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 22.4, Problem 1E
Program Plan Intro
To show the vertices order created by topological sort under the DAG.
Expert Solution & Answer
Want to see the full answer?
Check out a sample textbook solutionStudents have asked these similar questions
using R
Use the igraph library to define and plot graphs (multigraph and digraph).
setting up the arcs/edges algorithmically. (the list of arcs/edges or the adjacency
matrix should be implemented and the code should be able to deal with a different list of vertices and still produce graphs satisfying the stated rules.)
5. (This question goes slightly beyond what was covered in the lectures, but you can solve it by combining algorithms that we have described.)
A directed graph is said to be strongly connected if every vertex is reachable from every other vertex; i.e., for every pair of vertices u, v, there is a directed path from u to v and a directed path from v to u.
A strong component of a graph is then a maximal subgraph that is strongly connected. That is all vertices in a strong component can reach each other, and any other vertex in the directed graph either cannot reach the strong component or cannot be reached from the component.
(Note that we are considering directed graphs, so for a pair of vertices u and v there could be a path from u to v, but no path path from v back to u; in that case, u and v are not in the same strong component, even though they are connected by a path in one direction.)
Given a vertex v in a directed graph D, design an algorithm for com- puting the strong connected…
Please Answer this in Python language:
You're given a simple undirected graph G with N vertices and M edges. You have to assign, to each
vertex i, a number C; such that 1 ≤ C; ≤ N and Vi‡j, C; ‡ Cj.
For any such assignment, we define D; to be the number of neighbours j of i such that C; < C₁.
You want to minimise maai[1..N) Di - mini[1..N) Di.
Output the minimum possible value of this expression for a valid assignment as described above,
and also print the corresponding assignment.
Note:
The given graph need not be connected.
• If there are multiple possible assignments, output anyone.
• Since the input is large, prefer using fast input-output methods.
Input
1
57
12
13
14
23
24
25
35
Output
2
43251
Q
Chapter 22 Solutions
Introduction to Algorithms
Ch. 22.1 - Prob. 1ECh. 22.1 - Prob. 2ECh. 22.1 - Prob. 3ECh. 22.1 - Prob. 4ECh. 22.1 - Prob. 5ECh. 22.1 - Prob. 6ECh. 22.1 - Prob. 7ECh. 22.1 - Prob. 8ECh. 22.2 - Prob. 1ECh. 22.2 - Prob. 2E
Ch. 22.2 - Prob. 3ECh. 22.2 - Prob. 4ECh. 22.2 - Prob. 5ECh. 22.2 - Prob. 6ECh. 22.2 - Prob. 7ECh. 22.2 - Prob. 8ECh. 22.2 - Prob. 9ECh. 22.3 - Prob. 1ECh. 22.3 - Prob. 2ECh. 22.3 - Prob. 3ECh. 22.3 - Prob. 4ECh. 22.3 - Prob. 5ECh. 22.3 - Prob. 6ECh. 22.3 - Prob. 7ECh. 22.3 - Prob. 8ECh. 22.3 - Prob. 9ECh. 22.3 - Prob. 10ECh. 22.3 - Prob. 11ECh. 22.3 - Prob. 12ECh. 22.3 - Prob. 13ECh. 22.4 - Prob. 1ECh. 22.4 - Prob. 2ECh. 22.4 - Prob. 3ECh. 22.4 - Prob. 4ECh. 22.4 - Prob. 5ECh. 22.5 - Prob. 1ECh. 22.5 - Prob. 2ECh. 22.5 - Prob. 3ECh. 22.5 - Prob. 4ECh. 22.5 - Prob. 5ECh. 22.5 - Prob. 6ECh. 22.5 - Prob. 7ECh. 22 - Prob. 1PCh. 22 - Prob. 2PCh. 22 - Prob. 3PCh. 22 - Prob. 4P
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
- Write a program for depth-first traversal on the following graph using the algo-rithm defined in this weekarrow_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_forwardLet V= {cities of Metro Manila} and E = {(x; y) | x and y are adjacent cities in Metro Manila.} (a) Draw the graph G defined by G = (V; E). You may use initials to name a vertex representing a city. (b) Apply the Four-Color Theorem to determine the chromatic number of the vertex coloring for G.arrow_forward
