Introduction to Algorithms
3rd Edition
ISBN: 9780262033848
Author: Thomas H. Cormen, Ronald L. Rivest, Charles E. Leiserson, Clifford Stein
Publisher: MIT Press
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Chapter 26.1, Problem 3E
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To show that there a maximum flow f in G such that
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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).
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 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
Chapter 26 Solutions
Introduction to Algorithms
Ch. 26.1 - Prob. 1ECh. 26.1 - Prob. 2ECh. 26.1 - Prob. 3ECh. 26.1 - Prob. 4ECh. 26.1 - Prob. 5ECh. 26.1 - Prob. 6ECh. 26.1 - Prob. 7ECh. 26.2 - Prob. 1ECh. 26.2 - Prob. 2ECh. 26.2 - Prob. 3E
Ch. 26.2 - Prob. 4ECh. 26.2 - Prob. 5ECh. 26.2 - Prob. 6ECh. 26.2 - Prob. 7ECh. 26.2 - Prob. 8ECh. 26.2 - Prob. 9ECh. 26.2 - Prob. 10ECh. 26.2 - Prob. 11ECh. 26.2 - Prob. 12ECh. 26.2 - Prob. 13ECh. 26.3 - Prob. 1ECh. 26.3 - Prob. 2ECh. 26.3 - Prob. 3ECh. 26.3 - Prob. 4ECh. 26.3 - Prob. 5ECh. 26.4 - Prob. 1ECh. 26.4 - Prob. 2ECh. 26.4 - Prob. 3ECh. 26.4 - Prob. 4ECh. 26.4 - Prob. 5ECh. 26.4 - Prob. 6ECh. 26.4 - Prob. 7ECh. 26.4 - Prob. 8ECh. 26.4 - Prob. 9ECh. 26.4 - Prob. 10ECh. 26.5 - Prob. 1ECh. 26.5 - Prob. 2ECh. 26.5 - Prob. 3ECh. 26.5 - Prob. 4ECh. 26.5 - Prob. 5ECh. 26 - Prob. 1PCh. 26 - Prob. 2PCh. 26 - Prob. 3PCh. 26 - Prob. 4PCh. 26 - Prob. 5PCh. 26 - Prob. 6P
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- For a directed graph G = (V,E) (source and sink in V denoted by s and t respec- tively) with capacities c: E→+, and a flow f: E→, the support of the flow f on G is the set of edges E:= {e E| f(e) > 0}, i.e. the edges on which the flow function is positive. Show that for any directed graph G = (V,E) with non-negative capacities e: Ethere always exists a maximum flow f*: E→+ whose support has no directed cycle.arrow_forwardSuppose 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.arrow_forwardAmong 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?arrow_forward
- 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.arrow_forwardConsider a directed graph G with a starting vertex s, a destination t, and nonnegative edge lengths. Under what conditions is the shortest s-t path guaranteed to be unique? a) When all edge lengths are distinct positive integers. b) When all edge lengths are distinct powers of 2. c) When all edge lengths are distinct positive integers and the graph G contains no directed cycles. d) None of the other options are correct.arrow_forwardTrue or false: let G be an arbitrary connected, undirected graph with a distinct cost c(e) on every edge e. suppose e* is the cheapest edge in G; that is, c(e*) <c(e) for every edge e is not equal to e*. Any minimum spanning tree T of G contains the edge e*arrow_forward
- Algorithm: Network Flow(Maximu Flow, Ford-Fulkerson) and Application of Flow (Minimum Cuts, Bipartite Matching) Consider a flow network and an arbitrary s, t-cut (S, T). We know that by definition s must always be on the S "side" of a cut and t is always going to be on the T "side" of the cut. Obviously, this is true for any cut. Now, consider minimum cuts. This is obviously still true for s and t, but what about other vertices in the flow network? Are there vertices that will always be on one side or the other in every minimum cut? Let's define these notions more concretely. • We say a vertex v is source-docked if v ∈ S for all minimum cuts (S, T). • We say a vertex v is sink-docked if v ∈ T for all minimum cuts (S, T). • We say a vertex v is undocked if v is neither source-docked nor sink-docked. That is, there exist minimum cuts (S, T) and (S 0 , T0 ) such that v ∈ S and v ∈ T' Give an algorithm that takes as input a flow network G and assigns each vertex to one of the three…arrow_forwardConsider 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.arrow_forwardg) 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).arrow_forward
- Let G = (V, E) be an undirected graph and each edge e ∈ E is associated with a positive weight ℓ(e).For simplicity we assume weights are distinct. Is the following statement true or false? Let P be the shortest path between two nodes s, t. Now, suppose we replace each edge weight ℓ(e) withℓ(e)^2, then P is still a shortest path between s and t.arrow_forwardFor 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?arrow_forwarda) 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).arrow_forward
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