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Published in:
Limits? What Limits? Read chapter Read first chapter

2016 | OriginalPaper | Chapter

13. Complexity Classes

Author : Bernhard Reus

Published in: Limits of Computation

Publisher: Springer International Publishing

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Abstract

In this chapter the notion of runtime measure for programs is lifted to problems. We classify problems according to the time it takes to decide them. The notion of time bounds is introduced which abstracts away from concrete input expressing runtime simply in terms of the size of the input. The focus will be mainly on linear, polynomial and exponential time. The Big-O and little-o notation are introduced to describe the order of growth of a function. independent from constant factors. This is in line with asymptotic worst-case complexity which we will be using throughout.

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previous chapter Measuring Time Usage
next chapter Robustness of P
Footnotes
1
This is of course sufficient since we are interested in runtime of decision procedures.
 
2
It seems humans can much better grasp large amounts of time than large natural numbers. Maybe this is due to evolution?
 
3
In [1] the classes are called PTIME, LINTIME, NPTIME (there is no EXPTIME), respectively.
 
4
This notation as well as “little-o” discussed later, is by Edmund Georg Hermann Landau (14 February 1877–19 February 1938), a German mathematician who worked mainly in the field of analytic number theory. “\(\mathscr {O}\)” stands for order.
 
5
As the reader may already have guessed there is also a notation for lower bounds \(\Omega \) (called “Big-Omega”), dual to \(\mathop {}\mathopen {}\mathscr {O}\mathopen {}\left( \_\right) \), but we don’t need it for our considerations.
 
Literature
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Metadata
Title
Complexity Classes
Author
Bernhard Reus
Copyright Year
2016
Book
Limits of Computation

Print ISBN: 978-3-319-27887-2

Electronic ISBN: 978-3-319-27889-6

Copyright Year: 2016

DOI
https://doi.org/10.1007/978-3-319-27889-6_13

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