Derandomizing LOGSPACE

Based on a paper by

Russell Impagliazo, Noam Nissan and Avi Wigderson

Presented by Amir Rosenfeld

Derandomization & Pseudorandomness

Pseudorandomness is about understanding the minimum amount of randomness actually required by a probabilistic model of computation.

A Pseudorandom Generator takes m<<n random bits and deterministically stretches them into n pseudorandom bits.

A Pseudorandom Generator is said to “fool” a computational model, when n truly random bits used by the model can be replaced by n pseudorandom bits created by the generator without significant difference in its behavior.

If we can “fool” a probabilistic algorithm by using m<<n truly random bits, then we can derandomize it by trying out all 2m possibilities.

Relevant Computational Models The generator described by INW fools every

computational model which can be described as a network of probabilistic processors.

The number of truly random bits required by the generator (m) depends on the the communication bandwidth of the algorithm run by the network.

No assumptions are made about the computational power of the processors.

Communication Network

v1 v2 v3 v4 v5

Each processor receives part of

the input.

Each processor requires a preset

number of random coin flips

Each processor sends and receives

at most c bits of information

Each processor calculates some function of its input, random bits

and received information.

Some Intuition In a network algorithm that uses probabilistic processors, we

can reuse the same random bits on many processors provided their communication is limited enough such that a processor cannot learn much about the random bits of the others.

Alice Bob

r random bits are required by both Alice and Bob

c bits of communication

So Bob can reuse most of the

entropy in the bits given to Alice

The entropy of Alice's bits as seen here is r-c

Yao (1979) described a communication complexity model for computation where two parties are required to compute a function

The communication complexity was described as the minimum number of bits that need to be communicated between the parties until one of them outputs the answer.

The Basic Two-Party Model

}1,0{: NMf

Defining Protocol The network algorithm uses a specific protocol to communicate

between the parties.

We call a protocol “normal” if the total amount of information sent/received by a party equals the total number of bits that it sent/received. (I.e. length of messages and timing are known in advance)

A c-protocol is a normal protocol in which on any input to the network and every random choice, every party sends and receives at most c bits of communication.

Definition of the Basic Generator

.}1,0{in uniformly chosen is and

,}1,0{in randomat uniformly chosen are , where

accepts ))((Praccepts ),(Pr

ioncommunicat of bits most at of

protocolparty -every twofor if ,parameter

with complexityion communicat

for generator ompseudorand a called is

{0,1} {0,1}}1,0{:function A

21

21

rr

m

r

m

x

yy

xgPyyP

c

P

c

g

"" a is

then 2 If

rc-generato

gc

The Basic Generator

Fix an expander graph H=(V,E), with 2r vertices and degree D=2d.

The input is a name of a random directed edge in E. It therefore requires m=r+d random bits.

The output are the two vertices on the edge. Thus it produces two r bit output strings.

What does it Fool?

Theorem 1

g is a c-generator, i.e. it fools 2-party

c-protocols, for c = (d - log λ)/2

where λ is the second largest eigenvalue

of H.

Recall that g consists of an expander graph H with degree D=2d.

Proof of Theorem 1 For every graph H=(V,E) of degree D and

second largest eigenvalue λ, and for every S,T V the following inequality holds:

DV

T

V

S

E

TSE

),(

This is the Mixing Lemma that was presented earlier in the course.

Proof of Theorem 1 (cont.)

generator, theof definition by the Thus

. rectanglesfor accept protocol Let the

. rectanlges

2 into inputs thepartition only can A

Ii

TS

c-protocol

ii

c

Ii

cc

iiii

DV

T

V

S

E

TSE

xgPyyP

2 2

),(

accepts ))((Praccepts ),(Pr 21

Extractor – An Alternative View

The same construction could be achieved with an extractor:

r d

Seed = m random bits

r E(r,d)

The auxiliary bits used by the extractor

Bounded communication – r still contains much entropy to other party

Expanding the Model

The communication network is a graph H=(V,E) where the nodes are parties/processors and directional edges represent communication lines between them.

Each processor has unlimited power and it can use any input information and any communicated information it received.

We are concerned with networks algorithms using c-protocols.

Partition Trees A partition tree T of a graph H=(V,E) is a rooted binary

tree with a one-to-one onto mapping of V to the leaves of T.

