Download presentation

Presentation is loading. Please wait.

Published bySophia Harper Modified over 4 years ago

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].

Similar presentations

OK

Complexity and Efficient Algorithms Group / Department of Computer Science Testing the Cluster Structure of Graphs Christian Sohler joint work with Artur.

Complexity and Efficient Algorithms Group / Department of Computer Science Testing the Cluster Structure of Graphs Christian Sohler joint work with Artur.

© 2018 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Ppt on history of computer virus Ppt on teamviewer 9 Ppt on economic order quantity equation Ppt on limits and continuity in calculus Ppt on renewable energy in bangladesh Ppt on story of cricket Convert free pdf to ppt online conversion Download ppt on fdi in retail in india Ppt on word association test saturday Download ppt on phase controlled rectifier inversion