Undirected ST-Connectivity In Log Space

Omer ReingoldSlides by Sharon Bruckner

Today

bull Some history

bull What are we adding to the mixndash Connectivity in expandersndash Making expanders powering it up and

cutting it downndash Putting it together in Log-Space

bull In conclusion

What is ST-Connectivity

bull For an instance (Gst)ndash YES if there is a s-t path in graph Gndash NO otherwise

bull Two flavors ndash STCON G is directedndash USTCON G is undirected

bull Today wersquoll talk about USTCON

What do we know about itbull Solved easily with BFS

ndash But ndash polynomial space

bull USTCON is in NL (=NSPACE(log))ndash Just guess the path

bull USTCON is in DSPACE(log2n) (Savitch)ndash Guess a vertex on the path recurse

bull USTCON is complete for SLbull USTCON is in RL

ndash Random walkndash Randomness is a resource

What are we trying to accomplish

We would like an algorithm for USTCON which is

deterministic and works in space logarithmic in the size

of the graph

What are we adding to the mix

Bits Nrsquo Pieces

bull The adjacency matrix of a graph G

bull Wersquoll use D regular graphs and the normalized adjancecy matrix (1D)M

1 i j EM i j

0 otherwise

Bits Nrsquo Pieces

bull M is a stochastic matrix the ldquorandom walkrdquo matrix

bull Largest eigenvalue of M is 1 with eigenvector (11hellip1)єRn 2nd largest eigenvalue is λ(G)

bull Define (NDλ) a D regular graph over N vertices with λ(G) le λ

Expanders ndash a Reminder

Two equivalent definitions of expanderbull (ND λ) is an expander iff the

spectral gap 1- λ gt 0

bull (ND λ) is an expander if εgt0 st for at least (1+ ε)|S| vertices of G are connected to some vertex in S

S V S V 2

Expanders ndash a Reminder

2

22

GD G

D

2

min

V

S V S

E S SEdge Expansion G

S

2

| min

VS V S

v u u v SVertex Expansion G

S

CS vertices w neighbor in S c 1 S

D

If there are c|S| edges out of S

Connectivity in Expanders

What would happen if each connected component of our graph was an expander

We could decide USTCON in logspace

Every expander has a O(log(N)) diameter

TheoremFor any s and t in an expander therersquos

a path from s to t of length O(log(N))

Proof

i

i

n(1 )

2n

log 1 log2

log n 1ilog 1

Logspace Algorithm for paths in an expander

For (ND λ) there is a a space O(logDlogN) algorithm which decides USTCONIdeabullFrom any vertex s there are Dl=O(logN) different paths bullenumerate and check if any of them reach t

t

How much does it cost

bullAt each vertex we have a choice of D vertices log(D) to represent 1hellipD

bullEach path is log(N) long

bull Need log(D) at each stage of a log(N) path altogether O(logDlogN)

So far

We now know that if our graph G was an expander graph USTCON can be solved in Log-Space

How can we turn G into a graph of expanders

Making expanders powering it up and cutting it down

What do we want from the expander graph Grsquo

1 There is an s-t path in G iff there is an srsquo-trsquo path in Grsquo

2 Each connected component is an expander with constant expansion

3 Construction in Log-Space

The Plan

bull New tools for our toolboxndash Rotation Mapsndash Poweringndash Zig Zag products

bull The actual constructionbull Why we got what we wantedbull Why we got it in Log-Space

Overviewbull Series of powering by 8 and zig zag

products one increases the spectral gap and the other one decreases it but not by much

8

O(Log(N))

Zig Zag Powering

Rotation Maps

bull Graph representation wersquoll use

GRot N D N D

G is D regular undirected The rotation map

GRot vi w j

If the edge (vw) exists and is the ith edge coming out of v and the jth edge coming out of w

Rotation Map Example

1 23

4

17

hellip

hellip54

55

4

3 21

28

5554

hellip

hellipu

v

G

G

Rot u17 v28

Rot v28 u17

PoweringNo limitations on the degree rarr can expenderize with powering

The k-th power of the D-regular graph G is Gk where there is an edge (uv) iff there is a path of length k between u and v in G

=k

Powering

What is it good forbull GN is an N-clique an expanderbull If G = (ND λ) then Gk is a (NDk λk) λ(G)

The spectral gapbull Why is this not enough

Zig Zag Product

If G is a D-regular graph with N vertices and H is a d-regular graph with D vertices replace each v in G with a copy of H Hv Our new graph has [N]x[D] vertices

H

Part of G

v is now Hv

v

Zig-Zag ProductThe resulting graph is a d2 regular

graph with edges as follows

G

v

w

arsquo

brsquo

Hv

Hw

a

brsquo

G Z H

a

b

i

j

arsquoiirsquo

bj

jrsquo

Zig-Zag Product

Lemma

If G is an (NDλ) graph and H is a (Dd) graph

211 (G z H ) 1 1

2

The zig-zag product doesnrsquot hurt the spectral gap too badly

22 2 2 21 1last week we saw (G z H ) 1 1 4

2 2

Today

bull Some history

bull What are we adding to the mixndash Connectivity in expandersndash Making expanders powering it up and

cutting it downndash Putting it together in Log-Space

bull In conclusion

What is ST-Connectivity

bull For an instance (Gst)ndash YES if there is a s-t path in graph Gndash NO otherwise

bull Two flavors ndash STCON G is directedndash USTCON G is undirected

bull Today wersquoll talk about USTCON

What do we know about itbull Solved easily with BFS

ndash But ndash polynomial space

bull USTCON is in NL (=NSPACE(log))ndash Just guess the path

bull USTCON is in DSPACE(log2n) (Savitch)ndash Guess a vertex on the path recurse

bull USTCON is complete for SLbull USTCON is in RL

ndash Random walkndash Randomness is a resource

What are we trying to accomplish

We would like an algorithm for USTCON which is

deterministic and works in space logarithmic in the size

of the graph

What are we adding to the mix

Bits Nrsquo Pieces

bull The adjacency matrix of a graph G

bull Wersquoll use D regular graphs and the normalized adjancecy matrix (1D)M

1 i j EM i j

0 otherwise

Bits Nrsquo Pieces

bull M is a stochastic matrix the ldquorandom walkrdquo matrix

bull Largest eigenvalue of M is 1 with eigenvector (11hellip1)єRn 2nd largest eigenvalue is λ(G)

bull Define (NDλ) a D regular graph over N vertices with λ(G) le λ

Expanders ndash a Reminder

Two equivalent definitions of expanderbull (ND λ) is an expander iff the

spectral gap 1- λ gt 0

bull (ND λ) is an expander if εgt0 st for at least (1+ ε)|S| vertices of G are connected to some vertex in S

S V S V 2

Expanders ndash a Reminder

2

22

GD G

D

2

min

V

S V S

E S SEdge Expansion G

S

2

| min

VS V S

v u u v SVertex Expansion G

S

CS vertices w neighbor in S c 1 S

D

If there are c|S| edges out of S

Connectivity in Expanders

What would happen if each connected component of our graph was an expander

We could decide USTCON in logspace

Every expander has a O(log(N)) diameter

TheoremFor any s and t in an expander therersquos

a path from s to t of length O(log(N))

Proof

i

i

n(1 )

2n

log 1 log2

log n 1ilog 1

Logspace Algorithm for paths in an expander

For (ND λ) there is a a space O(logDlogN) algorithm which decides USTCONIdeabullFrom any vertex s there are Dl=O(logN) different paths bullenumerate and check if any of them reach t

t

How much does it cost

bullAt each vertex we have a choice of D vertices log(D) to represent 1hellipD

bullEach path is log(N) long

bull Need log(D) at each stage of a log(N) path altogether O(logDlogN)

So far

We now know that if our graph G was an expander graph USTCON can be solved in Log-Space

How can we turn G into a graph of expanders

Making expanders powering it up and cutting it down

What do we want from the expander graph Grsquo

1 There is an s-t path in G iff there is an srsquo-trsquo path in Grsquo

2 Each connected component is an expander with constant expansion

3 Construction in Log-Space

The Plan

bull New tools for our toolboxndash Rotation Mapsndash Poweringndash Zig Zag products

bull The actual constructionbull Why we got what we wantedbull Why we got it in Log-Space

Overviewbull Series of powering by 8 and zig zag

products one increases the spectral gap and the other one decreases it but not by much

8

O(Log(N))

Zig Zag Powering

Rotation Maps

bull Graph representation wersquoll use

GRot N D N D

G is D regular undirected The rotation map

GRot vi w j

If the edge (vw) exists and is the ith edge coming out of v and the jth edge coming out of w

Rotation Map Example

1 23

4

17

hellip

hellip54

55

4

3 21

28

5554

hellip

hellipu

v

G

G

Rot u17 v28

Rot v28 u17

PoweringNo limitations on the degree rarr can expenderize with powering

The k-th power of the D-regular graph G is Gk where there is an edge (uv) iff there is a path of length k between u and v in G

=k

Powering

What is it good forbull GN is an N-clique an expanderbull If G = (ND λ) then Gk is a (NDk λk) λ(G)

The spectral gapbull Why is this not enough

Zig Zag Product

If G is a D-regular graph with N vertices and H is a d-regular graph with D vertices replace each v in G with a copy of H Hv Our new graph has [N]x[D] vertices

H

Part of G

v is now Hv

v

Zig-Zag ProductThe resulting graph is a d2 regular

graph with edges as follows

G

v

w

arsquo

brsquo

Hv

Hw

a

brsquo

G Z H

a

b

i

j

arsquoiirsquo

bj

jrsquo

Zig-Zag Product

Lemma

If G is an (NDλ) graph and H is a (Dd) graph

211 (G z H ) 1 1

2

The zig-zag product doesnrsquot hurt the spectral gap too badly

22 2 2 21 1last week we saw (G z H ) 1 1 4

2 2

What is ST-Connectivity

bull For an instance (Gst)ndash YES if there is a s-t path in graph Gndash NO otherwise

bull Two flavors ndash STCON G is directedndash USTCON G is undirected

bull Today wersquoll talk about USTCON

What do we know about itbull Solved easily with BFS

ndash But ndash polynomial space

bull USTCON is in NL (=NSPACE(log))ndash Just guess the path

bull USTCON is in DSPACE(log2n) (Savitch)ndash Guess a vertex on the path recurse

bull USTCON is complete for SLbull USTCON is in RL

ndash Random walkndash Randomness is a resource

What are we trying to accomplish

We would like an algorithm for USTCON which is

deterministic and works in space logarithmic in the size

of the graph

What are we adding to the mix

Bits Nrsquo Pieces

bull The adjacency matrix of a graph G

bull Wersquoll use D regular graphs and the normalized adjancecy matrix (1D)M

1 i j EM i j

0 otherwise

Bits Nrsquo Pieces

bull M is a stochastic matrix the ldquorandom walkrdquo matrix

bull Largest eigenvalue of M is 1 with eigenvector (11hellip1)єRn 2nd largest eigenvalue is λ(G)

bull Define (NDλ) a D regular graph over N vertices with λ(G) le λ

Expanders ndash a Reminder

Two equivalent definitions of expanderbull (ND λ) is an expander iff the

spectral gap 1- λ gt 0

bull (ND λ) is an expander if εgt0 st for at least (1+ ε)|S| vertices of G are connected to some vertex in S

S V S V 2

Expanders ndash a Reminder

2

22

GD G

D

2

min

V

S V S

E S SEdge Expansion G

S

2

| min

VS V S

v u u v SVertex Expansion G

S

CS vertices w neighbor in S c 1 S

D

If there are c|S| edges out of S

Connectivity in Expanders

What would happen if each connected component of our graph was an expander

We could decide USTCON in logspace

Every expander has a O(log(N)) diameter

TheoremFor any s and t in an expander therersquos

a path from s to t of length O(log(N))

Proof

i

i

n(1 )

2n

log 1 log2

log n 1ilog 1

Logspace Algorithm for paths in an expander

For (ND λ) there is a a space O(logDlogN) algorithm which decides USTCONIdeabullFrom any vertex s there are Dl=O(logN) different paths bullenumerate and check if any of them reach t

t

How much does it cost

bullAt each vertex we have a choice of D vertices log(D) to represent 1hellipD

bullEach path is log(N) long

bull Need log(D) at each stage of a log(N) path altogether O(logDlogN)

So far

We now know that if our graph G was an expander graph USTCON can be solved in Log-Space

How can we turn G into a graph of expanders

Making expanders powering it up and cutting it down

What do we want from the expander graph Grsquo

1 There is an s-t path in G iff there is an srsquo-trsquo path in Grsquo

2 Each connected component is an expander with constant expansion

3 Construction in Log-Space

The Plan

bull New tools for our toolboxndash Rotation Mapsndash Poweringndash Zig Zag products

bull The actual constructionbull Why we got what we wantedbull Why we got it in Log-Space

Overviewbull Series of powering by 8 and zig zag

products one increases the spectral gap and the other one decreases it but not by much

8

O(Log(N))

Zig Zag Powering

Rotation Maps

bull Graph representation wersquoll use

GRot N D N D

G is D regular undirected The rotation map

GRot vi w j

If the edge (vw) exists and is the ith edge coming out of v and the jth edge coming out of w

Rotation Map Example

1 23

4

17

hellip

hellip54

55

4

3 21

28

5554

hellip

hellipu

v

G

G

Rot u17 v28

Rot v28 u17

PoweringNo limitations on the degree rarr can expenderize with powering

The k-th power of the D-regular graph G is Gk where there is an edge (uv) iff there is a path of length k between u and v in G

