An Elementary Construction of Constant-Degree Expanders Noga Alon *, Oded Schwartz * and Asaf Shapira ** *Tel-Aviv University, Israel **Microsoft Research, USA TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AA A SODA 2007

2 Expanders l Sparse l Highly connected [n,d,  ]- expander n vertices, d regular For every |S| ≤ n/2  d|S | ≤ |E(S,V \ S)|

3 Applications of Expanders l Robust networks l Derandomization l Error Correcting Codes l PCP l Hardness of approximations l …

4 Eigenvalues of Graphs G: [n, d,  ]- expander Adjacency matrix A: A i,j = #(i,j) edges Eigenvalues  1 ¸ 2 ¸ … ¸ n 1 = d, v 1 =1 n = max i  1 {| i |}

5 Expansion vs. Eigenvalues G : [n, d,  ]- expander  iff < d Thm: [Alon-Milman84, Dodziuk84, Alon86] (Actually 2 )

6 Existence [Pinsker73]: 9  > 0, s. t. 8 d ¸ 3 and (even) n, 9 [n, d,  ]- expander.

7 Constructions Explicit: Construct G »E.g, for reductions »Time = Poly(n) »or even Space = log n »Which n ’s? –8n.–8n. Fully Explicit: Find the i ’th neighbor of v »E.g, for derandomization »Time = Poly(|v|) = Poly(log n) »or even Space = loglog n »Which n ’s? –|G|=Poly(n) usually suffices.

8 Some Previous Results l [Margulis73], [Gabber-Galil81] Algorithm for Constant degree expanders. l …[Lubotzky-Phillips-Sarnak88], [Margulis88] Ramanujan Graphs. l [Alon-Roichman94] Polylog degree. l [Reingold-Vadhan-Wigderson02] Iterative construction.

9 Our Contribution Thm: 9  8 n : Explicit [Poly(n), O(1),  ]- expander »Simple to construct »Simple to analyze Applying the above we obtain: 1. Fully Explicit [Poly(n),O(1),  ]- expander 2. Explicit [n, O(1),  ]- expander

10 G2G2 Replacement Product: G 1 ® G 2 G 1 : [n, d 1, δ 1 ]-expander G 2 : [d 1, d 2, δ 2 ]-expander G1G1 For all v of G 1 : 1. Split 2. Install G 2 3. Duplicate edges

11 G 1 ® G 2 : Properties G 1 : [n, d 1, δ 1 ]-expander G 2 : [d 1, d 2, δ 2 ]-expander G2G2 G1G1 G 1 ® G 2 : [nd 1, 2d 2, δ(δ 1,δ 2 )]-expander

12 G 1 ® G 2 : Some History G 1 : [n, d 1, δ 1 ]-expander G 2 : [d 1, d 2, δ 2 ]-expander G2G2 G1G1 G 1 ® G 2 : [nd 1, 2d 2, δ(δ 1,δ 2 )]-expander [Gromov83] : Spectral Analysis [Reingold-Vadhan-Wigderson02] : via Zig-Zag [Papadimitriou-Yannakakis91] : for Inapprox. [Dinur05] : for PCP

13 G 1 ® G 2 : Expansion G 1 : [n, d 1, δ 1 ]-expander G 2 : [d 1, d 2, δ 2 ]-expander G2G2 G1G1 Thm:G 1 ® G 2 is a [nd 1, 2d 2, δ 1 2 ¢ δ 2 /80)]-expander [here]: Simple combinatorial proof.

