Topics in Probabilistic Combinatorics and Algorithms Winter, 2016

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1 Topics in Probabilistic Combinatorics and Algorithms Winter, Epander Graphs Review: verte boundary δ() = {v, v u }. edge boundary () = {{u, v} E, u, v / }. Remark. Of special interset are (, β) epander, which relate to the Cheeger ratios as defined below. Cheegers ratio for a subset V is defined as follows: () G: d-regular, () G: not regular, From now on, we will use definition (). Cheeger s constant: h () = δ h() = d h () = vol(δ) h() = h G = min h(). h G = min h (). Cheeger s inequality: h G λ h G.

2 Remark. Many equivalent definition for λ : < g, Lg > λ = sup g φ 0 < g, g > = sup f f()d=0 = sup f f()d=0 = sup f f()d=0 = inf sup f = inf f = inf f c < f, (D A)f > < f, Df > (φ 0 = D ) () () f (3) ()d y (f () + f (y)) (4) (f() (5) c) d ( ( f = f()d ) f() f) d d (6) (f() (7) f(y)) d d y Now, we prove Cheeger s inequality. Proof. (i) h G λ : ince λ R(f) for all f satisfies f()d = 0, where the Rayleigh quotient R(f) is defined to be R(f) = f. ()d et Then R(f) = This implies λ h G. ( f() = { + ) () + () if v if v / = h G. It remains to show (ii) λ h G. From equation (3), there eists f satisfying λ = R(f) = sup f: f()d=0 f. ()d

3 We can relabel the veritices so that f(v ) f(v ) f(v n ), and define i = {v j, j i}, j { = {v k : k j}, α = min h(i ), h( j) }. i,j uffices to show λ α since by the definition of h G, we have α h G and therefore λ h G. Let r denote the largest integer such that vol( r ) < vol( r+ ), and { f(v) f(vr ) if f(v) f(v g + (v) = r ) 0 otherwise. { f(vr ) f(v) if f(v) < f(v g (v) = r ) 0 otherwise. We consider λ = = f ()d (f() f(v r)) d ((g + () g + (y)) + (g () g (y)) ) y ( g + () + g () ) d (g +() g + (y)) ( g + () ) (use the fact: d (g +() g + (y)) (g +() + g + (y)) g +()d (g +() + g + (y)) a + b c + d min{a c, b d } w.l.o.g.) ( y g +() g +(y) ) ( ) by the Cauchy-chwarz inequality, g +()d (i = g +(v i ) g+(v i+ ) i ) ( ) by counting g +()d ( ( i g + (v i ) g+(v i+ ) ) ) α i ( ) by the definition of α g +()d (i = α g +(v i ) ( i i ) ) ) summation by parts ( g +()d = α. 3

4 Remark. G is a k-regular Ramanujan graph, Definition. λ k k σ = ma i 0 λ. Remark: G is bipartite if and only if σ =. Theorem (verte epansion). G = (V, E), V, vol ( δ) ε, (i) vol (δ) λ λ + ε. (ii) If λ ε vol (δ) ( λ + ε). Proof. Proof of (i): uppose λ R(f) = We set Then f where f satisfies ()d f()d = 0. f() = volt volt if T if T λ R(f) = T volt = f ()d ( ) T volt + volt ( volt + volt = T volt volt λ ( volt = λ volt ) λ ( ɛ) ) 4

5 Proof of (ii): Let vol (δ) ( δ) + λ ( ε)( + volδ) + λ ( ε) λ ( ε) λ ( ε) λ λ + ε. ρ = λ, f = s + ρ δ g = f c where c = f()d. Note that d g()d = 0. First, consider ince c = ( ) () f()d δ () vol( δ) f ()d By Cauchy-chwarz () ε f ()d (8) < g, (D A)g > λ R(g) = < g, Dg > < g, Ag > λ < g, Dg > we have < g, Ag > ( λ ) < g, Dg >= ρ < g, Dg >. (9) And < f, Df > = < + ρ δ, D( + ρ δ ) > = <, D > +ρ < δ, D δ > = + ρ vol(δ) < g, Dg > = < f c, D(f ) > = < f, Df > c < f, D > +c + ρ vol(δ) c (0) 5

