Theorem 26.4.1 Algorithm 4 builds an auxiliary graph G' = (V', E') in O(log m) time with O(m) processors and O(m) space on EREW PRAM.
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- COM. Given a graph G and a vertex x of G, present an O(m+n) time algorithm that uses DFS to compute and return the number of vertices that are reachable from x in G via a path; x is reachable from itself. Present a complete algorithmRecall the Floyd-Warshall algorithm. For this problem, we are interested in the number of paths between each pair of vertices i and j in a directed acyclic graph. Suppose we know the number of paths between each pair of vertices where we restrict the intermediate vertices to be chosen from 1, 2, . . . , k − 1, show how we can extend the result to allow vertex k as an intermediate vertex as well. To conclude what would its complexity be?Let G = (X ∪ Y, E) be a bipartite graph such that the vertices are partitioned into two groups Xand Y , and each edge has one end point in X and one end point in Y .A 2-1 generalized matching is a set of edges S ⊂ E satisfying the following two conditions:1. Every vertex in X belongs to at most two edges in S.2. Every vertex in Y belongs to at most one edge in S.Give an algorithm to find the size (number of edges) of maximum 2-1 generalized matching
- Given an undirected, weighted graph G(V, E) with n vertices and m edges, design an (O(m + n)) algorithm to compute a graph G1 (V, E1 ) on the same set of vertices, where E1 subset of E and E1 contains the k edges with the smallest edge weights , where k < m.- If an algorithm for a graph problem with |V | vertices and |E| edges has two stages where the first stage is O(|E|) and the second stage which follows after this is O(|V |) then what is the best big-O set for the algorithm overall in terms of |V | on sparse graphs? Consider sparse graphs to have |E| = O(|V |).2 Describe an algorithm (in words or pseudo code) for the following computationalproblem:Input: A weighted undirected graph G = (V;E) with weights w(u; v) for every edge(u; v) and an edge e = (u1; v1) in the graph G (assume that different edges in G havedifferent weights).Output: Yes if e is in the minimum spanning tree of G; Otherwise, output No.
- please answer both of the questions. 7. The Bellman-Ford algorithm for single-source shortest paths on a graph G(V,E) as discussed in class has a running time of O|V |3, where |V | is the number of vertices in the given graph. However, when the graph is sparse (i.e., |E| << |V |2), then this running time can be improved to O(|V ||E|). Describe how how this can be done.. 8. Let G(V,E) be an undirected graph such that each vertex has an even degree. Design an O(|V |+ |E|) time algorithm to direct the edges of G such that, for each vertex, the outdegree is equal to the indegree. Please give proper explanation and typed answer only.Construct an example of a graph on n vertices such that running this greedy algorithm on the vertices in one order yields an independent set of size 1, and processing the vertices in a different order yields an independent set of size n−1, which is maximal.Let G be an undirected edge-weighted graph with V vertices and O(V2) edges. Describe an efficient algorithm to compute a minimum spanning tree that runs in O(V2) time. You are only allowed to use data structures such as arrays and priority queues. Do not use complex data structures such as Fibonacci heaps. Give an efficient algorithm (to the best of your knowledge) and a formal proof of correctness for your algorithm. An answer without any formal proof will be considered incomplete. Analyze the running time of the algorithm. You must write down your algorithm as pseudocode
- 3 Question 4 Let G=(V,E) be undirected graph and: (u,v)∈E. (A) Build an algorithm in complexity O (V + E) that finds a spanning tree of G that includes the (u, v). (B) Build an efficient algorithm that finds all the ribs of G that are common to all the spanning trees of G.We discussed how Prim’s algorithm runs on a graph G = (V,E)in O(|V|2). However, we also noted that an alternative implementation exists using heaps that runs in O(|E| log |V|). You will need to execute Prim’s algorithm on a large, complete graph. Which of the two implementations should you choose based solely on the Big-Oh complexity analyses listed above? You must justify your answer using Big-Oh notation in terms of |E|and |V|.Please state if the next two statements about an undirected and connected graph P are true or false. Justify your answer. (a) The shortest path between two nodes is always part of some minimum spanning tree. (b)Prim’s algorithm will work properly if P has negative edge weights.