14 E 1 : [n, log 2 n, ¼ ]- expander [Special case of Alon-Roichman] E 2 : [log 2 n, (loglogn) 2, ¼ ]- expander [A&R again] E 3 : [2(loglog n) 2, 3,  ]- expander (Pinsker, exhaustive search) Return E = (E 1 ® E 2 ) ® E 3 New Construction (roughly) Constant Degree Expander Algorithm E: [Poly(n), 6,  ’ ]- expander Proof

15 G 1 ® G 2 : Combinatorial Proof G 1 : [n, d 1, δ 1 ]- expander G 1 ® G 2 : [nd 1, 2d 2, δ 1 2 ¢ δ 2 /80)]- expander G 2 : [d 1, d 2, δ 2 ] -expander S V \ S G 1 ® G 2

16 G2G2 G2G2 G2G2 G2G2 G2G2 G2G2 G2G2 G2G2 G2G2 G 1 ® G 2 : Combinatorial Proof G2G2 G1G1 G2G2 G 1 : [n, d 1, δ 1 ]- expander G 1 ® G 2 : [nd 1, 2d 2, δ 1 2 ¢ δ 2 /80)]- expander G 2 : [d 1, d 2, δ 2 ] -expander G2G2 X G2G2 ? G 1 ® G 2

17 G 1 ® G 2 : Combinatorial Proof G 1 : [n, d 1, δ 1 ]-expander G 1 ® G 2 : [nd 1, 2d 2, δ 1 2 ¢ δ 2 /80)]-expander G 2 : [d 1, d 2, δ 2 ]-expander G2G2 G2G2 G2G2 G2G2 G2G2 G2G2 G2G2 G2G2 G2G2 G2G2 G1G1 G2G2 G2G2 X G2G2 ? + G2G2 G2G2 G2G2 G2G2 - - X G 1 ® G 2

18 A&R: Proof (a special case of [Alon-Roichman94]). 8 r, q=2 t, LD(q,r): Vertices: (a 0,…,a r )a i 2 F 2 t = [q] Edges: (x,y) 2 [q] 2 The (x,y) neighbor of (a 0,…,a r ) is: (a 0,…,a r )+y ¢ (1,x,x 2,…,x r ).

19 LD(q,r): Properties LD(q,r) : n = q r+1 vertices d = q 2 -regular Thm: [Alon-Roichman94]: (LD(q,r)) · rq For r = q/2  ¸ ½(1- / d) ¸ ¼ log 2 (n) ¸ (½q log q) 2 > q 2 = d LD(q,r) : [n, O(log 2 n), ¼]- expander.

20 LD(q,r): Eigenvalues M: n £ n matrix of LD(2 t,r) L : 2 t ! {0,1}L(x 0,…x t-1 )=x 0 Eigenvectors:

21 LD(q,r): Eigenvalues 1. v a are orthogonal 2. v a are eigenvectors (what eigenvalues?) + v a are all eigenvectors

22 LD(q,r): Eigenvalues 1. v a are orthogonal Proof: =  b v a (b) v a’ (b) W.l.o.g, a 0  a’ 0. All values

23 LD(q,r): Eigenvalues 2. v a are eigenvectors Proof: v a (b + c) = v a (b) v a (c) (M v a )(b) =  c 2 F r+1 M bc ¢ v a (c) =  x,y 2 F v a (b+y(1,x,...,x r )) = (  x,y 2 F v a (y,yx,...,yx r )) ¢ v a (b) The (x,y) neighbor of b a

24 LD(q,r): Eigenvalues 2. v a are eigenvectors! What are a ? a =  x,y 2 F v a ((y,yx,...,yx r )) p a (x) =  i r =0 a i x i

25 LD(q,r): Eigenvalues 2. v a are eigenvectors. What are a ? =1, 8 y= |F| = q all values = 0 a = (0,…0) ) a = q 2 a  (0,…0), ) p a has at most r roots ) a · rq

26 Variants »For every n »Fully explicit

27 For Every n LD(q,r) : n = q r+1 vertices d = q 2 -regular E 1 : [ q 2,3,  ]-expander Search E 2 : [ q 6,q 2,1/4 ]-expander LD(q,5) E 3 : [ q 4(r+1),q 8,1/4 ]-expander LD(q 4,r)q 4 /100≤ r ≤ q 4 /2 E 4 = E 3 ® ( E 2 ® E 1 ) E 4 : [q 4r+12,12,  ’]- expander

