time complexity p vs np. consider a deterministic turing machine which decides a language
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Time ComplexityP vs NP
ML
Consider a deterministic Turing Machinewhich decides a language
For any string the computation ofterminates in a finite amount of transitions
w M
Acceptor Reject w
Initialstate
Acceptor Reject w
Decision Time = #transitions
Initialstate
Consider now all strings of lengthn
)(nTM = maximum time required to decide any string of length n
)(nTM
Max time to accept a string of length n
1 2 3 4 n STRING LENGTH
TIM
E
Time Complexity Class: ))(( nTTIME
All Languages decidable by a deterministic Turing Machinein time
1L 2L
3L
))(( nTO
Example: }0:{1 nbaL n
This can be decided in time)(nO
)(nTIME
}0:{1 nbaL n
)(nTIME
}0:{1 nbaL n
}0,:{ knabaabn
}even is :{ nbn
Other example problems in the same class
}3:{ knbn
)( 2nTIME
}0:{ nba nn
Examples in class:
}},{:{ bawwwR
}},{:{ bawww
)( 3nTIMEExamples in class:
} grammar free-context
by generated is :,{2
G
wwGL
} and
matrices :,,{
321
3213
MMM
nnMMML
CYK algorithm
Matrix multiplication
Polynomial time algorithms: )( knTIME
Represents tractable algorithms:
for small we can decidethe result fast
k
constant 0k
)( knTIME
)( 1knTIME
)()( 1 kk nTIMEnTIME It can be shown:
0
)(
k
knTIMEP
The Time Complexity Class P
•“tractable” problems
•polynomial time algorithms
Represents:
P
CYK-algorithm
}{ nnba }{ww
Class
Matrix multiplication
}{ ban
Exponential time algorithms: )2(knTIME
Represent intractable algorithms:
Some problem instancesmay take centuries to solve
Example: the Hamiltonian Path Problem
Question: is there a Hamiltonian path from s to t?
s t
s t
YES!
)2()!(knTIMEnTIMEL
Exponential time
Intractable problem
A solution: search exhaustively all paths
L = {<G,s,t>: there is a Hamiltonian path in G from s to t}
The clique problem
Does there exist a clique of size 5?
The clique problem
Does there exist a clique of size 5?
Example: The Satisfiability Problem
Boolean expressions inConjunctive Normal Form:
ktttt 321
pi xxxxt 321
Variables
Question: is the expression satisfiable?
clauses
)()( 3121 xxxx
Satisfiable: 1,1,0 321 xxx
1)()( 3121 xxxx
Example:
2121 )( xxxx
Not satisfiable
Example:
e}satisfiabl is expression :{ wwL
)2(knTIMEL
Algorithm: search exhaustively all the possible binary values of the variables
exponential
Non-Determinism
Language class: ))(( nTNTIME
A Non-Deterministic Turing Machine decides each string of length in time
1L
2L3L
))(( nTO
n
Non-Deterministic Polynomial time algorithms:
)( knNTIMEL
0
)(
k
knNTIMENP
The class NP
Non-Deterministic Polynomial time
Example: The satisfiability problem
Non-Deterministic algorithm:
•Guess an assignment of the variables
e}satisfiabl is expression :{ wwL
•Check if this is a satisfying assignment
Time for variables: n
)(nO
e}satisfiabl is expression :{ wwL
Total time:
•Guess an assignment of the variables
•Check if this is a satisfying assignment )(nO
)(nO
e}satisfiabl is expression :{ wwL
NPL
The satisfiability problem is an - ProblemNP
Observation:
NPP
DeterministicPolynomial
Non-DeterministicPolynomial
Open Problem: ?NPP
WE DO NOT KNOW THE ANSWER
Example: Does the Satisfiability problem have a polynomial time deterministic algorithm?
WE DO NOT KNOW THE ANSWER
Open Problem: ?NPP
P versus NPP versus NP (polynomial versus
nondeterministic polynomial) refers to a theoretical question presented in 1971 by Leonid Levin and Stephen Cook, concerning mathematical problems that are easy to solve (P type) as opposed to problems that are difficult to solve (NP type).
Any P type problem can be solved in "polynomial time.“ A P type problem is a polynomial in the number of bits
that it takes to describe the instance of the problem at hand. An example of a P type problem is finding the way from point A to point B on a map. An NP type problem requires vastly more time to solve than it takes to describe the problem.
An example of an NP type problem is breaking a 128-bit digital cipher.
The P versus NP question is important in communications, because it may ultimately determine the effectiveness (or ineffectiveness) of digital encryption methods.
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