=k

Powering

What is it good forbull GN is an N-clique an expanderbull If G = (ND λ) then Gk is a (NDk λk) λ(G)

The spectral gapbull Why is this not enough

Zig Zag Product

If G is a D-regular graph with N vertices and H is a d-regular graph with D vertices replace each v in G with a copy of H Hv Our new graph has [N]x[D] vertices

H

Part of G

v is now Hv

v

Zig-Zag ProductThe resulting graph is a d2 regular

graph with edges as follows

G

v

w

arsquo

brsquo

Hv

Hw

a

brsquo

G Z H

a

b

i

j

arsquoiirsquo

bj

jrsquo

Zig-Zag Product

Lemma

If G is an (NDλ) graph and H is a (Dd) graph

211 (G z H ) 1 1

2

The zig-zag product doesnrsquot hurt the spectral gap too badly

22 2 2 21 1last week we saw (G z H ) 1 1 4

2 2

What do we know about itbull Solved easily with BFS

ndash But ndash polynomial space

bull USTCON is in NL (=NSPACE(log))ndash Just guess the path

bull USTCON is in DSPACE(log2n) (Savitch)ndash Guess a vertex on the path recurse

bull USTCON is complete for SLbull USTCON is in RL

ndash Random walkndash Randomness is a resource

What are we trying to accomplish

We would like an algorithm for USTCON which is

deterministic and works in space logarithmic in the size

of the graph

What are we adding to the mix

Bits Nrsquo Pieces

bull The adjacency matrix of a graph G

bull Wersquoll use D regular graphs and the normalized adjancecy matrix (1D)M

1 i j EM i j

0 otherwise

Bits Nrsquo Pieces

bull M is a stochastic matrix the ldquorandom walkrdquo matrix

bull Largest eigenvalue of M is 1 with eigenvector (11hellip1)єRn 2nd largest eigenvalue is λ(G)

bull Define (NDλ) a D regular graph over N vertices with λ(G) le λ

Expanders ndash a Reminder

Two equivalent definitions of expanderbull (ND λ) is an expander iff the

spectral gap 1- λ gt 0

bull (ND λ) is an expander if εgt0 st for at least (1+ ε)|S| vertices of G are connected to some vertex in S

S V S V 2

Expanders ndash a Reminder

2

22

GD G

D

2

min

V

S V S

E S SEdge Expansion G

S

2

| min

VS V S

v u u v SVertex Expansion G

S

CS vertices w neighbor in S c 1 S

D

If there are c|S| edges out of S

Connectivity in Expanders

What would happen if each connected component of our graph was an expander

We could decide USTCON in logspace

Every expander has a O(log(N)) diameter

TheoremFor any s and t in an expander therersquos

a path from s to t of length O(log(N))

Proof

i

i

n(1 )

2n

log 1 log2

log n 1ilog 1

Logspace Algorithm for paths in an expander

For (ND λ) there is a a space O(logDlogN) algorithm which decides USTCONIdeabullFrom any vertex s there are Dl=O(logN) different paths bullenumerate and check if any of them reach t

t

How much does it cost

bullAt each vertex we have a choice of D vertices log(D) to represent 1hellipD

bullEach path is log(N) long

bull Need log(D) at each stage of a log(N) path altogether O(logDlogN)

So far

We now know that if our graph G was an expander graph USTCON can be solved in Log-Space

How can we turn G into a graph of expanders

Making expanders powering it up and cutting it down

What do we want from the expander graph Grsquo

1 There is an s-t path in G iff there is an srsquo-trsquo path in Grsquo

2 Each connected component is an expander with constant expansion

3 Construction in Log-Space

The Plan

bull New tools for our toolboxndash Rotation Mapsndash Poweringndash Zig Zag products

bull The actual constructionbull Why we got what we wantedbull Why we got it in Log-Space

Overviewbull Series of powering by 8 and zig zag

products one increases the spectral gap and the other one decreases it but not by much

8

O(Log(N))

Zig Zag Powering

Rotation Maps

bull Graph representation wersquoll use

GRot N D N D

G is D regular undirected The rotation map

GRot vi w j

If the edge (vw) exists and is the ith edge coming out of v and the jth edge coming out of w

Rotation Map Example

1 23

4

17

hellip

hellip54

55

4

3 21

28

5554

hellip

hellipu

v

G

G

Rot u17 v28

Rot v28 u17

PoweringNo limitations on the degree rarr can expenderize with powering

The k-th power of the D-regular graph G is Gk where there is an edge (uv) iff there is a path of length k between u and v in G

=k

Powering

What is it good forbull GN is an N-clique an expanderbull If G = (ND λ) then Gk is a (NDk λk) λ(G)

The spectral gapbull Why is this not enough

Zig Zag Product

If G is a D-regular graph with N vertices and H is a d-regular graph with D vertices replace each v in G with a copy of H Hv Our new graph has [N]x[D] vertices

H

Part of G

v is now Hv

v

Zig-Zag ProductThe resulting graph is a d2 regular

graph with edges as follows

G

v

w

arsquo

brsquo

Hv

Hw

a

brsquo

G Z H

a

b

i

j

arsquoiirsquo

bj

jrsquo

Zig-Zag Product

Lemma

If G is an (NDλ) graph and H is a (Dd) graph

211 (G z H ) 1 1

2

The zig-zag product doesnrsquot hurt the spectral gap too badly

22 2 2 21 1last week we saw (G z H ) 1 1 4

2 2

What are we trying to accomplish

We would like an algorithm for USTCON which is

deterministic and works in space logarithmic in the size

of the graph

What are we adding to the mix

Bits Nrsquo Pieces

bull The adjacency matrix of a graph G

bull Wersquoll use D regular graphs and the normalized adjancecy matrix (1D)M

1 i j EM i j

0 otherwise

Bits Nrsquo Pieces

bull M is a stochastic matrix the ldquorandom walkrdquo matrix

bull Largest eigenvalue of M is 1 with eigenvector (11hellip1)єRn 2nd largest eigenvalue is λ(G)

bull Define (NDλ) a D regular graph over N vertices with λ(G) le λ

Expanders ndash a Reminder

Two equivalent definitions of expanderbull (ND λ) is an expander iff the

spectral gap 1- λ gt 0

bull (ND λ) is an expander if εgt0 st for at least (1+ ε)|S| vertices of G are connected to some vertex in S

S V S V 2

Expanders ndash a Reminder

2

22

GD G

D

2

min

V

S V S

E S SEdge Expansion G

S

2

| min

VS V S

v u u v SVertex Expansion G

S

CS vertices w neighbor in S c 1 S

D

If there are c|S| edges out of S

Connectivity in Expanders

What would happen if each connected component of our graph was an expander

We could decide USTCON in logspace

Every expander has a O(log(N)) diameter

TheoremFor any s and t in an expander therersquos

a path from s to t of length O(log(N))

Proof

i

i

n(1 )

2n

log 1 log2

log n 1ilog 1

Logspace Algorithm for paths in an expander

For (ND λ) there is a a space O(logDlogN) algorithm which decides USTCONIdeabullFrom any vertex s there are Dl=O(logN) different paths bullenumerate and check if any of them reach t

t

How much does it cost

bullAt each vertex we have a choice of D vertices log(D) to represent 1hellipD

bullEach path is log(N) long

bull Need log(D) at each stage of a log(N) path altogether O(logDlogN)

So far

We now know that if our graph G was an expander graph USTCON can be solved in Log-Space

How can we turn G into a graph of expanders

Making expanders powering it up and cutting it down

What do we want from the expander graph Grsquo

1 There is an s-t path in G iff there is an srsquo-trsquo path in Grsquo

2 Each connected component is an expander with constant expansion

3 Construction in Log-Space

The Plan

bull New tools for our toolboxndash Rotation Mapsndash Poweringndash Zig Zag products

bull The actual constructionbull Why we got what we wantedbull Why we got it in Log-Space

Overviewbull Series of powering by 8 and zig zag

products one increases the spectral gap and the other one decreases it but not by much

8

O(Log(N))

Zig Zag Powering

Rotation Maps

bull Graph representation wersquoll use

GRot N D N D

G is D regular undirected The rotation map

GRot vi w j

If the edge (vw) exists and is the ith edge coming out of v and the jth edge coming out of w

Rotation Map Example

1 23

4

17

hellip

hellip54

55

4

3 21

28

5554

hellip

hellipu

v

G

G

Rot u17 v28

Rot v28 u17

PoweringNo limitations on the degree rarr can expenderize with powering

The k-th power of the D-regular graph G is Gk where there is an edge (uv) iff there is a path of length k between u and v in G

=k

Powering

What is it good forbull GN is an N-clique an expanderbull If G = (ND λ) then Gk is a (NDk λk) λ(G)

The spectral gapbull Why is this not enough

Zig Zag Product

If G is a D-regular graph with N vertices and H is a d-regular graph with D vertices replace each v in G with a copy of H Hv Our new graph has [N]x[D] vertices

H

Part of G

v is now Hv

v

Zig-Zag ProductThe resulting graph is a d2 regular

graph with edges as follows

G

v

w

arsquo

brsquo

Hv

Hw

a

brsquo

G Z H

a

b

i

j

arsquoiirsquo

bj

jrsquo

Zig-Zag Product

Lemma

If G is an (NDλ) graph and H is a (Dd) graph

211 (G z H ) 1 1

2

The zig-zag product doesnrsquot hurt the spectral gap too badly

22 2 2 21 1last week we saw (G z H ) 1 1 4

2 2

What are we adding to the mix

Bits Nrsquo Pieces

bull The adjacency matrix of a graph G

bull Wersquoll use D regular graphs and the normalized adjancecy matrix (1D)M

1 i j EM i j

0 otherwise

Bits Nrsquo Pieces

bull M is a stochastic matrix the ldquorandom walkrdquo matrix

bull Largest eigenvalue of M is 1 with eigenvector (11hellip1)єRn 2nd largest eigenvalue is λ(G)

bull Define (NDλ) a D regular graph over N vertices with λ(G) le λ

Expanders ndash a Reminder

Two equivalent definitions of expanderbull (ND λ) is an expander iff the

spectral gap 1- λ gt 0

bull (ND λ) is an expander if εgt0 st for at least (1+ ε)|S| vertices of G are connected to some vertex in S

S V S V 2

Expanders ndash a Reminder

2

22

GD G

D

2

min

V

S V S

E S SEdge Expansion G

S

2

| min

VS V S

v u u v SVertex Expansion G

S

CS vertices w neighbor in S c 1 S

D

If there are c|S| edges out of S

Connectivity in Expanders

What would happen if each connected component of our graph was an expander

We could decide USTCON in logspace

Every expander has a O(log(N)) diameter

TheoremFor any s and t in an expander therersquos

a path from s to t of length O(log(N))

Proof

i

i

n(1 )

2n

log 1 log2

log n 1ilog 1

Logspace Algorithm for paths in an expander

For (ND λ) there is a a space O(logDlogN) algorithm which decides USTCONIdeabullFrom any vertex s there are Dl=O(logN) different paths bullenumerate and check if any of them reach t

t

How much does it cost

bullAt each vertex we have a choice of D vertices log(D) to represent 1hellipD

bullEach path is log(N) long

bull Need log(D) at each stage of a log(N) path altogether O(logDlogN)

So far

We now know that if our graph G was an expander graph USTCON can be solved in Log-Space

How can we turn G into a graph of expanders

Making expanders powering it up and cutting it down

What do we want from the expander graph Grsquo

1 There is an s-t path in G iff there is an srsquo-trsquo path in Grsquo

2 Each connected component is an expander with constant expansion

3 Construction in Log-Space

The Plan

bull New tools for our toolboxndash Rotation Mapsndash Poweringndash Zig Zag products

bull The actual constructionbull Why we got what we wantedbull Why we got it in Log-Space

Overviewbull Series of powering by 8 and zig zag

products one increases the spectral gap and the other one decreases it but not by much

8

O(Log(N))

Zig Zag Powering

Rotation Maps

bull Graph representation wersquoll use

GRot N D N D

G is D regular undirected The rotation map

GRot vi w j

If the edge (vw) exists and is the ith edge coming out of v and the jth edge coming out of w

Rotation Map Example

1 23

4

17

hellip

hellip54

55

4

3 21

28

5554

hellip

hellipu

v

G

G

Rot u17 v28

Rot v28 u17

PoweringNo limitations on the degree rarr can expenderize with powering

The k-th power of the D-regular graph G is Gk where there is an edge (uv) iff there is a path of length k between u and v in G

=k

Powering

What is it good forbull GN is an N-clique an expanderbull If G = (ND λ) then Gk is a (NDk λk) λ(G)

The spectral gapbull Why is this not enough

Zig Zag Product

If G is a D-regular graph with N vertices and H is a d-regular graph with D vertices replace each v in G with a copy of H Hv Our new graph has [N]x[D] vertices

H

Part of G

v is now Hv

v

Zig-Zag ProductThe resulting graph is a d2 regular

graph with edges as follows

G

v

w

arsquo

brsquo

Hv

Hw

a

brsquo

G Z H

a

b

i

j

arsquoiirsquo

bj

jrsquo

Zig-Zag Product

Lemma

If G is an (NDλ) graph and H is a (Dd) graph

211 (G z H ) 1 1

2

The zig-zag product doesnrsquot hurt the spectral gap too badly

22 2 2 21 1last week we saw (G z H ) 1 1 4

2 2

Bits Nrsquo Pieces

bull The adjacency matrix of a graph G

bull Wersquoll use D regular graphs and the normalized adjancecy matrix (1D)M

1 i j EM i j

0 otherwise

Bits Nrsquo Pieces

bull M is a stochastic matrix the ldquorandom walkrdquo matrix

bull Largest eigenvalue of M is 1 with eigenvector (11hellip1)єRn 2nd largest eigenvalue is λ(G)

bull Define (NDλ) a D regular graph over N vertices with λ(G) le λ

Expanders ndash a Reminder

Two equivalent definitions of expanderbull (ND λ) is an expander iff the

spectral gap 1- λ gt 0

bull (ND λ) is an expander if εgt0 st for at least (1+ ε)|S| vertices of G are connected to some vertex in S

