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 1E
Program Plan Intro
To show the splitting an edge in a flow network yields an equivalent network.
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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.
In order to plot a graph of f(x)=z and g(y)=z in the same graph, with t as a
parameter. The function used is
mesh(z) O
subplot(x,y,z) O
plot(x,y,z) O
plot3(x,y,z) O
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22
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VA
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Each edge is annotated with the current flow (initially zero) and the edge's
capacity. In general, a flow of x along an edge with capacity y is shown as x/y.
(a) Show the residual graph that will be created from this network with the
given (empty) flow. In drawing a residual graph, to show a forward edge
with capacity x and a backward edge with capacity y, annotate the original
edge *; y.
(b) What is the bottleneck edge of the path (S, V₁, V3, V5, t) in the residual
graph you have given in answer to part (a) ?
(c) Show the network with the flow (s, V₁, V3, V5, t) that results from
augmenting the flow based on the path of the residual graph you have
given in answer to part (a).
(d) Show the residual graph for the network flow given in answer to part (c).
(e) What is the bottleneck edge of the path (s, v3, v4, t) in the residual graph
you have given in answer to part (d) ?
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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- Suppose you are given a directed graph G = (V, E) with a positive integer capacity Ce on each edge e, a designated source s E V, and a designated sink tE V. You are also given an integer maximum s-t flow value fe on each edge e. Now suppose we pick a specific edge e belongs E and increase its capacity by one unit. Show how to find a maximum flow in the resulting capacitated graph in O(m + n), where m is the number of edges in G and n is the number on nodes.arrow_forwardOne can manually count path lengths in a graph using adjacency matrices. Using the simple example below, produces the following adjacency matrix: A B A 1 1 B 1 0 This matrix means that given two vertices A and B in the graph above, there is a connection from A back to itself, and a two-way connection from A to B. To count the number of paths of length one, or direct connections in the graph, all one must do is count the number of 1s in the graph, three in this case, represented in letter notation as AA, AB, and BA. AA means that the connection starts and ends at A, AB means it starts at A and ends at B, and so on. However, counting the number of two-hop paths is a little more involved. The possibilities are AAA, ABA, and BAB, AAB, and BAA, making a total of five 2-hop paths. The 3-hop paths starting from A would be AAAA, AAAB, AABA, ABAA, and ABAB. Starting from B, the 3-hop paths are BAAA, BAAB, and BABA. Altogether, that would be eight 3-hop paths within this graph. Write a program…arrow_forwardWe are given an undirected connected graph G = (V, E) and vertices s and t.Initially, there is a robot at position s and we want to move this robot to position t by moving it along theedges of the graph; at any time step, we can move the robot to one of the neighboring vertices and the robotwill reach that vertex in the next time step.However, we have a problem: at every time step, a subset of vertices of this graph undergo maintenance andif the robot is on one of these vertices at this time step, it will be destroyed (!). Luckily, we are given theschedule of the maintenance for the next T time steps in an array M [1 : T ], where each M [i] is a linked-listof the vertices that undergo maintenance at time step i.Design an algorithm that finds a route for the robot to go from s to t in at most T seconds so that at notime i, the robot is on one of the maintained vertices, or output that this is not possible. The runtime ofyour algorithm should ideally be O((n + m) ·T ) but you will…arrow_forward
- 1. Recall that a flow network is a directed graph G = (V, E) with a source s, a sink t, and a capacity function c: V x V + Rj that is positive on E and 0 outside E.We only consider finite graphs here. Also, note that every flow network has a maximum flow. This sounds obvious but requires a proof (and we did not prove it in the video lecture). Which of the following statements are true for all flow networks (G, s,t,c)? O IfG = (V,E) has as cycle then it has at least two different maximum flows. (Recall: two flows f, f' are different if they are different as functions V x V R. That is, if f(u, v) # f'(u, v) for some u, ve V. The number of maximum flows is at most the number of minimum cuts. The number of maximum flows is at least the number of minimum cuts. If the value of f is 0 then f(u, v) = 0 for all u, v. The number of maximum flows is 1 or infinity. The number of minimum cuts is finite. Need help, as you can see the checked boxes is not the right answer, something is missing…arrow_forwardLet f be an s,t-flow in an s, t-network D = (V, A) with capacities c : A → R>o, and assume that there is no augmenting path. Let X be the set of vertices that can be reached from s by unsaturated paths. Let (a, b) E A(X,V\ X). Explain why f(a,b) = c(a, b).arrow_forwardConsider a directed graph G = (V, E), and two distinct vertices u, v V. Recall that a set of U-V paths is non-overlapping if they have no edges in common among them, and a set C of edges disconnects from U if in the graph (V, E-C) there is no path from U to V. Suppose we want to show that for any set of non-overlapping paths P and any disconnecting set C, |P| ≤ |C|. Consider the proof that defines A = P, B = C and f(path q) = qC, and applies the Pigeonhole Principle to obtain the result. True or False: f is a well-defined function (i.e. it satisfies the 3 properties of a well- defined function). True Falsearrow_forward
- Let G = (V, E) be a connected graph with a positive length function w. Then (V, d) is a finite metric space, where the distance function d is defined asarrow_forward2. Let G = (V, E) be a directed weighted graph with the vertices V = {A, B, C, D, E, F) and the edges E= {(A, B, 12), (A, D, 17), (B, C, 8), (B, D, 13), (B, E, 15), (B, F, 13), (C, E, 12), (C, F, 25)}, where the third components is the cost. (a) Write down the adjacency list representation the graph G = (V, E).arrow_forwardGiven N cities represented as vertices V₁, V2, un on an undirected graph (i.e., each edge can be traversed in both directions). The graph is fully-connected where the edge eij connecting any two vertices vį and vj is the straight-line distance between these two cities. We want to search for the shortest path from v₁ (the source) to VN (the destination). ... Assume that all edges have different values, and €₁,7 has the largest value among the edges. That is, the source and destination have the largest straight-line distance. Compare the lists of explored vertices when we run the uniform-cost search and the A* search for this problem. Hint: The straight-line distance is the shortest path between any two cities. If you do not know how to start, try to run the algorithms by hand on some small cases first; but remember to make sure your graphs satisfy the conditions in the question.arrow_forward
- 4. Let G=(V, E) be the following undirected graph: V = {1, 2, 3, 4, 5, 6, 7, 8} E = {(1, 7) (1, 4), (1, 6), (6, 4), (7, 5), (2, 5), (2, 4), (4, 3), (4, 8), (8, 3), (3, 5)}. a. Draw G. b. Is G connected? c. Give the adjacency matrix for the graph G given above. d. Determine whether the graph has a Hamiltonian cycle. If the graph does have a Hamiltonian cycle, specify the nodes of one. If it does not, prove that it does not have one. e. Determine whether the graph has a Euler cycle. If the graph does have a Euler cycle, specify the nodes of one. If it does not, prove that it does not have one.arrow_forwardSay that a graph G has a path of length three if there exist distinct vertices u, v, w, t with edges (u, v), (v, w), (w, t). Show that a graph G with 99 vertices and no path of length three has at most 99 edges.arrow_forwardTrue 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).arrow_forward
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