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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. - PowerPoint PPT Presentation
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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!