T is called balanced if the depth of T is O(log |V|).

Every internal node ν of T partitions V into 3 sets: Aν,

Bν and Cν ; these are the vertices of V residing in the

leaves of the left child of ν, right child of ν and the remaining leaves of T.

A Partition Tree

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T3

T2

T1

T4

A Partition Tree

v1 v2 v3 v4 v5

T3T4

T2

T1

Partition Trees (cont.) cut(ν) is the subset of E which connect vertices

in two different sets.

The width of ν is the smallest number of vertices which cover all of the edges in the cut.

The width of a tree is the minimal width of an internal node.

The width of a graph H is the the smallest width of a balanced partition tree for H.

A Partition Tree

v1 v2 v3 v4 v5

T3T4

T2

T1 cut(T2) = 4

k-measurement A k-measurement M on a protocol is a

k-bit function where each of the bits can be computed at the end of the protocol by a single processor (at least).

The entire k bit measurement

Processor 1

Processor 2

Processor n

The Required Generator

,

any and any for if ,on for

k),( called is }1,0{}1,0{:

graph.ion communicat a be Let

Mentk-measurem

Pc-protocolHsc-protocol

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Constructing the Generator

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Constructing the Generator (2)

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rightleft

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The Generator (3)

)(xg left

S

R

)(xg right

x (truly random seed)

v1 v2 v3 v4 v5

What can the Generator Do?

strings. ompseudorand into bits random

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converts

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Main Theorem:

Random LOGSPACE

Input Tape of length n

Work Tape of length O(log n)

Random Bits Tape

The class of problems that are decidable by a Turing machine with the following characteristics:

Non Uniform Machine

The Turing machine just described is for a uniform model of computation.

We can build a non-uniform machine by creating a specific hardwired machine for every input.

This machine has the random bit tape as the only input.

Non-Uniform Machine (2) We are now left with an OBDD that has to

Accept or Reject according to the random tape.

R

A

Random Bit Tape

Max Poly(n) Width resulting from LOGSPACE bound

The OBDD implies read-once access of the random bits, but this is only a simplification for the purpose of explanation.

The Communication Network

Reducing the OBDD to a network and protocol:

Each random-bit cell is a processor in the network

Whenever the head moves from cell Ai to its

neighbor Ai+1 , the entire state of the machine is sent

from processor i to processor i+1.

The Resulting Line Protocol

P1 P2 P3 Pr-1 Pr

System StateO(log n) bits

At each random cell transition a processor sends the system state to its neighbor.

Tree Width of the Line Protocol

P1 P2 P3 P4 P5

T1

T1

T1

T1

Constant Width !

What do we need to fool?

A processor sends the state at most constant number of times. Thus it is an O(S)-protocol.

The tree-width of the network is O(1)

The k-measurement is actually a 1-measurement:

Accept or Reject by the last processor.

The total state of the machine is held in O(log n) bits.

Therefore, our generator requires only:

random bits, allowing us to derandomize the algorithm with nlog n input strings.

Derandomizing LOGSPACE Bounded Read-Multiplicity Machines

)(log

)log)1)/log()(log((2 nO

nrnnOrO

Proof (for LOGSPACE Machines)

S

SDrandom

at rooted subtree theto

inputs random with worksmachine when the

M(P) ofon distributi thebe )( Define

.at rooted subtree for the inputs produce to

input random a with G uses machine when the

M(P) ofon distributi thebe )( Define

S

SDS

pseudo

nSSDSD randompseudo

|| )()(

that induction by prove We

Proof (cont.)

Induction base: For a leaf – the distributions are identical.

For the induction step we create hybrid distributions and prove that their combined distance from the fully random distribution meets the goal.

The Hybrid DistributionsR

R R

RRRR

R

R R

RRGG

R

G G

GGGG

R

R R

GGGG

By induction hypothesis, and averaging over possible values of right side

By induction hypothesis, and averaging over possible values of right side

Summary

We showed a generator that can fool randomized network algorithms.

We showed a reduction of LOGSPACE machines to a Network Algorithm

We proved that the generator works for the networks that result from that reduction

This proves that we can derandomize LOGSPACE by order nlog n random bits.

The END