1. (1)Use dynamic programming technique to solve the all-pairs shortest paths problem in a directed graph. ( So called Floyd-Warshall algorithm ) (Define the object function; state the recursive formula, initial condition, and answer. Give the time complexity and space complexity without explanation.) (2)Suppose that some of the edge weights are negative, what can you say about the algorithm given in part (1)? Do you still get the all-pairs shortest simple path correct? How do you detect the existence of a negative weight directed cycle? How do you find one such cycle?

1. (1)Use dynamic programming technique to solve the all-pairs shortest paths problem in a directed graph. ( So called Floyd-Warshall algorithm ) (Define the object function; state the recursive formula, initial condition, and answer. Give the time complexity and space complexity without explanation.) (2)Suppose that some of the edge weights are negative, what can you say about the algorithm given in part (1)? Do you still get the all-pairs shortest simple path correct? How do you detect the existence of a negative weight directed cycle? How do you find one such cycle?

Database System Concepts

7th Edition

ISBN:9780078022159

Author:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Publisher:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Chapter1: Introduction

Section: Chapter Questions

Problem 1PE

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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.
PLEASE TYPE ONLY IF NOT TYPE THAN DO NOT DO IT*** ONLY DO EXCERCISE B ON 8.4.3***** PLEASE DO ALL THREE PROBLEMS*** D AND E FOR 6.1.4 & B FOR 8.4.3 Exercise 6.1.4: Reasoning about common graphs. (d) For what value of n is Q2 = Cn? (e) Is Qn a regular graph for n ≥ 1? If so, what is the degree of the vertices in Qn?
A) Develop the Adjacency Matrix. B)List the nodes of the graph in alphabetical order using method 1 of breadth-first traversal starting from vertex F. C)List the nodes of the graph in alphabetical order using method 1 of depth-firsttraversal starting from vertex F.
PROBLEM 2. Let G be an undirected graph whose vertices are the integers 1 through 8, and let the adjacent vertices of each vertex be given by the table below: Assume that, in a traversal of G, the adjacent vertices of a given vertex are returned in the same order as they are listed in the table above.a. Draw G.b. Give the sequence of vertices of G visited using a DFS traversal starting at vertex 1.c. Give the sequence of vertices visited using a BFS traversal starting at vertex 1.
The following is a graph theory problem: Suppose we have a graph A = (X, Y) and a subgraph of A, which is expressed as B = (x, y). x is an element of X and y is an element of Y. A subgraph is plainly described as a graph obtained from A by removing vertices &/or edges. For every following statement determine whether it is true or false. Explain or provide counterexample. Be careful when answering the following questions to leave edges that have their endpoints in B. (i) If A is connected then B is connected (and vice versa: If B... then A) (ii) If A is simple graph then B is simple graph (and vice versa: If B... then A) (iii) If A contains a cycle then B contains a cycle (and vice versa: If B... then A) (iv) If A is a tree then B is a tree (and vice versa: If B... then A)
7. Consider the clique problem: given a graph G and a positive integer k, deter- mine whether the graph contains a clique of size k, i.e., a complete subgraph of k vertices. Design an exhaustive-search algorithm for this problem.
1. Let G = (V, E) be a DAG with weights associated with verticesand edges. Design a Dynamic Programming Algorithm for solvingthe longest path problem.
Please show written work with answer please!! A vertex cover of a graph is a set of vertices that includes at least one endpoint of every edge of the graph. Finding the minimum size vertex cover set is a NP problem, find an greedy approximation algorithm to find a close to optimum solution in polynomial time.
I want best algorithm for the problem. ALSO write its complexity and prove correctness of algorithm Given an undirected graph, with positive integer lengths c(e) on edges. Say that we replace c(s) by (c(2)^2) . Does this operation can make a shortest path between two vertees, into a path between vertices that is no longer a shortest path? Explain your answer precisely.
Define polynomial reduction in the context of NP completeness. Prove that clique is an NPC problem, given that independent set is a well-known NPC problem. Demonstrate the process with the help of a four vertices complete graph.
Problem 3: Complete Graph [a] Prove by mathematical induction that the complete graph Kn has n(n-1)/2 edges. [b] Is K4 planar? Explain why or why not. [c] Is K5 bi-partite? Explain why or why not. [d] Is K6 Eulerian? Explain why or why not.
Statement 1: The Floyd Warshal's Algorithm can be used to also detect negative weight cycles in a graph. Statement II: The Bellman Ford Algorithm can be modified to detect all pairs shortest paths.