chapter 7 . time complexity

Chapter 7. Time Complexity

CS314: FORMAL LANGUAGES AND AUTOMATA THEORYL. NADA ALZABEN

2

Quick Note don’t forget to read

chapter 6section 6.1 and 6.2

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Computer Science Department

Quick revision The theory of computation is divided over

three parts: Part1: Automata Part2: Computability Part3: Complexity.

→ Complexity

→ Computabilit

y → Automata

Computer problems (hard or

easy)Ch:7- end of

book

Problem can be solved

(true – false) Ch:3-6

Deal with definitions

and properties of mathematic

al model (FA – CFG)

Ch:1-2

Lecture #16

Time complexity Even when a problem is decidable and

computationally solvable it may not be solvable in practice if the solution requires an inordinate amount of time and space.

Complexity theory: is an investigation of the time, memory or other resources required for solving computational problems.

7.1 measuring complexity How much time does a Turing Machine

tape need to decide language A?

Describe TM ,M1, at a low level and count the number of steps that m1 uses when it runs.

7.1 measuring complexity The number of steps that an algorithm uses on

a particular input may depend on several parameters but for simplicity we compute as a function of the length of the string representing the inputs.

In worst-case analysis consider the longest running time of all inputs of a particular length.

In average-case analysis consider the average running time of all inputs of a particular length.

7.1 measuring complexity

Big-O and Small-O notation Asymptotic analysis one convenient

form of estimation. By just considering the highest order term of the expression and ignore any coefficient and any lower order term .

For example:

The asymptotic notation or Big-O notation is

Big-O and Small-O notation

Big-O and Small-O notation Example:

Example:

Example: Let f3(n) be the function

Big-O and Small-O notation

Example:

Analyzing algorithms Lecture #17

Analyzing algorithms

Time () contains all languages that can be decided with time complexity O () so we can say for the language A:

Analyzing algorithms Machine M2 uses different algorithm to

decide on language A to give a shorter time than M1.

Analyzing algorithms Machine M3 uses second tape and uses a

different algorithm to decide on language A (linear time )

Complexity Relationships among models

The choice of computational model can effect the time complexity of languages.

Three models used:1. The single tape TM.2. The multitape TM.3. Nondeterministic TM

Complexity Relationships among models

Complexity Relationships among models

Complexity Relationships among models

7.2 The Class P Previous theorems show two important

distinction: Polynomial difference in running time is small Exponential difference in running time is large. Ex: when n=1000.

Polynomial Time: Algorithms are fast enough for many

purposes unlike the Exponential Time. (ex. Brute-force search)

Polynomially equivalent. (ex. Deterministic single and multi tape TM’s)

Lecture #18

7.2 The Class P

The class P plays an important role because:1. P is invariant for all models of computation that are

Polynomially equivalent to the deterministic single-tape TM.

2. P roughly corresponds to the class of problems that are realistically solvable on a computer.

7.2 The Class P Giving a polynomial time algorithm in a high-level

description with out referring to a particular computation model or even details of tapes and head motion.

Algorithms are described with numbered stages. Stage of an algorithm might be implemented by several

steps in TM. To give the polynomial time of an algorithm :1. Give the polynomial upper bound on the number of

stages that an algorithm uses when running on input of length n.

2. Examine the individual stages to be sure that each can be implemented in polynomial time on a deterministic model.

7.2 The Class P Examples of problems in P: Encoding of graphs – an encoding of a graph is a list

of its nodes and edges. (adjacency matrix : the (i,j)th entry is 1 if there is an edge from I to j) and 0 when not.

The running time on graphs is computed by the number of nodes instead of the size of the node. So it gives a polynomial time on the number of nodes and then it is polynomial in the size of the input.

In the directed graph G contains nodes s and t the PATH problem is to find a directed path between the nodes s and t. as in the figure.

7.2 The Class P Examples of problems in P: PATH

problem is there a path from s to t?

7.2 The Class P

7.2 The Class P Examples of problems in P: RELPRIME

problem is deciding if two numbers are relatively prime?

7.2 The Class P

Greatest common divisor gcd(x,y)

7.2 The Class P Using a powerful technique called

dynamic programming

7.2 The Class P

7.3 The Class NP We previously produced the polynomial time

and stated its is faster than the exponential one.

Some problems was solved by brute-force search and we had an alternative solution to have it in polynomial time

Sometimes this can not be obtain. Example: Hamiltonian path in directed

graph G is a directed path that goes through each node exactly once.

7.3 The Class NP

7.3 The Class NP By modifying the PATH algorithm that was solved by

brute-force search (just before it was solved by a polynomial time) we can obtain an algorithm that can decide if the path is Hamiltonian.

The HAMPATH problem have feature called polynomial verifiability which is used for its complexity.

Although deciding if a path is Hamiltonian requires an exponential time it may be solved and proven to be existed. This is called verifying.

Verifying the existence of Hamiltonian path is easier than determining its existence.

Example: of polynomial verifiability is compositeness. (composite number is a none prime number)

7.3 The Class NP

HAMPATH and COMPOSITES are members of NP.

COMPOSITES are members of P too. P is a subset of NP.

7.3 The Class NP NTM that decides HAMPATH problem in NP:

Although each stage is decided in a polynomial time this algorithm runs in nondeterministic polynomial time.

7.3 The Class NP

7.3 The Class NP Example of problem in NP: A clique in an undirected graph is a sub-

graph where every two nodes are connected by an edge.

A k-clique is a clique that contain k nodes.

7.3 The Class NP Example

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