6 We consider < f, Af > = < g + c, A(g + c) > And < g, Ag > +c ρ < g, Dg > +c by equation (9) ρ( + ρ vol(δ) c ) + c by equation (0) = ρ( + ρ vol(δ)) + ( ρ)c ρ( + ρ vol(δ)) + ( ρ)ε( + ρ vol(δ)) by equation (8) From above, we have (ρ + ε)( + ρ vol(δ)) < f, Af > e(, ) + ρe(, δ) = ( ρ)e(δ) + ρ() ρ ρ (ρ + ε)( + ρ vol(δ)) (ρ ε) (ρ + ε)ρ vol(δ) vol(δ) ρ ε (ρ + ε)ρ Theorem. G : (V, E), X, Y V, dist(x, Y ) = e(x, Y ) volxvoly volxvoly volxvoly σ σ volxvoly () Proof. ince e(x, Y ) =, A Y =, D (I L)D Y = D, (I L)D Y D X = i a i φ i a 0 = D X, D X = volx D Y = i b i φ i b 0 = D Y, D Y = voly 6

7 Then e(x, Y ) a 0 b 0 = i 0( λ i )a i b i σ a i b i i 0 i 0 ( = σ volx volx volxvolxvoly voly = σ. ) ( voly voly ) Remark. If take Y = X, the above theorem becomes ( ) volx dist(x, X) X volx σvolx volx Review: L = I D AD with eigenvalues 0 = λ 0 λ λ n and eigenvalues φ 0, φ φ n, respectively. n L = λ i φ i φ i i=0 Theorem 3 (hitting / sampling theorem). G : (V, E), undiredted, V, µ =, σ = λ. The probability that a random walk with a random stationary verte (v 0, v, v,, v t ), P r(v 0 = v) = dv = π(v) always remain in is P r[ v 0, v,, v t stay in ] (µ + σ( µ)) t. Proof. Let P = D A: transition probability matri, I s = I s, M = M. Claim: D AD λ I φ 0φ 0 I s + ( λ )I. Proof: ( ) D AD = I D AD I = I (I L)I = I φ 0φ 0 + λ i )φ i 0( i φ i I I (λ φ 0φ 0 + i ( λ )φ i φ i )I λ I φ 0φ 0 I + ( λ )I 7

8 From the claim, we have D AD λ I φ 0 +( λ ) Now, we can prove our theorem. = λ + ( λ ) (φ 0 = D ) = λ µ + σ = µ + σ( µ) P r[ v 0, v,, v t stay in ] = D I (P I ) t = D I = D ( D AI ) t ( D AD ) t D D D AD = (µ + σ( µ))t (µ + σ( µ)) t t Theorem 4 (Chernoff bound for random walks). G : (V, E), f : V [0, ], µ f = f()d. A random walk from a random stationary verte is denoted by v d 0, v,, v t, P r[ t f(vi ) µ f σ + ε ] e ε t 4 Proof. Let X i = f(v i ), X = i X i, P : transition probability matri and define T to be a diagonal matri satisfying { e rf() if = y T (, y) = 0 otherwise. Consider E(e rx ) = D T (P T )t = D ( T D T T D AD T ) t T D T D AD T t 8

9 ince D T v = d ve rf(v) v d v( + rf(v) + r ) = + rµ f + r e rµ f +r (use the fact: + e r + + if [0, ]) And T D AD T = T φ 0φ 0 + λ i )φ i 0( i φ i T T ( λ φ 0φ 0 + ( λ ) i φ i φ i ) T We have Then T D AD T λ T φ0 +( λ ) T λ ( + rµ r + r ) + σ( + r + r ) < + r(µ f + σ) + r e r(µ f +σ)+r E(e rx ) e rµ f +r e (t )(r(µ f +σ)+r ) e t(r(µ f +σ)+r ) P r[ t f(vi ) µ f σ + ε ] E(e rx ) e rt(µ f +σ+ε) e rtε+tr e ε t 4 (choose r = ε ) Applications:. Error Reduction: L BPP: L P r[ A() accepts ] 3 / L P r[ A() accepts ] 3 9

10 uppose each test takes m bit, Number of Repetitions Number of Random Bits Independent Repetitions O(k) O(km) Pairwise Independent Repetitions O( k ) O(m + k) Epander Random Walks O(k) O(m + k). ampling: f : {0, } m [0, ]. Approimate µ f within additive error ε with error probability δ. Number of amples Number of Random Bits Truly Independent O((/ε ) log(/δ)) O(m (/ε ) log(/δ)) Pairwise Independent O((/ε ) (/δ)) O(m + log(/ε) + log(/δ)) Epander Random Walk O((/ε ) log(/δ)) O(m + (/ε ) log(/δ) + log t) Eplict Construction Goal: Construct an infinite family of graphs {G i } of d regular graphs with eigenvalue λ γ. A couple of alternatives for defining eplicit constructions of epanders on a graph G = (V, E) with n vertices are: Mildy Eplict: G can be represented in time poly(log N). Fully Eplict: G can be represented in time poly(log log N). Eample 5. [0, ], =., i {0,,, b } is called a base b epansion. Known: Almost all [0, ] is normal.(in Lebesque measure) ( is normal if b in base b epansion of, each digit (0,,, b ) appear asymptotical equally often.) Open: No one can name a normal value.(e: π, e, ) Eample 6 (The problem of Ramsey numbers). The party problem, also known as the maimum clique problem, asks to find the minimum number of guests that must be invited so that at least k of them will know each other or at least k of them will not know each other. The Ramsey number R(k, k) gives the solution to the party problem: R(k, k) = min n such that every -color of K n contain monochromatic Known: 4 k R(k, k) k. K k 0

