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ONNECT. = {(C is a connected graph} TRIANGLE-FREE = {(G) | G is a graph with no triangle- BIPARTITE = {(G) | G is a bipartite graph} TREE = {{G) | G is a tree, i.e. a connected acyclic graph} %3D that a graph is bipartite if its nodes may be divided into

ONNECT. = {(C is a connected graph} TRIANGLE-FREE = {(G) | G is a graph with no triangle- BIPARTITE = {(G) | G is a bipartite graph} TREE = {{G) | G is a tree, i.e. a connected acyclic graph} %3D that a graph is bipartite if its nodes may be divided into

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...
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Prove that the following problems are in P. (a) CONNECTED {(G) | G is a connected graph} %3D (b) TRIANGLE-FREE = {(G) | G is a graph with no triangles, i.e. cycles of length 3} (c) BIPARTITE = {(G) | G is a bipartite graph} (d) TREE = {(G) | G is a tree, i.e. a connected acyclic graph} %3D Note that a graph is bipartite if its nodes may be divided into two sets s.t. all edges go between the sets, and that a graph is bipartite if and only if it does not contain an odd length cycle.
Transcribed Image Text:Prove that the following problems are in P. (a) CONNECTED {(G) | G is a connected graph} %3D (b) TRIANGLE-FREE = {(G) | G is a graph with no triangles, i.e. cycles of length 3} (c) BIPARTITE = {(G) | G is a bipartite graph} (d) TREE = {(G) | G is a tree, i.e. a connected acyclic graph} %3D Note that a graph is bipartite if its nodes may be divided into two sets s.t. all edges go between the sets, and that a graph is bipartite if and only if it does not contain an odd length cycle.
The class P: definition Definition: P is the lass of languages that are decidable in polynomial time on a deterministic single tape Turing machine. That is P=UTIME(n* ). k • The class P plays an important role in our theory and is important because •Pis invariant for all models of computation that are polynomially equivalent to the deterministic single tape TM, ånd •P roughly corresponds to the class of problems that are realistically solvable on a computer. • When we analyze an algorithm to show that it runs in polynomial time, we need to do two things • First, give a polynomial upper bound (usually in big-O notation) on the number of stages that the algorithm uses when it runs on input of length n. • Then, examine the individual stages in the description of the algorithm to be sure thạt each can be implemented in polynomial time on a reasonable deterministic model. • When both tasks have been done, we can conclude that it runs in polynomial time because we have demonstrated thạt it runs for a polynomiąl number of stages, each of which can be done in polynomial time, and the composition of polynomials is a polynomial.
Transcribed Image Text:The class P: definition Definition: P is the lass of languages that are decidable in polynomial time on a deterministic single tape Turing machine. That is P=UTIME(n* ). k • The class P plays an important role in our theory and is important because •Pis invariant for all models of computation that are polynomially equivalent to the deterministic single tape TM, ånd •P roughly corresponds to the class of problems that are realistically solvable on a computer. • When we analyze an algorithm to show that it runs in polynomial time, we need to do two things • First, give a polynomial upper bound (usually in big-O notation) on the number of stages that the algorithm uses when it runs on input of length n. • Then, examine the individual stages in the description of the algorithm to be sure thạt each can be implemented in polynomial time on a reasonable deterministic model. • When both tasks have been done, we can conclude that it runs in polynomial time because we have demonstrated thạt it runs for a polynomiąl number of stages, each of which can be done in polynomial time, and the composition of polynomials is a polynomial.
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