S V S V 2

Expanders ndash a Reminder

2

22

GD G

D

2

min

V

S V S

E S SEdge Expansion G

S

2

| min

VS V S

v u u v SVertex Expansion G

S

CS vertices w neighbor in S c 1 S

D

If there are c|S| edges out of S

Connectivity in Expanders

What would happen if each connected component of our graph was an expander

We could decide USTCON in logspace

Every expander has a O(log(N)) diameter

TheoremFor any s and t in an expander therersquos

a path from s to t of length O(log(N))

Proof

i

i

n(1 )

2n

log 1 log2

log n 1ilog 1

Logspace Algorithm for paths in an expander

For (ND λ) there is a a space O(logDlogN) algorithm which decides USTCONIdeabullFrom any vertex s there are Dl=O(logN) different paths bullenumerate and check if any of them reach t

t

How much does it cost

bullAt each vertex we have a choice of D vertices log(D) to represent 1hellipD

bullEach path is log(N) long

bull Need log(D) at each stage of a log(N) path altogether O(logDlogN)

So far

We now know that if our graph G was an expander graph USTCON can be solved in Log-Space

How can we turn G into a graph of expanders

Making expanders powering it up and cutting it down

What do we want from the expander graph Grsquo

1 There is an s-t path in G iff there is an srsquo-trsquo path in Grsquo

2 Each connected component is an expander with constant expansion

3 Construction in Log-Space

The Plan

bull New tools for our toolboxndash Rotation Mapsndash Poweringndash Zig Zag products

bull The actual constructionbull Why we got what we wantedbull Why we got it in Log-Space

Overviewbull Series of powering by 8 and zig zag

products one increases the spectral gap and the other one decreases it but not by much

8

O(Log(N))

Zig Zag Powering

Rotation Maps

bull Graph representation wersquoll use

GRot N D N D

G is D regular undirected The rotation map

GRot vi w j

If the edge (vw) exists and is the ith edge coming out of v and the jth edge coming out of w

Rotation Map Example

1 23

4

17

hellip

hellip54

55

4

3 21

28

5554

hellip

hellipu

v

G

G

Rot u17 v28

Rot v28 u17

PoweringNo limitations on the degree rarr can expenderize with powering

The k-th power of the D-regular graph G is Gk where there is an edge (uv) iff there is a path of length k between u and v in G

=k

Powering

What is it good forbull GN is an N-clique an expanderbull If G = (ND λ) then Gk is a (NDk λk) λ(G)

The spectral gapbull Why is this not enough

Zig Zag Product

If G is a D-regular graph with N vertices and H is a d-regular graph with D vertices replace each v in G with a copy of H Hv Our new graph has [N]x[D] vertices

H

Part of G

v is now Hv

v

Zig-Zag ProductThe resulting graph is a d2 regular

graph with edges as follows

G

v

w

arsquo

brsquo

Hv

Hw

a

brsquo

G Z H

a

b

i

j

arsquoiirsquo

bj

jrsquo

Zig-Zag Product

Lemma

If G is an (NDλ) graph and H is a (Dd) graph

211 (G z H ) 1 1

2

The zig-zag product doesnrsquot hurt the spectral gap too badly

22 2 2 21 1last week we saw (G z H ) 1 1 4

2 2

Bits Nrsquo Pieces

bull M is a stochastic matrix the ldquorandom walkrdquo matrix

bull Largest eigenvalue of M is 1 with eigenvector (11hellip1)єRn 2nd largest eigenvalue is λ(G)

bull Define (NDλ) a D regular graph over N vertices with λ(G) le λ

Expanders ndash a Reminder

Two equivalent definitions of expanderbull (ND λ) is an expander iff the

spectral gap 1- λ gt 0

bull (ND λ) is an expander if εgt0 st for at least (1+ ε)|S| vertices of G are connected to some vertex in S

S V S V 2

Expanders ndash a Reminder

2

22

GD G

D

2

min

V

S V S

E S SEdge Expansion G

S

2

| min

VS V S

v u u v SVertex Expansion G

S

CS vertices w neighbor in S c 1 S

D

If there are c|S| edges out of S

Connectivity in Expanders

What would happen if each connected component of our graph was an expander

We could decide USTCON in logspace

Every expander has a O(log(N)) diameter

TheoremFor any s and t in an expander therersquos

a path from s to t of length O(log(N))

Proof

i

i

n(1 )

2n

log 1 log2

log n 1ilog 1

Logspace Algorithm for paths in an expander

For (ND λ) there is a a space O(logDlogN) algorithm which decides USTCONIdeabullFrom any vertex s there are Dl=O(logN) different paths bullenumerate and check if any of them reach t

t

How much does it cost

bullAt each vertex we have a choice of D vertices log(D) to represent 1hellipD

bullEach path is log(N) long

bull Need log(D) at each stage of a log(N) path altogether O(logDlogN)

So far

We now know that if our graph G was an expander graph USTCON can be solved in Log-Space

How can we turn G into a graph of expanders

Making expanders powering it up and cutting it down

What do we want from the expander graph Grsquo

1 There is an s-t path in G iff there is an srsquo-trsquo path in Grsquo

2 Each connected component is an expander with constant expansion

3 Construction in Log-Space

The Plan

bull New tools for our toolboxndash Rotation Mapsndash Poweringndash Zig Zag products

bull The actual constructionbull Why we got what we wantedbull Why we got it in Log-Space

Overviewbull Series of powering by 8 and zig zag

products one increases the spectral gap and the other one decreases it but not by much

8

O(Log(N))

Zig Zag Powering

Rotation Maps

bull Graph representation wersquoll use

GRot N D N D

G is D regular undirected The rotation map

GRot vi w j

If the edge (vw) exists and is the ith edge coming out of v and the jth edge coming out of w

Rotation Map Example

1 23

4

17

hellip

hellip54

55

4

3 21

28

5554

hellip

hellipu

v

G

G

Rot u17 v28

Rot v28 u17

PoweringNo limitations on the degree rarr can expenderize with powering

The k-th power of the D-regular graph G is Gk where there is an edge (uv) iff there is a path of length k between u and v in G

=k

Powering

What is it good forbull GN is an N-clique an expanderbull If G = (ND λ) then Gk is a (NDk λk) λ(G)

The spectral gapbull Why is this not enough

Zig Zag Product

If G is a D-regular graph with N vertices and H is a d-regular graph with D vertices replace each v in G with a copy of H Hv Our new graph has [N]x[D] vertices

H

Part of G

v is now Hv

v

Zig-Zag ProductThe resulting graph is a d2 regular

graph with edges as follows

G

v

w

arsquo

brsquo

Hv

Hw

a

brsquo

G Z H

a

b

i

j

arsquoiirsquo

bj

jrsquo

Zig-Zag Product

Lemma

If G is an (NDλ) graph and H is a (Dd) graph

211 (G z H ) 1 1

2

The zig-zag product doesnrsquot hurt the spectral gap too badly

22 2 2 21 1last week we saw (G z H ) 1 1 4

2 2

Expanders ndash a Reminder

Two equivalent definitions of expanderbull (ND λ) is an expander iff the

spectral gap 1- λ gt 0

bull (ND λ) is an expander if εgt0 st for at least (1+ ε)|S| vertices of G are connected to some vertex in S

S V S V 2

Expanders ndash a Reminder

2

22

GD G

D

2

min

V

S V S

E S SEdge Expansion G

S

2

| min

VS V S

v u u v SVertex Expansion G

S

CS vertices w neighbor in S c 1 S

D

If there are c|S| edges out of S

Connectivity in Expanders

What would happen if each connected component of our graph was an expander

We could decide USTCON in logspace

Every expander has a O(log(N)) diameter

TheoremFor any s and t in an expander therersquos

a path from s to t of length O(log(N))

Proof

i

i

n(1 )

2n

log 1 log2

log n 1ilog 1

Logspace Algorithm for paths in an expander

For (ND λ) there is a a space O(logDlogN) algorithm which decides USTCONIdeabullFrom any vertex s there are Dl=O(logN) different paths bullenumerate and check if any of them reach t

t

How much does it cost

bullAt each vertex we have a choice of D vertices log(D) to represent 1hellipD

bullEach path is log(N) long

bull Need log(D) at each stage of a log(N) path altogether O(logDlogN)

So far

We now know that if our graph G was an expander graph USTCON can be solved in Log-Space

How can we turn G into a graph of expanders

Making expanders powering it up and cutting it down

What do we want from the expander graph Grsquo

1 There is an s-t path in G iff there is an srsquo-trsquo path in Grsquo

2 Each connected component is an expander with constant expansion

3 Construction in Log-Space

The Plan

bull New tools for our toolboxndash Rotation Mapsndash Poweringndash Zig Zag products

bull The actual constructionbull Why we got what we wantedbull Why we got it in Log-Space

Overviewbull Series of powering by 8 and zig zag

products one increases the spectral gap and the other one decreases it but not by much

8

O(Log(N))

Zig Zag Powering

Rotation Maps

bull Graph representation wersquoll use

GRot N D N D

G is D regular undirected The rotation map

GRot vi w j

If the edge (vw) exists and is the ith edge coming out of v and the jth edge coming out of w

Rotation Map Example

1 23

4

17

hellip

hellip54

55

4

3 21

28

5554

hellip

hellipu

v

G

G

Rot u17 v28

Rot v28 u17

PoweringNo limitations on the degree rarr can expenderize with powering

The k-th power of the D-regular graph G is Gk where there is an edge (uv) iff there is a path of length k between u and v in G

=k

Powering

What is it good forbull GN is an N-clique an expanderbull If G = (ND λ) then Gk is a (NDk λk) λ(G)

The spectral gapbull Why is this not enough

Zig Zag Product

If G is a D-regular graph with N vertices and H is a d-regular graph with D vertices replace each v in G with a copy of H Hv Our new graph has [N]x[D] vertices

H

Part of G

v is now Hv

v

Zig-Zag ProductThe resulting graph is a d2 regular

graph with edges as follows

G

v

w

arsquo

brsquo

Hv

Hw

a

brsquo

G Z H

a

b

i

j

arsquoiirsquo

bj

jrsquo

Zig-Zag Product

Lemma

If G is an (NDλ) graph and H is a (Dd) graph

211 (G z H ) 1 1

2

The zig-zag product doesnrsquot hurt the spectral gap too badly

22 2 2 21 1last week we saw (G z H ) 1 1 4

2 2

Expanders ndash a Reminder

2

22

GD G

D

2

min

V

S V S

E S SEdge Expansion G

S

2

| min

VS V S

v u u v SVertex Expansion G

S

CS vertices w neighbor in S c 1 S

D

If there are c|S| edges out of S

Connectivity in Expanders

What would happen if each connected component of our graph was an expander

We could decide USTCON in logspace

Every expander has a O(log(N)) diameter

TheoremFor any s and t in an expander therersquos

a path from s to t of length O(log(N))

Proof

i

i

n(1 )

2n

log 1 log2

log n 1ilog 1

Logspace Algorithm for paths in an expander

For (ND λ) there is a a space O(logDlogN) algorithm which decides USTCONIdeabullFrom any vertex s there are Dl=O(logN) different paths bullenumerate and check if any of them reach t

t

How much does it cost

bullAt each vertex we have a choice of D vertices log(D) to represent 1hellipD

bullEach path is log(N) long

bull Need log(D) at each stage of a log(N) path altogether O(logDlogN)

So far

We now know that if our graph G was an expander graph USTCON can be solved in Log-Space

How can we turn G into a graph of expanders

Making expanders powering it up and cutting it down

What do we want from the expander graph Grsquo

1 There is an s-t path in G iff there is an srsquo-trsquo path in Grsquo

2 Each connected component is an expander with constant expansion

3 Construction in Log-Space

The Plan

bull New tools for our toolboxndash Rotation Mapsndash Poweringndash Zig Zag products

bull The actual constructionbull Why we got what we wantedbull Why we got it in Log-Space

Overviewbull Series of powering by 8 and zig zag

products one increases the spectral gap and the other one decreases it but not by much

8

O(Log(N))

Zig Zag Powering

Rotation Maps

bull Graph representation wersquoll use

GRot N D N D

G is D regular undirected The rotation map

GRot vi w j

If the edge (vw) exists and is the ith edge coming out of v and the jth edge coming out of w

Rotation Map Example

1 23

4

17

hellip

hellip54

55

4

3 21

28

5554

hellip

hellipu

v

G

G

Rot u17 v28

Rot v28 u17

PoweringNo limitations on the degree rarr can expenderize with powering

The k-th power of the D-regular graph G is Gk where there is an edge (uv) iff there is a path of length k between u and v in G

=k

Powering

What is it good forbull GN is an N-clique an expanderbull If G = (ND λ) then Gk is a (NDk λk) λ(G)

The spectral gapbull Why is this not enough

Zig Zag Product

If G is a D-regular graph with N vertices and H is a d-regular graph with D vertices replace each v in G with a copy of H Hv Our new graph has [N]x[D] vertices

