Provide an algorithm with the same time complexity as Bellman-Ford so that: it sets dist [w] to −∞for all vertices w for which there is a vertex that belongs to a negative-weighted cycle on some path from the source vertex to w, and outputs False if there is no such vertex w.
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vertices w for which there is a vertex that belongs to a negative-weighted cycle on some path from the source
vertex to w, and outputs False if there is no such vertex w.
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- Given an undirected weighted graph G with n nodes and m edges, and we have used Prim’s algorithm to construct a minimum spanning tree T. Suppose the weight of one of the tree edge ((u, v) ∈ T) is changed from w to w′, design an algorithm to verify whether T is still a minimum spanning tree. Your algorithm should run in O(m) time, and explain why your algorithm is correct. You can assume all the weights are distinct. (Hint: When an edge is removed, nodes of T will break into two groups. Which edge should we choose in the cut of these two groups?)Consider the Minimum-Weight-Cycle Problem: Input: A directed weighted graph G (V, E) (where the weight of edge e is w(e)) and an integer k. Output: TRUE if there is a cycle with total weight at most k and FALSE if there is no cycle with total weight at most k. Remember, a cycle is a list of vertices such that each vertex has an edge to the next and the final vertex has an edge to the first vertex. Each vertex can only occur once in the cycle. A vertex with a self-loop forms a cycle by itself. (a) Assume that all edge weights are positive. Give a polynomial-time algorithm for the Minimum-Weight-Cycle Problem. For full credit, you should: - Give a clear description of your algorithm. If you give pseudocode, you should support it with an expla- nation of what the algorithm does. Give the running time of your algorithm in terms of the number of vertices n and the number of edges m. You do not need to prove the correctness of your algorithm or the correctness of your running time analysis.Provide an eficient algorithm that given a directed graph G with n vertices and m edges as input, finds the outdegree of each vertex in G. Note that outdegree of a vertex u is the number of edges directed from u to some other vertex v. Discuss the running-time of your algorithm and Provide an algorithm that given a directed graph G with n vertices and m edges as input, nds the indegree of each vertex in G. Note that indegree of a vertex u is the number of edges directed into u from some other vertex v. Discuss the running-time of your algorithm.
- Consider the Minimum-Weight-Cycle Problem: Input: A directed weighted graph G = :(V, E) (where the weight of edge e is w(e)) and an integer k. Output: TRUE if there is a cycle with total weight at most k and FALSE if there is no cycle with total weight at most k. Remember, a cycle is a list of vertices such that each vertex has an edge to the next and the final vertex has an edge to the first vertex. Each vertex can only occur once in the cycle. A vertex with a self-loop forms a cycle by itself. (a) Assume that all edge weights are positive. Give a polynomial-time algorithm for the Minimum-Weight-Cycle Problem. For full credit, you should: Give a clear description of your algorithm. If you give pseudocode, you should support it with an expla- nation of what the algorithm does. Give the running time of your algorithm in terms of the number of vertices n and the number of edges m. - You do not need to prove the correctness of your algorithm or the correctness of your running time…The Triangle Vertex Deletion problem is defined as follows: Given: an undirected graph G = (V, E) , with IVI=n, and an integer k>= 0. Is there a set of at most k vertices in G whose deletion results in deleting all triangles in G? (a) Give a simple recursive backtracking algorithm that runs in O(3^k * ( p(n))) where p(n) is a low-degree polynomial corresponding to the time needed to determine whether a certain vertex belongs to a triangle in G. (b) Selecting a vertex that belong to two different triangles can result in a better algorithm. Using this idea, provide an improved algorithm whose running time is O((2.562^n) * p(n)) where 2.652 is the positive root of the equation x^2=x+4Let G be a directed acyclic graph. You would like to know if graph G contains directed path that goes through every vertex exactly once. Give an algorithm that tests this property. Provide justification of the correctness and analyze running time complexity of your algorithm. Your algorithm must have a running time in O(|V | + |E|). Detailed pseudocode is required.
- An algorithm X of an undirected graph G returns, Vc. Vc is a set of vertices of G and every edge in G has at least one of its endpoints in Vc. In other words, the algorithm finds a set of the fewest vertices such that every edge includes at least one of the vertices of G in the set, Vc. The following is a pseudo-code of a proposed algorithm for the vertex cover problem using the degree of each vertex. Vc = { } //Vc is a set of vertices covering edges V' = All the vertices in G E' = All the edges in G While E' is not empty Find a vertex v of the highest possible degree in E' Add v to Vc. Remove vertex v from V' and remove all the edges that hit v from E' return Vc. Vc needs to cover all of the edges in G. This proposed algorithm is based on the idea that selecting a vertex including the most edges at each step of the while loop will result in covering the most edges. With that in mind, please answer the following three questions: What type (approach) of the algorithm is the…Given a graph that is a tree (connected and acyclic). (I) Pick any vertex v.(II) Compute the shortest path from v to every other vertex. Let w be the vertex with the largest shortest path distance.(III) Compute the shortest path from w to every other vertex. Let x be the vertex with the largest shortest path distance. Consider the path p from w to x. Which of the following are truea. p is the longest path in the graphb. p is the shortest path in the graphc. p can be calculated in time linear in the number of edges/verticesYou are given a connected, undirected graph G. Devise an algorithm that produces a path that traverses each edge in G exactly once in each direction. A vertex may occur multiple times on the path. Provide a short justification about why your algorithm is correct, and analyze its efficiency.
- (3) Question 3: Given an undirected graph G and an edge uv in G. Design an algorithm that runs in O(|E|+|V)) time that decides if there is a cycle that contains UV. (4) Question 4: Problem 3 in page 245 of (Erickson]. Hint: Use topological sort. O S PC D00 吕口 F3 000 F4 F5 F7 F8 F9 F10 F2 # $ & 一 3 4 5 6 7 8 9. W E R T Y U FIL S D F H J K L V В N M and command B.Given a graph G = (V, E), let us call G an almost-tree if G is connected and G contains at most n + 12 edges, where n = |V |. Each edge of G has an associated cost, and we may assume that all edge costs are distinct. Describe an algorithm that takes as input an almost-tree G and returns a minimum spanning tree of G. Your algorithm should run in O(n) time.Consider a weighted, directed graph G with n vertices and m edges that have integer weights. A graph walk is a sequence of not-necessarily-distinct vertices v1, v2, ... , Vk such that each pair of consecutive vertices Vi, Vi+1 are connected by an edge. This is similar to a path, except a walk can have repeated vertices and edges. The length of a walk in a weighted graph is the sum of the weights of the edges in the walk. Let s, t be given vertices in the graph, and L be a positive integer. We are interested counting the number of walks from s to t of length exactly L. Assume all the edge weights are positive. Describe an algorithm that computes the number of graph walks from s to t of length exactly L in O((n+ m)L) time. Prove the correctness and analyze the running time. (Hint: Dynamic Programming solution)