Database System Concepts
7th Edition
ISBN: 9780078022159
Author: Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
Publisher: McGraw-Hill Education
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Transcribed Image Text:Iterative Improvement
Apply the shortest-augmenting path Ford-Fulkerson algorithm to find a
maximum flow in the following network:
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- Hello, assistance with this question would be greatly appreciated. Thank you!!arrow_forwardusa java coding in needed to explain asap, please be clear, thanksarrow_forwardA Hamiltonian path on a directed graph G = (V, E) is a path that visits each vertex in V exactly once. Consider the following variants on Hamiltonian path: (a) Give a polynomial-time algorithm to determine whether a directed graph G contains either a cycle or a Hamiltonian path (or both). Given a directed graph G, your algorithm should return true when a cycle or a Hamiltonian path or both and returns false otherwise. (b) Show that it is NP-hard to decide whether a directed graph G’ contains both a cycle and a Hamiltonian Path, by giving a reduction from the HAMILTONIAN PATH problem: given a graph G, decide whether it has a Hamiltonian path. (Recall that the HAMILTONIAN PATH problem is NP-complete.)arrow_forward
- Show the distance matrix D(k) for 0 ≤ k ≤ n that results from applying the Floyd-Warshall algorithm for the graph shown in Figure 3.arrow_forwardSuppose you are given a connected undirected weighted graph G with a particular vertex s designated as the source. It is also given to you that weight of every edge in this graph is equal to 1 or 2. You need to find the shortest path from source s to every other vertex in the graph. This could be done using Dijkstra’s algorithm but you are told that you must solve this problem using a breadth-first search strategy. Design a linear time algorithm (Θ(|V | + |E|)) that will solve your problem. Show that running time of your modifications is O(|V | + |E|). Detailed pseudocode is required. Hint: You may modify the input graph (as long as you still get the correct shortest path distances).arrow_forwardFind the all pair shortest paths matrix for the above graph using the Floyd-Warshall algorithm. Show the 5 matrixes that will be generated by the algorithm in each step.arrow_forward
- Write a pseudocode to find all pairs shortest paths using the technique used in Bellman-Ford's algorithm so that it will produce the same matrices like Floyd-Warshall algorithm produces. Also provide the algorithm to print the paths for a source vertex and a destination vertex. For the pseudocode consider the following definition of the graph - Given a weighted directed graph, G = (V, E) with a weight function wthat maps edges to real-valued weights. w(u, v) denotes the weight of an edge (u, v). Assume vertices are labeled using numbers from1 to n if there are n vertices.arrow_forwardDesign and Analysis of Algorithms Solve the all-pairs shortest-path problem for the digraph with the given weight matrix DO (find the distance of the shortest path between every pair of vertices). DO = 0 2 5 1 6 0 2 4 4 30 1 1 3 2 0arrow_forwardGive code to a Java example to solve Dijkstra's Shortest Path Algorithm using Adjacency Matrixarrow_forward
- Prove that The number of augmenting paths needed in the shortest-augmenting-path implementation of the Ford-Fulkerson maxflow algorithm for a flow network with V vertices and E edges is at most EV /2.arrow_forwardCan you provide me the answers in a simple way possible? Thank youarrow_forwardThe following solution designed from a problem-solving strategy has been proposed for finding a minimum spanning tree (MST) in a connected weighted graph G: Randomly divide the vertices in the graph into two subsets to form two connected weighted subgraphs with equal number of vertices or differing by at most Each subgraph contains all the edges whose vertices both belong to the subgraph’s vertex set. Find a MST for each subgraph using Kruskal’s Connect the two MSTs by choosing an edge with minimum wight amongst those edges connecting Is the final minimum spanning tree found a MST for G? Justify your answer.arrow_forward
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