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
Question
Chapter 35, Problem 4P
a.
Program Plan Intro
To show that a maximal matching need not be a maximum matching by drawing an undirected graph G.
b.
Program Plan Intro
To show that the linear-time greedy
c.
Program Plan Intro
To show that the size of a maximum matching in G is a lower bound on the size of any vertex cover for G.
d.
Program Plan Intro
To prove that the number of bins used by the first-fit heuristic is never more than
e.
Program Plan Intro
To prove an approximation ratio of 2 for the first-fit heuristic.
f.
Program Plan Intro
To give an efficient implementation of the first-fit heuristic, and analyze its running time.
Expert Solution & Answer
Want to see the full answer?
Check out a sample textbook solution
Students have asked these similar questions
In the figure below there is a weighted graph, dots represent vertices, links
represent edges, and numbers represent edge weights.
S
2
1
2
1
2
3
T
1
1
2
4
(a) Find the shortest path from vertex S to vertex T, i.e., the path of
minimum weight between S and T.
(b) Find the minimum subgraph (set of edges) that connects all vertices
in the graph and has the smallest total weight (sum of edge weights).
2.
3.
Q2
a
Let G be a graph. We say that a set of vertices C form a vertex cover if every edge of G is
incident to at least one vertex in C. We say that a set of vertices I form an independent set if
no edge in G connects two vertices from I.
For example, if G is the graph above, C = [b, d, e, f, g, h, j] is a vertex cover since each of
the 20 edges in the graph has at least one endpoint in C, and I = [a, c, i, k] is an
independent set because none of these edges appear in the graph: ac, ai, ak, ci, ck, ik.
In the example above, notice that each vertex belongs to the vertex cover C or the independent
set I. Do you think that this is a coincidence?
In the above graph, clearly explain why the maximum size of an independent set is 5. In other
words, carefully explain why there does not exist an independent set with 6 or more vertices.
The minimum vertex cover problem is stated as follows: Given an undirected graph
G = (V, E) with N vertices and M edges. Find a minimal size subset of vertices X
from V such that every edge (u, v) in E is incident on at least one vertex in X. In
other words you want to find a minimal subset of vertices that together touch all the
edges.
For example, the set of vertices X = {a,c} constitutes a minimum vertex cover for the
following graph:
a---b---c---g
d
e
Formulate the minimum vertex cover problem as a Genetic Algorithm or another
form of evolutionary optimization. You may use binary representation, OR any repre-
sentation that you think is more appropriate. you should specify:
• A fitness function. Give 3 examples of individuals and their fitness values if you
are solving the above example.
• A set of mutation and/or crossover and/or repair operators. Intelligent operators
that are suitable for this particular domain will earn more credit.
• A termination criterion for the…
Chapter 35 Solutions
Introduction to Algorithms
Ch. 35.1 - Prob. 1ECh. 35.1 - Prob. 2ECh. 35.1 - Prob. 3ECh. 35.1 - Prob. 4ECh. 35.1 - Prob. 5ECh. 35.2 - Prob. 1ECh. 35.2 - Prob. 2ECh. 35.2 - Prob. 3ECh. 35.2 - Prob. 4ECh. 35.2 - Prob. 5E
Ch. 35.3 - Prob. 1ECh. 35.3 - Prob. 2ECh. 35.3 - Prob. 3ECh. 35.3 - Prob. 4ECh. 35.3 - Prob. 5ECh. 35.4 - Prob. 1ECh. 35.4 - Prob. 2ECh. 35.4 - Prob. 3ECh. 35.4 - Prob. 4ECh. 35.5 - Prob. 1ECh. 35.5 - Prob. 2ECh. 35.5 - Prob. 3ECh. 35.5 - Prob. 4ECh. 35.5 - Prob. 5ECh. 35 - Prob. 1PCh. 35 - Prob. 2PCh. 35 - Prob. 3PCh. 35 - Prob. 4PCh. 35 - Prob. 5PCh. 35 - Prob. 6PCh. 35 - Prob. 7P
Knowledge Booster
Similar questions
- 5. 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_forwardRequired 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_forwardBe G=(V, E)a connected graph and u, vEV. The distance Come in u and v, denoted by d(u, v), is the length of the shortest path between u'and v, Meanwhile he width from G, denoted as A(G), is the greatest distance between two of its vertices. a) Show that if A(G) 24 then A(G) <2. b) Show that if G has a cut vertex and A(G) = 2, then Ġhas a vertex with no neighbors.arrow_forward
- Let G be a graph with n vertices. If the maximum size of an independent set in G is k, clearly explain why the minimum size of a vertex cover in G is n - k.arrow_forwardGiven a graph that is a tree (connected and acyclic). (I) Pick any vertex v.(II) Compute the shortest path from v to every other vertex. Let w be the vertex with the largest shortest path distance.(III) Compute the shortest path from w to every other vertex. Let x be the vertex with the largest shortest path distance. Consider the path p from w to x. Which of the following are truea. p is the longest path in the graphb. p is the shortest path in the graphc. p can be calculated in time linear in the number of edges/verticesarrow_forwardIn graph theory, graph coloring is a special case of graph labeling; it is an assignment of labels traditionally called "colors" to elements of a graph subject to certain constraints. In its simplest form, it is a way of coloring the vertices of a graph such that no two adjacent vertices share the same color; this is called a vertex coloring. The chromatic number of a graph is the least mumber of colors required to do a coloring of a graph. Example Here in this graph the chromatic number is 3 since we used 3 colors The degree of a vertex v in a graph (without loops) is the number of edges at v. If there are loops at v each loop contributes 2 to the valence of v. A graph is connected if for any pair of vertices u and v one can get from u to v by moving along the edges of the graph. Such routes that move along edges are known by different names: edge progressions, paths, simple paths, walks, trails, circuits, cycles, etc. a. Write down the degree of the 16 vertices in the graph below: 14…arrow_forward
