4. (i)Show how the analysis of the flows in a network with several sources and sinks can be reduced to the standard case by the addition of a new 'source vertex' and 'sink vertex'. (ii) Illustrate your answer to part (i) with reference to the network in the following figure. v 3 A 3 1 W 6 B W 2 5 3 v C W 7 5

4. (i)Show how the analysis of the flows in a network with several sources and sinks can be reduced to the standard case by the addition of a new 'source vertex' and 'sink vertex'. (ii) Illustrate your answer to part (i) with reference to the network in the following figure. v 3 A 3 1 W 6 B W 2 5 3 v C W 7 5

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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True or false. If true, give a brief explanation justifying the statement. If false, provide a counterexample. - If you are given a flow network (labeled F), let (L,R) be a minimum capacity cut in the flow graph. If the capacity of all of the graph edges is increased by 1, then (L,R) is still a minimum capacity cut in our new modified graph.
For any given connected graph, G, if many different spanning trees can be obtained, is there any method or condition setting that allows the DFS spanning tree of G to only produce a unique appearance? can you give me some simple opinion?
a) Now suppose the graph above is a residual graph after processing two augmenting paths. What were those paths and what were their flows? b) Draw the original directed acyclic graph and provide the maximum flow of this graph below. You may use either Ford-Fulkerson or Edmunds Karp to find the solution. Please show your work
Question 1: In graph theory, a graph X is a "complement" of a graph F if which of the following is true? Select one: a. If X is isomorph to F, then X is a complement of F. b. If X has half of the vertices of F (or if F has half of the vertices of X) then X is a complement of F. c. If X has the same vertex set as F, and as its edges ONLY all possible edges NOT contained in F, then X is a complement of F. d. If X is NOT isomorph to F, then X is a complement of F. Question 2: Which statement is NOT true about Merge Sort Algorithm: Select one: a. Merge Sort time complexity for worst case scenarios is: O(n log n) b. Merge Sort is a quadratic sorting algorithm c. Merge Sort key disadvantage is space overhead as compared to Bubble Sort, Selection Sort and Insertion Sort. d. Merge Sort adopts recursive approach
a) Draw the connected subgraph of the given graph above which contains only four nodes ACGB and is also a minimum spanning tree with these four nodes. What is its weighted sum? Draw the adjacency matrix representation of this subgraph (use boolean matrix with only 0 or 1, to show its adjacency in this case).b) Find the shortest path ONLY from source node D to destination node G of the given graph above, using Dijkstra’s algorithm. Show your steps with a table as in our course material, clearly indicating the node being selected for processing in each step.c) Draw ONLY the shortest path obtained above, and indicate the weight in each edge in the diagram. Also determine the weighted sum
Among all pairs of nodes in a directed network that are connected by an edge, half are connected in only one direction and the rest are connected in both directions. What is the reciprocity of the network?
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.
A network is considered to be biconnected if every pair of its vertices is linked by two distinct paths. A vertex that, if it and its surrounding edges were removed, would result in the graph becoming unconnected is known as an articulation point in a linked network. show any graph without articulation points that it is biconnected. Use the fact that none of the vertices on the path is an articulation point to construct two disjoint paths connecting s and t given a set of vertices s and t and a path connecting them.
Q1) how many graphs are there on 20 nodes? (To make this question precise, we have to make sure we known what it mean that two graphs are the same . For the purpose of this exercise,we consider the nodes given and labeled,say,asAlice ,Bob,...... The graph consisting of a single edge connecting Alice and Bob is different from the graph consisting of a single edge connecting Eve and frank.) Q2) Formulate the following assertion as a theorem about graphs and prove it :At every party one can find two people who know the same number of other people (like Bob and Eve in our first example).
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.
a) Show the residual graph for the network flow given in answer to part (c). b) What is the bottleneck edge of the path (?, ?3, ?4, ?) in the residual graph you have given in answer to part (d) ? c) Show the network with the flow that results from augmenting the flow based on the path (?, ?3, ?4, ?) of the residual graph you have given in answer to part (d).
Let’s consider the directed graph with unweighted links presented in Figure 4.12. Thisgraph is similar to the previous graph except by one missing link between A and E.Write networking code to describes how to create the new links dataset and then search for the existing cycles within the directed graph. ans in 20 min.or asap