3. Suppose that we want to know which agents are connected by a walk of length two in a given network. That is, want to know the set of ordered pairs (i, j) such that there exists a walk of length 2 between any i and j. Define n as the number of nodes and m as the number of links. (a) One way to do this is to input the adjacency matrix A, calculate A2, then output all pairs (i, j) for which A(i) > 0. Defining scalar multiplication and addition as "basic operations," up to a constant c, how many "basic operations" are required to calculate A², in terms of n and m? (Hint: First think about how many operations are required to calculate each entry of A2, then multiply by the number of entries.) (b) Suppose that we want to know the number of agents who are connected by a walk of length k in a given network, where k 21. Up to a constant, how many basic operations are required to calculate Ak, in terms of n and m? (c) (Harder) We showed in class (Lecture 4) that a simple adaptation of Breadth-First Search (BFS) will calculate the length of the shortest path from any starting node i to all other nodes j in O(m) time. Taking this result as given, suggest an algorithm that runs in O(nm) time that will output all pairs (i, j) such that there is a path of length 2 between them, and briefly discuss why it runs in O(nm) time.
3. Suppose that we want to know which agents are connected by a walk of length two in a given network. That is, want to know the set of ordered pairs (i, j) such that there exists a walk of length 2 between any i and j. Define n as the number of nodes and m as the number of links. (a) One way to do this is to input the adjacency matrix A, calculate A2, then output all pairs (i, j) for which A(i) > 0. Defining scalar multiplication and addition as "basic operations," up to a constant c, how many "basic operations" are required to calculate A², in terms of n and m? (Hint: First think about how many operations are required to calculate each entry of A2, then multiply by the number of entries.) (b) Suppose that we want to know the number of agents who are connected by a walk of length k in a given network, where k 21. Up to a constant, how many basic operations are required to calculate Ak, in terms of n and m? (c) (Harder) We showed in class (Lecture 4) that a simple adaptation of Breadth-First Search (BFS) will calculate the length of the shortest path from any starting node i to all other nodes j in O(m) time. Taking this result as given, suggest an algorithm that runs in O(nm) time that will output all pairs (i, j) such that there is a path of length 2 between them, and briefly discuss why it runs in O(nm) time.
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
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Problem R1RQ: What is the difference between a host and an end system? List several different types of end...
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Transcribed Image Text:3. Suppose that we want to know which agents are connected by a walk of length two in a given
network. That is, want to know the set of ordered pairs (i, j) such that there exists a walk
of length 2 between any i and j. Define n as the number of nodes and m as the number of
links.
(a) One way to do this is to input the adjacency matrix A, calculate A2, then output all
pairs (i, j) for which A(i) > 0. Defining scalar multiplication and addition as "basic
operations," up to a constant c, how many "basic operations" are required to calculate
A², in terms of n and m? (Hint: First think about how many operations are required
to calculate each entry of A²2, then multiply by the number of entries.)
(b) Suppose that we want to know the number of agents who are connected by a walk of
length k in a given network, where k≥ 1. Up to a constant, how many basic operations
are required to calculate Ak, in terms of n and m?
(c) (Harder) We showed in class (Lecture 4) that a simple adaptation of Breadth-First
Search (BFS) will calculate the length of the shortest path from any starting node i to
all other nodes j in O(m) time. Taking this result as given, suggest an algorithm that
runs in O(nm) time that will output all pairs (i, j) such that there is a path of length 2
between them, and briefly discuss why it runs in O(nm) time.
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