1. Undirected ST-Connectivity in Log-Space Omer Reingold Weizmann Institute

2. Undirected ST-Connectivity

3. How to explore a maze? Random Mouse Wall Follower Recursive Backtracker Tremaux's Algorithm Dead End Filler Cul-de-sac Filler Blind Alley Sealer Blind Alley Filler Collision Solver Shortest Paths Finder Shortest Path Finder http://www.astrolog.org/ labyrnth/algrithm.htmRandom Mouse Wall Follower Recursive Backtracker Tremaux's Algorithm Dead End Filler Cul-de-sac Filler Blind Alley Sealer Blind Alley Filler Collision Solver Shortest Paths Finder Shortest Path Finder http://www.astrolog.org/ labyrnth/algrithm.htm http://www.astrolog.org/ labyrnth/algrithm.htm http://www.astrolog.org/ labyrnth/algrithm.htm Exploring a maze in polynomial time for dummies This work, in particular, gives (universal exploration sequence [Koucky 01]):

4. Undirected ST-Connectivity (USTCON) Basic graph problem. Extensively studied. Time complexity – well understood: Two linear time algorithms, BFS and DFS, are known and used at least since the 1960s (context of AI, mazes, wiring of circuits, …). Work also for the directed case. … G st

5. What About Space (Memory)? Our focus: space complexity of USTCON. BFS and DFS require linear space. Best we could hope for – logarithmic space (needed for holding even a single vertex!) [Savitch 70] st-connectivity in space O(log 2 N). [Aleliunas, Karp, Lipton, Lovasz, Rackoff 79] Randomized space O(log N) for USTCON. The algorithm: take a, polynomially long, random walk from s and see if you reach t.

6. Can Randomness Save Space? Thrown into the context of derandomization: Can the random walk algorithm be derandomized without substantial increase in space? Gain additional motivation as an important test case for the more general question: What is the tradeoff between these two central resources of computations? ).Gain the tools of derandomization (particularly, pseudorandom generators that fool space- bounded algorithms [AKS87, BNS89, Nisan90, INW94]).

7. Log-Space Complexity classes L – deterministic space O(log N) NL – non deterministic space O(log N) Complete problem: st-connectivity in directed graphs. SL – Symmetric, non deterministic space O(logN) Complete problem: undirected st-connectivity. RL – Randomized space O(log N). Conclusion: L SL RL NL L 2 Open problems: SL=L ? RL=L ? (NL=L ?)

8. Few Highlights of Previous Work [Nisan 90] – powerful pseudorandom generator for space bounded computations. [Nisan 92] Poly-time, space-O(log 2 N) algorithm for RL. [Nisan,Szemerédi,Wigderson 92] SL L 3/2 [Nisan,Ta-Shma 95] SL closed under complement many more interesting problems in SL (an hierarchy of classes collapses to SL). [Saks,Zhou 95] RL L 3/2 [Armoni,Ta-Shma,Wigderson,Zhou 97]: SL L 4/3

9. Main Result of This Work Previously known: L SL L 4/3 ; SL RL Here: Undirected st-connectivity in deterministic log-space (and polynomial time) SL = L Independently: [Trifonov 04] SL in deterministic space O(log N loglog N) Additional results: universal traversal and exploration sequences, pseudorandom generators

10. Symmetric LogSpace [ Lewis, Papadimitriou 82] Quite a few problems in SL [Reference - Alvarez, Greenlaw 96]: Bounded degree planarity 2-Colorability (Nonbipartiteness) Chordal graph Interval graph Split graph Permutation graph … By this work, all of these problems in L

11. This Talk: Undirected Connectivity G is connected if every two vertices u,v are connected by a path. … G uv

12. Our Approach If you want to solve a connectivity problem on your input graph, first improve its connectivity. Give log-space transformation that turns a connected graph into a constant degree expander (= sparse but highly connected graph). Overview of the algorithm: First transform your input graph G into a constant degree expander G. Verify that G is connected. (Easy)

13. Dual Defs of Expander Graphs S ½ n For every set S of ( ½ n ) vertices Neighbors(S) (1+α) S Neighbors(S) (1+α) S Expansion parameter (G) : 2 nd largest eigenvalue (in abs. value) of normalized adjacency matrix. Duality due to [Tan84, AM84,Alo86]. For this talk enough to remember: 0 1 ; = 1 = 00 1 ; ( = 1 not an expander; = 0 perfect expander) 1/2< 1 1/2 an expander (any constant < 1 will do).

14. Connectivity in Expander Graphs Expander graphs have a logarithmic diameter (i.e., every two vertices are connected by a short path) For a constant degree G, easy to separate two cases: G is an expander. G is not connected. The algorithm: for every two vertices u,v check all logarithmically long paths from u and verify that one of them reaches v (only polynomial number of such paths).

