Lecture 1

COMS4236: Introduction to

Computational Complexity

Summer 2014

Mihalis Yannakakis

Lecture 1

Course Information

Lectures:

Monday, Wednesday 2:40-3:55

535 Mudd

Web site:

Courseworks

TA: Dimitris Paparas

Course Work

Homeworks, Final

Policies

Late Homework: 10% penalty per late day

Lowest hw does not count in grade

Collaboration policy

Grading: Homeworks 60%, Final 40%

Homework 0: Info about yourself due Monday 1/27

Textbook

Required:

Computational Complexity, by C. Papadimitriou

Other:

-Computational Complexity: A Modern Approach, by S. Arora,

B. Barak

-Introduction to the Theory of Computation, by M. Sipser

-Introduction to Automata Theory, Languages and

Computation, by Hopcroft, Motwani, Ullman

-Computers and Intractability: A Guide to the Theory of NP-

completeness, by Garey, Johnson

Problems, Algorithms, Models

Problem: What is to be computed

Model: Who computes it

Algorithm: How to compute it

Problems

Problem: What needs to be computed Input Output

Example: Graph Reachability problem

input = (graph G, nodes s,t)

output = Yes if G has path from s to t, No otherwise

Factoring problem

input = positive integer

output = list of its prime factors

Set of instances (possible inputs)

For each instance a set of acceptable (correct) outputs

Problem: Given an instance, compute an acceptable output

Algorithms, Models

Model of Computation

eg. sequential (von Neumann) computer parallel computer

quantum computer

Boolean circuit

..

Algorithm: Method for solving the problem on the model of computation

Resources & Complexity

Computational resources: Time, Space, Processors (for parallel computation), Communication (for distributed

computation), Area (for a VLSI implementation), .

R-Complexity of an algorithm: amount of resource R that it uses (R=time, space,)

R-Complexity of a problem: lowest complexity of an algorithm that solves the problem wrt resource R of

interest (time, space, etc)

Goals of Complexity Theory

1. Characterize the complexity of problems

and classify them into complexity classes:

classes of problems that can be solved within

certain complexity bounds

- Problems are studied not only in isolation but

also as they relate to each other

Central notions of

- Reduction between problems

- Completeness for a class

Goals of Complexity Theory ctd.

2. Relations between

- resources (time, space etc.)

- complexity classes

- models of computation

- Power and limitations of different modes/paradigms of computation, eg. nondeterminism, randomization, parallelism etc.

- Laws of Computation

Focus of this course

Basic Resources: Time, Space mainly

Basic Models: Turing machines mainly (and briefly RAM, Boolean combinational circuits)

Deterministic, nondeterministic, probabilistic

Problems: Various types: Decision, search, optimization, games, counting problems

Graph Reachability

Input: Directed Graph G=(N,E), nodes s, t

Question: Is there a path in G from s to t?

i.e. Output= Yes (if there is a path) or No (if there is no path)

Decision problem: output = Yes/No

Graph may be given by its adjacency matrix A or

by adjacency lists: a list Adj[u] for each u in N

that lists all nodes v to which u has an edge

Example

1

2 3

4

5

0 1 0 1 0

0 0 1 0 0

0 0 0 1 1

0 1 0 0 1

0 0 0 0 0

Graph Adjacency Matrix

A[i,j]=1 iff ij

Node 1 can reach node 5

Node 4 cannot reach node 1

Graph Search

for each vN-{s} do {mark[v]=0}

mark[s]=1 ;

Q = {s}

while Q do

{ u= select and delete a node from Q

for each v N do

if (A[u,v]=1 and mark[v]=0) then

{mark[v]=1; Q=Q{v} }

}

If mark[t]=1 then print Yes else print No

If Q implemented as a queue (i.e. we delete from front, insert into back), then Breadth-First-Search

Time Complexity

For every input graph with n nodes, the time complexity is proportional to n: Time=O(n)

Some important points to note:

Complexity is expressed as a function of a parameter n measuring the size of the input (here the number of

nodes)

Worst-case complexity: maximum over all inputs of size n

Expressed complexity in terms of its rate of growth: asymptotic notation O(.) suppresses constant factors

Asymptotic Notation Review:

Theta, Big-Oh, Omega

Big-Oh:

(Order)

))(()( iteusually wr We :Convention

})()(0

:s.t.and0constant |)({))(( 00

ngOnf

ngcnf

nnncnfngO

f(n) cg(n)

Example:

)(5 2nOn

but not vice-versa

Caution: = here denotes membership, not equality

0n

Asymptotic Notations:


Omega:


})()(

:s.t.and0constant |)({))(( 00

ngnf

nfngc

nnncnfng

f(n)

c g(n)

Example:

)(5 2 nn

but not vice-versa

0n



Theta:


})()()(

:s.t.and0, constants|)({))((

21

0021

ngnf

ngcnfngc

nnnccnfng

f(n)

)(1 ngc

)(2 ngc

0n


little-oh, little-omega

little-oh:

})()(0

:s.t.0constant |)({))((

})()(0

:s.t.0constant |)({))((

00

00

nfngc

nnncnfng

ngcnf

nnncnfngo

little-omega:

f(n)=o(g(n) means that for large n, function f is smaller

than any constant fraction of g

f(n)=(g(n) means that for large n, function f is larger than any constant multiple of g, i.e., g=o(f(n))

Example: )(5),(5 22 nnnon

Asymptotic Notations Summary

Notation Ratio f(n)/g(n) for large n

0)(/)())(()(

)(/)())(()(

)(/)())(()(

)(/)())(()(

)(/)())(()(

21

ngnfngonf

cngnfngOnf

cngnfcngnf

ngnfcngnf

ngnfngnf

Examples

For every polynomial p(n) of degree d with a positive leading term, p(n)=(nd). For example, 4n-3n+7=(n)

Every constant is O(1), for example 1000=O(1)

logan = (logcn) for any two constant bases a, c,

For example log10n = (log2n)

Every constant power of logn is little-oh of every constant root of n, for example

Every power of n is little-oh of every exponential function of n, for example

10 5(log ) ( )n o n

10 (1.1 )nn o

Some common functions

2 3

2 3

log log log

log

2 3 !n n

n n n

n n n n n n

n

(polynomial)

(exponential)

Lecture 1

Documents

Transcript of Lecture 1