Undirected ST-Undirected ST-Connectivity in Log-SpaceConnectivity in Log-Space

Omer ReingoldOmer ReingoldWeizmannWeizmann InstituteInstitute

Undirected ST-Connectivity Undirected ST-Connectivity

Undirected ST-Connectivity Undirected ST-Connectivity

How to explore a maze?How to explore a maze?

• Random MouseRandom Mouse Wall Follower Wall Follower Recursive Backtracker Recursive Backtracker Tremaux's Algorithm Tremaux's Algorithm Dead End Filler Dead End Filler Cul-de-sac Filler Cul-de-sac Filler Blind Alley Sealer Blind Alley Sealer Blind Alley Filler Blind Alley Filler Collision Solver Collision Solver Shortest Paths FinderShortest Paths FinderShortest Path FinderShortest Path Finder

http://http://www.astrolog.org/labyrntwww.astrolog.org/labyrnth/algrithm.htmh/algrithm.htm

Exploring a Exploring a maze in maze in polynomial polynomial timetimefor dummiesfor dummies

This work, in particular, This work, in particular, gives (universal gives (universal exploration sequence exploration sequence [Koucky 01]):[Koucky 01]):

Undirected ST-Connectivity Undirected ST-Connectivity (USTCON) (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 1960’s (context of AI, mazes, wiring of circuits, …).

Work also for the directed case.

……GG ss tt

What About Space What About Space (Memory)?(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(log2 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.

Can Randomness Save Can Randomness Save Space?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]).).

Log-Space Complexity Log-Space Complexity classesclasses

• 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 L2

• Open problems: SL=L ? RL=L ? (NL=L ?)

Few Highlights of Previous Few Highlights of Previous WorkWork

• [Nisan 90] – powerful pseudorandom generator for space bounded computations.

• [Nisan 92] Poly-time, space-O(log2 N) algorithm for RL.

• [Nisan,Szemerédi,Wigderson 92] SL L3/2 • [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 L3/2 • [Armoni,Ta-Shma,Wigderson,Zhou 97]: SL

L4/3

Main Result of This WorkMain Result of This Work

• Previously known: L SL L4/3 ; 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

Symmetric LogSpace [Symmetric LogSpace [Lewis, Lewis, Papadimitriou 82]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

This Talk: Undirected ConnectivityThis Talk: Undirected Connectivity

• G is connected if every two vertices u,v are connected by a path.

……GG uu vv

Our ApproachOur 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)

Dual Defs of Expander GraphsDual Defs of Expander Graphs

For every set SS of ( ½ n ½ n) vertices

Neighbors(S)Neighbors(S) ≥≥ (1+α) (1+α)SS

• Expansion parameter (G): 2nd largest eigenvalue (in abs. value) of normalized adjacency matrix. Duality due to [Tan84, AM84,Alo86].

• For this talk enough to remember:• 0 0 1 ; 1 ; ( = 1= 1 not an expander; = 0= 0 perfect expander) 1/21/2 an expander (any constant << 1 1 will do).

Connectivity in Expander Connectivity in Expander GraphsGraphs

• 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).

Turning Into a Constant degree Turning Into a Constant degree Expander –Warm Up Expander –Warm Up

• Phase 0: obtaining any constant degree D 3 3

Replacement product with a cycle:

uuuu

11

77 22

33

66

5544

(u,(u,11))

(u,(u,77)) (u,(u,22))

(u,(u,33))(u,(u,66))

(u,(u,55))(u,(u,44))

First attempt: Graph PoweringFirst attempt: Graph Powering

• Important: G is (D-regular), connected and

non-bipartite G is a “slight expander”. [Alon Sudakov] (G) 1-1/(DN1-1/(DN22))

• Squaring a graph: G G2: for every path of length 2 in G, put an edge in G2 (allow parallel edges). Similarly, raising to a power c.

• Easy fact: (Gc) = ((G))c

• Conclusion: Squaring O(log N) times will turn G into an expander.

• Problem: no longer constant degree.

Short Detour: Two Graph ProductsShort 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?

Reducing Degree, Preserving Reducing Degree, Preserving Expansion 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.

Balanced Replacement ProductBalanced 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

uuuu

88

77 22

33

66

5544

(u,(u,88))

(u,(u,77)) (u,(u,22))

(u,(u,33))(u,(u,66))

(u,(u,55))(u,(u,44))

11 (u,1)(u,1)

HH

Putting It All TogetherPutting 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) G0.• Phase i: Set Gi+1=(Gi ® H)10.• For L=O(log N), verify that GL is connected.

• Correctness: (Gi) = 1- (Gi ® H) ≤ 1-/4 (Gi+1) ≤ (1-/4)10 ≤ max {1/2, ((Gi))2}

( If G is connected) (GL) ≤ 1/2

Logarithmic SpaceLogarithmic Space• Phase 0: transform the input graph G (on N

vertices) into a D-regular (non-bipartite) G0.

• Phase i: Set Gi+1=(Gi ® H)10.

• For L=O(log N), verify that GL is connected.

• Each phase (i>0) costs a constant amount in space: Powering and replacement very local and simple. Gi+1 is not much more complicated than Gi.

• A step on Gi+1 is composed of a constant number of operations, either a step on Gi or require constant space (e.g. a step on H).

Technical Comparison With Technical Comparison With Previous TechniquesPrevious 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.

Profiteroles: Universal Traversal and Profiteroles: Universal Traversal and Exploration SequencesExploration 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-Nlog N UTS.• Explicit polynomial-length UTS only for very

few and limited cases (e.g., cycles [Istrail88]).

Profiteroles: Universal Traversal and Profiteroles: Universal Traversal and Exploration SequencesExploration 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 Exploring a maze in maze in polynomial polynomial timetimefor dummiesfor dummies

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.

22 3344

11 11

44

44

33

22

Open Problems (Cont.)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(L2) or even NL = L. • New tradeoffs between time and space (a rich

area of research we didn’t discuss).

In particular, optimizing the constants. Better analysis of the zig-zag and

replacement products. Some progress: [Rozenman,Vadhan].