Complexity ©D.Moshkovitz 1 Introduction. Complexity ©D.Moshkovitz 2 Introduction Objectives: –To introduce some of the basic concepts of complexity theory.

Complexity©D.Moshkovitz

1

Introduction


2

Introduction

• Objectives:– To introduce some of the basic concepts of

complexity theory.

• Overview:– Tea parties and computations– Growth rate and reasonability– Reducibility– … And more…


3

The Crazy Tea PartyProblem To seat all guests at a round table, so

people who sit in adjacent seats like each other.

John Mary Bob Jane Alice

John

Mary

Bob

Jane

Alice


4

Solution for the Example

Alice

Bob

Jane

John

Problem To seat all guests at a round table, so people who sit an adjacent seats like each other.

Mary


5

Naive Algorithm

• For each ordering of the guests around the table– Verify each guest likes the guest

sitting in the next seat.


6

How Much Time Should This Take? (worse case)guests steps

n (n-1)!

5 24

15 87178291200

100 9·10155

say our computer is capable of 1010

instructions per second, this will still take 3·10138 years!


7

ToursProblem Plan a trip that visits every site exactly

once.


8

Solution for the ExampleProblem Plan a trip that visits every site exactly

once.


9

Naive Algorithm (Backtracking)

• For each site– Try out all reachable sites, which

have not been visited yet– Backtrack and retry – Repeat the process until stuck


10

How Much Time Should This Take?

sites steps

n n!

5 120

15 1307674368000

100 9·10157

If our computer can carry out 10,000 instructions per

second, than it will take us 4 years!


11

Is a Problem Tractable?

• YES! And here’s the efficient algorithm for it

• NO! and I can prove it

and what if neither is the case?


12

Food for Thought…

• Can you design an efficient algorithm for the tour problem if the sites’ map contains no cycles?


13

Growth Rate: Sketch

10n

n2

2n

n! =2O(n lg

n)

input length

time


14

The World According to Complexity

reasonable unreasonable

polynomial nO(1)

exponential 2nO(1)


15

Could One be Fundamentally Harder

than the Other?

Tour

Seating

?


16

Relations Between Problems

Assuming an efficient procedure for problem A, there is an efficient procedure for

problem B

B cannot be radically harder than A


17

Reductions

B p A

B cannot be radically harder than A

In other words: A is at least as hard as B


18

Which One is Harder?

Tour

Seating

?


19

Reduce Tour to SeatingFirst Observation: The problems aren’t so

different

site guest

“directly reachable from…”

“liked by…”


20

Reduce Tour to SeatingSecond Observation: Completing the circle

• Let’s invite to our party a very popular guest,• i.e one who can sit next to everyone else.


21

Reduce Tour to Seating

• If there is a tour, there is also a way to seat all the imagined guests around the table.

. . . . . .

popular guest


22

Reduce Tour to Seating

• If there is a seating, we can easily find a tour path (no tour, no seating).

. . .

popular guest

. . .


23

Bottom Line

The seating problem is at least as hard as the tour problem


24

Discussion

• Although we couldn’t come up with an efficient algorithm for the problems

• Nor to prove they don’t have one,• We managed to show a very

powerful claim regarding the relation between their hardness


25

Discussion

• Interestingly, we can also reduce the seating problem to the tour problem.

• Moreover, there is a whole class of problems, which can be pairwise efficiently reduced to each other.


26

Food for Thought…

• Can you reduce the seating problem to the tour problem?


27

NPC

NPC

Contains thousands of distinct problem

exponential algorithms

efficient algorithms

?

each reducible to all others


28

How Can Studying Complexity Make You a Millionaire?

• P vs NP is the most fundamental open question of computer science.

• Resolving it would get you great honor

• … as well as substantial fortune…www.claymath.org/prizeproblems/pvsnp.htm

• Philosophical: if P=NP – Human ingenuity unnecessary– No need for mathematicians!!

• Is nature nondeterministic?

http://www.claymath.org/prizeproblems/pvsnp.htm

http://www.claymath.org/prizeproblems/pvsnp.htm


29

What More?

• Let’s review some more questions to be discussed in this course.


30

Generalized Tour Problem

• Add prices to the roads of the tour problem• Ask for the least costly tour

$8

$10

$13$12

$19

$3$17

$13


31

Approximation

• How about approximating the optimal tour? • I.e – finding a tour which costs, say, no more

than twice as much as the least costly.

$8

$10

$13$12

$19

$3$17

$13


32

Is Running Time the Only Precious Resource?

• How about memory (space)?• Are there other resources?


33

Games

• Each player picks a word, which begins with the letter that ended the previous word.

Apple EggGirl LordDog GuyYard Deer


34

Dictionary

Apple

Egg Girl

Ear

LordLordDog

Red


35

Game Tree

Egg Girl DogApple Ear LordRed

GirlEgg EarDog DogLord Girl Red

Lord Lord

Dog

Dog

Girl

Lord

Dog Girl

Lord

Girl Girl Red Lord Dog Lord Dog Girl

Apple

Egg Girl

Ear

LordLordDog

Red


36

Winning Games

• How can we find a winning strategy for this game?

• How much space would it take to compute it using game trees?


37

Summary

• We got to know two problems:– The seating problem (HAMILTONIAN-CYCLE)– The tour problem (HAMILTONIAN-PATH)

• Although we could say very little about them, we managed to show their computational hardness are related.

• Moreover, we claimed they are part of a large class of problems, called NPC.


38

Summary

• We also mentioned some of the topics we will discuss later in the course:– Approximation– Space-bounded computations

Complexity ©D.Moshkovitz 1 Introduction. Complexity ©D.Moshkovitz 2 Introduction Objectives: –To introduce some of the basic concepts of complexity theory.

Documents

complexity d

b slide

mary slide

introduction slide

stuck slide

n o1 slide

popular guest slide

tour problem