Consider the flow network G shown in figure 1 with source s and sink t. The edge capacities arethe numbers given near each edge.(a) Find a maximum flow in this network. Once you have done this, draw a copy of the originalnetwork G and clearly indicate the flow on each edge of G in your maximum flow.(b) Find a minimum s-t cut in the network, i.e. name the two (non-empty) sets of vertices thatdefine a minimum cut. Also, say what its capacity is.gC436db3Fig.1 A flow network, with sources and sink t. The numbers next to the edges are the capacities.

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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 Let G be an arbitrary flow network, with a source s, a sink t, and a positiveinteger capacity ceon every edge e. If f is a maximum s −t flow in G, then f saturates every edge out of s with flow (i.e., for all edges e out of s, we have f (e) = ce).
Suppose you were allowed to add a single edge to the given network, connecting one of nodes c or d to any one node that it is not currently connected to. Could you do this in such a way that now behavior A, starting from S and spreading with a threshold of 2/5, would reach all nodes? Give a brief explanation for your answer
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?
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.
Consider the following flow network. The figure describes a flow ? and the capacity of the edges: if (?, ?) appears next to an edge ?, then the capacity of the edge ?ₑ is ?, and the flow ?ₑ that goes through ? in ? is ?. For example, if ? = (?, ?₁), then ?ₑ = 2 and ?ₑ = 1. 1. Draw the residual network ? & of the above flow ?. [Draw a graph containing all the nodes, edges, and the values on the edges]. 2. Find an augmenting path that will increase the flow by 1. You only need to list the vertices in the path and indicate the resulting flow in the following figure (using the same notation as the above figure). 3. Find a minimum ?-? cut in the graph (where the weight of an edge is its capacity). Briefly justify why the cut you found is a minimum cut.
Consider the first image, a weighted directed network with nodes named A, B, C, D, E, F : Which of the following most accurately describes the connectedness of this network? Strongly Weakly Disconnected None of the What is the in-strength of node D? What is the out-strength of node C? How many nodes are in the largest strongly connected component? Consider the second network numbered from 1 to 6: Which of the following most accurately describes the connectedness of this network? Strongly Weakly Disconnected None of the When discussing path lengths on a weighted graph, one must first define how the weights are related to the length of a path between two nodes is then the sum of the distances of the links in that path. Consider the previous network and assume that the link weights represent distances. Using this distance metric, what is the shortest path between nodes 1 and 6?
Consider a network that is a rooted tree, with the root as its source, the leaves as its sinks, and all the edges directed along the paths from the root to the leaves. Design an efficient algorithm for finding a maximum flow in such a network. What is the time efficiency of your algorithm? Describe your algorithm step by step.
Consider the network depicted in Figure 1; suppose that each node starts with the behavior B, and each node has a threshold of q = 1/2 for switching to behavior A. Find a cluster of density greater than 1 − q = 1/2 in the part of the graph outside S that blocks behavior A from spreading to all nodes, starting from S, at threshold q
Consider an arbitrary connected undirected graph network, with unique identifiers for nodes. We run a deterministic distributed algorithm, such that each node retains an edge only to its lowest indexed neighbour. An edge is retained, if at least one of its two endpoints retains it. A node communicates to that neighbour with which it is retaining its edge, to that effect. Then, the retained sub network
g) Show the residual graph for the network flow given in answer to part (c). What is the bottleneck edge of the path (?, ?2, ?3, ?1, ?4, ?) in the residual graph you have given in answer to part (g) ? Show the network with the flow that results from augmenting the flow based on the path (?, ?2, ?3, ?1, ?4, ?) of the residual graph you have given in answer to part (g).
Develop an SP client that performs a sensitivity analysis on the edge-weighted digraph’s edges with respect to a given pair of vertices s and t: Compute a V-by-V boolean matrix such that, for every v and w, the entry in row v and column w is true if v->w is an edge in the edge-weighted digraphs whose weight can be increasedwithout the shortest-path length from v to w being increased and is false otherwise.
Suppose that the road network is defined by the undirected graph, where the vertices represent cities and edges represent the road between two cities. The Department of Highways (DOH) decides to install the cameras to detect the bad driver. To reduce the cost, the cameras are strategically installed in the cities that a driver must pass through in order to go from one city to another city. For example, if there are two cities A and B such that the path that goes from A to B and the path that goes from B to A must pass the city C, then C must install the camera. Suppose that there are m cities and n roads. Write an O (m + n) to list all cities where cameras should be installed.

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