H

Part of G

v is now Hv

v

Zig-Zag ProductThe resulting graph is a d2 regular

graph with edges as follows

G

v

w

arsquo

brsquo

Hv

Hw

a

brsquo

G Z H

a

b

i

j

arsquoiirsquo

bj

jrsquo

Zig-Zag Product

Lemma

If G is an (NDλ) graph and H is a (Dd) graph

211 (G z H ) 1 1

2

The zig-zag product doesnrsquot hurt the spectral gap too badly

22 2 2 21 1last week we saw (G z H ) 1 1 4

2 2

Connectivity in Expanders

What would happen if each connected component of our graph was an expander

We could decide USTCON in logspace

Every expander has a O(log(N)) diameter

TheoremFor any s and t in an expander therersquos

a path from s to t of length O(log(N))

Proof

i

i

n(1 )

2n

log 1 log2

log n 1ilog 1

Logspace Algorithm for paths in an expander

For (ND λ) there is a a space O(logDlogN) algorithm which decides USTCONIdeabullFrom any vertex s there are Dl=O(logN) different paths bullenumerate and check if any of them reach t

t

How much does it cost

bullAt each vertex we have a choice of D vertices log(D) to represent 1hellipD

bullEach path is log(N) long

bull Need log(D) at each stage of a log(N) path altogether O(logDlogN)

So far

We now know that if our graph G was an expander graph USTCON can be solved in Log-Space

How can we turn G into a graph of expanders

Making expanders powering it up and cutting it down

What do we want from the expander graph Grsquo

1 There is an s-t path in G iff there is an srsquo-trsquo path in Grsquo

2 Each connected component is an expander with constant expansion

3 Construction in Log-Space

The Plan

bull New tools for our toolboxndash Rotation Mapsndash Poweringndash Zig Zag products

bull The actual constructionbull Why we got what we wantedbull Why we got it in Log-Space

Overviewbull Series of powering by 8 and zig zag

products one increases the spectral gap and the other one decreases it but not by much

8

O(Log(N))

Zig Zag Powering

Rotation Maps

bull Graph representation wersquoll use

GRot N D N D

G is D regular undirected The rotation map

GRot vi w j

If the edge (vw) exists and is the ith edge coming out of v and the jth edge coming out of w

Rotation Map Example

1 23

4

17

hellip

hellip54

55

4

3 21

28

5554

hellip

hellipu

v

G

G

Rot u17 v28

Rot v28 u17

PoweringNo limitations on the degree rarr can expenderize with powering

The k-th power of the D-regular graph G is Gk where there is an edge (uv) iff there is a path of length k between u and v in G

=k

Powering

What is it good forbull GN is an N-clique an expanderbull If G = (ND λ) then Gk is a (NDk λk) λ(G)

The spectral gapbull Why is this not enough

Zig Zag Product

If G is a D-regular graph with N vertices and H is a d-regular graph with D vertices replace each v in G with a copy of H Hv Our new graph has [N]x[D] vertices

H

Part of G

v is now Hv

v

Zig-Zag ProductThe resulting graph is a d2 regular

graph with edges as follows

G

v

w

arsquo

brsquo

Hv

Hw

a

brsquo

G Z H

a

b

i

j

arsquoiirsquo

bj

jrsquo

Zig-Zag Product

Lemma

If G is an (NDλ) graph and H is a (Dd) graph

211 (G z H ) 1 1

2

The zig-zag product doesnrsquot hurt the spectral gap too badly

22 2 2 21 1last week we saw (G z H ) 1 1 4

2 2

Every expander has a O(log(N)) diameter

TheoremFor any s and t in an expander therersquos

a path from s to t of length O(log(N))

Proof

i

i

n(1 )

2n

log 1 log2

log n 1ilog 1

Logspace Algorithm for paths in an expander

For (ND λ) there is a a space O(logDlogN) algorithm which decides USTCONIdeabullFrom any vertex s there are Dl=O(logN) different paths bullenumerate and check if any of them reach t

t

How much does it cost

bullAt each vertex we have a choice of D vertices log(D) to represent 1hellipD

bullEach path is log(N) long

bull Need log(D) at each stage of a log(N) path altogether O(logDlogN)

So far

We now know that if our graph G was an expander graph USTCON can be solved in Log-Space

How can we turn G into a graph of expanders

Making expanders powering it up and cutting it down

What do we want from the expander graph Grsquo

1 There is an s-t path in G iff there is an srsquo-trsquo path in Grsquo

2 Each connected component is an expander with constant expansion

3 Construction in Log-Space

The Plan

bull New tools for our toolboxndash Rotation Mapsndash Poweringndash Zig Zag products

bull The actual constructionbull Why we got what we wantedbull Why we got it in Log-Space

Overviewbull Series of powering by 8 and zig zag

products one increases the spectral gap and the other one decreases it but not by much

8

O(Log(N))

Zig Zag Powering

Rotation Maps

bull Graph representation wersquoll use

GRot N D N D

G is D regular undirected The rotation map

GRot vi w j

If the edge (vw) exists and is the ith edge coming out of v and the jth edge coming out of w

Rotation Map Example

1 23

4

17

hellip

hellip54

55

4

3 21

28

5554

hellip

hellipu

v

G

G

Rot u17 v28

Rot v28 u17

PoweringNo limitations on the degree rarr can expenderize with powering

The k-th power of the D-regular graph G is Gk where there is an edge (uv) iff there is a path of length k between u and v in G

=k

Powering

What is it good forbull GN is an N-clique an expanderbull If G = (ND λ) then Gk is a (NDk λk) λ(G)

The spectral gapbull Why is this not enough

Zig Zag Product

If G is a D-regular graph with N vertices and H is a d-regular graph with D vertices replace each v in G with a copy of H Hv Our new graph has [N]x[D] vertices

H

Part of G

v is now Hv

v

Zig-Zag ProductThe resulting graph is a d2 regular

graph with edges as follows

G

v

w

arsquo

brsquo

Hv

Hw

a

brsquo

G Z H

a

b

i

j

arsquoiirsquo

bj

jrsquo

Zig-Zag Product

Lemma

If G is an (NDλ) graph and H is a (Dd) graph

211 (G z H ) 1 1

2

The zig-zag product doesnrsquot hurt the spectral gap too badly

22 2 2 21 1last week we saw (G z H ) 1 1 4

2 2

Proof

i

i

n(1 )

2n

log 1 log2

log n 1ilog 1

Logspace Algorithm for paths in an expander

For (ND λ) there is a a space O(logDlogN) algorithm which decides USTCONIdeabullFrom any vertex s there are Dl=O(logN) different paths bullenumerate and check if any of them reach t

t

How much does it cost

bullAt each vertex we have a choice of D vertices log(D) to represent 1hellipD

bullEach path is log(N) long

bull Need log(D) at each stage of a log(N) path altogether O(logDlogN)

So far

We now know that if our graph G was an expander graph USTCON can be solved in Log-Space

How can we turn G into a graph of expanders

Making expanders powering it up and cutting it down

What do we want from the expander graph Grsquo

1 There is an s-t path in G iff there is an srsquo-trsquo path in Grsquo

2 Each connected component is an expander with constant expansion

3 Construction in Log-Space

The Plan

bull New tools for our toolboxndash Rotation Mapsndash Poweringndash Zig Zag products

bull The actual constructionbull Why we got what we wantedbull Why we got it in Log-Space

Overviewbull Series of powering by 8 and zig zag

products one increases the spectral gap and the other one decreases it but not by much

8

O(Log(N))

Zig Zag Powering

Rotation Maps

bull Graph representation wersquoll use

GRot N D N D

G is D regular undirected The rotation map

GRot vi w j

If the edge (vw) exists and is the ith edge coming out of v and the jth edge coming out of w

Rotation Map Example

1 23

4

17

hellip

hellip54

55

4

3 21

28

5554

hellip

hellipu

v

G

G

Rot u17 v28

Rot v28 u17

PoweringNo limitations on the degree rarr can expenderize with powering

The k-th power of the D-regular graph G is Gk where there is an edge (uv) iff there is a path of length k between u and v in G

=k

Powering

What is it good forbull GN is an N-clique an expanderbull If G = (ND λ) then Gk is a (NDk λk) λ(G)

The spectral gapbull Why is this not enough

Zig Zag Product

If G is a D-regular graph with N vertices and H is a d-regular graph with D vertices replace each v in G with a copy of H Hv Our new graph has [N]x[D] vertices

H

Part of G

v is now Hv

v

Zig-Zag ProductThe resulting graph is a d2 regular

graph with edges as follows

G

v

w

arsquo

brsquo

Hv

Hw

a

brsquo

G Z H

a

b

i

j

arsquoiirsquo

bj

jrsquo

Zig-Zag Product

Lemma

If G is an (NDλ) graph and H is a (Dd) graph

211 (G z H ) 1 1

2

The zig-zag product doesnrsquot hurt the spectral gap too badly

22 2 2 21 1last week we saw (G z H ) 1 1 4

2 2

Logspace Algorithm for paths in an expander

For (ND λ) there is a a space O(logDlogN) algorithm which decides USTCONIdeabullFrom any vertex s there are Dl=O(logN) different paths bullenumerate and check if any of them reach t

t

How much does it cost

bullAt each vertex we have a choice of D vertices log(D) to represent 1hellipD

bullEach path is log(N) long

bull Need log(D) at each stage of a log(N) path altogether O(logDlogN)

So far

We now know that if our graph G was an expander graph USTCON can be solved in Log-Space

How can we turn G into a graph of expanders

Making expanders powering it up and cutting it down

What do we want from the expander graph Grsquo

1 There is an s-t path in G iff there is an srsquo-trsquo path in Grsquo

2 Each connected component is an expander with constant expansion

3 Construction in Log-Space

The Plan

bull New tools for our toolboxndash Rotation Mapsndash Poweringndash Zig Zag products

bull The actual constructionbull Why we got what we wantedbull Why we got it in Log-Space

Overviewbull Series of powering by 8 and zig zag

products one increases the spectral gap and the other one decreases it but not by much

8

O(Log(N))

Zig Zag Powering

Rotation Maps

bull Graph representation wersquoll use

GRot N D N D

G is D regular undirected The rotation map

GRot vi w j

If the edge (vw) exists and is the ith edge coming out of v and the jth edge coming out of w

Rotation Map Example

1 23

4

17

hellip

hellip54

55

4

3 21

28

5554

hellip

hellipu

v

G

G

Rot u17 v28

Rot v28 u17

PoweringNo limitations on the degree rarr can expenderize with powering

The k-th power of the D-regular graph G is Gk where there is an edge (uv) iff there is a path of length k between u and v in G

=k

Powering

What is it good forbull GN is an N-clique an expanderbull If G = (ND λ) then Gk is a (NDk λk) λ(G)

The spectral gapbull Why is this not enough

Zig Zag Product

If G is a D-regular graph with N vertices and H is a d-regular graph with D vertices replace each v in G with a copy of H Hv Our new graph has [N]x[D] vertices

H

Part of G

v is now Hv

v

Zig-Zag ProductThe resulting graph is a d2 regular

graph with edges as follows

G

v

w

arsquo

brsquo

Hv

Hw

a

brsquo

G Z H

a

b

i

j

arsquoiirsquo

bj

jrsquo

Zig-Zag Product

Lemma

If G is an (NDλ) graph and H is a (Dd) graph

211 (G z H ) 1 1

2

The zig-zag product doesnrsquot hurt the spectral gap too badly

22 2 2 21 1last week we saw (G z H ) 1 1 4

2 2

How much does it cost

bullAt each vertex we have a choice of D vertices log(D) to represent 1hellipD

bullEach path is log(N) long

bull Need log(D) at each stage of a log(N) path altogether O(logDlogN)

So far

We now know that if our graph G was an expander graph USTCON can be solved in Log-Space

How can we turn G into a graph of expanders

Making expanders powering it up and cutting it down

What do we want from the expander graph Grsquo

1 There is an s-t path in G iff there is an srsquo-trsquo path in Grsquo

2 Each connected component is an expander with constant expansion

3 Construction in Log-Space

The Plan

bull New tools for our toolboxndash Rotation Mapsndash Poweringndash Zig Zag products

bull The actual constructionbull Why we got what we wantedbull Why we got it in Log-Space

Overviewbull Series of powering by 8 and zig zag

products one increases the spectral gap and the other one decreases it but not by much

8

O(Log(N))

Zig Zag Powering

Rotation Maps

bull Graph representation wersquoll use

GRot N D N D

G is D regular undirected The rotation map

GRot vi w j

If the edge (vw) exists and is the ith edge coming out of v and the jth edge coming out of w

Rotation Map Example

1 23

4

17

hellip

hellip54

55

4

3 21

28

5554

hellip

hellipu

v

G

G

Rot u17 v28

Rot v28 u17

PoweringNo limitations on the degree rarr can expenderize with powering

The k-th power of the D-regular graph G is Gk where there is an edge (uv) iff there is a path of length k between u and v in G

=k

Powering

What is it good forbull GN is an N-clique an expanderbull If G = (ND λ) then Gk is a (NDk λk) λ(G)

The spectral gapbull Why is this not enough

Zig Zag Product

If G is a D-regular graph with N vertices and H is a d-regular graph with D vertices replace each v in G with a copy of H Hv Our new graph has [N]x[D] vertices

