Time complexity: The amount of computer
program needs to run it to completion.
time
the
Space complexity: The amount of memory it needs to run
to completion.
BASIC DIFFERENCES BETWEEN SPACE
COMPLEXITY SPACE COMPLEXITY:
COMPLEXITY
AND
TIME
The space complexity of an algorithm is the amount of memory
it requires to run to completion.
The space needed by a program contains the following components:
1) Instruction space: -stores the executable version of programs
and is generally fixed.
2) Data space: It contains: a) Space required by constants and
simple variables. Its space is fixed. b) Space needed by fixed
size structure variables such as array and structures. c)
Dynamically allocated space. This space is usually variable.
3) Enviorntal stack: -Needed to stores information required to
reinvoke suspended processes or functions. The following data is
saved on the stack - return address. -Value of all local
variables -value of all formal parameters in the function.
TIME COMPLEXITY: The time complexity of an algorithm is the
amount of time it needs to run to completion. Namely space to
measure the time complexity we can count all operations
performed in an algorithm and if we know the time taken for each
operation then we can easily compute the total time taken by the
algorithm. This time varies from system to system. Our intention
is to estimate execution time of an algorithm irrespective of
the computer on which it will be used. Hence identify the key
operation and count such operation performed till the program
completes its execution. The time complexity can be expressed as
a function of a key operation performed. The space and time
complexity is usually expressed in the form of function f (n),
where n is the input size for a given instance of a problem
being solved. F (n) helps us to predict the rate of growth of
complexity that will increase as size of input to the problem
increases. F (1) also helps us to predict complexity of two or
more algorithms in order or find which is more efficient.
The O-notation
In other words: c is not really important for the description of
the running time! To take this circumstance into account,
running time complexities are always specified in the socalled O-notation in computer science. One says: The sorting
method has running time O(n2). The expression O is also
called Landau's symbol.
Mathematically speaking, O(n2) stands for a set of functions,
exactly for all those functions which, “in the long run”, do not
grow faster than the function n2, that is for those functions for
which the function n2 is an upper bound (apart from a constant
factor.) To be precise, the following holds true: A function f
is an element of the set O(n2) if there are a factor c and an
integer number n0 such that for all n equal to or greater than
this n0 the following holds:
f(n) ≤ c·n2.
The function n2 is then called an asymptotically upper bound for
f. Generally, the notation f(n)=O(g(n)) says that the function f
is asymptotically bounded from above by the function g.[12]
A function f from O(n2) may grow considerably more slowly than
n2 so
that,
mathematically
speaking,
the
quotient
f
/
2
n converges to 0 with growing n. An example of this is the
function f(n)=n. However, this does not hold for the function f
which describes the running time of our sorting method. This
method always requires n2 comparisons (apart from a constant
factor of 1/2). n2 is therefore also an asymptotically lower
bound for f. This f behaves in the long run exactly like n2.
Expressed mathematically: There are factors c1 and c2 and an
integer number n0 such that for all n equal to or larger than
n0 the following holds:
c1·n2 ≤ f(n) ≤ c2·n2.
So f is bounded by n2 from above and from below. There also is a
notation of its own for the set of these functions: Θ(n2).
Figure 2.9 contrasts a function f which is bounded from above by
O(g(n)) to a function whose asymptotic behavior is described by
Θ(g(n)): The latter one lies in a tube around g(n), which
results from the two factors c1 and c2.
The time complexity of an algorithm is commonly expressed
using big O notation, which excludes coefficients and lower
order terms. When expressed this way, the time complexity is
said to be described asymptotically, i.e., as the input size
goes to infinity. For example, if the time required by an
algorithm on all inputs of size n is at most 5n3 + 3n, the
asymptotic time complexity is O(n3).
Complexity classes
The concept of polynomial time leads to several complexity
classes in computational complexity theory. Some important
classes defined using polynomial time are the following.






P: The complexity class of decision problems that can be
solved on a deterministic Turing machine in polynomial time.
NP: The complexity class of decision problems that can be
solved on a non-deterministic Turing machine in polynomial
time.
ZPP: The complexity class of decision problems that can be
solved with zero error on a probabilistic Turing machine in
polynomial time.
RP: The complexity class of decision problems that can be
solved with 1-sided error on a probabilistic Turing machine in
polynomial time.
BPP: The complexity class of decision problems that can be
solved with 2-sided error on a probabilistic Turing machine in
polynomial time.
BQP: The complexity class of decision problems that can be
solved with 2-sided error on a quantum Turing machine in
polynomial time.
