Undirected ST-Connectivity in Log-Space. Omer Reingold. Presented by: Thang N. Dinh CISE, University of Florida, Fall, 2010
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1 Undirected ST-Connectivity in Log-Space Omer Reingold Presented by: Thang N. Dinh CISE, University of Florida, Fall, 2010
2 Undirected s-t connectivity(ustcon) Is t reachable from s in a undirected graph G? BFS, DFS take O(n) space USTCON NL (nondeterministic logspace Turing machine) Are we connected on Facebook?
3 SL Complexity class SL = problems log-space reducible to USTCON Defined by Papadimitriou in 1982 via Symmetric Turing machine L SL NL L 2 ( SPACE( log 2 n) ) Problems in SL: Vertex-disjoint paths: are there k vertex-disoint paths between s and t? Is a graph a bipartite? Is there a cycle containing a given edge? Exclusive or 2-satisfiability
4 Sequential of Works Savitch s Theorem USTCON L 2 Nisan, Szemeredi and Wigderson 1989 USTCON L 3/2 Armoni, Ta-Shma, Wigderson, and S. Zhou 2000 USTCON L 4/3 Trifonov 2005 USTCON SPACE( O(log n log log n) )
5 Important Results Omer Reingold paper All problems in SL are now in L. L = SL NL Showing that a problems are not in L, by showing that such no log-space reduction from it to USTCON exist Best paper award STOC Godel s prize 2009 Trifonov got Best student paper award in STOC 2005
6 Outline I. Introduction Space Complexity Connectivity is NL-complete Savitch s Theorem (NL L 2 ) Immerman s theorem (NL=coNL) II. Transforming to (N,D, λ)-graph Expander Graph Transforming into an expander
7 Space Complexity 7
8 Motivation Complexity classes correspond to bounds on resources One such resource is space: the number of tape cells a TM uses when solving a problem 8
9 Space Complexity Classes For any function f:n N, we define: SPACE(f(n))={ L : L is decidable by a deterministic O(f(n)) space TM} NSPACE(f(n))={ L : L is decidable by a non-deterministic O(f(n)) space TM} 9
10 Low Space Classes Definitions (logarithmic space classes): L = SPACE(logn) NL = NSPACE(logn) 10
11 Problem! How can a TM use only logn space if the input itself takes n cells?!!? 11
12 3Tape Machines a a b a b b _... Only the size of the work tape is counted for complexity purposes b b b b b... input readonly work read/ write output writeonly b a a b a... 12
13 Example Question: How much space would a TM that decides {a n b n n>0} require? Note: to count up to n, we need logn bits 13
14 Graph Connectivity CONN Instance: a directed graph G=(V,E) and two vertices s,t V Problem: To decide if there is a path from s to t in G? 14
15 Graph Connectivity s t 15
16 CONN is in NL Start at s For i = 1,.., V { Non-deterministically choose a neighbor and jump to it Accept if you get to t } If you got here reject! Counting up to V requires log V space Storing the current position requires log V space 16
17 Configurations Which objects determine the configuration of a TM of the new type? The content of the work tape The machine s state The head position on the input tape The head position on the work tape The head position on the output tape If the TM uses logarithmic space, there are polynomially many configurations Complexity D.Moshkovits 17
18 Log-Space Reductions Definition: A is log-space reducible to B, written A L B, if there exists a log space TM M that, given input w, outputs f(w) s.t. w A iff f(w) B the reduction 18
19 Do Log-Space Reductions Imply what they should? Suppose A 1 L A 2 and A 2 L; how to construct a log space TM which decides A 1? Wrong Solution: w Too Large! f(w) Use the TM for A 2 to decide if f(w) A 2 19
20 Log-Space reductions Claim: if 1. A 1 L A 2 f is the log-space reduction 2. A 2 L M is a log-space machine for A 2 Then, A 1 is in L Proof: on input x, in or not-in A 1 : Simulate M and whenever M reads the i th symbol of its input tape run f on x and wait for the i th bit to be outputted 20
21 NL Completeness Definition: A language B is NL-Complete if 1. B NL 2. For every A NL, A L B. If (2) holds, B is NL-hard 21
22 Savitch s Theorem Theorem: S(n) log(n) NSPACE(S(n)) SPACE(S(n) 2 ) Proof: First we ll prove NL SPACE(log 2 n) then, show this implies the general case 22
23 Savitch s Theorem Theorem: NSPACE(logn) SPACE(log 2 n) Proof: 1. First prove CONN is NL-complete (under logspace reductions) 2. Then show an algorithm for CONN that uses log 2 n space 23
24 CONN is NL-Complete Theorem: CONN is NL-Complete Proof: by the following reduction: s L t Does M accept x? Is there a path from s to t? 24
25 Configurations Graph A Computation of a NTM M on an input x can be described by a graph G M,x : the start configuration s A vertex per configuration t (u,v) E if M can move from u to v in one step the accepting configuration 25
