Homework Help is Here – Start Your Trial Now!
Engineering
Computer Engineering
On a set of subsets, the three operations amount to: - Create a set of n disjoint subsets with every node in its own subset - Test whether A and B are in the same subset - If A and B are in the same subset, then do nothing, else unify the subsets to which A and B belong The most efficient way to implement the operation is: - For every A it is possible to ask for the index of the subset to which A belongs - This will be denoted as find(A). And we have A~B == find(A) == find(B). The operation of adding a relation between A and B will be denoted by union(A, B) (even though nothing needs to happen). Thus, union(A, B) one performs Solution: - Keep track of the size of each equivalence class. - Let every element point to the representative of its set. On “union", change the name of the smaller equivalence class to the larger. - Thus "union(A,B)" takes O(min(|A], |B|) time. The total time spent for k "union" operations is O(k logk) C O(k logN). (Why ?) - In other words, the Amortized time complexity of "union" is O(log N).

On a set of subsets, the three operations amount to: - Create a set of n disjoint subsets with every node in its own subset - Test whether A and B are in the same subset - If A and B are in the same subset, then do nothing, else unify the subsets to which A and B belong The most efficient way to implement the operation is: - For every A it is possible to ask for the index of the subset to which A belongs - This will be denoted as find(A). And we have A~B == find(A) == find(B). The operation of adding a relation between A and B will be denoted by union(A, B) (even though nothing needs to happen). Thus, union(A, B) one performs Solution: - Keep track of the size of each equivalence class. - Let every element point to the representative of its set. On “union", change the name of the smaller equivalence class to the larger. - Thus "union(A,B)" takes O(min(|A], |B|) time. The total time spent for k "union" operations is O(k logk) C O(k logN). (Why ?) - In other words, the Amortized time complexity of "union" is O(log N).

Computer Networking: A Top-Down Approach (7th Edition)
Computer Networking: A Top-Down Approach (7th Edition)
7th Edition
ISBN:9780133594140
Author:James Kurose, Keith Ross
Publisher:James Kurose, Keith Ross
Chapter1: Computer Networks And The Internet
Section: Chapter Questions
Problem R1RQ: What is the difference between a host and an end system? List several different types of end...
See similar textbooks
icon
Related questions
Question

Prove that the amortized complexity of the union operation in the solution is O(log n). Consider the number of times a given element may change its set.

Problem: On a set of subsets, the three operations amount to: - Create a set of n disjoint subsets with every node in its own subset Test whether A and B are in the same subset - If A and B are in the same subset, then do nothing, else unify the subsets to which A and B belong The most efficient way to implement the operation is: - For every A it is possible to ask for the index of the subset to which A belongs - This will be denoted as find(A). And we have A~B == find(A) == find(B). The operation of adding a relation between A and B will be denoted by union(A, B) (even though nothing needs to happen). Thus, union(A, B) one performs Solution: - Keep track of the size of each equivalence class. - Let every element point to the representative of its set. - On “union", change the name of the smaller equivalence class to the larger. Thus "union(A,B)" takes O(min(|A], |B|) time. The total time spent for k "union" operations is O(k logk) C O(k logN). (Why ?) - In other words, the Amortized time complexity of "union" is O(log N).
Transcribed Image Text:Problem: On a set of subsets, the three operations amount to: - Create a set of n disjoint subsets with every node in its own subset Test whether A and B are in the same subset - If A and B are in the same subset, then do nothing, else unify the subsets to which A and B belong The most efficient way to implement the operation is: - For every A it is possible to ask for the index of the subset to which A belongs - This will be denoted as find(A). And we have A~B == find(A) == find(B). The operation of adding a relation between A and B will be denoted by union(A, B) (even though nothing needs to happen). Thus, union(A, B) one performs Solution: - Keep track of the size of each equivalence class. - Let every element point to the representative of its set. - On “union", change the name of the smaller equivalence class to the larger. Thus "union(A,B)" takes O(min(|A], |B|) time. The total time spent for k "union" operations is O(k logk) C O(k logN). (Why ?) - In other words, the Amortized time complexity of "union" is O(log N).
Representatives 3 4 5 6 7 8 Find:O(1) 9. 1 3 4 5 6 7 8 9 3 1 1 1 1 Sizes of the sets 22
Transcribed Image Text:Representatives 3 4 5 6 7 8 Find:O(1) 9. 1 3 4 5 6 7 8 9 3 1 1 1 1 Sizes of the sets 22
Expert Solution
steps

Step by step

Solved in 2 steps

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer
Recommended textbooks for you
Computer Networking: A Top-Down Approach (7th Edi…
Computer Networking: A Top-Down Approach (7th Edi…
Computer Engineering
ISBN:
9780133594140
Author:
James Kurose, Keith Ross
Publisher:
PEARSON
Computer Organization and Design MIPS Edition, Fi…
Computer Organization and Design MIPS Edition, Fi…
Computer Engineering
ISBN:
9780124077263
Author:
David A. Patterson, John L. Hennessy
Publisher:
Elsevier Science
Network+ Guide to Networks (MindTap Course List)
Network+ Guide to Networks (MindTap Course List)
Computer Engineering
ISBN:
9781337569330
Author:
Jill West, Tamara Dean, Jean Andrews
Publisher:
Cengage Learning
Concepts of Database Management
Concepts of Database Management
Computer Engineering
ISBN:
9781337093422
Author:
Joy L. Starks, Philip J. Pratt, Mary Z. Last
Publisher:
Cengage Learning
Prelude to Programming
Prelude to Programming
Computer Engineering
ISBN:
9780133750423
Author:
VENIT, Stewart
Publisher:
Pearson Education
Sc Business Data Communications and Networking, T…
Sc Business Data Communications and Networking, T…
Computer Engineering
ISBN:
9781119368830
Author:
FITZGERALD
Publisher:
WILEY
SEE MORE TEXTBOOKS