H

Part of G

v is now Hv

v

Zig-Zag ProductThe resulting graph is a d2 regular

graph with edges as follows

G

v

w

arsquo

brsquo

Hv

Hw

a

brsquo

G Z H

a

b

i

j

arsquoiirsquo

bj

jrsquo

Zig-Zag Product

Lemma

If G is an (NDλ) graph and H is a (Dd) graph

211 (G z H ) 1 1

2

The zig-zag product doesnrsquot hurt the spectral gap too badly

22 2 2 21 1last week we saw (G z H ) 1 1 4

2 2

So far

We now know that if our graph G was an expander graph USTCON can be solved in Log-Space

How can we turn G into a graph of expanders

Making expanders powering it up and cutting it down

What do we want from the expander graph Grsquo

1 There is an s-t path in G iff there is an srsquo-trsquo path in Grsquo

2 Each connected component is an expander with constant expansion

3 Construction in Log-Space

The Plan

bull New tools for our toolboxndash Rotation Mapsndash Poweringndash Zig Zag products

bull The actual constructionbull Why we got what we wantedbull Why we got it in Log-Space

Overviewbull Series of powering by 8 and zig zag

products one increases the spectral gap and the other one decreases it but not by much

8

O(Log(N))

Zig Zag Powering

Rotation Maps

bull Graph representation wersquoll use

GRot N D N D

G is D regular undirected The rotation map

GRot vi w j

If the edge (vw) exists and is the ith edge coming out of v and the jth edge coming out of w

Rotation Map Example

1 23

4

17

hellip

hellip54

55

4

3 21

28

5554

hellip

hellipu

v

G

G

Rot u17 v28

Rot v28 u17

PoweringNo limitations on the degree rarr can expenderize with powering

The k-th power of the D-regular graph G is Gk where there is an edge (uv) iff there is a path of length k between u and v in G

=k

Powering

What is it good forbull GN is an N-clique an expanderbull If G = (ND λ) then Gk is a (NDk λk) λ(G)

The spectral gapbull Why is this not enough

Zig Zag Product

If G is a D-regular graph with N vertices and H is a d-regular graph with D vertices replace each v in G with a copy of H Hv Our new graph has [N]x[D] vertices

H

Part of G

v is now Hv

v

Zig-Zag ProductThe resulting graph is a d2 regular

graph with edges as follows

G

v

w

arsquo

brsquo

Hv

Hw

a

brsquo

G Z H

a

b

i

j

arsquoiirsquo

bj

jrsquo

Zig-Zag Product

Lemma

If G is an (NDλ) graph and H is a (Dd) graph

211 (G z H ) 1 1

2

The zig-zag product doesnrsquot hurt the spectral gap too badly

22 2 2 21 1last week we saw (G z H ) 1 1 4

2 2

Making expanders powering it up and cutting it down

What do we want from the expander graph Grsquo

1 There is an s-t path in G iff there is an srsquo-trsquo path in Grsquo

2 Each connected component is an expander with constant expansion

3 Construction in Log-Space

The Plan

bull New tools for our toolboxndash Rotation Mapsndash Poweringndash Zig Zag products

bull The actual constructionbull Why we got what we wantedbull Why we got it in Log-Space

Overviewbull Series of powering by 8 and zig zag

products one increases the spectral gap and the other one decreases it but not by much

8

O(Log(N))

Zig Zag Powering

Rotation Maps

bull Graph representation wersquoll use

GRot N D N D

G is D regular undirected The rotation map

GRot vi w j

If the edge (vw) exists and is the ith edge coming out of v and the jth edge coming out of w

Rotation Map Example

1 23

4

17

hellip

hellip54

55

4

3 21

28

5554

hellip

hellipu

v

G

G

Rot u17 v28

Rot v28 u17

PoweringNo limitations on the degree rarr can expenderize with powering

The k-th power of the D-regular graph G is Gk where there is an edge (uv) iff there is a path of length k between u and v in G

=k

Powering

What is it good forbull GN is an N-clique an expanderbull If G = (ND λ) then Gk is a (NDk λk) λ(G)

The spectral gapbull Why is this not enough

Zig Zag Product

If G is a D-regular graph with N vertices and H is a d-regular graph with D vertices replace each v in G with a copy of H Hv Our new graph has [N]x[D] vertices

H

Part of G

v is now Hv

v

Zig-Zag ProductThe resulting graph is a d2 regular

graph with edges as follows

G

v

w

arsquo

brsquo

Hv

Hw

a

brsquo

G Z H

a

b

i

j

arsquoiirsquo

bj

jrsquo

Zig-Zag Product

Lemma

If G is an (NDλ) graph and H is a (Dd) graph

211 (G z H ) 1 1

2

The zig-zag product doesnrsquot hurt the spectral gap too badly

22 2 2 21 1last week we saw (G z H ) 1 1 4

2 2

What do we want from the expander graph Grsquo

1 There is an s-t path in G iff there is an srsquo-trsquo path in Grsquo

2 Each connected component is an expander with constant expansion

3 Construction in Log-Space

The Plan

bull New tools for our toolboxndash Rotation Mapsndash Poweringndash Zig Zag products

bull The actual constructionbull Why we got what we wantedbull Why we got it in Log-Space

Overviewbull Series of powering by 8 and zig zag

products one increases the spectral gap and the other one decreases it but not by much

8

O(Log(N))

Zig Zag Powering

Rotation Maps

bull Graph representation wersquoll use

GRot N D N D

G is D regular undirected The rotation map

GRot vi w j

If the edge (vw) exists and is the ith edge coming out of v and the jth edge coming out of w

Rotation Map Example

1 23

4

17

hellip

hellip54

55

4

3 21

28

5554

hellip

hellipu

v

G

G

Rot u17 v28

Rot v28 u17

PoweringNo limitations on the degree rarr can expenderize with powering

The k-th power of the D-regular graph G is Gk where there is an edge (uv) iff there is a path of length k between u and v in G

=k

Powering

What is it good forbull GN is an N-clique an expanderbull If G = (ND λ) then Gk is a (NDk λk) λ(G)

The spectral gapbull Why is this not enough

Zig Zag Product

If G is a D-regular graph with N vertices and H is a d-regular graph with D vertices replace each v in G with a copy of H Hv Our new graph has [N]x[D] vertices

H

Part of G

v is now Hv

v

Zig-Zag ProductThe resulting graph is a d2 regular

graph with edges as follows

G

v

w

arsquo

brsquo

Hv

Hw

a

brsquo

G Z H

a

b

i

j

arsquoiirsquo

bj

jrsquo

Zig-Zag Product

Lemma

If G is an (NDλ) graph and H is a (Dd) graph

211 (G z H ) 1 1

2

The zig-zag product doesnrsquot hurt the spectral gap too badly

22 2 2 21 1last week we saw (G z H ) 1 1 4

2 2

The Plan

bull New tools for our toolboxndash Rotation Mapsndash Poweringndash Zig Zag products

bull The actual constructionbull Why we got what we wantedbull Why we got it in Log-Space

Overviewbull Series of powering by 8 and zig zag

products one increases the spectral gap and the other one decreases it but not by much

8

O(Log(N))

Zig Zag Powering

Rotation Maps

bull Graph representation wersquoll use

GRot N D N D

G is D regular undirected The rotation map

GRot vi w j

If the edge (vw) exists and is the ith edge coming out of v and the jth edge coming out of w

Rotation Map Example

1 23

4

17

hellip

hellip54

55

4

3 21

28

5554

hellip

hellipu

v

G

G

Rot u17 v28

Rot v28 u17

PoweringNo limitations on the degree rarr can expenderize with powering

The k-th power of the D-regular graph G is Gk where there is an edge (uv) iff there is a path of length k between u and v in G

=k

Powering

What is it good forbull GN is an N-clique an expanderbull If G = (ND λ) then Gk is a (NDk λk) λ(G)

The spectral gapbull Why is this not enough

Zig Zag Product

If G is a D-regular graph with N vertices and H is a d-regular graph with D vertices replace each v in G with a copy of H Hv Our new graph has [N]x[D] vertices

H

Part of G

v is now Hv

v

Zig-Zag ProductThe resulting graph is a d2 regular

graph with edges as follows

G

v

w

arsquo

brsquo

Hv

Hw

a

brsquo

G Z H

a

b

i

j

arsquoiirsquo

bj

jrsquo

Zig-Zag Product

Lemma

If G is an (NDλ) graph and H is a (Dd) graph

211 (G z H ) 1 1

2

The zig-zag product doesnrsquot hurt the spectral gap too badly

22 2 2 21 1last week we saw (G z H ) 1 1 4

2 2

Overviewbull Series of powering by 8 and zig zag

products one increases the spectral gap and the other one decreases it but not by much

8

O(Log(N))

Zig Zag Powering

Rotation Maps

bull Graph representation wersquoll use

GRot N D N D

G is D regular undirected The rotation map

GRot vi w j

If the edge (vw) exists and is the ith edge coming out of v and the jth edge coming out of w

Rotation Map Example

1 23

4

17

hellip

hellip54

55

4

3 21

28

5554

hellip

hellipu

v

G

G

Rot u17 v28

Rot v28 u17

PoweringNo limitations on the degree rarr can expenderize with powering

The k-th power of the D-regular graph G is Gk where there is an edge (uv) iff there is a path of length k between u and v in G

=k

Powering

What is it good forbull GN is an N-clique an expanderbull If G = (ND λ) then Gk is a (NDk λk) λ(G)

The spectral gapbull Why is this not enough

Zig Zag Product

If G is a D-regular graph with N vertices and H is a d-regular graph with D vertices replace each v in G with a copy of H Hv Our new graph has [N]x[D] vertices

H

Part of G

v is now Hv

v

Zig-Zag ProductThe resulting graph is a d2 regular

graph with edges as follows

G

v

w

arsquo

brsquo

Hv

Hw

a

brsquo

G Z H

a

b

i

j

arsquoiirsquo

bj

jrsquo

Zig-Zag Product

Lemma

If G is an (NDλ) graph and H is a (Dd) graph

211 (G z H ) 1 1

2

The zig-zag product doesnrsquot hurt the spectral gap too badly

22 2 2 21 1last week we saw (G z H ) 1 1 4

2 2

Rotation Maps

bull Graph representation wersquoll use

GRot N D N D

G is D regular undirected The rotation map

GRot vi w j

If the edge (vw) exists and is the ith edge coming out of v and the jth edge coming out of w

Rotation Map Example

1 23

4

17

hellip

hellip54

55

4

3 21

28

5554

hellip

hellipu

v

G

G

Rot u17 v28

Rot v28 u17

PoweringNo limitations on the degree rarr can expenderize with powering

The k-th power of the D-regular graph G is Gk where there is an edge (uv) iff there is a path of length k between u and v in G

=k

Powering

What is it good forbull GN is an N-clique an expanderbull If G = (ND λ) then Gk is a (NDk λk) λ(G)

The spectral gapbull Why is this not enough

Zig Zag Product

If G is a D-regular graph with N vertices and H is a d-regular graph with D vertices replace each v in G with a copy of H Hv Our new graph has [N]x[D] vertices

H

Part of G

v is now Hv

v

Zig-Zag ProductThe resulting graph is a d2 regular

graph with edges as follows

G

v

w

arsquo

brsquo

Hv

Hw

a

brsquo

G Z H

a

b

i

j

arsquoiirsquo

bj

jrsquo

Zig-Zag Product

Lemma

If G is an (NDλ) graph and H is a (Dd) graph

211 (G z H ) 1 1

2

The zig-zag product doesnrsquot hurt the spectral gap too badly

22 2 2 21 1last week we saw (G z H ) 1 1 4

2 2

Rotation Map Example

1 23

4

17

hellip

hellip54

55

4

3 21

28

5554

hellip

hellipu

v

G

G

Rot u17 v28

Rot v28 u17

PoweringNo limitations on the degree rarr can expenderize with powering

The k-th power of the D-regular graph G is Gk where there is an edge (uv) iff there is a path of length k between u and v in G

=k

Powering

What is it good forbull GN is an N-clique an expanderbull If G = (ND λ) then Gk is a (NDk λk) λ(G)

The spectral gapbull Why is this not enough

Zig Zag Product

If G is a D-regular graph with N vertices and H is a d-regular graph with D vertices replace each v in G with a copy of H Hv Our new graph has [N]x[D] vertices

H

Part of G

v is now Hv

v

Zig-Zag ProductThe resulting graph is a d2 regular

graph with edges as follows

G

v

w

arsquo

brsquo

Hv

Hw

a

brsquo

G Z H

a

b

i

j

arsquoiirsquo

bj

jrsquo

Zig-Zag Product

Lemma

If G is an (NDλ) graph and H is a (Dd) graph

211 (G z H ) 1 1

2

The zig-zag product doesnrsquot hurt the spectral gap too badly

22 2 2 21 1last week we saw (G z H ) 1 1 4

2 2

PoweringNo limitations on the degree rarr can expenderize with powering

The k-th power of the D-regular graph G is Gk where there is an edge (uv) iff there is a path of length k between u and v in G

=k

Powering

What is it good forbull GN is an N-clique an expanderbull If G = (ND λ) then Gk is a (NDk λk) λ(G)

The spectral gapbull Why is this not enough

Zig Zag Product

If G is a D-regular graph with N vertices and H is a d-regular graph with D vertices replace each v in G with a copy of H Hv Our new graph has [N]x[D] vertices