28 Poly(n) E 4 : [q 4r+12,12,  ’]- expander q 4 /100≤ r ≤ q 4 /2 q = 2 t n = q ^  (q 4 )  q 4 =  (lg n / lg lg n) t’ = t + 1  q’ = 2q, r’ = 16r n’ ≤ (2q) 4r’+12 = Poly(n) r’ = r - 1  n’ = n/ q 4 q 4(1/2 q^4)+12 > (2q) 4(16 q^4 /100) +12  n,n lglgn / lgn ≤ n 0 ≤ n

29  (n)  n,n lglgn / lgn ≤ n 0 ≤ n a = n / n 0 < lgn / lglgn if a < 100 – done. E 5 : [12a, 3,  ]- expander search : (lgn/lglgn)  (lg n / lglg n) = Poly(n) l Duplicate edges l E = E 4 ® E 5 E:[12n, 6,  ’’]- expander

30 Fully Explicit Vertex naming: (x,y) The i ’th neighbor of (x,y) »If i ≤ d 2 then (x, y’) : y’ is neighbor i in G 2 » else (x’, y): x’ is neighbor y in G 1 G2G2 G1G1 G 1 : [n, d 1, δ 1 ]- expander G 2 : [d 1, d 2, δ 2 ]- expander G 1 ® G 2 : [nd 1, 2d 2, δ(δ 1,δ 2 )]- expander

31 Connection Scheme G 2 is d 2 - edge-colorable + G 1 ® G 2 is 2d 2 - edge-colorable. G 1 : [n, d 1, δ 1 ]- expander G 2 : [d 1, d 2, δ 2 ]- expander G2G2 G1G1 G 1 ® G 2 : [nd 1, 2d 2, δ(δ 1,δ 2 )]- expander Connection scheme? (rotation map) e.g : d 1 - edge-coloring for G 1.

32 Fully Explicit l G 1 ® G 2 »Fully Explicit »D-edge-colorable l Alon-Roichman: »Fully Explicit »D-edge-colorable l Pinsker »Fully Explicit »D-edge-colorable

33 A&R: Edge Colorability (a special case of [Alon-Roichman94]). 8 r, q=2 t, LD(q,r): Vertices: (a 0,…,a r )a i 2 F 2 t = [q] Edges: (x,y) 2 [q] 2 the (x,y) neighbor of (a 0,…,a r ): (a 0,…,a r )+y ¢ (1,x,x 2,…,x r ).

34 Conclusions and Open Problems l A (Fully) explicit constant degree expander »Simple to construct »Simple to analyze l Exploiting the simplicity? l Simple combinatorial proof »for [n, poly-log n, ¼ ]-expander? »for graph powering? »for (near) Ramanujan graphs?

35 Expanders Construction Thank You.

36 G2G2 G2G2 G2G2 G2G2 G2G2 G2G2 G2G2 G2G2 G2G2 G 1 ® G 2 : Combinatorial Proof G2G2 G1G1 G2G2 G 1 : [n, d 1, δ 1 ]- expander G 1 ® G 2 : [nd 1, 2d 2, δ 1 2 ¢ δ 2 /80)]- expander G 2 : [d 1, d 2, δ 2 ] -expander

37 G 1 ® G 2 : Combinatorial Proof G 1 : [n, d 1, δ 1 ]-expander G 1 ® G 2 : [nd 1, 2d 2, δ 1 2 ¢ δ 2 /80)]-expander G 2 : [d 1, d 2, δ 2 ]-expander (1-  1 /4)d 1 Each contributes:  1 d 1 /4 ¢  2 d 2 If  ( small ) ¸  1 |S|/10 #sets ¸  1 |S|/10d 1 Total :  1 d 1 /4 ¢  2 d 2 ¢  1 |S|/10d 1 =  1 2  2 /80 ¢ 2d 2 ¢ |S| Else  small) <  1 |S|/10  ( Large ) ¸ (1-  1 /10) |S|

38 G 1 ® G 2 : Combinatorial Proof G 1 : [n, d 1, δ 1 ]-expander G 1 ® G 2 : [nd 1, 2d 2, δ 1 2 ¢ δ 2 /80)]-expander G 2 : [d 1, d 2, δ 2 ]-expander Else  small) <  1 |S|/10  ( Large ) ¸ (1-  1 /10) |S|

