Lecture 10: May 6, 2013

Transcription

1 TTIC/CMSC Mathematcal Toolkt Sprng 013 Madhur Tulsan Lecture 10: May 6, 013 Scrbe: Wenje Luo In today s lecture, we manly talked about random walk on graphs and ntroduce the concept of graph expander, as well as an applcaton of random walk to show ts effectveness. 1 Cheerger s Inequalty Recap Gven a d-regular graph wth ts adjacent matrx A, we defne the Laplacan of ths graph wth adjacent matrx N = I 1 d A. Assume A has egenvalues: µ 1, µ,..., µ n, N has egenvalues: λ 1, λ,..., λ n, we know: Then the Cheeger s Inequalty gves that: where the Φ G s the expanson of the graph. d = µ 1 µ... µ n d 0 = λ 1 λ... λ n λ Φ G λ Random Walk on Graphs.1 Basc Idea Frst of all, we defne the random walk as follows: Have a startng vertex. At every step, go to a random neghbor of the current vertex. we use x R n to represent the current status, e.g., f we are at th vertex, than x = 1 wth other tems beng 0. Then we defne the random walk matrx to be M, s.t. x t+1 = Mx t. For a d-regular graph, t s easy to check that M = 1 da. We can see that ths defnton also apples to x beng a dstrbuton. For general graph, we have the random walk matrx to be: M = D 1 A, where D = deg(), D j = 0 j. Notce that ths matrx has smlar egenvalues to D 1/ AD 1/. Now consderng more about d-regular graph, we can see that unform dstrbuton s a statonary status: M 1 n1 = 1 n1 snce 1 s a egenvector of M. So the queston now s: If we start wth any random dstrbuton, how quckly do we converge to the unform dstrbuton(statonary dstrbuton)? Explctly, t progresses as follows: x Mx... M x 1

2 We splt x to be x = x µ + x where x µ share the drecton wth statonary dstrbuton and x s orthogonal to t. Then we have: x µ = 1 < x, 1 > 1 1 = 1 n n n Mx = M(x µ + x ) = x µ + Mx Thus, the quantty we care about s M x x µ. When t s gettng close to 0, we are convergng to statonary dstrbuton. Consder one step, we have: Mx x µ = Mx µ d x = µ d x x µ, where µ = max{µ, µ n }. Then after l steps: M l (µ) l x x µ x xµ ( µ ) l d d If we set a up-bound: M l x x µ ε, then ( µ ) ε l = Ω ( log ε ) d log µ d So wthn l steps, we can get the dstrbuton very close to statonary dstrbuton.. Lazy Random Walk The prevous random walk process wll go to a new status n each step. Now let s look at another lazy one, where t wll stay n the same dstrbuton wth 0.5 probablty and walk to a new dstrbuton wth 0.5 probablty. Then the random walk matrx s thus when we have egenvalues for M: we have egenvalues for M to be: M = 1 I + 1 M 1 = µ 1 d µ 1 d... µ 1 d 1 1 = 1 + µ 1 d 1 + µ d µ n d 0 One convenence of ths s that we don t need to consder about both µ and µ n, snce ths tme: µ = max{µ, µ n } = 1 d + µ

3 .3 Expanders Now we ntroduce the concept of Expander Graph : an expander graph s a sparse graph that has strong connectvty propertes, quantfed usng vertex, edge or spectral expanson. Expander constructons have spawned research n pure and appled mathematcs, wth several applcatons to complexty theory, desgn of robust computer networks, and the theory of error-correctng codes[wkpeda]. For example, here we use expanson of a graph Φ G 0.1, then we have: Φ G 0.1 λ 1 10 λ 1 00 thus 1 µ d 1 00 µ d whch s equvalent to say µ c d, (c < 1). We hereby gve another defnton of Cheeger s Inequalty: µ = max{µ, µ n } c d, (c < 1) For more nformaton about expander graph and ther applcatons, please refer to ths survey[hoory06]. 3 Applcaton of Random Walk Here we ll use random walk to desgn an algorthm wth less random bts but equvalent performance. 3.1 Problem Setup Frst suppose we now have a randomzed algorthm whch can output whether x s n L for any gven any x as: wth nput: x, r x L, P r [Algo(x, r) = YES] 1/ x / L, P r [Algo(x, r) = YES] = 0 *class of L for whch above algorthm exst s called RP(Randomzed Polynomal Tme). Then our objectve s to apply expander graph to mprove the above nequalty of 1/ to somethng close to Basc Idea A basc dea for dong that s Run the algorthm wth l ndependent r 1, r,.. r n 3

