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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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