Give a linear time algorithm via pseudo code that takes as input a directed acyclic graph G = (V, E) and two vertices u and v, that returns the number of simple paths from u to v in G. Your algorithm needs only to count the simple paths, not list them. Explain why your code runs in linear time.
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- 6. In Java create an algorithm for an undirected graph with n vertices and m edges that...- Takes as a parameter some integer k- Returns the maximum induced subgraph where each vertex of that subgraph has a degree greater than or equal to k (or returns null if no such subgraph exists)- Operates in O(n + m) timeLet 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.Computer Science Let V be a fixed set of n vertices. Consider a sequence of m undirected edges e1, e2, ..., em. For 1 ≤ i ≤ m, let Gi denote the graph with vertex set V and edge set Ei = {e1, ..., ei}. Let ci denote the number of connected components of Gi . Design an algorithm to compute ci for all i. Your algorithm should be asymptotically as fast as possible. What is the running time of your algorithm?
- We are given an undirected connected graph G = (V, E) and vertices s and t.Initially, there is a robot at position s and we want to move this robot to position t by moving it along theedges of the graph; at any time step, we can move the robot to one of the neighboring vertices and the robotwill reach that vertex in the next time step.However, we have a problem: at every time step, a subset of vertices of this graph undergo maintenance andif the robot is on one of these vertices at this time step, it will be destroyed (!). Luckily, we are given theschedule of the maintenance for the next T time steps in an array M [1 : T ], where each M [i] is a linked-listof the vertices that undergo maintenance at time step i.Design an algorithm that finds a route for the robot to go from s to t in at most T seconds so that at notime i, the robot is on one of the maintained vertices, or output that this is not possible. The runtime ofyour algorithm should ideally be O((n + m) ·T ) but you will…please send handwritten solution for Q3 part aSuppose we represent a graph G = (V,E) as an adjacency matrix. Give a simple Implementation via pseudo code of Prim's algorithm for this case that runs in 0(V²) time. Explain why your code the running time has the upper bound as 0(v²).
- The given inputs consist of two nodes (s, t) and a directed graph G = (V, E). In addition, each edge of the graph is either blue or red. The goal is to find a path from point s to point t such that red edges always follow blue edges. There need not be any red or blue borders on the route, but if there are, the red ones should follow the blue ones. Develop an algorithm that does the task in O(n + m) time and analyze its performance.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…Consider an undirected graph with n nodes and m edges. The goal is to find a path between two specified nodes u and v that maximizes the minimum weight of any edge along the path. Assume that all edge weights are positive and distinct. Design an algorithm to solve this problem with a time complexity of O(m log n).
- ou are given a directed graph G = (V, E) and two vertices s and t. Moreover, each edge of this graph is colored either blue or red. Your goal is to find whether there is at least one path from s to t such that all red edges in this path appear after all blue edges (the path may not contain any blue edges or any red edges, but if it has both types of edges, all red edges should appear after all blue edges). Design and analyze an algorithm for solving this problem in O(n + m) time.3. We are given a weighted undirected graph G containing exactly 10 cycles. Write an algorithm to compute the MST of G. Your algorithm should have a runtime of O(V + E).w describe the Python code for the 2-approximation algorithm. The main part of the code deals with deleting the randomly selected edge from the graph along with its neighboring edges. Input to the procedure VC_Approx is the incidence matrix B(G) of the graph G. The unmarked edges are held in the list edges and the endpoints of a selected edge are included in the list V C at each iteration. The incidence matrix B of the graph is input and we find adjacent edges to the matched edge by iterating through rows of B. Each found edge is deleted from the list edges and main while loop iterates until this list becomes empty. We check adjacent edges to the matched edge from this matrix by deleting adjacent edges at each endpoint.