Question 7.. We have computed the shortest path between all the nodes in a weighted graph. Then all the edges increase their costs by adding the same value c > 0. Will the shortest paths change? If your answer is NO, prove so, if the answer is YES, give an example of a graph where at least one shortest path changes.
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- A weighted graph consists of 5 nodes and the connections are described by: i. ii. iii. iv. Node 1 is connected with nodes 2 and 3 using weights 2 and 6 respectively.Node 2 is connected with nodes 3, 4 and 5 using weights 7, 3 and 4 respectively.Node 3 is connected with nodes 4 and 5 using weights -2 and 2 respectively.Node 4 is connected with node 5 using weight -1. Solve the problem for finding the minimum path to reach each nodes from the first nodeAn aircraft company has their flight data as shown in the table below, where a forward flight from A to B will take 4 miles and a return B to A will take 3 miles. A B C D A 4 3 1 B 3 3 C 3 3 3 D 2 5 2 11. With the above information provided, draw a graph for the data provided. Indicate the weights on them. 12. Produce the adjacency matrix for your graph drawn 13. Find the shortest path in your graph and show the vertices and edges SECTION C An artificial intelligence system was design to forecastthe financial trading market and predict change based on observed variables from the business environment. The variables to be observed are i. Population(A) ii. Poverty rate(B) iii. Inflation (C) iv. Available resources(D) Theses variables can affect each other or impact upon one another. Below is a table of the variables with their effect on each other. A B C C A 0.1 0.3 0.3 0.3 B 0.2 0.2 0.2 0.4 C 0.2 0.5 0.1 0.2 D 0.1 0.2 0.2 0.5 14. Draw the Markov chain state…Given a directed graph w/ nonnegative edge lengths & two distinct nodes, a and z. Let X denote a shortest path from a to z. If we add 10 to the length of every edge in the graph, then: i) X definitely remains a shortest a-z path. ii) X definitely does not remain a shortest a-z path. iii) X might or might not remain a shortest a-z path (depending on the graph). iv) If X has only one edge, then X definitely remains a shortest a-z path. O i and iv only O iv only O None of the choices O i only O i and iv only O iii only O ii only
- Given an undirected graph G=(V,E) with positive edge weights, solve the following problem. Let SG(u,v) signify the length of the shortest path between u and v in the graph G for any two nodes u and v. Is the following statement correct or incorrect? SG(u,v)8G(u,w) denotes SG" (u,v)8G" (u,w), where G" denotes the graph produced by doubling each of G's edge weights.Daniel and Ria are taking a road trip from Somerville to Vancouver (that’s in Canada). Because it’s a 52-hour drive, Daniel and Ria decide to switch off driving at each rest stop they visit; however, because Ria has a better sense of direction than Daniel, she should be driving both when they depart and when they arrive (to navigate the city streets). Given a route map represented as a weighted undirected graph G = (V, E, w) with positive edge weights, where vertices represent rest stops and edges represent routes between rest stops, devise an efficient algorithm to find a route (if possible) of minimum distance between Somerville and Vancouver such that Daniel and Ria alternate edges and Ria drives the first and last edge. Specify the space and time complexity.A weighted graph is given below. Compute the following metrics for this graph. You can use a calculator if necessary. 1) Eccentricity for each vertex; 2) Average length of the shortest paths from each vertex; 3) Count of triangles of each vertex; 4) Count of triples of each vertex; 5) Radius of the graph; 6) Diameter of the graph; 7) Average path length of the graph; 8) Characteristic path length of the graph; 9) Network transitivity of the graph; 10) Network density of the graph.
- We know that when we have a graph with negative edge costs, Dijkstra’s algorithm is not guaranteed to work. (a) Does Dijkstra’s algorithm ever work when some of the edge costs are negative? Explain why or why not. (b) Find an algorithm that will always find a shortest path between two nodes, under the assumption that at most one edge in the input has a negative weight. Your algorithm should run in time O(m log n), where m is the number of edges and n is the number of nodes. That is, the runnning time should be at most a constant factor slower than Dijkstra’s algorithm. To be clear, your algorithm takes as input (i) a directed graph, G, given in adjacency list form. (ii) a weight function f, which, given two adjacent nodes, v,w, returns the weight of the edge between them. For non-adjacent nodes v,w, you may assume f(v,w) returns +1. (iii) a pair of nodes, s, t. If the input contains a negative cycle, you should find one and output it. Otherwise, if the graph contains at least one…Consider the problem of finding the length of a "longest" path in a weighted, not necessarily connected, dag. We assume that all weights are positive, and that a "longest" path is a path whose edge weights add up to the maximal possible value. For example, for the following graph, the longest path is of length 15: 9. 9. (h) 2 3 Use a dynamic programming approach to the problem of finding longest path in a weighted dag.A weighted, directed graph is a suitable representation to represent the daily airline routes flown by a small airline. The airline have the following daily flights: - Three flights from Cape Town to Johannesburg, - Two flights from Johannesburg to Cape Town. - Four flights from Johannesburg to Durban. - Three flights from Durban to Johannesburg. - One flight from Johannesburg to George. - One flight from George to Johannesburg. Draw a graph that represents this problem
- We have a Directed Weighted Graph with positive edge weights. Let us think the current shortest path in the graph is p to q. Suppose we change each edge weight in the graph by taking cube root of each weight. Will the shortest path remain the same or will it change for the new graph? Give an argument for or counterexamples for this. Could someone please help me answer this with explanation and examples along with a graph diagram.Thank youThis is an algorithmic graph problem. Consider a set of movies M1, M2, ... , Mk. There is a set of customers, each one of which indicates the two movies they would like to see this weekend. Movies are shown on Saturday evening and Sunday evening. Multiple movies may be screened at the same time. You must decide which movies should be televised on Saturday and which on Sunday, so that every customer gets to see the two movies they desire. Is there a schedule where each movie is shown at most once? Design an efficient algorithm to find such a schedule if one exists.Given N cities represented as vertices V₁, V2,..., UN on an undirected graph (i.e., each edge can be traversed in both directions). The graph is fully-connected where the edge eij connecting any two vertices v; and vj is the straight-line distance between these two cities. We want to search for the shortest path from v₁ (the source) to VN (the destination). Assume that all edges have different values, and e₁, has the largest value among the edges. That is, the source and destination have the largest straight-line distance. Compare the lists of explored vertices when we run the uniform-cost search and the A* search for this problem. Hint: The straight-line distance is the shortest path between any two cities. If you do not know how to start, try to run the algorithms by hand on some small cases first; but remember to make sure your graphs satisfy the conditions in the question.