S-T Connectivity on Digraphs with a Known Stationary Distribution - PowerPoint PPT Presentation

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S-T Connectivity on Digraphs with a Known Stationary Distribution. Kai-Min Chung Harvard University Joint work with: Omer Reingold (Weizmann Inst. of Science) Salil Vadhan (Harvard University). Outline. RL vs. L and s-t connectivity Our main result Derandomization settings

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S-T Connectivity on Digraphs with a Known Stationary Distribution

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S-T Connectivity on Digraphs with a Known Stationary Distribution

Kai-Min Chung

Harvard University

Joint work with:

Omer Reingold (Weizmann Inst. of Science)

Outline

• RL vs. L and s-t connectivity

• Our main result

• Derandomization settings

• Algorithm outline

• Main step

• Conclusion

Space

Space Bounded Complexity Classes

L : deterministic logspace

Lk : deterministic space

O(logk n)

RL: randomized logspace

polynomial time

one-sided error

NL: non-deterministic

logspace

RL vs. L

Does randomness help

space bounded computation?

..

..

..

..

..

Input:

Work:

R/W

M

Random

coins

0/1

..

..

..

write only

Output:

One way

Results on derandomizing space

• How much space is sufficient?

• RL ⊆ NL ⊆ L2 [Sav70]

• RL ⊆L3/2 [Nis92,SZ95]

• How many bits can we derandomize?

• polylog(n) bits [NZ93]

• 2√logn bits with restrictions [RR99]

• Make complexity assumptions

• RL = L by hardness assumptions[KvM99]

• Study s-t connectivity problem

G

t

s

S-T Connectivity (STConn)

n: # vertices

d: out-degree

• Input: (G,s,t)

• G – (n,d)-digraph

• s,t – vertices in G

• Output: does s connect to t?

STConnand logspace computation

• Arbitrary digraph

• NL-complete

• Poly-Mixing digraph

• RL-complete [RTV06]

• Undirected graph - USTConn

• SL-complete [LP82]

• USTConn ∈ RL [AKLLR79]

• USTConn ∈ L [Rei05]

G

t

s

Random walk algorithm for USTConn [AKLLR79]

• Random Walk Algorithm:

• Do poly(n) steps random walk from s

• Accept if ends at t

π(t) is non-negligible

converge to stat dist π

Poly-Mixing STConn[RTV06]

• Input: (G,s,t,1k)

• k: mixing time parameter

• YES instance:

• s and t are connected

• Random walk from s converges to a stationary dist πin k steps

• π(s), π(t) ≥ 1/k

• NO instance:

• No path from s to t

Complete for (promise) RL

Graph classes and RL vs. L

regular

reversible

stationary distribution

• For what graph class can we solve s-t connectivity in deterministic logspace?

• [Rei05] USTConn ∈ L

• [RTV06] STConn on regular digraph ∈ L

• Our result:

Known-Stationary STConn ∈ L

δ-Known-Stationary STConn

• Input: (G,s,t,1k,p1,…,pn)

• pv: estimate of a stationary dist π(v)

(input stationary distribution)

• YES instance:

• s and t are connected

• |pv - π(v)| ≤ δ for all v that can reach s

• Random walk from s converges to a stationary dist πs in k steps.

• π(s), π(t) ≥ 1/k

• No instance: no path from s to t

p can be arbitrary

δ-Known-Stationary Find-Path

• Input: (G,s,t,1k,p1,…,pn)

• YES instance:

• s and t are connected

• |pv - π(v)| ≤ δ for all v that can reach s

• Random walk from s converges to a stationary dist πs in k steps.

• π(s), π(t) ≥ 1/k

• Output: path from s to t

Inspired by [RR99]:

estimate of state distribution

is available

Our result

δ-Known-Stationary STConn

and

δ-Known-Stationary Find-Path

Both in deterministic logspace

where δ =1/poly(n,d,k)

Outline

• RL vs. L and s-t connectivity

• Our main result

• Derandomization settings

• Algorithm outline

• Main step

• Conclusion

Explicit vs. oblivious derandomization

• Explicit setting

• Given input graph explicitly

• Oblivious setting

• CANNOT look at the graph

• Only know parameters of graph

• n: #vertices

• d: (out-)degree

• k: mixing time parameter

• Construct PRG for Random Walk Algorithm

Results in explicit settings

• Explicit setting

• RL ⊆L3/2 [SZ95]

• USTConn ∈ L [Rei05]

• Known-Stationary STConn ∈ L [Our result]

Results in oblivious settings

• Oblivious setting

• Nisan’s PRG: O(log2 n) seed length [Nis92]

• RTV’s PRG: O(log n) seed length, only for

regular digraph w/ “consistent labelling” (based on Reingold’s approach) [RTV06]

• PRG for

regular digraph w/ arbitrary labelling

⇒ RL = L [RTV06]

• Other applications [Ind00,KNO05,…]

Outline

• RL vs. L and s-t connectivity

• Our main result

• Derandomization settings

• Algorithm outline

• Main step

• Conclusion

δ-Known-Stationary Find-Path

• Input: (G,s,t,1k,p1,…,pn)

• YES instance:

• s and t are connected

• |pv - π(v)| ≤ δ for all v that can reach s

• Random walk from s converges to a stationary dist πs in k steps.

