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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

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
Outline

  • RL vs. L and s-t connectivity

  • Our main result

  • Derandomization settings

  • Algorithm outline

    • Main step

  • Conclusion


Space bounded complexity classes

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?

..

..

..

... read only

..

..

Input:

Work:

R/W

M

Random

coins

0/1

..

..

..

write only

Output:

One way


Results on derandomizing space
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


S t connectivity stconn

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?


Stconn and logspace computation
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]


Random walk algorithm for ustconn akllr79

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
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
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


Known stationary stconn
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


Known stationary find path
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
Our result

δ-Known-Stationary STConn

and

δ-Known-Stationary Find-Path

Both in deterministic logspace

where δ =1/poly(n,d,k)


Outline1
Outline

  • RL vs. L and s-t connectivity

  • Our main result

  • Derandomization settings

  • Algorithm outline

    • Main step

  • Conclusion


Explicit vs oblivious derandomization
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
Results in explicit settings

  • Explicit setting

    • RL ⊆L3/2 [SZ95]

    • USTConn ∈ L [Rei05]

    • Known-Stationary STConn ∈ L [Our result]


Results in oblivious settings
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,…]



Outline2
Outline

  • RL vs. L and s-t connectivity

  • Our main result

  • Derandomization settings

  • Algorithm outline

    • Main step

  • Conclusion


Known stationary find path1
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
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
Algorithm

Logspace

G

G’

G’’

ε

ε

nearly regular

Gcon

Greg

Analysis

Step 1: use input stationary distribution

to convert G to nearly regularG’


Algorithm1
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
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
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
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.


Outline3
Outline

  • RL vs. L and s-t connectivity

  • Our main result

  • Derandomization settings

  • Algorithm outline

    • Main step

  • Conclusion


Stationary dist as network flow
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
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 construction1
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 construction2
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 construction3
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
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
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


Visiting length

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
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
Recap

Apply PRG

Logspace

G

G’

G’’

ε

ε

Gcon

Greg

Analysis

Apply PRG to G’ to find a path


Outline4
Outline

  • RL vs. L and s-t connectivity

  • Our main result

  • Derandomization settings

  • Algorithm outline

  • Conclusion


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?


Thank you

Thank you!

Questions?


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