39 G 1 ® G 2 : Combinatorial Proof G 1 : [n, d 1, δ 1 ]-expander G 1 ® G 2 : [nd 1, 2d 2, δ 1 2 ¢ δ 2 /80)]-expander G 2 : [d 1, d 2, δ 2 ]-expander G2G2 G2G2 G2G2 G2G2 G2G2 G2G2 G2G2 G2G2 G2G2 G2G2 G1G1 G2G2

40 G 1 ® G 2 : Combinatorial Proof G2G2 G2G2 G2G2 G2G2 G2G2 G2G2 G2G2 G2G2 G2G2 G2G2 G2G2 G 1 : [n, d 1, δ 1 ]-expander G 1 ® G 2 : [nd 1, 2d 2, δ 1 2 ¢ δ 2 /80)]-expander G 2 : [d 1, d 2, δ 2 ]-expander G2G2 G2G2 G2G2 G2G2 G 1 : Total ¸ ½  1 d 1 #large-sets G 1 ® G 2 : ½  1 d 1 d 2 #large-sets - corrections

41 G 1 ® G 2 : Combinatorial Proof G2G2 G2G2 G2G2 G2G2 G2G2 G2G2 G2G2 G2G2 G2G2 G2G2 G2G2 G 1 : [n, d 1, δ 1 ]-expander G 1 ® G 2 : [nd 1, 2d 2, δ 1 2 ¢ δ 2 /80)]-expander G 2 : [d 1, d 2, δ 2 ]-expander G2G2 G2G2 G2G2 G2G2 At most: ¼  1 d 1 d 2 #large-sets

42 G 1 ® G 2 : Combinatorial Proof G2G2 G2G2 G2G2 G2G2 G2G2 G2G2 G2G2 G2G2 G2G2 G2G2 G2G2 G 1 : [n, d 1, δ 1 ]-expander G 1 ® G 2 : [nd 1, 2d 2, δ 1 2 ¢ δ 2 /80)]-expander G 2 : [d 1, d 2, δ 2 ]-expander At most: d 2  ( small ) < d 2  1 |S| /10 ·  1 d 2 /10 ¢ #large-sets d 1 10/(10-  1 ) · 1/9  1 d 1 d 2 #large-sets G2G2 G2G2 G2G2 G2G2

43 G 1 ® G 2 : Combinatorial Proof G2G2 G2G2 G2G2 G2G2 G2G2 G2G2 G2G2 G2G2 G2G2 G2G2 G2G2 G 1 : [n, d 1, δ 1 ]-expander G 1 ® G 2 : [nd 1, 2d 2, δ 1 2 ¢ δ 2 /80)]-expander G 2 : [d 1, d 2, δ 2 ]-expander G2G2 G2G2 G2G2 G2G2 ¼  1 d 1 d 2 #large-sets ½  1 d 1 d 2 #large-sets - corrections 1/9  1 d 1 d 2 #large-sets

44 Compared to Random Walk G2G2 G2G2 G2G2 G2G2 G2G2 G2G2 G2G2 G2G2 G2G2 G2G2 G1G1 G2G2 G 1 : [n, d 1, δ 1 ]-expander G 1 ® G 2 : [nd 1, 2d 2, δ 1 2 ¢ δ 2 /80)]-expander G 2 : [d 1, d 2, δ 2 ]-expander

45 [n,polylog n,  ]-expander (a special case of [Alon-Roichman94]). 8 r, q=2 t, LD(q,r): Vertices: (a 0,…,a r )a i 2 F 2 t = [q] Edges: (x,y) 2 [q] 2 the (x,y) neighbor of (a 0,…,a r ): (a 0,…,a r )+y ¢ (1,x,x 2,…,x r ).

46 LD(q,r): Properties LD(q,r): n = q r+1 vertices d = q 2 -regular q 2 - edge-colorable Thm: [Alon-Roichman94]: (LD(q,r)) · rq Recall ½(1- / d) ·  If r · q/2 then LD(q,r) is a [q r+1, q 2, ¼]- expander.