Download presentation

Presentation is loading. Please wait.

1
Time Complexity P vs NP

2
**Consider a deterministic Turing Machine**

which decides a language

3
**For any string the computation of**

terminates in a finite amount of transitions Initial state Accept or Reject

4
**Decision Time = #transitions**

Initial state Accept or Reject

5
**Consider now all strings of length**

= maximum time required to decide any string of length

6
TIME STRING LENGTH Max time to accept a string of length

7
**Time Complexity Class:**

All Languages decidable by a deterministic Turing Machine in time

8
Example: This can be decided in time

9
**Other example problems in the same class**

10
Examples in class:

11
Examples in class: CYK algorithm Matrix multiplication

12
**Polynomial time algorithms:**

constant Represents tractable algorithms: for small we can decide the result fast

13
It can be shown:

14
**The Time Complexity Class**

Represents: polynomial time algorithms “tractable” problems

15
Class CYK-algorithm Matrix multiplication

16
**Exponential time algorithms:**

Represent intractable algorithms: Some problem instances may take centuries to solve

17
**Example: the Hamiltonian Path Problem**

s t Question: is there a Hamiltonian path from s to t?

18
s t YES!

19
**A solution: search exhaustively all paths**

L = {<G,s,t>: there is a Hamiltonian path in G from s to t} Exponential time Intractable problem

20
The clique problem Does there exist a clique of size 5?

21
The clique problem Does there exist a clique of size 5?

22
**Example: The Satisfiability Problem**

Boolean expressions in Conjunctive Normal Form: clauses Variables Question: is the expression satisfiable?

23
Example: Satisfiable:

24
Example: Not satisfiable

25
exponential Algorithm: search exhaustively all the possible binary values of the variables

26
**Non-Determinism Language class: A Non-Deterministic Turing Machine**

decides each string of length in time

27
**Non-Deterministic Polynomial time algorithms:**

28
The class Non-Deterministic Polynomial time

29
Example: The satisfiability problem Non-Deterministic algorithm: Guess an assignment of the variables Check if this is a satisfying assignment

30
Time for variables: Guess an assignment of the variables Check if this is a satisfying assignment Total time:

31
**The satisfiability problem is an - Problem**

32
Observation: Deterministic Polynomial Non-Deterministic Polynomial

33
Open Problem: WE DO NOT KNOW THE ANSWER

34
Open Problem: Example: Does the Satisfiability problem have a polynomial time deterministic algorithm? WE DO NOT KNOW THE ANSWER

35
P versus NP P 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).

36
**Any P type problem can be solved in "polynomial time**

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.

Similar presentations

OK

Prof. Busch - LSU1 Time Complexity. Prof. Busch - LSU2 Consider a deterministic Turing Machine which decides a language.

Prof. Busch - LSU1 Time Complexity. Prof. Busch - LSU2 Consider a deterministic Turing Machine which decides a language.

© 2017 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Ppt on conservation of natural vegetation and wildlife Ppt on ectopic pregnancy Ppt on non communicable diseases Ppt on history of basketball Ppt on operation research in linear programming Ppt on operating system download Ppt on balanced diet completes physical health Ppt on mass production Ppt on conservation of momentum calculator Ppt on hard gelatin capsule composition