H

Part of G

v is now Hv

v

Zig-Zag ProductThe resulting graph is a d2 regular

graph with edges as follows

G

v

w

arsquo

brsquo

Hv

Hw

a

brsquo

G Z H

a

b

i

j

arsquoiirsquo

bj

jrsquo

Zig-Zag Product

Lemma

If G is an (NDλ) graph and H is a (Dd) graph

211 (G z H ) 1 1

2

The zig-zag product doesnrsquot hurt the spectral gap too badly

22 2 2 21 1last week we saw (G z H ) 1 1 4

2 2

Powering

What is it good forbull GN is an N-clique an expanderbull If G = (ND λ) then Gk is a (NDk λk) λ(G)

The spectral gapbull Why is this not enough

Zig Zag Product

If G is a D-regular graph with N vertices and H is a d-regular graph with D vertices replace each v in G with a copy of H Hv Our new graph has [N]x[D] vertices

H

Part of G

v is now Hv

v

Zig-Zag ProductThe resulting graph is a d2 regular

graph with edges as follows

G

v

w

arsquo

brsquo

Hv

Hw

a

brsquo

G Z H

a

b

i

j

arsquoiirsquo

bj

jrsquo

Zig-Zag Product

Lemma

If G is an (NDλ) graph and H is a (Dd) graph

211 (G z H ) 1 1

2

The zig-zag product doesnrsquot hurt the spectral gap too badly

22 2 2 21 1last week we saw (G z H ) 1 1 4

2 2

Zig Zag Product

If G is a D-regular graph with N vertices and H is a d-regular graph with D vertices replace each v in G with a copy of H Hv Our new graph has [N]x[D] vertices

H

Part of G

v is now Hv

v

Zig-Zag ProductThe resulting graph is a d2 regular

graph with edges as follows

G

v

w

arsquo

brsquo

Hv

Hw

a

brsquo

G Z H

a

b

i

j

arsquoiirsquo

bj

jrsquo

Zig-Zag Product

Lemma

If G is an (NDλ) graph and H is a (Dd) graph

211 (G z H ) 1 1

2

The zig-zag product doesnrsquot hurt the spectral gap too badly

22 2 2 21 1last week we saw (G z H ) 1 1 4

2 2

Zig-Zag ProductThe resulting graph is a d2 regular

graph with edges as follows

G

v

w

arsquo

brsquo

Hv

Hw

a

brsquo

G Z H

a

b

i

j

arsquoiirsquo

bj

jrsquo

Zig-Zag Product

Lemma

If G is an (NDλ) graph and H is a (Dd) graph

211 (G z H ) 1 1

2

The zig-zag product doesnrsquot hurt the spectral gap too badly

22 2 2 21 1last week we saw (G z H ) 1 1 4

2 2

Zig-Zag Product

Lemma

If G is an (NDλ) graph and H is a (Dd) graph

211 (G z H ) 1 1

2

The zig-zag product doesnrsquot hurt the spectral gap too badly

22 2 2 21 1last week we saw (G z H ) 1 1 4

2 2

Question Is a connected component in G a connected component in G z H

Technicalities

Rotation Map for Powering

u w

v

G

3

12 5

8

G

G

Rot u3 v12

Rot v5 w8

u w

G2

2GRot u 35 w 812

35812

u whellipa1 a2 ak

b1b(k-1)bk

G

G 1 1

G k k

Rot ua vb

Rot v a wb

u wa1ak

bkb1

Gk

k 1 2 k k k 1 1GRot u a a a w b b b

Rotation Map for ig ag

Hw

(wbrsquo) (wb)

G

v

w

arsquo

brsquo

Hv

(va) (varsquo)i irsquo

j jrsquo

Rot ((va) (ij)) = zG H

H

G

H

wb j i where

a i Rot a i

wb Rot va

b j Rot b j

The Actual Construction

Show a transformation that turns every connected component of a graph into an expander

ButNot any graph but a D16 regular graph

How to construct such a graph - later

Main Transformation

0

8

i i 1

G G

G G z H

Gi is a D16 regular graph with N(D16)i vertices

On input G and H where G is a D16 regular graph on N vertices and H is a D-regular graph on D16 vertices

define GH G where O log N

Technical Lemma

For D-reg connected non-bipartite G on [N]

Proof

2 GD

2D

2

1G 1

DN

2

1 2

2

min

N

S V S

E

N

S

SG

SG

N

2

4not normalized D

N 2 O 1

2 2

O 1 O 1normalized 1 G G 1

N N

Details

bullSet l to be the smallest integer st If D is constant then this number is O(logN)

bullTransforming the entire graph is equivalent to transforming each connected component separately

2

2

1 11DN 2

Technicalities

Rotation Map for Powering

u w

v

G

3

12 5

8

G

G

Rot u3 v12

Rot v5 w8

u w

G2

2GRot u 35 w 812

35812

u whellipa1 a2 ak

b1b(k-1)bk

G

G 1 1

G k k

Rot ua vb

Rot v a wb

u wa1ak

bkb1

Gk

k 1 2 k k k 1 1GRot u a a a w b b b

Rotation Map for ig ag

Hw

(wbrsquo) (wb)

G

v

w

arsquo

brsquo

Hv

(va) (varsquo)i irsquo

j jrsquo

Rot ((va) (ij)) = zG H

H

G

H

wb j i where

a i Rot a i

wb Rot va

b j Rot b j

The Actual Construction

Show a transformation that turns every connected component of a graph into an expander

ButNot any graph but a D16 regular graph

How to construct such a graph - later

Main Transformation

0

8

i i 1

G G

G G z H

Gi is a D16 regular graph with N(D16)i vertices

On input G and H where G is a D16 regular graph on N vertices and H is a D-regular graph on D16 vertices

define GH G where O log N

Technical Lemma

For D-reg connected non-bipartite G on [N]

Proof

2 GD

2D

2

1G 1

DN

2

1 2

2

min

N

S V S

E

N

S

SG

SG

N

2

4not normalized D

N 2 O 1

2 2

O 1 O 1normalized 1 G G 1

N N

Details

bullSet l to be the smallest integer st If D is constant then this number is O(logN)

bullTransforming the entire graph is equivalent to transforming each connected component separately

2

2

1 11DN 2

Rotation Map for Powering

u w

v

G

3

12 5

8

G

G

Rot u3 v12

Rot v5 w8

u w

G2

2GRot u 35 w 812

35812

u whellipa1 a2 ak

b1b(k-1)bk

G

G 1 1

G k k

Rot ua vb

Rot v a wb

u wa1ak

bkb1

Gk

k 1 2 k k k 1 1GRot u a a a w b b b

Rotation Map for ig ag

Hw

(wbrsquo) (wb)

G

v

w

arsquo

brsquo

Hv

(va) (varsquo)i irsquo

j jrsquo

Rot ((va) (ij)) = zG H

H

G

H

wb j i where

a i Rot a i

wb Rot va

b j Rot b j

The Actual Construction

Show a transformation that turns every connected component of a graph into an expander

ButNot any graph but a D16 regular graph

How to construct such a graph - later

Main Transformation

0

8

i i 1

G G

G G z H

Gi is a D16 regular graph with N(D16)i vertices

On input G and H where G is a D16 regular graph on N vertices and H is a D-regular graph on D16 vertices

define GH G where O log N

Technical Lemma

For D-reg connected non-bipartite G on [N]

Proof

2 GD

2D

2

1G 1

DN

2

1 2

2

min

N

S V S

E

N

S

SG

SG

N

2

4not normalized D

N 2 O 1

2 2

O 1 O 1normalized 1 G G 1

N N

Details

bullSet l to be the smallest integer st If D is constant then this number is O(logN)

bullTransforming the entire graph is equivalent to transforming each connected component separately

2

2

1 11DN 2

Rotation Map for ig ag

Hw

(wbrsquo) (wb)

G

v

w

arsquo

brsquo

Hv

(va) (varsquo)i irsquo

j jrsquo

Rot ((va) (ij)) = zG H

H

G

H

wb j i where

a i Rot a i

wb Rot va

b j Rot b j

The Actual Construction

Show a transformation that turns every connected component of a graph into an expander

ButNot any graph but a D16 regular graph

How to construct such a graph - later

Main Transformation

0

8

i i 1

G G

G G z H

Gi is a D16 regular graph with N(D16)i vertices

On input G and H where G is a D16 regular graph on N vertices and H is a D-regular graph on D16 vertices

define GH G where O log N

Technical Lemma

For D-reg connected non-bipartite G on [N]

Proof

2 GD

2D

2

1G 1

DN

2

1 2

2

min

N

S V S

E

N

S

SG

SG

N

2

4not normalized D

N 2 O 1

2 2

O 1 O 1normalized 1 G G 1

N N

Details

bullSet l to be the smallest integer st If D is constant then this number is O(logN)

bullTransforming the entire graph is equivalent to transforming each connected component separately

2

2

1 11DN 2

The Actual Construction

Show a transformation that turns every connected component of a graph into an expander

ButNot any graph but a D16 regular graph

How to construct such a graph - later

Main Transformation

0

8

i i 1

G G

G G z H

Gi is a D16 regular graph with N(D16)i vertices

On input G and H where G is a D16 regular graph on N vertices and H is a D-regular graph on D16 vertices

define GH G where O log N

Technical Lemma

For D-reg connected non-bipartite G on [N]

Proof

2 GD

2D

2

1G 1

DN

2

1 2

2

min

N

S V S

E

N

S

SG

SG

N

2

4not normalized D

N 2 O 1

2 2

O 1 O 1normalized 1 G G 1

N N

Details

bullSet l to be the smallest integer st If D is constant then this number is O(logN)

bullTransforming the entire graph is equivalent to transforming each connected component separately

2

2

1 11DN 2

Main Transformation

0

8

i i 1

G G

G G z H

Gi is a D16 regular graph with N(D16)i vertices

On input G and H where G is a D16 regular graph on N vertices and H is a D-regular graph on D16 vertices

define GH G where O log N

Technical Lemma

For D-reg connected non-bipartite G on [N]

Proof

2 GD

2D

2

1G 1

DN

2

1 2

2

min

N

S V S

E

N

S

SG

SG

N

2

4not normalized D

N 2 O 1

2 2

O 1 O 1normalized 1 G G 1

N N

Details

bullSet l to be the smallest integer st If D is constant then this number is O(logN)

bullTransforming the entire graph is equivalent to transforming each connected component separately

2

2

1 11DN 2

Technical Lemma

For D-reg connected non-bipartite G on [N]

Proof

2 GD

2D

2

1G 1

DN

2

1 2

2

min

N

S V S

E

N

S

SG

SG

N

2

4not normalized D

N 2 O 1

2 2

O 1 O 1normalized 1 G G 1

N N

Details

bullSet l to be the smallest integer st If D is constant then this number is O(logN)

bullTransforming the entire graph is equivalent to transforming each connected component separately

2

2

1 11DN 2

Details

bullSet l to be the smallest integer st If D is constant then this number is O(logN)

bullTransforming the entire graph is equivalent to transforming each connected component separately

2

2

1 11DN 2

More Details

S1

S2

S3

S1

S2

S3

Why

Show that this is true for a single iteration

1

m

S

S

0

0

Adjacency matrix sorted by connected components

Both powering and zig zag maintain the block structure

Why is this an expander

G and H are inputs of above If and G is connected and non-bipartite then

Which means that the output of is indeed an expander

H 1 2

GH 1 2

Proof

G0=G connected and non-bipartite

Itrsquos enough to show and from the lemma we get

l2 N2 21 1 DN 1 1 DN 1 2

2

i i 1i 0 G max G 1 2

H 1 2

i 1 i 1 i 1G z H 1 38 1 G 1 1 3 1 G

20G 1 1 DN

Proof Continued

Recall if G = (NDλ) then Gt = (NDtλt) =We can bound λ(Gi) by

For each i one of two cases occurs

Otherwise itrsquos always true that

8

i 11 1 3 1 G

8

i 1 i

1 5 1G G

2 6 2

41 1 3 1 x x

4 2

i 1 i 1 i i 11 1 3 1 G G G G

Why

Show that this is true for a single iteration

1

m

S

S

0

0

Adjacency matrix sorted by connected components

Both powering and zig zag maintain the block structure

Why is this an expander

G and H are inputs of above If and G is connected and non-bipartite then

Which means that the output of is indeed an expander

H 1 2

GH 1 2

Proof

G0=G connected and non-bipartite

Itrsquos enough to show and from the lemma we get

l2 N2 21 1 DN 1 1 DN 1 2

2

i i 1i 0 G max G 1 2

H 1 2

i 1 i 1 i 1G z H 1 38 1 G 1 1 3 1 G

20G 1 1 DN

Proof Continued

Recall if G = (NDλ) then Gt = (NDtλt) =We can bound λ(Gi) by

For each i one of two cases occurs

Otherwise itrsquos always true that

8

i 11 1 3 1 G

8

i 1 i

1 5 1G G

2 6 2

41 1 3 1 x x

4 2

i 1 i 1 i i 11 1 3 1 G G G G

Why is this an expander

G and H are inputs of above If and G is connected and non-bipartite then

Which means that the output of is indeed an expander

H 1 2

GH 1 2

Proof

G0=G connected and non-bipartite

Itrsquos enough to show and from the lemma we get

l2 N2 21 1 DN 1 1 DN 1 2

2

i i 1i 0 G max G 1 2

H 1 2

i 1 i 1 i 1G z H 1 38 1 G 1 1 3 1 G

20G 1 1 DN

Proof Continued

Recall if G = (NDλ) then Gt = (NDtλt) =We can bound λ(Gi) by

For each i one of two cases occurs

Otherwise itrsquos always true that

8

i 11 1 3 1 G

8

i 1 i

1 5 1G G

2 6 2

41 1 3 1 x x

4 2

i 1 i 1 i i 11 1 3 1 G G G G

Proof

G0=G connected and non-bipartite

Itrsquos enough to show and from the lemma we get

l2 N2 21 1 DN 1 1 DN 1 2

2

i i 1i 0 G max G 1 2

H 1 2

i 1 i 1 i 1G z H 1 38 1 G 1 1 3 1 G

20G 1 1 DN

Proof Continued

Recall if G = (NDλ) then Gt = (NDtλt) =We can bound λ(Gi) by

For each i one of two cases occurs

Otherwise itrsquos always true that

8

i 11 1 3 1 G

8

i 1 i

1 5 1G G

2 6 2

41 1 3 1 x x

4 2

i 1 i 1 i i 11 1 3 1 G G G G

Proof Continued

Recall if G = (NDλ) then Gt = (NDtλt) =We can bound λ(Gi) by

For each i one of two cases occurs

Otherwise itrsquos always true that

8

i 11 1 3 1 G

8

i 1 i

1 5 1G G

2 6 2

41 1 3 1 x x

4 2

i 1 i 1 i i 11 1 3 1 G G G G

What do we want from the expander graph Grsquo

1 There is an s-t path in G iff there is an srsquo-trsquo path in Grsquo

2 Each connected component is an expander

3 Construction in Log-Space

Log Space Construction

Prove that for any D can be calculated in space O(log(N))

