Derandomizing LOGSPACE
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Transcript of Derandomizing LOGSPACE
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Derandomizing LOGSPACE
Based on a paper by
Russell Impagliazo, Noam Nissan and Avi Wigderson
Presented by Amir Rosenfeld
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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.
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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.
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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.
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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
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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
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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.
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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
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r
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c
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then 2 If
rc-generato
gc
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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.
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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.
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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.
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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
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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
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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.
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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.
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A Partition Tree
v1 v2 v3 v4 v5
T3
T2
T1
T4
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A Partition Tree
v1 v2 v3 v4 v5
T3T4
T2
T1
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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.
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A Partition Tree
v1 v2 v3 v4 v5
T3T4
T2
T1 cut(T2) = 4
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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
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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
dom-pseudoranG
(V,E)HVrm
)))((())(( xGPMyPM
)()(21
:difference lstatistica denotes
YPXPYX
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Constructing the Generator
))/log(( Define
||Let
. width of for treepartition balanced a be Let
knwcp
Vn
wHT
.in leaves tomapped
are that of vertices theofsubset theis where
,}1,0{ of range a have will , of subtree
afor y whererecursivel defined isgenerator The)(
S
GV(S)
GTSSVrS
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Constructing the Generator (2)
. of bits first theis )( tree, theof leaf aFor xrxGf f
ion.concatenat denotes and , basic a of
output theof halves two theare and where
, ))(())(()(
ly,respective
and subtreesright andleft with of subtree aFor
grp-generato
gg
xgGxgGxG
RLTS
rightleft
rightRleftLS
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L
The Generator (3)
)(xg left
S
R
)(xg right
x (truly random seed)
v1 v2 v3 v4 v5
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What can the Generator Do?
strings. ompseudorand into bits random
)log))/log(((
)log(
converts
.on for , is
r-bitn
nknwcrO
npOrm
G
Hsc-protocolandom k)-pseudor(εG
T
T
Main Theorem:
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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:
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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.
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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.
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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.
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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.
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Tree Width of the Line Protocol
P1 P2 P3 P4 P5
T1
T1
T1
T1
Constant Width !
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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.
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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
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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
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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.
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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
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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.
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The END