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Polynomial time

Efficient algorithms

• The running time of an

algorithm depends on

the input

• For longer inputs, we

allow more time

• Efficiency is measured

as a function of input

size

ATM

PCP

Input representation

• Since we measure efficiency in terms of input size, how

the input is represented will make a difference

• For us, any “reasonable” representation will be okay

The number 17 17 10001 ( 17 in base two) 11111111111111111

OK OK NO

This graph

0000,0010,0001,

1 2

3 4

OK (2,3),(3,4) (^) OK

Measuring running time

• What does it mean when we say:

• One step in

all mean different things!

“This algorithm runs in 1000 steps”

java RAM machine Turing Machine if (x > 0) y = 5*y + x;

write r3; δ(q^3 , a) = (q^7 , b, R)

Efficiency and the Church-Turing thesis

• The Church-Turing thesis says all these models

are equivalent in power…

… but not in running time!

java

RAM machine

Turing Machine

multitape TM

UNIVAC

The Cobham-Edmonds thesis

• However, there is an extension to the Church-

Turing thesis that says

For any realistic models of computation M 1 and

M 2 :

• So any task that takes time T on M 1 can be done in

time (say) T^2 or T^3 on M 2

M 1 can be simulated on M 2 with at most

polynomial slowdown

Example of efficient simulation

• Recall simulating multiple tapes on a single tape

M

0 1 0 …

0 1 …

1 0 0 …

S^0 1 0 #^0 1 #^1 00 # …

  

Γ = {0, 1, ☐, 0, 1,^ ^ ^ ☐ , #}

Running time of simulation

• Each move of the multiple tape TM might

require traversing the whole single tape

after t steps

O ( s ) steps of single tape TM s = rightmost cell ever visited

s ≤ 3 t + 4

1 step of 3-tape TM

t steps of 3-tape O ( ts ) = O ( t^2 ) single tape steps

multi-tape TM single tape TM

quadratic slowdown

Running time of nondeterministic

TM

• What about nondeterministic TMs?

• For ordinary TMs, the running time of M on

input x is the number of transitions M makes

before it halts

• But a nondeterministic TM can run for a

different time on different “computation

paths”

Example

• Definition of running time for nondeterministic

TM

qacc q 0 1/1R q 1 0/0R

what is the running time?

qrej

running time =

computation path: any possible sequence of transitions

max length of any computation path

Simulation slowdown for

For all k > 0 nondeterminism

For all possible strings a of length k Copy x to z. Simulate N on input z using a as choices If a specifies an invalid choice or simulation loops/rejects, abandon simulation. If N enters its accept state, accept and halt. If N rejected on all a s of length k , reject and halt.

running time of N is t simulation will halt when k = t

running time of simulation= (running time for specific a ) × (number of a s of length ≤ t ) = O( t ) × 2 O( t )^ = 2^ O( t )

Simulation slowdown

multi-tape java TM

RAM machine single tape TM

O ( t ) (^) O ( t ) O ( t^2 )

O ( t^2 ) O ( t ) O ( t )

nondeterministic TM

2 O( t )

Do nondeterministic TM violate the Cobham-Edmonds thesis?

Example

• Recall the scheduling problem

• Scheduling with nondeterminism:

CSC 3230 CSC 2110

CSC 3130 CSC 3160

Can you schedule final exams so that there are no conflicts?

Exams → vertices

Slots → colors

Conflicts → edges Y R B

schedule(int n, Edges edges) { for i := 1 to n: choose { c[i] := Y; } or { c[i] := R; } or { c[i] := B; } for all e in edges: if c[e.left] == c[e.right] reject; accept; }

Example

... but if we had it, we could schedule in linear

time!

schedule(int n, Edges edges) { for i := 1 to n: choose { c[i] := Y; } or { c[i] := R; } or { c[i] := B; } for all e in edges: if c[e.left] == c[e.right] reject; accept; }

In reality, programming languages don’t allow us to choose

We have to tell the computer how to make these choices

Nondeterminism does not seem like a realistic

feature of a programming language or computer