RotG RotHInput Tape

Output Tape 16 16G

edge numbervertex in G

v a Rot v a | v N D a D

The Work Tape

8i i 1G G z H

16 16 16

1

2

1 2 1Vertex in G vertex in H vertex in H vertex in H edge in G

D D D

Vertex in G

Vertx in G

Vertex in G

v a a a a

v a 1 2 16k k k

A step in H

1 H 1 1a k Rot a k a k

Use the last vertex of H and the first instruction

16 16

1

16

1 2 1Vertex in G vertex in H vertex in H

D D

Vertex in G

1edge in G

Vertex in G

New vertex in HD

v a a a aa

This is the direction

61 2 1 k kk

A step in Gl-1

bull Use the same variables for the recursion

11 2 1 G 1 2 1va a a Rot va a a a

16 16

1

16

1 2 1Vertex in G vertex in H vertex in H

D D

New Vertex in G

1edge in Gnew vertex in H

Vertex n

G

D

i

v a a a a a

Another Step in H

2 H 2a k Rot a k

16 16

1

16

1 2 1Vertex in G vertex in H vertex in H

D D

Vertex in G

1edge in G

Ve

New vertex i

rtex i

n H

n

D

G

v a a a aa

The next instruction

21 6 1k kk

Use the next instruction and the new vertex to finish the zig zag

All Together

1 2 3

Repeat 8 times to get powering

1 H 1 1a k Rot a k a k

11 2 1 G 1 2 1va a a Rot va a a a

2 H 2a k Rot a k

8

Why is this LogSpace

IdeaAll evaluations are done in placeWe use the same l+1 variablesv O(logN)ai O(log(D16)) = constcounters O(logN)

O log N log N const O log O ogN l N

Piecing it together

Convert G into a

D16 regular graph

Transform into an expander graph with

Check for path between

(s1hellip1) and (t1hellip1)

Vertex v cycle length N Add self loops as needed

Vertex v cycle length N Add self loops as needed

8i i 1G G z H

Enumerate all Dl paths

Enumerate all Dl paths

O(logN)

O(logN)

O(logN)

To Conclude

Composition of a Constant number of logspace

machines is in logspaceWe decide USTCON in logspace

• Undirected ST-Connectivity In Log Space
• Today
• What is ST-Connectivity
• What do we know about it
• What are we trying to accomplish
• What are we adding to the mix
• Bits Nrsquo Pieces
• Slide 8
• Expanders ndash a Reminder
• Slide 10
• Connectivity in Expanders
• Every expander has a O(log(N)) diameter
• Proof
• Logspace Algorithm for paths in an expander
• How much does it cost
• So far
• Slide 17
• What do we want from the expander graph Grsquo
• The Plan
• Overview
• Rotation Maps
• Rotation Map Example
• Powering
• Slide 24
• Zig Zag Product
• Zig-Zag Product
• Slide 27
• Question
• Slide 29
• Rotation Map for Powering
• Rotation Map for ig ag
• The Actual Construction
• Main Transformation
• Technical Lemma
• Details
• More Details
• Why
• Why is this an expander
• Slide 39
• Proof Continued
• Slide 41
• Slide 42
• Log Space Construction
• The Work Tape
• A step in H
• A step in Gl-1
• Another Step in H
• All Together
• Why is this LogSpace
• Piecing it together
• Slide 51
• To Conclude

Log Space Construction

Prove that for any D can be calculated in space O(log(N))

RotG RotHInput Tape

Output Tape 16 16G

edge numbervertex in G

v a Rot v a | v N D a D

The Work Tape

8i i 1G G z H

16 16 16

1

2

1 2 1Vertex in G vertex in H vertex in H vertex in H edge in G

D D D

Vertex in G

Vertx in G

Vertex in G

v a a a a

v a 1 2 16k k k

A step in H

1 H 1 1a k Rot a k a k

Use the last vertex of H and the first instruction

16 16

1

16

1 2 1Vertex in G vertex in H vertex in H

D D

Vertex in G

1edge in G

Vertex in G

New vertex in HD

v a a a aa

This is the direction

61 2 1 k kk

A step in Gl-1

bull Use the same variables for the recursion

11 2 1 G 1 2 1va a a Rot va a a a

16 16

1

16

1 2 1Vertex in G vertex in H vertex in H

D D

New Vertex in G

1edge in Gnew vertex in H

Vertex n

G

D

i

v a a a a a

Another Step in H

2 H 2a k Rot a k

16 16

1

16

1 2 1Vertex in G vertex in H vertex in H

D D

Vertex in G

1edge in G

Ve

New vertex i

rtex i

n H

n

D

G

v a a a aa

The next instruction

21 6 1k kk

Use the next instruction and the new vertex to finish the zig zag

All Together

1 2 3

Repeat 8 times to get powering

1 H 1 1a k Rot a k a k

11 2 1 G 1 2 1va a a Rot va a a a

2 H 2a k Rot a k

8

Why is this LogSpace

IdeaAll evaluations are done in placeWe use the same l+1 variablesv O(logN)ai O(log(D16)) = constcounters O(logN)

O log N log N const O log O ogN l N

Piecing it together

Convert G into a

D16 regular graph

Transform into an expander graph with

Check for path between

(s1hellip1) and (t1hellip1)

Vertex v cycle length N Add self loops as needed

Vertex v cycle length N Add self loops as needed

8i i 1G G z H

Enumerate all Dl paths

Enumerate all Dl paths

O(logN)

O(logN)

O(logN)

To Conclude

Composition of a Constant number of logspace

machines is in logspaceWe decide USTCON in logspace

• Undirected ST-Connectivity In Log Space
• Today
• What is ST-Connectivity
• What do we know about it
• What are we trying to accomplish
• What are we adding to the mix
• Bits Nrsquo Pieces
• Slide 8
• Expanders ndash a Reminder
• Slide 10
• Connectivity in Expanders
• Every expander has a O(log(N)) diameter
• Proof
• Logspace Algorithm for paths in an expander
• How much does it cost
• So far
• Slide 17
• What do we want from the expander graph Grsquo
• The Plan
• Overview
• Rotation Maps
• Rotation Map Example
• Powering
• Slide 24
• Zig Zag Product
• Zig-Zag Product
• Slide 27
• Question
• Slide 29
• Rotation Map for Powering
• Rotation Map for ig ag
• The Actual Construction
• Main Transformation
• Technical Lemma
• Details
• More Details
• Why
• Why is this an expander
• Slide 39
• Proof Continued
• Slide 41
• Slide 42
• Log Space Construction
• The Work Tape
• A step in H
• A step in Gl-1
• Another Step in H
• All Together
• Why is this LogSpace
• Piecing it together
• Slide 51
• To Conclude

The Work Tape

8i i 1G G z H

16 16 16

1

2

1 2 1Vertex in G vertex in H vertex in H vertex in H edge in G

D D D

Vertex in G

Vertx in G

Vertex in G

v a a a a

v a 1 2 16k k k

A step in H

1 H 1 1a k Rot a k a k

Use the last vertex of H and the first instruction

16 16

1

16

1 2 1Vertex in G vertex in H vertex in H

D D

Vertex in G

1edge in G

Vertex in G

New vertex in HD

v a a a aa

This is the direction

61 2 1 k kk

A step in Gl-1

bull Use the same variables for the recursion

11 2 1 G 1 2 1va a a Rot va a a a

16 16

1

16

1 2 1Vertex in G vertex in H vertex in H

D D

New Vertex in G

1edge in Gnew vertex in H

Vertex n

G

D

i

v a a a a a

Another Step in H

2 H 2a k Rot a k

16 16

1

16

1 2 1Vertex in G vertex in H vertex in H

D D

Vertex in G

1edge in G

Ve

New vertex i

rtex i

n H

n

D

G

v a a a aa

The next instruction

21 6 1k kk

Use the next instruction and the new vertex to finish the zig zag

All Together

1 2 3

Repeat 8 times to get powering

1 H 1 1a k Rot a k a k

11 2 1 G 1 2 1va a a Rot va a a a

2 H 2a k Rot a k

8

Why is this LogSpace

IdeaAll evaluations are done in placeWe use the same l+1 variablesv O(logN)ai O(log(D16)) = constcounters O(logN)

O log N log N const O log O ogN l N

Piecing it together

Convert G into a

D16 regular graph

Transform into an expander graph with

Check for path between

(s1hellip1) and (t1hellip1)

Vertex v cycle length N Add self loops as needed

Vertex v cycle length N Add self loops as needed

8i i 1G G z H

Enumerate all Dl paths

Enumerate all Dl paths

O(logN)

O(logN)

O(logN)

To Conclude

Composition of a Constant number of logspace

machines is in logspaceWe decide USTCON in logspace

• Undirected ST-Connectivity In Log Space
• Today
• What is ST-Connectivity
• What do we know about it
• What are we trying to accomplish
• What are we adding to the mix
• Bits Nrsquo Pieces
• Slide 8
• Expanders ndash a Reminder
• Slide 10
• Connectivity in Expanders
• Every expander has a O(log(N)) diameter
• Proof
• Logspace Algorithm for paths in an expander
• How much does it cost
• So far
• Slide 17
• What do we want from the expander graph Grsquo
• The Plan
• Overview
• Rotation Maps
• Rotation Map Example
• Powering
• Slide 24
• Zig Zag Product
• Zig-Zag Product
• Slide 27
• Question
• Slide 29
• Rotation Map for Powering
• Rotation Map for ig ag
• The Actual Construction
• Main Transformation
• Technical Lemma
• Details
• More Details
• Why
• Why is this an expander
• Slide 39
• Proof Continued
• Slide 41
• Slide 42
• Log Space Construction
• The Work Tape
• A step in H
• A step in Gl-1
• Another Step in H
• All Together
• Why is this LogSpace
• Piecing it together
• Slide 51
• To Conclude

A step in H

1 H 1 1a k Rot a k a k

Use the last vertex of H and the first instruction

16 16

1

16

1 2 1Vertex in G vertex in H vertex in H

D D

Vertex in G

1edge in G

Vertex in G

New vertex in HD

v a a a aa

This is the direction

61 2 1 k kk

A step in Gl-1

bull Use the same variables for the recursion

11 2 1 G 1 2 1va a a Rot va a a a

16 16

1

16

1 2 1Vertex in G vertex in H vertex in H

D D

New Vertex in G

1edge in Gnew vertex in H

Vertex n

G

D

i

v a a a a a

Another Step in H

2 H 2a k Rot a k

16 16

1

16

1 2 1Vertex in G vertex in H vertex in H

D D

Vertex in G

1edge in G

Ve

New vertex i

rtex i

n H

n

D

G

v a a a aa

The next instruction

21 6 1k kk

Use the next instruction and the new vertex to finish the zig zag

All Together

1 2 3

Repeat 8 times to get powering

1 H 1 1a k Rot a k a k

11 2 1 G 1 2 1va a a Rot va a a a

2 H 2a k Rot a k

8

Why is this LogSpace

IdeaAll evaluations are done in placeWe use the same l+1 variablesv O(logN)ai O(log(D16)) = constcounters O(logN)

O log N log N const O log O ogN l N

Piecing it together

Convert G into a

D16 regular graph

Transform into an expander graph with

Check for path between

(s1hellip1) and (t1hellip1)

Vertex v cycle length N Add self loops as needed

Vertex v cycle length N Add self loops as needed

8i i 1G G z H

Enumerate all Dl paths

Enumerate all Dl paths

O(logN)

O(logN)

O(logN)

To Conclude

Composition of a Constant number of logspace

machines is in logspaceWe decide USTCON in logspace

• Undirected ST-Connectivity In Log Space
• Today
• What is ST-Connectivity
• What do we know about it
• What are we trying to accomplish
• What are we adding to the mix
• Bits Nrsquo Pieces
• Slide 8
• Expanders ndash a Reminder
• Slide 10
• Connectivity in Expanders
• Every expander has a O(log(N)) diameter
• Proof
• Logspace Algorithm for paths in an expander
• How much does it cost
• So far
• Slide 17
• What do we want from the expander graph Grsquo
• The Plan
• Overview
• Rotation Maps
• Rotation Map Example
• Powering
• Slide 24
• Zig Zag Product
• Zig-Zag Product
• Slide 27
• Question
• Slide 29
• Rotation Map for Powering
• Rotation Map for ig ag
• The Actual Construction
• Main Transformation
• Technical Lemma
• Details
• More Details
• Why
• Why is this an expander
• Slide 39
• Proof Continued
• Slide 41
• Slide 42
• Log Space Construction
• The Work Tape
• A step in H
• A step in Gl-1
• Another Step in H
• All Together
• Why is this LogSpace
• Piecing it together
• Slide 51
• To Conclude

