Basic Concepts of Non Deter Minis Tic Algorithms

8/8/2019 Basic Concepts of Non Deter Minis Tic Algorithms

1/20

By: Sunil Nakum,

NITTTR,Chandigarh.

Basic Concepts of Nondeterministic

Algorithms

By: Sunil Nakum, NITTTR, Chandigarh1


2/20

Topics To Be Covered


y Introduction

y Nondeterministic functions

y Searching Example

y Sorting Example

yDeterministic interpretation of nondeterministic algorithm

y Time Analysis

y Clique Decision Problem

y References


3/20

Deterministic Algorithm


y Algorithms with property that result of every operation in uniquely

defined.

y They behaves predictably.

y Given a particular input, it will always produce the same output,

and the underlying machine will always pass through the same

sequence of states.

y For some problems deterministic algo. are hard to find.

y Sometimes we dont want results to be predictable.

yCertain problems cant be solved by fast deterministic algorithms.

y The above need given rise to nondeterministic algorithm.


4/20

Nondeterministic Algorithm


y In theoretical framework we can remove this restriction on

outcome of every operation.

y We can allow algorithms to contain operations whose outcomes are

not uniquely defined but are limited to specified set of possibilities.

y Machine executing such operations is allowed to choose any one of

these outcomes subject to a termination condition.

y This leads to concept of nondeterministic algorithm.


5/20

Nondeterministic Functions


1. Choice ( S )

y

Arbitrarily chooses one of the elements of set S.y x = choice(1,n) can result in x being assigned any of the integers in the

range [1, n], in a completely arbitrary manner.

y No rule to specify how this choice is to be made.

2. Failure()

y Signals unsuccessful completion of a computation

y Cannot be used as a return value

3. Success()

y Signals successful completion of a computation.

y Cannot be used as a return value.

y If there is a set of choices that leads to a successful completion, then onechoice from this set must be made.


6/20

Nondeterministic Functions (Cont.)


y Whenever there is set of choices that leads to successful completion,

then one such set of choice is always made and algorithm terminatessuccessfully.

y Nondeterministic algorithm fails iff there exists no set of choices

leading to a successful signal.

y

Computing time of Choice, Success and Failure are taken to beO(1).

y Machines capable of executing nondeterministic algorithm is called

nondeterministic machine.

yNondeterministic machine do not exists in practice.


7/20

Searching Example


1. Algorithm Search(A,n,x)

2. {

3. j:=Choice(1,n);

4. if A[j]=x then {write(j);Success();}

5. write(0);Failure();6. }

y Here number 0 can be output iff there is no j such that A[j]=x.

y Nondeterministic complexity of O(1).


8/20

Sorting Example


1. Algorithm NSort(A,n)

2. {//sort n positive integers.

3. for i:=1 to n do B[i]:=0; //Initialize B[]

4. for i:=1 to n do

5. {

6. j:=Choice(1,n);

7. if B[j]0 then Failure();8. B[j]:=A[i];

9. }

10. for i:=1 to n-1 do //verify order

if B[i]>B[i+1] then Failure();

12. write(B[1:n]);

13. Success();

14. }


9/20

Sorting Operation


y Auxiliary array B[1:n] is used for convenience.

y

Line 3 initializes B to 0, any value different from all A[i] will do.y for loop of line 4 to 9 each A[i] is assigned to a position in B.

y Line 6 non deterministically determines this position.

y Line 7 ascertains that B[j] has not already used.

y Order of numbers in B is some permutation of the initial order in A.

y for loop of lines 10 and 11 verifies that B is sorted in non decreasing

order.

y Its complexity is O(n).


10/20

Deterministic interpretation of

nondeterministic algorithm


y Possible by allowing unbounded parallelism in computation .y Imagine making n copies of the search instance above, all running in

parallel and searching at different index values for x.

y The first copy to reach Success() terminates all other copies.

y If a copy reaches Failure(), only that copy is terminated.

y In abstract terms, nondeterministic machine has the capability to

recognize the correct solution from a set of allowable choices, without

making copies of the program.


11/20

How correct element is defined


y A correct element is defined relative to a shortest sequence of choices

that leads to a successful termination.y In case if there is no sequence of choices leading to a successful

termination we assume that algorithm terminates in one unit of time

with output Unsuccessful computation.

y

Whenever successful termination is possible a nondeterministicmachine makes a sequence of choices that is a shortest sequence

leading to successful termination.

y Since machine we are defining is untrue not necessary for us to

concern ourselves with how machine can make correct choice at eachstep.


12/20

Decision vs Optimization


y Any problem with answers 0 or 1 is called decision problem.

y

Algorithm for decision problem is called decision algorithm.y Any problem that involves the identification of an optimal (either

minimum or maximum) value of given cost function is known as

optimization problem. Optimization algorithm is used to solve these

kind of problems.y In NSort algo. if the number of A[i] is not distinct then many different

permutations will result in sorted sequence. We only concern with

those nondeterministic algorithms that generate unique outputs. i.e.

nondeterministic decision algorithms.y Output statement is implicit in Signals Success and Failure.


13/20

Time Analysis


y


14/20

Binary representation of input length


y


15/20

Clique


y Clique of undirected graph G=(V,E) is vertex set C V, such that forevery two vertices in C, there exists an edge connecting the two. This is

equivalent to saying that the sub graph. induced by C is complete.

y Size of Clique is number of vertices in it.

y In some cases, the term clique may also refer to the sub graph.

yA maximal clique is a clique that cannot be extended by includingone more adjacent vertex, that is, a clique which does not exist

exclusively within the vertex set of a larger clique.

y A maximum clique is a clique of the largest possible size in a given

graph.


16/20

Input to Clique Decision Problem


y


17/20

Clique Decision Problem Algorithm


1. Algorithm DCK(G,n,k)

2. {

3. S:=; //S is initially an empty set.4. for i:=1 to k do

5. {

6. t:=Choice(1,n);

7. if t S then Failure();8. S:=SU{t}; //Add t to Set S

9. }

10. //At this point S contains k distinct vertex indices

11. for all pairs (i,j) such that i,j S and ij do

12. if (i,j) is not an edge of G then Failure();13. Success();

14. }


18/20

Clique Decision Problem


y


19/20

References


Books

y Fundamentals of Computer Algorithmd By Ellis Horowitz, SarrajSahani, Rajasekaran.

Websites

y http://www.authorstream.com/Presentation/ankush85-159135-nphard-nphard171-175-education-ppt-powerpoint/

y http://en.wikipedia.org/wiki/Deterministic_algorithm

y http://www.jiahenglu.net/course/algorithm2010/slides/np.pdf

y http://en.wikipedia.org/wiki/Nondeterministic_algorithm

y http://en.wikipedia.org/wiki/Clique_(graph_theory)

y http://www.ecst.csuchico.edu/~amk/foo/csci650/notes/ch11/NP2.html


20/20

Thank You


Date post:	10-Apr-2018
Category:	Documents
Upload:	nakumsunil
View:	217 times
Download:	0 times

Basic Concepts of Non Deter Minis Tic Algorithms

Documents