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

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

3. 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.

4. Part 1: Basic Idea An (N,D, λ)-graph G has diameter O(logN), thus ST-Connectivity can be solved in O(logN) space.

5. 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.

6. Rotation Map For D-regular undirected graph G, Rot G : [N]x[D]  [N]x[D] Rot G (v1,2)=(v3,1) Rot G (v2,1)=(v3,2) v1v1 v2v2 v3v3 v4v4 v1v1 1011 v2v2 0010 v3v3 1101 v4v4 1011

7. Normalized Adjacency Matrix M n G: D-regular undirected graph M n (v i, v j )=M(v i,v j )/D M (D=3) M n v1v1 v2v2 v3v3 v4v4 v1v1 1/302/30 v2v2 01/3 v3v3 2/31/300 v4v4 0 02/3 v1v1 v2v2 v3v3 v4v4 v1v1 1020 v2v2 0111 v3v3 2100 v4v4 0102

8. (N,D, λ)-graph 1 N =(1,…,1) is an eigenvector of M n with eigenvalue 1 since M n *1 N =1* 1 N |λ|<=1 for any other eigenvalue λ of M n --λ(G): the second largest eigenvalue of M n --(N,D, λ)-graph: a D-regular graph G on N vertices such that λ(G)<= λ.

9. Vertex Expansion For an (N,D, λ)-graph G For every λ 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.

10. 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.

11. 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.

12. 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.

13. Powering G: an (N,D, λ)-graph G by rotation map Rot G. The t’th power G t of G is: Rot G (v 0,(a 1,…,a t ))=(v t,(b 1,…b t )). i.e., there is path v 0 -a 1 -b 1 -v 1 -a 2 …v t-1 -a t -b t -v t. G: an (N,D, λ)-graph G t :an (N,D t, λ t )-graph

14. Zig-zag Graph Product H: an (D, d, α)-graph a (ND, d 2, λ’)-graph G: an (N,D, λ)-graph v1v1 h1h1 h2h2 h3h3 h4h4 v2v2 24 h1h1 h2h2 h3h3 h4h4 h1h1 h2h2 h3h3 h4h4 v1v1 v2v2

15. Main Transformation Input: H: a (D 16,D,1/2)-graph and G: a (N,D 16,λ)-graph where λ≤1-1/(DN 2 ) Processing: for i=1 to l=O(logN) do G i =(G i-1 z H) 8 Output: G l =: a (N poly, D 16, 1/2)-graph

16. Transforming G to a regular Graph Input: a ((De) 16,De,1/2)-graph H and a graph G. Output: a (N 2,(De) 16 )-graph G reg from G. v1v1 v2v2 G (v 1,v 1 ) (v 1,v 2 ) (v 1,v 3 ) (v2,v1) (v 2,v 2 ) (v 2,v 3 ) 4 1 2 3 … (De) 16 G reg

17. Algorithm Transform G to a G reg (a regular graph) Transform Greg to G exp (an (N,D,)-graph) Solve the s’t’-Connectivity on G exp

18. Questions Thank You!