• π(s), π(t) ≥ 1/k

• Output: path from s to t

Overview

• Main idea:

• Use stationary distribution to convert G to nearlyregular, “consistently labelled” graph so that we can apply RTV’s PRG

• Challenges:

• Logspace construction

• Maintain mixing time

Algorithm

Logspace

G

G’

G’’

ε

ε

nearly regular

Gcon

Greg

Analysis

Step 1: use input stationary distribution

to convert G to nearly regularG’

Algorithm

Apply PRG

Logspace

G

G’

G’’

ε

ε

a path

from s to t

Gcon

Greg

Analysis

Step 2: apply PRG to G’, and

project path to G

Actually…

Apply PRG

Logspace

G

G’

G’’

ε

ε

consistently labelled

Gcon

Greg

Analysis

Need to obtain “consistent labelling”

But this is easy!

Analysis: PRG can find a path on G’

Logspace

Error:

ε⋅#step

G

G’

G’’

Apply PRG

PRG works!

ε

ε

Gcon

Greg

Analysis

G’ ≈ Greg

PRG works for Greg ⇒ PRG works for G’

Maintain mixing time

Logspace

G

G’

G’’

Maintain mixing-time

Maintain mixing-time

ε

ε

Gcon

Greg

Analysis

Mixing time: G ≈ G’ ≈ Greg

New mixing time measure: visiting length.

Outline

• RL vs. L and s-t connectivity

• Our main result

• Derandomization settings

• Algorithm outline

• Main step

• Conclusion

Stationary dist as network flow

• Vertex v has mass π(v)

• Edge (v,w) carries π(v)/d flow

• Regular ⇔ uniform dist is stationary ⇔ vertices have equal mass &

edges carry equal flow

0.2

0.1

0.1

0.2

0.2

0.1

0.1

0.2

0.2

0.1

Ideal construction

• Blow up each v to cloud Cvof size π(v)⋅N

• Each v’ ∈Cv has equal mass 1/N

G

4/7

1/7

1/7

2/7

1/7

1/7

1/7

1/7

1/7

1/7

Ideal construction

• Blow up each v to cloud Cv of size π(v)⋅N

• Each v’ ∈Cv has equal mass 1/N

• Each vertex has a copy of outgoing edges

1/7

1/7

1/7

1/7

1/7

1/7

1/7

1/7

Ideal construction

• Blow up each v to cloud Cv of size π(v)⋅N

• Each v’ ∈Cv has equal mass 1/N

• Each vertex has a copy of outgoing edges

• Split each outgoing edge into D edges, distributed evenly to target cloud

• Each edge shares equal 1/(dDN) flow

Regular!

G’

1/7

1/7

1/7

1/7

1/7

1/7

1/7

1/7

Ideal construction

rounding error

rounding error

• Blow up each v to cloud Cv of size π(v)⋅N

• Each v’ ∈Cv has equal mass 1/N

• Each vertex has a copy of outgoing edges

• Split each outgoing edge into D edges, distributed evenly to the target cloud

• Each edge shares equal 1/(dDN) flow

negligible probability

G’

1/7

1/7

1/7

1/7

1/7

1/7

1/7

1/7

G’ is nearly regular

• There are ε-fraction of bad vertices with small degree

• For every good vertex

• (1- ε)dD ≤ outdeg ≤ dD

• (1- ε)dD ≤ indeg ≤ (1+ ε)dD

G’ maintains mixing time

• Visiting length k

• A mixing time measure

• The construction maintains the visiting length

• visiting len. of G’ = visiting len. of G

G

S

v

Visiting length

• k(S) : visiting length of a vertex set S

• For every v reachable from S

Pr[ v visits S in k(S) steps ] ≥ ½

Lemma: mix in time poly(k)⋅log(nd) if S is a clique.

G ⇒ G’ maintains visiting length

• visiting length k(Cs) in G’ =

visiting length k(s) in G

• On the cloud level, G’ = G

• Pr[ Cv to Cw in G’] = Pr[ v to w in G]

⇒G’ has short mixing time

• independent of ε

G’ vs. G

Recap

Apply PRG

Logspace

G

G’

G’’

ε

ε

Gcon

Greg

Analysis

Apply PRG to G’ to find a path

Outline

• RL vs. L and s-t connectivity

• Our main result

• Derandomization settings

• Algorithm outline

• Conclusion

Conclusion

• Key to Reingold’s approach

• Oblivious: consistent labelling of edges

• Explicit: knowledge of stationary dist

• Toward RL = L

• Focus on unknown stationary dist

• Put RL in L1.4?

Questions?