Lecture 8 - Space - Complexity Notes | EduRev

Created by: Stuti Nagpal

: Lecture 8 - Space - Complexity Notes | EduRev

 Page 1


Space Complexity
CS60001: Foundations of Computing Science
Pallab Dasgupta
Professor, Dept. of Computer Sc. & Engg.,
Indian Institute of Technology Kharagpur
Page 2


Space Complexity
CS60001: Foundations of Computing Science
Pallab Dasgupta
Professor, Dept. of Computer Sc. & Engg.,
Indian Institute of Technology Kharagpur
Indian Institute of Technology Kharagpur
Pallab Dasgupta
Introduction
? Definition
¦ Let M be a deterministic Turing machine that halts on all inputs. The space 
complexity of M is the function f: N ? N, where f(n) is the maximum 
number of tape cells that M scans on any input of length n. If the space 
complexity of M is f(n), we also say that M runs in space f(n)
¦ If M is a non-deterministic Turing machine wherein all branches halt on all 
inputs, we define its space complexity f(n) to be the maximum number of 
tape cells that M scans on any branch of its computation for any input of 
length n
2
Page 3


Space Complexity
CS60001: Foundations of Computing Science
Pallab Dasgupta
Professor, Dept. of Computer Sc. & Engg.,
Indian Institute of Technology Kharagpur
Indian Institute of Technology Kharagpur
Pallab Dasgupta
Introduction
? Definition
¦ Let M be a deterministic Turing machine that halts on all inputs. The space 
complexity of M is the function f: N ? N, where f(n) is the maximum 
number of tape cells that M scans on any input of length n. If the space 
complexity of M is f(n), we also say that M runs in space f(n)
¦ If M is a non-deterministic Turing machine wherein all branches halt on all 
inputs, we define its space complexity f(n) to be the maximum number of 
tape cells that M scans on any branch of its computation for any input of 
length n
2
Indian Institute of Technology Kharagpur
Pallab Dasgupta
Space Complexity Classes
? Definition: Let f: N ? R
+
be a function. The space complexity classes,
SPACE(f(n)) and NSPACE(f(n)), are defined as follows:
¦ SPACE(f(n)) =   {L | L is a language decided by O(f(n)) space
deterministic Turing machine}
¦ NSPACE(f(n)) = {L | L is a language decided by an O(f(n)) space
non-deterministic Turing machine}
? Examples
¦ SAT can be solved with a linear space algorithm [Space complexity = O(n)]
¦ Testing whether a non-deterministic finite automaton accepts all strings, 
i.e. ALL
NFA
= {<A> | A is a NFA and L(A) = S*}
? Non-deterministic space complexity = O(n)
? SAVITCH’S Theorem
¦ For any function f: N ? R
+
, where f(n) = n,
NSPACE(f(N))      SPACE(f
2
(n))
3
UI
Page 4


Space Complexity
CS60001: Foundations of Computing Science
Pallab Dasgupta
Professor, Dept. of Computer Sc. & Engg.,
Indian Institute of Technology Kharagpur
Indian Institute of Technology Kharagpur
Pallab Dasgupta
Introduction
? Definition
¦ Let M be a deterministic Turing machine that halts on all inputs. The space 
complexity of M is the function f: N ? N, where f(n) is the maximum 
number of tape cells that M scans on any input of length n. If the space 
complexity of M is f(n), we also say that M runs in space f(n)
¦ If M is a non-deterministic Turing machine wherein all branches halt on all 
inputs, we define its space complexity f(n) to be the maximum number of 
tape cells that M scans on any branch of its computation for any input of 
length n
2
Indian Institute of Technology Kharagpur
Pallab Dasgupta
Space Complexity Classes
? Definition: Let f: N ? R
+
be a function. The space complexity classes,
SPACE(f(n)) and NSPACE(f(n)), are defined as follows:
¦ SPACE(f(n)) =   {L | L is a language decided by O(f(n)) space
deterministic Turing machine}
¦ NSPACE(f(n)) = {L | L is a language decided by an O(f(n)) space
non-deterministic Turing machine}
? Examples
¦ SAT can be solved with a linear space algorithm [Space complexity = O(n)]
¦ Testing whether a non-deterministic finite automaton accepts all strings, 
i.e. ALL
NFA
= {<A> | A is a NFA and L(A) = S*}
? Non-deterministic space complexity = O(n)
? SAVITCH’S Theorem
¦ For any function f: N ? R
+
, where f(n) = n,
NSPACE(f(N))      SPACE(f
2
(n))
3
UI
Indian Institute of Technology Kharagpur
Pallab Dasgupta
The Class PSPACE
? Definition
¦ PSPACE is the class of languages that are decidable in polynomial space 
on a deterministic Turing machine. In other words,
PSPACE = U SPACE(n
k
)
¦ NPSPACE is the class of languages that are decidable in polynomial space 
on a non-deterministic Turing machine. In other words,
NPSPACE = U NSPACE(n
k
)
? Relationship among the Complexity Classes
¦ P       NP       PSPACE = NPSPACE       EXPTIME
4
k
k
UI
UI
UI
P
NP
PSPACE
EXPTIME
Page 5


