path from node to another without cycles. Time complexity could be either O(n!) or O(2^n), or is there another answer? What is the space complexity? def dfs(currency_pairs, source, target): graph = defaultdict(dict) for s1, s2, rate_val1, rate_val2 in currency_pairs: graph[s1][s2] = rate_val1 graph[s2][s1] = rate_val2 def backtrack(current, seen): if current == target: return 1 product = 0 if currentingraph: neighbors = graph[current] for neighbor in neighbors: if neighbor not in seen: seen.add(neighbor) product = max(product, graph[current][neighbor] * backtrack(neighbor, seen)) seen.remove(neighbor) return product return backtrack(source,…
- Let’s consider the directed graph with unweighted links presented in Figure 4.12. Thisgraph is similar to the previous graph except by one missing link between A and E.Write networking code to describes how to create the new links dataset and then search for the existing cycles within the directed graph. ans in 20 min.or asapConsider the Omega network and Butterfly network from p nodes in the leftmost column to p nodes in the rightmost column for some p=2^k. The Omega network is defined in Chapter 2 of the text book such that Si is connected to element S j if j=2i for or j=2i+1-p for See Chapter 2 in text book for its definition. The Butterfly network is an interconnection network composed of log p levels (as the omega network). In a Butterfly network, each switching node i at a level l is connected to the identically numbered element at level l + 1 and to a switching node whose number differs from itself only at the lth most significant bit. Therefore, switching node Si is connected to element S j at level l if j = i or j . Prove that for each node Si in the leftmost column and a node Sj in the rightmost column, there is a path from Si to Sj in the Omega network. Prove that for each node Si in the leftmost and a node Sj in the rightmost, there is a path from Si to Sj in the Butterfly network.(c) Suppose that you have access to the sorted weight list in an ascending order in O(1), which of the minimum cost spanning tree algorithms (Prim or Kruskal) would you use to achieve a better running time performance for graph ? Justify why your selected algorithm is better suited for the above graph. Show the final result of the minimum cost spanning tree that you have obtained.
- Given the following adjacency matrix for a undirected, weighted graph: undirected, weighted graph data Source Node Destination Node Weight 0 3 2 3 4 3 3 7 6 4 7 8 0 4 7 1 4 9 1 5 3 4 8 1 7 8 2 0 1 4 4 5 2 5 8 3 8 9 2 5 9 10 1 2 5 5 2 6 5 6 7 2 6 8 9 6 9 Calculate each of the following searches assuming we start at node 4. Breadth-first search Depth-first search Show the order that each search will produce assuming it chooses a node with a higher number first when there are multiple choices. Remember unordered graphs allow for traveling from source to destination and from destination to source so to save space, an adjacency matrix may only provide from node A to B instead of including node B to A as another edge in the list.Given the following adjacency matrix for a undirected, weighted graph: undirected, weighted graph data Source Node Destination Node Weight 0 3 2 3 4 3 3 7 6 4 7 8 0 4 7 1 4 9 1 5 3 4 8 1 7 8 2 0 1 4 4 5 2 5 8 3 8 9 2 5 9 10 1 2 5 5 2 6 5 6 7 2 6 8 9 6 9 Calculate each of the following searches assuming we start at node 4. Breadth-first search Depth-first search Show the order that each search will produce in a visual graph assuming it chooses a node with a higher number first when there are multiple choices. Remember unordered graphs allow for traveling from source to destination and from destination to source so to save space, an adjacency matrix may only provide from node A to B instead of including node B to A as another edge in the list.Please explain Some engineers at Google want you to implement a graph structure of the subway system in Boston showing the connections between each station.They are wondering what data structure to use to represent the graph and use the least amount of memory, given that most stations do not connect to many other stations. Which graph data structure representation should you use and why? Group of answer choices a. Adjacency List -- Because the complexity is O(V+E), the data structure is better at representing sparse graphs since their are fewer connections. b. Adjacency Matrix -- Because the complexity is O(V+E), the data structure is better at representing sparse graphs since their are fewer connections.
- A star configuration of a network represents its structure as a graph with an edge from a single, central node to each remaining node. A point-to-point configuration represents a network as a complete graph. Draw a picture of an example of each kind of configuration with four nodes, and use big-O notation to state the efficiency of adding or removing a given node in each type of configuration. You can assume for now that removing each edge is a constant-time operation. *PythonGiven the above weighted graph (undirected): a) Determine and draw a Minimum Spanning Tree (MST) of the given graph above. Indicate the minimum weighted sum. No steps required. Students may choose whichever algorithm to find the MST b) Find the shortest path from source node D to destination node G of the given graph above, using Dijkstra’s algorithm. Show your steps with a table as in our course material, clearly indicating the node being selected for processing in each step. c) Draw ONLY the shortest path obtained above, from source node D to destination node G, and indicate its weighted sum.Here is a graph between two different campuses comprised of several location Traverse the graph using Breadth first search