4 Output YES f any run says YES, else output NO. Ths algorthm can gve us the followng concluson: x L, x / L, P [Algo (x, r) = YES] 1 1 r 1,...,r n l P [Algo (x, r) = YES] = 0 r 1,...,r n Then f r = R, Algo uses l R random bts. Actually we can select those r more wsely so that we can use less random bts to get the same concluson. We can acheve the same concluson wth just O(l + R) random bts. 3.3 Apply Random Walk on Graph The algorthm works as follows: Frst assume we have an access to an expander graph G wth R vertces and d = O(1), for example d = 10, thus µ = max{µ, µ n } 9d 10. Then we sample r as follows: r 1 : random vertex of G r : random neghbor of r 1. r n : random neghbor of r n 1 Thus the number of random bts we use s: R + log d (l 1) Now we need to prove that we acheve the same level of accuracy here(also equvalent to say random walk on expander graph s as good as unform samplng). Lemma 3.1 For adjacent matrx A, A l j = # walks of length l form j Proof: Frst of all, for l =, t s obvous that A j = k A ka kj Then by nducton, we can fnd that: A l j = A k1 A k1 k...a kl 1 j k 1,k,...,k l 1 wth rght hand sde to be exactly the number of walks of length l Now, let S V be a set s.t. S n Lemma 3. P [Random walk of length l never vsts S] = Ω(l) 4

5 Proof: Gven x L, defne that: S = {r : Algo (x, r) = YES} From lemma 3.1, we can see the total # walks of length l: (total number) l =,j A l j = 1A l 1 = d l 1 1 = d l n Then defne a matrx A s.t. { 0 f S or j S A j = otherwse. A j then # walks that avod S s 1 A l 1. If we can prove that all egenvalues of A are less than d, we are done. For any x, consder x Ax = z Az, where { 0 f S z = otherwse. x then x Ax = j A jx x j = j A jx x j = z Az. Smlarly as what we dd n secton.1, let z = z µ + z, then z µ = z n 1. Now: z Az = (z µ + z ) A(z µ + z ) = (z µ + z ) (dz µ + Az ) = z µ d+ < z, Az > z µ d + µ z = z µ d + µ ( z z µ ) whle at the same tme, we have: z µ = z m u = (n S ) ( z ) n (1 S n )( z ) ( z ) n recall: S n 1 1 z n thus, together we get: z Az = z µ d + µ ( z z µ ) ( 1 d + 1 µ) z d + µ x 5

6 whch s to say: x Ax d+µ x. Recall that µ = max{µ, µ n } c d (c < 1). Then we have: P [Random walk of length l never vsts S] = 1A l 1 1A l 1 ( d+µ ) l n d l = Ω(l) n Snce we know P [Algo answers NO] = P [Random walk never vsts S] thus from lemma 3., we can conclude that x L, P [Algo (x, r) = YES] 1 Ω(l) r 1,...,r n.e. we can acheve the same accuracy as before usng only R + log d (l 1) random bts. References [HOORY06] S. HOORY, N. LINIAL and A. WIGDERSON, Expander Graphs and Ther Applcatons, Bulletn of the Amercan Mathematcal Socety, Oct. 006, Vol. 43, pp