A step in Gl-1

bull Use the same variables for the recursion

11 2 1 G 1 2 1va a a Rot va a a a

16 16

1

16

1 2 1Vertex in G vertex in H vertex in H

D D

New Vertex in G

1edge in Gnew vertex in H

Vertex n

G

D

i

v a a a a a

Another Step in H

2 H 2a k Rot a k

16 16

1

16

1 2 1Vertex in G vertex in H vertex in H

D D

Vertex in G

1edge in G

Ve

New vertex i

rtex i

n H

n

D

G

v a a a aa

The next instruction

21 6 1k kk

Use the next instruction and the new vertex to finish the zig zag

All Together

1 2 3

Repeat 8 times to get powering

1 H 1 1a k Rot a k a k

11 2 1 G 1 2 1va a a Rot va a a a

2 H 2a k Rot a k

8

Why is this LogSpace

IdeaAll evaluations are done in placeWe use the same l+1 variablesv O(logN)ai O(log(D16)) = constcounters O(logN)

O log N log N const O log O ogN l N

Piecing it together

Convert G into a

D16 regular graph

Transform into an expander graph with

Check for path between

(s1hellip1) and (t1hellip1)

Vertex v cycle length N Add self loops as needed

Vertex v cycle length N Add self loops as needed

8i i 1G G z H

Enumerate all Dl paths

Enumerate all Dl paths

O(logN)

O(logN)

O(logN)

To Conclude

Composition of a Constant number of logspace

machines is in logspaceWe decide USTCON in logspace

• Undirected ST-Connectivity In Log Space
• Today
• What is ST-Connectivity
• What do we know about it
• What are we trying to accomplish
• What are we adding to the mix
• Bits Nrsquo Pieces
• Slide 8
• Expanders ndash a Reminder
• Slide 10
• Connectivity in Expanders
• Every expander has a O(log(N)) diameter
• Proof
• Logspace Algorithm for paths in an expander
• How much does it cost
• So far
• Slide 17
• What do we want from the expander graph Grsquo
• The Plan
• Overview
• Rotation Maps
• Rotation Map Example
• Powering
• Slide 24
• Zig Zag Product
• Zig-Zag Product
• Slide 27
• Question
• Slide 29
• Rotation Map for Powering
• Rotation Map for ig ag
• The Actual Construction
• Main Transformation
• Technical Lemma
• Details
• More Details
• Why
• Why is this an expander
• Slide 39
• Proof Continued
• Slide 41
• Slide 42
• Log Space Construction
• The Work Tape
• A step in H
• A step in Gl-1
• Another Step in H
• All Together
• Why is this LogSpace
• Piecing it together
• Slide 51
• To Conclude

Another Step in H

2 H 2a k Rot a k

16 16

1

16

1 2 1Vertex in G vertex in H vertex in H

D D

Vertex in G

1edge in G

Ve

New vertex i

rtex i

n H

n

D

G

v a a a aa

The next instruction

21 6 1k kk

Use the next instruction and the new vertex to finish the zig zag

All Together

1 2 3

Repeat 8 times to get powering

1 H 1 1a k Rot a k a k

11 2 1 G 1 2 1va a a Rot va a a a

2 H 2a k Rot a k

8

Why is this LogSpace

IdeaAll evaluations are done in placeWe use the same l+1 variablesv O(logN)ai O(log(D16)) = constcounters O(logN)

O log N log N const O log O ogN l N

Piecing it together

Convert G into a

D16 regular graph

Transform into an expander graph with

Check for path between

(s1hellip1) and (t1hellip1)

Vertex v cycle length N Add self loops as needed

Vertex v cycle length N Add self loops as needed

8i i 1G G z H

Enumerate all Dl paths

Enumerate all Dl paths

O(logN)

O(logN)

O(logN)

To Conclude

Composition of a Constant number of logspace

machines is in logspaceWe decide USTCON in logspace

• Undirected ST-Connectivity In Log Space
• Today
• What is ST-Connectivity
• What do we know about it
• What are we trying to accomplish
• What are we adding to the mix
• Bits Nrsquo Pieces
• Slide 8
• Expanders ndash a Reminder
• Slide 10
• Connectivity in Expanders
• Every expander has a O(log(N)) diameter
• Proof
• Logspace Algorithm for paths in an expander
• How much does it cost
• So far
• Slide 17
• What do we want from the expander graph Grsquo
• The Plan
• Overview
• Rotation Maps
• Rotation Map Example
• Powering
• Slide 24
• Zig Zag Product
• Zig-Zag Product
• Slide 27
• Question
• Slide 29
• Rotation Map for Powering
• Rotation Map for ig ag
• The Actual Construction
• Main Transformation
• Technical Lemma
• Details
• More Details
• Why
• Why is this an expander
• Slide 39
• Proof Continued
• Slide 41
• Slide 42
• Log Space Construction
• The Work Tape
• A step in H
• A step in Gl-1
• Another Step in H
• All Together
• Why is this LogSpace
• Piecing it together
• Slide 51
• To Conclude

All Together

1 2 3

Repeat 8 times to get powering

1 H 1 1a k Rot a k a k

11 2 1 G 1 2 1va a a Rot va a a a

2 H 2a k Rot a k

8

Why is this LogSpace

IdeaAll evaluations are done in placeWe use the same l+1 variablesv O(logN)ai O(log(D16)) = constcounters O(logN)

O log N log N const O log O ogN l N

Piecing it together

Convert G into a

D16 regular graph

Transform into an expander graph with

Check for path between

(s1hellip1) and (t1hellip1)

Vertex v cycle length N Add self loops as needed

Vertex v cycle length N Add self loops as needed

8i i 1G G z H

Enumerate all Dl paths

Enumerate all Dl paths

O(logN)

O(logN)

O(logN)

To Conclude

Composition of a Constant number of logspace

machines is in logspaceWe decide USTCON in logspace

• Undirected ST-Connectivity In Log Space
• Today
• What is ST-Connectivity
• What do we know about it
• What are we trying to accomplish
• What are we adding to the mix
• Bits Nrsquo Pieces
• Slide 8
• Expanders ndash a Reminder
• Slide 10
• Connectivity in Expanders
• Every expander has a O(log(N)) diameter
• Proof
• Logspace Algorithm for paths in an expander
• How much does it cost
• So far
• Slide 17
• What do we want from the expander graph Grsquo
• The Plan
• Overview
• Rotation Maps
• Rotation Map Example
• Powering
• Slide 24
• Zig Zag Product
• Zig-Zag Product
• Slide 27
• Question
• Slide 29
• Rotation Map for Powering
• Rotation Map for ig ag
• The Actual Construction
• Main Transformation
• Technical Lemma
• Details
• More Details
• Why
• Why is this an expander
• Slide 39
• Proof Continued
• Slide 41
• Slide 42
• Log Space Construction
• The Work Tape
• A step in H
• A step in Gl-1
• Another Step in H
• All Together
• Why is this LogSpace
• Piecing it together
• Slide 51
• To Conclude

Why is this LogSpace

IdeaAll evaluations are done in placeWe use the same l+1 variablesv O(logN)ai O(log(D16)) = constcounters O(logN)

O log N log N const O log O ogN l N

Piecing it together

Convert G into a

D16 regular graph

Transform into an expander graph with

Check for path between

(s1hellip1) and (t1hellip1)

Vertex v cycle length N Add self loops as needed

Vertex v cycle length N Add self loops as needed

8i i 1G G z H

Enumerate all Dl paths

Enumerate all Dl paths

O(logN)

O(logN)

O(logN)

To Conclude

Composition of a Constant number of logspace

machines is in logspaceWe decide USTCON in logspace

• Undirected ST-Connectivity In Log Space
• Today
• What is ST-Connectivity
• What do we know about it
• What are we trying to accomplish
• What are we adding to the mix
• Bits Nrsquo Pieces
• Slide 8
• Expanders ndash a Reminder
• Slide 10
• Connectivity in Expanders
• Every expander has a O(log(N)) diameter
• Proof
• Logspace Algorithm for paths in an expander
• How much does it cost
• So far
• Slide 17
• What do we want from the expander graph Grsquo
• The Plan
• Overview
• Rotation Maps
• Rotation Map Example
• Powering
• Slide 24
• Zig Zag Product
• Zig-Zag Product
• Slide 27
• Question
• Slide 29
• Rotation Map for Powering
• Rotation Map for ig ag
• The Actual Construction
• Main Transformation
• Technical Lemma
• Details
• More Details
• Why
• Why is this an expander
• Slide 39
• Proof Continued
• Slide 41
• Slide 42
• Log Space Construction
• The Work Tape
• A step in H
• A step in Gl-1
• Another Step in H
• All Together
• Why is this LogSpace
• Piecing it together
• Slide 51
• To Conclude

Piecing it together

Convert G into a

D16 regular graph

Transform into an expander graph with

Check for path between

(s1hellip1) and (t1hellip1)

Vertex v cycle length N Add self loops as needed

Vertex v cycle length N Add self loops as needed

8i i 1G G z H

Enumerate all Dl paths

Enumerate all Dl paths

O(logN)

O(logN)

O(logN)

To Conclude

Composition of a Constant number of logspace

machines is in logspaceWe decide USTCON in logspace

• Undirected ST-Connectivity In Log Space
• Today
• What is ST-Connectivity
• What do we know about it
• What are we trying to accomplish
• What are we adding to the mix
• Bits Nrsquo Pieces
• Slide 8
• Expanders ndash a Reminder
• Slide 10
• Connectivity in Expanders
• Every expander has a O(log(N)) diameter
• Proof
• Logspace Algorithm for paths in an expander
• How much does it cost
• So far
• Slide 17
• What do we want from the expander graph Grsquo
• The Plan
• Overview
• Rotation Maps
• Rotation Map Example
• Powering
• Slide 24
• Zig Zag Product
• Zig-Zag Product
• Slide 27
• Question
• Slide 29
• Rotation Map for Powering
• Rotation Map for ig ag
• The Actual Construction
• Main Transformation
• Technical Lemma
• Details
• More Details
• Why
• Why is this an expander
• Slide 39
• Proof Continued
• Slide 41
• Slide 42
• Log Space Construction
• The Work Tape
• A step in H
• A step in Gl-1
• Another Step in H
• All Together
• Why is this LogSpace
• Piecing it together
• Slide 51
• To Conclude

Convert G into a

D16 regular graph

Transform into an expander graph with

Check for path between

(s1hellip1) and (t1hellip1)

Vertex v cycle length N Add self loops as needed

Vertex v cycle length N Add self loops as needed

8i i 1G G z H

Enumerate all Dl paths

Enumerate all Dl paths

O(logN)

O(logN)

O(logN)

To Conclude

Composition of a Constant number of logspace

machines is in logspaceWe decide USTCON in logspace

• Undirected ST-Connectivity In Log Space
• Today
• What is ST-Connectivity
• What do we know about it
• What are we trying to accomplish
• What are we adding to the mix
• Bits Nrsquo Pieces
• Slide 8
• Expanders ndash a Reminder
• Slide 10
• Connectivity in Expanders
• Every expander has a O(log(N)) diameter
• Proof
• Logspace Algorithm for paths in an expander
• How much does it cost
• So far
• Slide 17
• What do we want from the expander graph Grsquo
• The Plan
• Overview
• Rotation Maps
• Rotation Map Example
• Powering
• Slide 24
• Zig Zag Product
• Zig-Zag Product
• Slide 27
• Question
• Slide 29
• Rotation Map for Powering
• Rotation Map for ig ag
• The Actual Construction
• Main Transformation
• Technical Lemma
• Details
• More Details
• Why
• Why is this an expander
• Slide 39
• Proof Continued
• Slide 41
• Slide 42
• Log Space Construction
• The Work Tape
• A step in H
• A step in Gl-1
• Another Step in H
• All Together
• Why is this LogSpace
• Piecing it together
• Slide 51
• To Conclude

To Conclude

Composition of a Constant number of logspace

machines is in logspaceWe decide USTCON in logspace

• Undirected ST-Connectivity In Log Space
• Today
• What is ST-Connectivity
• What do we know about it
• What are we trying to accomplish
• What are we adding to the mix
• Bits Nrsquo Pieces
• Slide 8
• Expanders ndash a Reminder
• Slide 10
• Connectivity in Expanders
• Every expander has a O(log(N)) diameter
• Proof
• Logspace Algorithm for paths in an expander
• How much does it cost
• So far
• Slide 17
• What do we want from the expander graph Grsquo
• The Plan
• Overview
• Rotation Maps
• Rotation Map Example
• Powering
• Slide 24
• Zig Zag Product
• Zig-Zag Product
• Slide 27
• Question
• Slide 29
• Rotation Map for Powering
• Rotation Map for ig ag
• The Actual Construction
• Main Transformation
• Technical Lemma
• Details
• More Details
• Why
• Why is this an expander
• Slide 39
• Proof Continued
• Slide 41
• Slide 42
• Log Space Construction
• The Work Tape
• A step in H
• A step in Gl-1
• Another Step in H
• All Together
• Why is this LogSpace
• Piecing it together
• Slide 51
• To Conclude