- The Graph Data Structure is made up of nodes and edges. (A Tree Data Structure is a special kind of a Graph Data Structure). A Graph may be represented by an Adjacency Matrix or an Adjacency List. Through this exercise, you should be able to have a better grasp the Adjacency Matrix concept. You are expected to read about the Adjacency Matrix concept as well as the Adjacency List concept. Suppose the vertices A, B, C, D, E, F, G and H of a Graph are mapped to row and column indices(0,1,2,3,4,5,6,and 7) of a matrix (i.e. 2-dimensional array) as shown in the following table. Vertex of Graph Index in the 2-D Array Adjacency Matrix Representation of Graph A B 2 F 6. H 7 Suppose further, that the following is an Adjacency Matrix representing the Graph. 3 4 5. 6. 7 0. 1 1 1 1 01 1 01 1. 3 14 1 1 1 6. 1 Exercise: Show/Draw the Graph that is represented by the above Adjacency matrix. Upload the document that contains your result. (Filename: AdjacencyMatrixExercise.pdf) Notes: -The nodes of the…arrow_forwardI have employed two different traversal algorithms, namely, Algo I and Algo Il on a graph. Then I have shown the output of the two traversal algorithms in Figure -I and Figure -II. In both cases, the traversal starts at the node shown by the block right arrow. The nodes are numbered/labeled in the sequence they are visited. Identify the traversal algorithm for Figure -I and Figure -II and justify my answer. 3 COLOR 02x02 202020 (XD) CO COZOZO 8282620 () COS Figure I Figure I 02.02arrow_forwardWrite a program RandomSparseGraph to generate random sparse graphs for a well-chosen set of values of V and E such that you can use it torun meaningful empirical tests on graphs drawn from the Erdös-Renyi modelarrow_forward
- Do some outside research on depth-first traversal as it relates to traversing graphs. Then answer the following questions: a. Suppose you have an arbitrary connected graph G, shown in the image below. Use the vertex A as your starting point. Write out the order in which the algorithm could traverse the graph with a depth-first search, and explain your reasoning (there are multiple correct answers, hence the need for an explanation). b. Use a proof by induction to prove that when a depth-first traversal is performed, every vertex v in your graph G will have been visited at least one time. B D H E A G с I FLarrow_forwardRun experiments to determine empirically the average number ofvertices that are reachable from a randomly chosen vertex, for various digraph modelsarrow_forwardLet G (V, E) be a digraph in which every vertex is a source, or a sink, or both a sink and a source. (a) Prove that G has neither self-loops nor anti-parallel edges.arrow_forward
- Required information NOTE: This is a multi-part question. Once an answer is submitted, you will be unable to return to this part. Consider the bipartite graph Km.n- Find the values of mand n if Km n has an Euler path. (Check all that apply.) Check All That Apply Km,n has an Euler path when both mand n are even. Km,n has an Euler path when both mand n are odd. Km, n has an Euler path if m=2 and n is odd. Km, n has an Euler path if n= 2 and m is odd. Km, n has an Euler path when m= n=1.arrow_forward5. Fleury's algorithm is an optimisation solution for finding a Euler Circuit of Euler Path in a graph, if they exist. Describe how this algorithm will always find a path or circuit if it exists. Describe how you calculate if the graph is connected at each edge removal. Fleury's Algorithm: The algorithm starts at a vertex of v odd degree, or, if the graph has none, it starts with an arbitrarily chosen vertex. At each step it chooses the next edge in the path to be one whose deletion would not disconnect the graph, unless there is no such edge, in which case it picks the remaining edge (a bridge) left at the current vertex. It then moves to the other endpoint of that edge and adds the edge to the path or circuit. At the end of the algorithm there are no edges left ( or all your bridges are burnt). (NOTE: Please elaborate on the answer and explain. Please do not copy-paste the answer from the internet or from Chegg.)arrow_forwardLinked lists are employed in a specific manner to express adjacence lists on a graph. Give an illustration of your point with an example. Does coding not require any prior knowledge?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