15. Turning Into a Constant degree Expander –Warm Up 3Phase 0 : obtaining any constant degree D 3 Replacement product with a cycle: u u 1 7 2 3 6 5 4 (u, 1 ) (u, 7 ) (u, 2 ) (u, 3 ) (u, 6 ) (u, 5 ) (u, 4 )

16. First attempt: Graph Powering Important: G is (D-regular), connected and non-bipartite G is a slight expander. 1-1/(DN 2 ) [Alon Sudakov] (G) 1-1/(DN 2 ) Squaring a graph: G G 2 : for every path of length 2 in G, put an edge in G 2 (allow parallel edges). Similarly, raising to a power c. Easy fact: (G c ) = ( ( G)) c Conclusion: Squaring O(log N) times will turn G into an expander. Problem: no longer constant degree.

17. Short Detour: Two Graph Products Context: a simple combinatorial construction w/simple analysis of constant degree expanders [Reingold, Vadhan, Wigderson 00]. Idea: start with a constant size expander and make it larger. This is easy, but degree becomes larger too. Sounds familiar?

18. Reducing Degree, Preserving Expansion [RVW 00]: a method to reduce the degree of a graph while not harming its expansion by much. For that, introduced a new graph product - the zig-zag product. Similarly for the balanced replacement product. For the balanced replacement product, also follows from a decomposition theorem for Markov chains, given independently by [Martin, Randall 00]. The setting of parameters is very different than ours, but analysis is sufficiently strong.

19. Balanced Replacement Product H is of degree d (G ® H) of degree 2d. [RVW00,MR00] If H is an expander then (G) 1 - ( G ® H ) 1 - /4 u u 8 7 2 3 6 5 4 (u, 8 ) (u, 7 ) (u, 2 ) (u, 3 ) (u, 6 ) (u, 5 ) (u, 4 ) 1 (u,1) H

20. Putting It All Together Let H be an expander on D= (2d) 10 vertices of degree d. Phase 0: transform the input graph G (on N vertices) into a D-regular (non-bipartite) G 0. Phase i: Set G i+1 =(G i ® H) 10. For L=O(log N), verify that G L is connected. Correctness: (G i ) = 1 - ( G i ® H ) 1 - /4 ( G i+1 ) (1 - /4 ) 10 max {1/2, ( (G i )) 2 } ( If G is connected) (G L ) 1/2

21. Logarithmic Space Phase 0: transform the input graph G (on N vertices) into a D-regular (non-bipartite) G 0. Phase i: Set G i+1 =(G i ® H) 10. For L=O(log N), verify that G L is connected. Each phase (i>0) costs a constant amount in space: Powering and replacement very local and simple. G i+1 is not much more complicated than G i. A step on G i+1 is composed of a constant number of operations, either a step on G i or require constant space (e.g. a step on H).

22. Technical Comparison With Previous Techniques [Savitch 70] reduces the diameter of the graph in O(log N) phases (constant degree not preserved). We work with a stronger parameter of connectivity. [NSW 92, ATWZ97] improve upon Savitch, by reducing the number of phases. Each phase shrinks the graph. We deviate in two ways: (a) Back to O(logN) phases (b) Enlarge the graph.

23. Profiteroles: Universal Traversal and Exploration Sequences Universal-traversal sequence (UTS) introduced by Cook in the late 70's with the motivation of proving SL=L. (N,D)-UTS: a sequence of edge labels in [D]. Guides a walk through all of the vertices of any D-regular graph on N vertices. [AKLLR79] poly-long UTS exist (probabilistic). [Nisan 90] Explicit, length-N log N UTS. Explicit polynomial-length UTS only for very few and limited cases (e.g., cycles [Istrail88]).

24. Profiteroles: Universal Traversal and Exploration Sequences UTS – move a pebble on the vertices. Our algorithm naturally implies: moving a pebble on the edges. In particular, get log-space constructible Universal Exploration Sequences (UXS) [Koucky 01]. Like UTS but directions are relative. Perfect for traversing a maze. Exploring a maze in polynomial time for dummies

25. First Open Problems - First Open Problems - Full Fledged Universal Traversal Sequences. Even for expanders: UTS known only for consistent labeling [Hoory,Wigderson 93]. Here: log-space constructible UTS for all graphs under restriction on the labeling. Restriction relaxed in [Reingold, Trevisan, Vadhan 05] to consistent labeling. 23 4 1 1 4 43 2

26. Open Problems (Cont.) RL=L – randomness cannot save memory. Some progress [Reingold,Trevisan,Vadhan 05] In short: God is in the labelling Improving Savitch in the directed case. NL = o(L 2 ) or even NL = L. New tradeoffs between time and space (a rich area of research we didnt discuss). In particular, optimizing the constants. Better analysis of the zig-zag and replacement products. Some progress: [Rozenman,Vadhan].