26 CONN is NL-Complete Corollary: CONN is NL-Complete Proof: We ve shown CONN is in NL. We ve also presented a reduction from any NL language to CONN which is computable in log space (Why?) 26
27 A Byproduct Claim: NL P Proof: Any NL language is log-space reducible to CONN Thus, any NL language is poly-time reducible to CONN CONN is in P Thus any NL language is in P. 27
28 What Next? We need to show CONN can be decided by a deterministic TM in O(log 2 n) space. 28
29 The Trick Is Is there a a path vertex from z, u so to there v of length is a path d? from u to z of size d/2 and one from z to v of size d/2? d/2 d/2 u... z v d 29
30 Example of Savitch s algorithm boolean PATH(a,b,d) { if there is an edge from a to b then return TRUE else { if (d=1) return FALSE for every vertex v (not a,b) { if PATH(a,v, d/2 ) and PATH(v,b, d/2 ) then return TRUE } return FALSE } } (a,b,c)=is there a path from a to b, that takes no more than c steps. (1,4,3)(1,3,2)TRUE (1,4,3)(1,2,2)TRUE (1,4,3)(2,4,1)FALSE (1,4,3)(1,3,2)(2,3,1)TRUE (1,4,3)(3,4,1)TRUE (1,4,3)(1,3,2)(1,2,1)TRUE 3Log 2 (d) 30
31 O(log 2 n) Space DTM Claim: There is a deterministic TM which decides CONN in O(log 2 n) space. Proof: To solve CONN, we invoke PATH(s,t, V ) The space complexity: S(n)=S(n/2)+O(logn)=O(log 2 n) 31
32 Conclusion Theorem: NSPACE(logn) SPACE(log 2 n) How about the general case NSPACE(S(n)) SPACE(S 2 (n))? 32
33 Formally s i (n) can be computed with space s i (n) Claim: For any two space constructible functions s 1 (n),s 2 (n) logn, f(n) n: simulation overhead NSPACE(s 1 (n)) SPACE(s 2 (n)) NSPACE(s 1 (f(n))) SPACE(s 2 (f(n))) E.g NSPACE(n) SPACE(n 2 ) NSPACE(n 2 ) SPACE(n 4 ) 33
34 Savitch: Generalized Version Theorem (Savitch): S(n) log(n) NSPACE(S(n)) SPACE(S(n) 2 ) 34
35 Corollary Corollary: PSPACE = NPSPACE Proof: Clearly, PSPACE NPSPACE. By Savitch s theorem, NPSPACE PSPACE. 35
36 NON-CONN Clearly, NON-CONN is conl-complete. (Because CONN is NL-Complete. See conp ) If we ll show it is also in NL, then NL=coNL. (Again, see the conp ) 36
37 An Algorithm for NON-CONN We ll see a log space algorithm for counting reachability 1. Count how many vertices are reachable from s. 2. Take out t and count again. 3. Accept if the two numbers are the same. 37
38 Immerman s Theorem Theorem[Immerman/Szelepcsenyi]: NL=coNL Proof: (1) NON-CONN is NL-Complete (2) NON-CONN NL Hence, NL=coNL. 38
39 Corollary Corollary: s(n) log(n), NSPACE(s(n))=coNSPACE(s(n)) 39
40 Summary Connectivity is NL-complete Savitch s Theorem (USTCONN NL L 2 ) Immerman s theorem (NL=coNL)
41 USTCONN is in log-space Low diameter property of Expander Connectivity Amplification Powering Zig-Zag product Log-space algorithm for USTCONN
42 Expander Combinatorial: no small cuts, high connectivity Probabilistic: rapid convergence of random walk Algebraic: small second eigenvalue Theorem. [Cheeger, Buser, Tanner, Alon-Milman, Alon, Jerrum-Sinclair, ]: All properties are equivalent!
43 Expander Graph D-regular graphs : Every vertex has degree D A (N, D, λ) is a D-regular graph of N vertices where λ(g) λ G = (N, D, λ) is an expander iff the spectral gap 1- λ > 0 G = (N, D, λ) is an expander if there exists ε >0 such that for any set S, S < ½ N, at least (1+ ε) S vertices of G are connected to some vertex in S
44 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 v 1 v 2 v 3 v 4 v v v v v 1 v 2 v 3 v 4 v 1 1/3 0 2/3 0 v 2 0 1/3 1/3 1/3 v 3 2/3 1/3 0 0 v 4 0 1/3 0 2/3
45 (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.
46 Diameter of a (N,D, λ)-graph Lemma: Diameter of a (N,D, λ) is bounded by O(log N) Proof: 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 2 l between any two vertices.
47 ST-Connectivity for (N,D, λ)-graph Can be determined in space O(Log D*logN) Enumerate all paths from s of length O(log N) Memory: log D for remembering an edge in the path, and at most O(log N) edges for a path.
48 Rotation Map For D-regular undirected graph G, Rot G : [N]x[D] [N]x[D] v 1 v 2 v 3 v 4 v v v Rot G (v1,2)=(v3,1) Rot G (v2,1)=(v3,2) v
49 Sketch of the algorithm Transform input graph into a D-regular nonbipartite graph (How? ) Amplify the connectivity without increasing too much the degree
50 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.
51 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 In rotation map notation, this means that GRotv,a.av,b.bwherev,bRotv,a k01k kk1 ii Gi1i
52 Zig-zag Graph Product H: an (D, d, α)-graph a (ND, d 2, λ )-graph h 1 h 2 v 1 h 1 h 2 h 3 h 4 h 3 h 4 G: an (N,D, λ)-graph v v 2 v 2 h 1 h 2 h 3 h 4
53 Zig-Zag Graph Product G an (n,m, )-graph. H an (m,d, ) -graph. Theorem. G z H is an (nm,d 2, + )-graph.
54 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
55 USTCONN is in log-space Theorem 1. USTCONN L Theorem 2. SL = L
56 Questions Thanks you for listening!
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