11 Construction.(discrete torus epanders). Let G = (V, E) be the graph with verte set V = Z M Z M, and edges from each node (, y) to the nodes (, y), ( +, y), (, y + ), (, + y), (y, ), where all arithmetic is modulo M. Construction.(p-cycle with inverse chords). This is the graph G = (V, E) with verte set V = Z p and edges that connect each node with the nodes: +,, and (where all arithmetic is mod p and we define 0 to be 0). Construction 3.(Ramanujan graphs). G = (V, E) is a graph with verte set V = F q { }, the finite field of prime order q s.t. q mod 4 plus one etra node representing infinity. The edges in this graph connect each node z with all z of the form: z = (a 0 + i )z + (a + ia 3 ) (a + ia 3 )z + (a 0 ia ) for a 0, a, a, a 3 Z such that a 0 + a + a + a 3 = p, a 0 is odd and positive, and a, a, a 3 are even, for some fied prime p q such that p mod 4, q is a square modulo p, and i F q such that i = mod q. Recall: s t connectivity problem: Given a graph G, does it eist a path from s to t? If we use random walks to solve it, the number of steps is approimately λ log n. Our goal is using the eplicit construction to construct a sequence of graphs {G i } such that G < G < < G k and G k has a better eigenvalues than G. Definition. G = (V, E) is a (n, d, γ)-graph if V (G) = n, maimum degree in G is d, λ γ. Now, we can define some products of graphs. Definition. quarzing: G is a (n, d, σ )-graph, where σ = λ. Tensor Product: G is a (n, d, γ)-graph and G is a (n, d, γ )-graph. Then G G = (V, E), V = {(v, v ) : v V (G), v V (G )}, E = {(v, v ) (u, u ) iff u v and u v }). Cartision Product: G G = (V, E), where V = {(v, v ) : v V (G), v V (G )} and E = {(u, u ) (v, v ) iff ((u = v, u v ) or (u = v, u v))}. Lemma 7. G is a (n, d, γ)-graph and G is a (n, d, γ )-graph, then G G is a (nn, dd, γ + γ γγ )-graph. Proof. Let F (u, u ) = f(u)g(u ), f and g are combinatorial eigenfunctions associated with λ in G and λ in G, respectively.

12 Fi u, u, F (v, v ) = f(v)g(v ) v u,v u v u,v u = f(v) g(v ) v:v u v :v u = ( λ)f(u)d u ( λ )g(u )d u = ( λ)( λ )f(u)g(u )d u d u = ( (λ + λ λλ ))f(u)g(u )d u d u F is the combinatorial eigenfunction associated with (λ + λ λλ ) in G G and the associated Laplacian eigenvalue of G G is λ + λ λλ. Definition (Zig-Zag Product Z ). Let G is a (n, d, γ )-graph and H is a (d, d, γ )-graph. Then G Z H = (V, E), V = {(u, i) : u V (G), i [d ] = V (H)}. For a, b [d ], the (a, b)th neighbor of a verte (u, i) is the verte (v, j) computed as follows: () Let i be the ath neighbor of i in H. () Let v be the i th neighbor of u in G. (3) j satisfies v j = u (4) Let j be the b th neighbor of j in H. Theorem 8. Let G is a (n, d, γ )-graph and H is a (d, d, γ )-graph. Then G Z H is a (nd, d, γ γ )-graph. Epander Construction Construction (Mildly Eplicit Epanders) Let H is a (d 4, d, 7 )-graph and define: 8 G = H G t+ = G t Z H Proposition. For all t, G t is a (d 4t, d, )-graph. Proof. By induction on t. Base case: G = H is a (d 4, d, γ ), and γ /. Induction tep: G t Z H is d4t, d, -graph. Consider G t+ : the number of vertices in G t+ is d 4t d 4 = d 4(t+) and λ(g t+ ) λ(g t ) + λ(h) (/) + (/8) = /.

13 Construction (Fully Eplicit Epanders) Let H is a (d 4, d, 7 )-graph and define: 8 G = H G t+ = (G t G t ) Z H Proposition. The number of vertices n t c t, λ t c where c is some absolute positive constant. 3