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the input, i.e. it is just the length of the string written on the tape in the start
configuration.
This is a simplification, since the number of steps required by an algorithm
may depend on several of the parameters defining the instance of the problem.
For example, in a graph problem, both the number of nodes and the number
of edges may have an effect on the running time of an algorithm. On the other
hand, the maximum number of edges in a graph with n nodes is n(n 1)/2,
−
20The extent to which quantum computation is stronger than classical is not yet fully
understood.
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so using the square of the number of nodes we get a reasonable measure of the
instance size. The idea is here that the instances are coded in some way on
the machine tape, and coding both node data and edge data, we take this into
account when we measure the instance size as the length of the string written
on the tape.
Asymptotic upper bounds
The”bigO”notation isusedtosetupper bounds onthe asymptoticbehavior
O
offunctions. Wewanttocapturethenotionthatthefunctionf isasymptotically
bounded by the function g.
Let f and g be two functions from the natural numbers to the positive real
numbers. f(n) is in the class of functions (g(n)), or simply f(n)= (g(n)) if
O O
there exist positive integers c and n such that f(n) cg(n) for every integer
0
≤
n n . Thissimply saysthatforsufficientlylargen,the functionf isbounded
0
≥
from above by the function g apart from a constant factor.
Asymptotic lower bounds
For lower bonds, the ”big Omega” Ω is used.
Againletf andg be twofunctions fromthe naturalnumberstothepositive
realnumbers. f(n)isintheclassoffunctionsΩ(g(n)),orsimplyf(n)=Ω(g(n))
ifthere exist positiveintegersc andn suchthatcg(n) f(n)for everyinteger
0
≤
n n . Thissimply saysthatforsufficientlylargen,the functionf isbounded
0
≥
from below by the function g apart from a constant factor.
Asymptotic behavior
If a function f is in both (g) and Ω(g), i.e. if it, apart from constant factors,
O
is bounded both fromabove and below by the same function g, then it behaves
asymptotically as g. The ”big Θ” notation is used to indicate this.
Thus, f(n) is in Θ(g(n)) if it is in both (g(n)) and Ω(g(n)).
O
Time complexity
The time complexity of a deterministic Turing machine M is a function
f : N N, where f is the maximum number of steps performed by M during
→
any computation with input length n.
This is also phrased in any of the following ways: f is the running time of
M, M runs in time f, M is a time f machine.
The time complexity class TIME(f(n)), is defined as
TIME(f(n))= L L is decided by an (f(n)) time machine . (2.25)
{ | O }
There is a corresponding notion of time complexity for non-deterministic
computations. The time complexity of a non-deterministic Turing machine M
53
is a function f : N N, where f is the maximum number of steps performed
→
by M on any branch of the computation with input length n.
Space complexity
The space complexity of a deterministic Turing machine M is a function
f : N N, where f is the maximum number of tape cells scanned by M
→
during any computation with input length n.
This is also phrased in any of the following ways: M runs in space f, M is
a space f machine.
The space complexity class SPACE(f(n)), is defined as
SPACE(f(n))= L L is decided by an (f(n)) space machine (2.26)
{ | O }
Analysis of algorithms
Algorithms areanalyzedby roughly estimating the number of steps requiredto
perform the parts of the algorithm. Textbooks on complexity theory normally
goes through the techniques of doing this. The details differs from model to
model depending on the programming primitive available. We will bypass this
topic here,andjust relyonourintuitioninthe simple models weareconcerned
with, Turing machines and circuits.
2.8.2 Complexity classes
For easy reference,we will briefly review the definitions of the basic complexity
classes.
The class P
The time complexity class P is the collection of all languages that are in
TIME(nk)forsomeconstantk. Thatis,alanguageis inPifitcanbe decided
by a deterministic Turing machine whose running time is bounded from above
by a polynomial in the number of steps.
The class NP
NP is an extremely important time complexity class. Its name is an abbre-
viation of Non-deterministic Polynomial. It is defined as the collection of all
languages that are in NTIME(nk) for some constant k. That is, a language is
in NP if it can be decided by a non-deterministic Turing machine whose run-
ning time is bounded from above by a polynomial in the number of steps. This
class is potentially larger than P, indeed P NP.
⊆
ThereisanothercharacterizationofNPthatdonotrefertonon-deterministic
computations. It is based on the easy (polynomial) verification of a ”yes” in-