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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)

Salil Vadhan (Harvard University)

- RL vs. L and s-t connectivity
- Our main result
- Derandomization settings
- Algorithm outline
- Main step

- Conclusion

Space

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?

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... read only

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Input:

Work:

R/W

M

Random

coins

0/1

..

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write only

Output:

One way

- 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

n: # vertices

d: out-degree

- Input: (G,s,t)
- G – (n,d)-digraph
- s,t – vertices in G

- Output: does s connect to t?

- 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:
- Do poly(n) steps random walk from s
- Accept if ends at t

π(t) is non-negligible

converge to stat dist π

- 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

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

- Input: (G,s,t,1k,p1,…,pn)
- pv: estimate of a stationary dist π(v)
(input stationary distribution)

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

- 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

δ-Known-Stationary STConn

and

δ-Known-Stationary Find-Path

Both in deterministic logspace

where δ =1/poly(n,d,k)

- RL vs. L and s-t connectivity
- Our main result
- Derandomization settings
- Algorithm outline
- Main step

- Conclusion

- 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

- Explicit setting
- RL ⊆L3/2 [SZ95]
- USTConn ∈ L [Rei05]
- Known-Stationary STConn ∈ L [Our result]

- 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,…]

- RL vs. L and s-t connectivity
- Our main result
- Derandomization settings
- Algorithm outline
- Main step

- Conclusion

- 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

- 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

Logspace

G

G’

G’’

ε

ε

nearly regular

Gcon

Greg

Analysis

Step 1: use input stationary distribution

to convert G to nearly regularG’

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

Apply PRG

Logspace

G

G’

G’’

ε

ε

consistently labelled

Gcon

Greg

Analysis

Need to obtain “consistent labelling”

But this is easy!

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’

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.

- RL vs. L and s-t connectivity
- Our main result
- Derandomization settings
- Algorithm outline
- Main step

- Conclusion

- 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

- 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

- 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

- 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

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

- 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

- Visiting length k
- A mixing time measure

- The construction maintains the visiting length
- visiting len. of G’ = visiting len. of G

G

S

v

- k(S) : visiting length of a vertex set S
- For every v reachable from S
Pr[ v visits S in k(S) steps ] ≥ ½

- For every v reachable from S

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

- 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

Apply PRG

Logspace

G

G’

G’’

ε

ε

Gcon

Greg

Analysis

Apply PRG to G’ to find a path

- RL vs. L and s-t connectivity
- Our main result
- Derandomization settings
- Algorithm outline
- 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!

Questions?