P is the smallest time-complexity class on a deterministic
machine which is robust in terms of machine model changes. (For
example, a change from a single-tape Turing machine to a multitape machine can lead to a quadratic speedup, but any algorithm
that runs in polynomial time under one model also does so on the
other.) Any given abstract machine will have a complexity class
corresponding to the problems which can be solved in polynomial
time on that machine.
CS Advanced Theory of Computation
Department of Computer Science, IIU
Spring 2011
Course Structure
Lectures = 1 [HEC Recommended 3]
Labs = 0
Credit hours = 3
Prerequisites
Graduate Standing and Consent of instructor. Previous coursework involving
proofs and some programming experience are needed.
Course Objective
To gain an understanding of the mathematics this underlies the theory of
computation. At the end of the course, the student should be able to formalize
mathematical models of computations; use these formalisms to explore the
inherent limitations of computations; and describe some major current
approaches to investigating feasible computation.
Rationale
The theory of computation is concerned with the theoretical limits of
computability. Several mathematical models of computation have been
formulated independently and under any such computational model, the
existence of well-defined but unsolvable problems can be formally shown.
These topics form part of the core of the mathematical foundations of computer
science that will provide students and researchers with a sound theoretical view
of the most fundamental concepts of computation. Specifically, this course
provides a rigorous introduction to the theoretical foundations of computer
science. It deals with a number of interconnected topics and tries to answer the
basic questions, "What is a computer?", "What is an algorithm?", and "What is
computable?". This course examines important theorems and proofs, establishes
a number of interesting assertions in order to expose the techniques used in the
area of theory of computation. Note that although this is not a "mathematics"
course, it does make significant use of mathematical structures, abstractions,
definitions, theorems, proofs, lemmas, corollaries, logical reasoning, inductive
proofs, and the like. If such concepts are difficult for you, you will find this
course very difficult but rewarding. I invite you to accept the challenge.
Course Outline
Automata theory, formal languages, Turing machines, computability theory and
reducibility, computational complexity, determinism, nondeterminism, time
hierarchy, space hierarchy, NP completeness, selected advanced topics.
Class Time
Mondays, 4:30 PM - 7:30 PM
Texts
Our Primary Source
Michael Sipser, Introduction to the Theory of Computation, PWS Publishing,
Boston, 1997.
Our Secondary Source
John E. Hopcroft and Jeffrey D. Ullman, Introduction to Automata Theory,
Languages, and Computation, Addison-Wesley, 1979.
John E. Hopcroft, Rajeev Motwani, and Jeffrey D. Ullman, Introduction to
Automata Theory, Languages, and Computation, Addison-Wesley, 2006.
Mikhail J. Atallah and Mariana Blanton (Eds.), Algorithms and Theory of
Computation Handbook: General Concepts and Techniques, CRC Press, New
York, 2009 (2nd Edition).
Thomas Cormen, Charles Leiserson, Ronald Rivest, and Cliff
Stein, Introduction to Algorithms, McGraw Hill Publishing Company and MIT
Press, 2009 (3rd Edition).
HEC Recommended Text Books/Reference Books
Michael Sipser, Introduction to the Theory of Computation, PWS Publishing,
Boston, 1997.
Christos Papadimitriou, Computational Complexity, 1994, Addison-Wesley.
John Hopcroft and Jeffery Ullman, Introduction to Automata Theory,
Languages, and Computation, 1979, Addison-Wesley.
Tao Jiang, Ming Li, and Bala Ravikumar, Formal models and Computability, in
Handbook of Computer Science, CRC Press, 1996.
T. H. Cormen, et al., Introduction to Algorithms, MIT Press and McGraw-Hill
Book Co., 1990.
Peter Linz, An Introduction to Formal Languages and Automata, Jones and
Bartlett Publishers, 2006.
Ethics
The Honor Code will be strictly enforced in the classroom. It is a violation to
represent joint work as your own or to let others use your work; always
acknowledge any assistance you received in preparing work that bears your
name. You are expected to work independently unless explicitly permitted to
collaborate on a particular assignment. It is not a violation to discuss
approaches to problems with others; however, it is a violation to use wording or
expressions in your assignments that have been written by others without
acknowledging the source.
Exams
Midterm
April 4,
2011
4:30 PM
Final
June 6,
2001
Evening
Session
Points Distribution
Quizzes
15%
Midterm
25%
Final
60%
Grades
A+
3
A
2
B+
1
B
1
C+
3
C
2
D+
1
D
0
F
4
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