Undirected ST-Connectivity in Log-Space Author: Omer Reingold Presented by: Yang Liu.

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Undirected ST- Connectivity in Log- Space Author: Omer Reingold Presented by: Yang Liu
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Transcript of Undirected ST-Connectivity in Log-Space Author: Omer Reingold Presented by: Yang Liu.

Undirected ST-Connectivity in Log-Space

Author: Omer Reingold

Presented by: Yang Liu

Outline

• Introduction• Part 1: basic ideas• Part 2: transforming a graph G to an (N,D, λ)-

graph G’• Questions

Introduction

• Undirected ST-Connectivity: decide whether s and t are connected in space O(logN).

• Can be done in linear time and linear space• This paper gives an algorithm to solve this

problem, thus solves an outstanding open problem and concludes that SL=L.

Part 1: Basic Idea

• An (N,D, λ)-graph G has diameter O(logN), thus ST-Connectivity can be solved in O(logN) space.

Original Input Graph

• Graph G adjacency matrix M• Undirected graph M is symmetric• D-regular the sum of entries in each row

(and column) of M is D.

Rotation Map

• For D-regular undirected graph G, RotG: [N]x[D][N]x[D]

RotG(v1,2)=(v3,1)

RotG(v2,1)=(v3,2)

v1 v2 v3 v4

v1 1 0 1 1

v2 0 0 1 0

v3 1 1 0 1

v4 1 0 1 1

Normalized Adjacency Matrix Mn

• G: D-regular undirected graph Mn(vi,vj)=M(vi,vj)/D

M (D=3) Mn

v1 v2 v3 v4

v1 1/3 0 2/3 0

v2 0 1/3 1/3 1/3

v3 2/3 1/3 0 0

v4 0 1/3 0 2/3

v1 v2 v3 v4

v1 1 0 2 0

v2 0 1 1 1

v3 2 1 0 0

v4 0 1 0 2

(N,D, λ)-graph

• 1N=(1,…,1) is an eigenvector of Mn with eigenvalue 1 since Mn *1N =1* 1N

• |λ|<=1 for any other eigenvalue λ of Mn

--λ(G): the second largest eigenvalue of Mn

--(N,D, λ)-graph: a D-regular graph G on N vertices such that λ(G)<= λ .

Vertex Expansion

• For an (N,D, λ)-graph G For every λ<1, there exists ε >0 such that

for any set S such that |S|<=|G|/2, at least (1+ε)|S| vertices of G are connected to some vertex in S.

Diameter of a (N,D, λ)-graph

• Diameter is bounded by O(logN)• Pick any vertex s, let l=O(logN). Then at least

(1+ε)l>=N/2 vertices are at distance at most l to s.

• Pick any two vertices s and t, then at least one vertex is of distance at most l from both s and t a path of length at most 2l between any two vertices.

ST-Connectivity for (N,D, λ)-graph

• Can be determined in space O(LogD*logN)• Enumerate all paths from s of length O(logN)• Memory: logD for remembering an edge in

the path, and at most O(logN) edges for a path.

Part 2: Transforming G to an (N,D, λ)-graph

• Idea: increase the connectivity of G by powering G

• Challenge: keeping degree be constant by using zig-zag expander.

Powering

• G: an (N,D, λ)-graph G by rotation map RotG. The t’th power Gt of G is:

RotG(v0,(a1,…,at))=(vt,(b1,…bt)).

i.e., there is path v0-a1-b1-v1-a2…vt-1-at-bt-vt.

G: an (N,D, λ)-graph Gt:an (N,Dt, λt)-graph

Zig-zag Graph Product

H: an (D, d, α)-graph a (ND, d2, λ’)-graph

G: an (N,D, λ)-graph

v1

h1h2

h3 h4

v2

2 4

h1h2

h3 h4

h1h2

h3 h4

v1

v2

Main Transformation

• Input: H: a (D16,D,1/2)-graph and G: a (N,D16,λ)-graph where λ≤1-1/(DN2)• Processing: for i=1 to l=O(logN) do Gi=(Gi-1 z H)8

• Output: Gl=: a (Npoly, D16, 1/2)-graph

Transforming G to a regular Graph

• Input: a ((De)16,De,1/2)-graph H and a graph G.

• Output: a (N2,(De)16)-graph Greg from G.

v1 v2

G

(v1,v1)

(v1,v2)

(v1,v3)(v2,v1)

(v2,v2)

(v2,v3)4

1

2

3

…(De)16

Greg

Algorithm

• Transform G to a Greg (a regular graph)

• Transform Greg to Gexp (an (N,D,)-graph)

• Solve the s’t’-Connectivity on Gexp

Questions

Thank You!