- 3. Kleinberg, Jon. Algorithm Design (p. 519, q. 28) Consider this version of the Independent Set Problem. You are given an undirected graph G and an integer k. We will call a set of nodes I "strongly independent" if, for any two nodes v, u € I, the edge (v, u) is not present in G, and neither is there a path of two edges from u to v. That is, there is no node w such that both (v, w) and (u, w) are present. The Strongly Independent Set problem is to decide whether G has a strongly independent set of size at least k. Show that the Strongly Independent Set Problem is NP-Complete.arrow_forwardIn the game of Nim, an arbitrary number of chips are divided into an arbitrary number of piles. Each player can remove as many chips as desired from any single pile. The last player to remove a chip wins. Consider a limited version of this game, in which three piles contain 3, 5, and 8 chips, respectively. You can represent this game as a directed graph. Each vertex in this graph is a possible configuration of the piles (chips in each pile). The initial configuration, for example, is (3, 5, 8). Each edge in the graph represents a legal move in the game. Write Java statements that will construct this directed graph. Discuss how a computer program might use this graph to play Nim. java programarrow_forwardInput: A directed graph G with n vertices. Task: If one can add at most one directed edge to G such that every vertex can reach all other vertices, then print 'Yes'. Otherwise, print 'Unsatisfiable'. Give an efficient algorithm (to the best of your knowledge) and a formal proof of correctness for your algorithm. An answer without any formal proof will be considered incomplete. Analyze the running time of the algorithm. You must write down your algorithm as pseudocodearrow_forward
- We recollect that Kruskal's Algorithm is used to fi nd the minimum spanning tree in a weighted graph. Given a weighted undirected graph G = (V , E, W), with n vertices/nodes, the algorithm will fi rst sort the edges in E according to their weights. It will then select (n-1) edges with smallest weights that do not form a cycle. (A cycle in a graph is a path along the edges of a graph that starts at a node and ends at the same node after visiting at least one other node and not traversing any of the edges more than once.) Use Kruskal's Algorithm to nd the weight of the minimum spanning tree for the following graph.arrow_forward3) The graph k-coloring problem is stated as follows: Given an undirected graph G= (V,E) with N vertices and M edges and an integer k. Assign to each vertex v in V a color c(v) such that 1arrow_forward4-Clique Problem The clique problem is to find cliques in a graph. A clique is a set of vertices that are all adjacent - connected - to each other. A 4-clique is a set of 4 vertices that are all connected to each other. So in this example of the 4-Clique Problem, we have a 7-vertex graph. A brute-force algorithm has searched every possible combination of 4 vertices and found a set that forms a clique: https://en.wikipedia.org/wiki/Clique_problem You should read the Wikipedia page for the Clique Problem (and then read wider if need be) if you need to understand more about it. Note that the Clique Problem is NP-Complete and therefore when the graph size is large a deterministic search is impractical. That makes it an ideal candidate for an evolutionary search. For this assignment you must suppose that you have been tasked to implement the 4-clique problem as an evolutionary algorithm for any graph with any number of vertices (an n-vertex graph). The algorithm succeeds if it finds a…arrow_forwardarrow_back_iosSEE MORE QUESTIONSarrow_forward_ios
Recommended textbooks for you
Database System ConceptsComputer ScienceISBN:9780078022159Author:Abraham Silberschatz Professor, Henry F. Korth, S. SudarshanPublisher:McGraw-Hill Education
Starting Out with Python (4th Edition)Computer ScienceISBN:9780134444321Author:Tony GaddisPublisher:PEARSON
Digital Fundamentals (11th Edition)Computer ScienceISBN:9780132737968Author:Thomas L. FloydPublisher:PEARSON
C How to Program (8th Edition)Computer ScienceISBN:9780133976892Author:Paul J. Deitel, Harvey DeitelPublisher:PEARSON
Database Systems: Design, Implementation, & Manag...Computer ScienceISBN:9781337627900Author:Carlos Coronel, Steven MorrisPublisher:Cengage Learning
Programmable 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