1. Undirected ST-Connectivity 2 DL Omer Reingold, STOC 2005: 376-385 Presented by: Fenghui Zhang 3. 24. 2006 CPSC 637 – paper presentation

2. Motivation  We have learned that DL µ NL µ D(log 2 n), a big question remains if DL=NL or not  S-T connectivity on directed graph is NL complete, is it in DL?  Undirected S-T connectivity A special case of directed connectivity Characterizes a new type of TM

3. Previous work  We know that it can be solved in deterministic log 2 (n) space  D(log 3/2 n) – M. Saks and S. Zhou, 1995  D(log 4/3 n) – R. Armoni, A. Ta-Shama, A. Wigderson and S. Zhou, 2000

4. Symmetric TM and SL  Symmetric TM A Nondeterministic Turing machine such that for any two configurations c i and c j, if there is a path from c i to c j, there will also be a path from c j to c i.  SL The classes of languages that are accepted by a Symmetric TM in log-space  DL µ SL µ RL µ NL  Undirected S-T connectivity is SL complete.

5. Undirect S-T connectivity  USTCON - Given a undirected graph G and two vertices s and t, is there a path in G from s to t? In SL µ RL µ NL µ P Complete for symmetric, non-deterministic log space (SL)  Is it in DL? (main contribution of this paper) Yes, thus SL=DL

6. Why is it important?  USTCON is the representative for SL, hence SL has another definition: SL contains all the languages that are log-space reducible to USTCON  The space complexity is thought critical for some networking problems Wireless network routing  We suspect that More space ~ routing overhead = extra energy consuming

7. Preliminary - Presentation of graphs  Adjacency matrices  Adjacency lists  Rotation map We order the edges incident to v somehow Rot(v,i)=(w,j) if the ith edge of v is the jth edge of j  We assume that the input graph is given by its adjacency matrix and use the rotation map to ``record” the intermittent graphs

8. How about some simpler cases?  Bounded degree  Expander Graph ``A well connected sparse graph” The vertex expansion can be measured by the spectral gap (1- ) of the normalized adjacency matrix of the graph  The largest eigenvalue is 1  The smaller the second largest eigenvalue is the better the graph is ``connected”

9. D-regular expander graph  Has diameter O(log n)  S-T connectivity on D-regular expander graphs can be checked in log space with a deterministic TM How?  If we can reduce our graph G to such a special graph using only log-space, then we are done.

10. Increase the connectivity of G  For each connected component of G, the second eigenvalue is less than 1  G t is a graph with second eigenvalue t  Problems Degrees increase also (we want constant degree) Space …

11. Reduce the degree  Goal – reduce the degree Without hurting the expansion too much Space…  Zig-zag graph product of G, H Roughly speaking, the resulting graph inherits the size of G, degree of H and expansion properties from both If H has low degrees and good expansion properties, the resulting graph will also be nice

12. Zig-zag graph product - definition  Given G, H, the vertex in the resulting graph GH will be (v, a) where v 2 G, a 2 H, edges are a subset of [N] £ [D] where |V(G)|=N and |V(H)|=D An edge (v, a)’s (i, j) is formed like  In H v, (a, i) → (a’, i’)  In GH, (v, a’) → (w, b’)  In H w, (b’, j) → (b, j’)

13. Zig-zag graph product - example

14. Zig-zag graph product - in the algorithm  If G is a D-regular graph and H is a d-regular graph of D vertices  The resulting graph Has degree d 2  Reduced a lot, given that d << D Of size N £ D Has spectral gap smaller than that of G by a factor that only depends on (H)  Not hurt too much

15. The basic algorithm  Produce a D e -regular graph H of size D e 16 and spectral gap 1/2  Transform G into a D e 16 -regular graph G’ Replace each vertex v by a cycle of size N, if the ith of v is the jth edge of w, we connect (v, i) to (w, j);If some vertices have degree less than D e 16, we add self-loops to make it so.  Turn G’ into an expander G exp Denote G §H the zig-zag product of G’ and H G i+1 =(G i §H) 8, until spectral gap is less than  Solve the st-connectivity in G exp, s’=(s,1), t’=(t,1)

16. How about the space?  We are talking about log-space, surely we can not really store any new graph  Fortunately we do not have to store them Like Dr. Chen did in the class for many times… Whenever some connection information is needed, we compute it from the original input Running time could be increased a lot but still polynomial

17. Correctness and running time  Intuitively, the connectivity of two vertices will be preserved during the transformations of the graphs  We omit the details here

18. Discussions  The techniques are not all new, yet it looks very complicated  The results are very important (DL=SL)  Further questions Is RL=DL?

19. Discussions – cont.  Still hard to apply it directly to localized routing algorithms The running time will be a high degree polynomial of N Localized algorithm has shortcomings comparing to log-space TM  We will not be able to reach the ``next node” in the localized algorithm unless a spanning tree has already been found and labeled nicely

20. Homework  Let B be the set of all bipartite graphs, i.e. for any graph G 2 B, we can partition its vertices into two parts: L and R, such that all edges of G are between vertices in L and in R. Prove that B can be accepted by a non- deterministic TM in log-space.  Due: 3/31(Friday) My office: 315D Email: fhzhang@cs.tamu.edu

21. The end Questions? Thank you very much!