Introduction to Algorithms
3rd Edition
ISBN: 9780262033848
Author: Thomas H. Cormen, Ronald L. Rivest, Charles E. Leiserson, Clifford Stein
Publisher: MIT Press
expand_more
expand_more
format_list_bulleted
Concept explainers
Question
Chapter 25.2, Problem 9E
Program Plan Intro
To explain the computation of the transitive closure of the directed acyclic graph using monotonic function of vertex and edge in
Expert Solution & Answer
Want to see the full answer?
Check out a sample textbook solutionStudents have asked these similar questions
Let G = (V, E) be an undirected graph with at least two distinct vertices a, b ∈ V . Prove that we can assign a direction to each edge e ∈ E as to form a directed acyclic graph G′ where a is a source and b is a sink.
If a graph G = (V, E), |V | > 1 has N strongly connected components, and an edge E(u, v) is removed, what are the upper and lower bounds on the number of strongly connected components in the resulting graph? Give an example of each boundary case.
Hall's theorem
Let d be a positive integer. We say that a graph is d-regular if every node has degree exactly d.
Show that every d-regular bipartite graph G = (L ∪ R, E) with bipartition classes L and R has |L| = |R|.
Show that every d-regular bipartite graph has a perfect matching by (directly) arguing that a minimum cut of the corresponding flow network has capacity |L|
Chapter 25 Solutions
Introduction to Algorithms
Ch. 25.1 - Prob. 1ECh. 25.1 - Prob. 2ECh. 25.1 - Prob. 3ECh. 25.1 - Prob. 4ECh. 25.1 - Prob. 5ECh. 25.1 - Prob. 6ECh. 25.1 - Prob. 7ECh. 25.1 - Prob. 8ECh. 25.1 - Prob. 9ECh. 25.1 - Prob. 10E
Ch. 25.2 - Prob. 1ECh. 25.2 - Prob. 2ECh. 25.2 - Prob. 3ECh. 25.2 - Prob. 4ECh. 25.2 - Prob. 5ECh. 25.2 - Prob. 6ECh. 25.2 - Prob. 7ECh. 25.2 - Prob. 8ECh. 25.2 - Prob. 9ECh. 25.3 - Prob. 1ECh. 25.3 - Prob. 2ECh. 25.3 - Prob. 3ECh. 25.3 - Prob. 4ECh. 25.3 - Prob. 5ECh. 25.3 - Prob. 6ECh. 25 - Prob. 1PCh. 25 - Prob. 2P
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, computer-science and related others by exploring similar questions and additional content below.Similar questions
- (a) Show that a graph G with at least three vertices is 2-connected if and only if any vertex and any edge of G lie on a common cycle of G: (b) Show that a graph G with at least three vertices is 2-connected if and only if any two edges of G lie on a common cyclearrow_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_forwardLet G = (V, E) be a graph with vertex-set V = {1, 2, 3, 4, 5} and edge-set E = {(1, 2), (3, 2), (4, 3), (1, 4), (2, 4), (1, 3)}. (a) Draw the graph. Find (b) maximal degree, i.e. ∆(G), (c) minimal degree, i.e. δ(G), (d) the size of biggest clique, i.e. ω(G),(e) the size of biggest independent set, i.e. α(G), ter(f) the minimal number of colours needed to color the graph, i.e. χ(G).arrow_forward
- Consider a graph with nodes and directed edges and let an edge from node a to node b berepresented by a fact edge (a,b). Define a binary predicate path that is true for nodes c and dif, and only if, there is a path from c to d in the graph.arrow_forwardA 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.arrow_forwardFor the following simple graphs G=(V,E) (described by their vertex and edge sets) decide whether they are bipartite or not. If G is bipartite, then give its bipartition, and if it is not explain why. 1) V={a,b,c,d,e,f,g}, E={ag,af,ae,bf,be,bc,dc,dg,df,de} 2) V={a,b,c,d,e,f,g,h}, E={ac,ad,ah,bc,bh,bd,ec,ef,eg,hf,hg}arrow_forward