Space Complexity
CS60001: Foundations of Computing Science
Pallab Dasgupta
Professor, Dept. of Computer Sc. & Engg.,
Indian Institute of Technology Kharagpur
Indian Institute of Technology Kharagpur
Pallab Dasgupta
Introduction
? Definition
¦ Let M be a deterministic Turing machine that halts on all inputs. The space 
complexity of M is the function f: N ? N, where f(n) is the maximum 
number of tape cells that M scans on any input of length n. If the space 
complexity of M is f(n), we also say that M runs in space f(n)
¦ If M is a non-deterministic Turing machine wherein all branches halt on all 
inputs, we define its space complexity f(n) to be the maximum number of 
tape cells that M scans on any branch of its computation for any input of 
length n
2
Indian Institute of Technology Kharagpur
Pallab Dasgupta
Space Complexity Classes
? Definition: Let f: N ? R
+
be a function. The space complexity classes,
SPACE(f(n)) and NSPACE(f(n)), are defined as follows:
¦ SPACE(f(n)) =   {L | L is a language decided by O(f(n)) space
deterministic Turing machine}
¦ NSPACE(f(n)) = {L | L is a language decided by an O(f(n)) space
non-deterministic Turing machine}
? Examples
¦ SAT can be solved with a linear space algorithm [Space complexity = O(n)]
¦ Testing whether a non-deterministic finite automaton accepts all strings, 
i.e. ALL
NFA
= {<A> | A is a NFA and L(A) = S*}
? Non-deterministic space complexity = O(n)
? SAVITCH’S Theorem
¦ For any function f: N ? R
+
, where f(n) = n,
NSPACE(f(N))      SPACE(f
2
(n))
3
UI
Indian Institute of Technology Kharagpur
Pallab Dasgupta
The Class PSPACE
? Definition
¦ PSPACE is the class of languages that are decidable in polynomial space 
on a deterministic Turing machine. In other words,
PSPACE = U SPACE(n
k
)
¦ NPSPACE is the class of languages that are decidable in polynomial space 
on a non-deterministic Turing machine. In other words,
NPSPACE = U NSPACE(n
k
)
? Relationship among the Complexity Classes
¦ P       NP       PSPACE = NPSPACE       EXPTIME
4
k
k
UI
UI
UI
P
NP
PSPACE
EXPTIME
Indian Institute of Technology Kharagpur
Pallab Dasgupta
PSPACE-Completeness
? Definition
¦ A language B is PSPACE-complete if it satisfies two conditions:
? B is in PSPACE, and
? Every A in PSPACE is polynomial time reducible to B
¦ If B merely satisfies condition-2, we say that it is PSPACE-hard
? Examples of PSPACE-complete Problems
¦ TQBF = {<F> | F is a true fully quantified Boolean formula}
¦ FORMULA-GAME = {<F> | Player E has a winning strategy in
the formula game associated with F}
¦ GENERALIZED-GEOGRAPHY =
{<G, b> | Player I has a winning strategy for the generalized
geography game played on the graph G starting at node b}
5
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