- A path of length two is denoted by P2. If a graph G does not contain P2 as induced subgraph, then: 1- G must be a clique (i.e., a complete graph). 2- Every vertex of G must of degree one. 3- Every connected component of G must be a clique. 4- Every connected component of G must consist of at most two vertices.arrow_forward(1) Given a weighted directed graph G = (V, E, w), where V = {1, 2, 3, 4}, E ={(1,3), (2,1), (2,4), (3,2), (4,1), (4,3)}, and w(1,3) = −2, w(2,1) = 1, w(2,4) = 2,w(3,2) = 4, w(4,1) = −3, w(4,3) = 3. (a) Represent the graph G graphically;(b) Run SLOW-ALL-PAIRS-SHORTEST-PATHS on the above graph and show the matricesL(k) that result for each iteration of the loop. (2) Discuss how to use the Floyd-Warshall algorithm to detect a negative-weight cycle. (3) The essence of Johnson’s algorithm is re-weighting so that a transformed graph has no negativeweight edges, which enables the use of Dijkstra’s algorithm. Let us modify Johnson’s algorithmsuch that G' = G and s be any vertex. (a) Give a counter example (i.e. a simple weighted and directed graph) to show thismodification is incorrect assuming ∞ − ∞ is undefined(b) Show (using logical arguments) this modification produces the correct results when a givenG is strongly connected.arrow_forwardSuppose that G is an unconnected graph that consists of 4 connected components. The first component is K4, the second is K2,2, the third is C4 and the fourth is a single vertex. Your job is to show how to add edges to G so that the graph has an Euler tour. Justify that your solution is the minimum number of edges added.arrow_forward
- Let G be a directed graph where each edge is colored either red, white, or blue. A walk in G is called a patriotic walk if its sequence of edge colors is red, white, blue, red, white, blue, and soon. Formally ,a walk v0 →v1 →...vk is a patriotic walk if for all 0≤i<k, the edge vi →vi+1 is red if i mod3=0, white if i mod3=1,and blue if i mod3=2. Given a graph G, you wish to find all vertices in G that can be reached from a given vertex v by a patriotic walk. Show that this can be accomplished by efficiently constructing a new graph G′ from G, such that the answer is determined by a single call to DFS in G′. Do not forget to analyze your algorithm.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_forwardLet G be a directed acyclic graph with exactly one source r such that for any other vertex v there exists a unique directed path from r to v. Let Gu be the undirected graph obtained by erasing the direction on each edge of G. Prove that (Gu,r) is a rooted tree.arrow_forward
arrow_back_ios
SEE MORE QUESTIONS
arrow_forward_ios
Recommended textbooks for you
- Database System ConceptsComputer ScienceISBN:9780078022159Author:Abraham Silberschatz Professor, Henry F. Korth, S. SudarshanPublisher:McGraw-Hill EducationStarting Out with Python (4th Edition)Computer ScienceISBN:9780134444321Author:Tony GaddisPublisher:PEARSONDigital Fundamentals (11th Edition)Computer ScienceISBN:9780132737968Author:Thomas L. FloydPublisher:PEARSON
- C How to Program (8th Edition)Computer ScienceISBN:9780133976892Author:Paul J. Deitel, Harvey DeitelPublisher:PEARSONDatabase Systems: Design, Implementation, & Manag...Computer ScienceISBN:9781337627900Author:Carlos Coronel, Steven MorrisPublisher:Cengage LearningProgrammable Logic ControllersComputer ScienceISBN:9780073373843Author:Frank D. PetruzellaPublisher:McGraw-Hill Education
Database System Concepts
Computer Science
ISBN:9780078022159
Author:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
Publisher:McGraw-Hill Education
Starting Out with Python (4th Edition)
Computer Science
ISBN:9780134444321
Author:Tony Gaddis
Publisher:PEARSON
Digital Fundamentals (11th Edition)
Computer Science
ISBN:9780132737968
Author:Thomas L. Floyd
Publisher:PEARSON
C How to Program (8th Edition)
Computer Science
ISBN:9780133976892
Author:Paul J. Deitel, Harvey Deitel
Publisher:PEARSON
Database Systems: Design, Implementation, & Manag...
Computer Science
ISBN:9781337627900
Author:Carlos Coronel, Steven Morris
Publisher:Cengage Learning
Programmable Logic Controllers
Computer Science
ISBN:9780073373843
Author:Frank D. Petruzella
Publisher:McGraw-Hill Education