1 Bref Introducton Ths memo reorts artal results regardng the task of testng whether a gven bounded-degree grah s an exander. The model s of testng gr

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1 On Testng Exanson n Bounded-Degree Grahs Oded Goldrech Det. of Comuter Scence Wezmann Insttute of Scence Rehovot, Israel oded@wsdom.wezmann.ac.l Dana Ron Det. of EE { Systems Tel Avv Unversty Ramat Avv, Israel danar@eng.tau.ac.l June 15, 000 Abstract We consder testng grah exanson n the bounded-degree grah model (as formulated n [1]). Seccally, we refer to algorthms for testng whether the grah has a second egenvalue bounded above by a gven threshold or s far from any grah wth such (or related) roerty. We resent a natural algorthm amed towards achevng the above task. The algorthm s gven a (normalzed) second egenvalue bound < 1, oracle access to a bounded-degree -vertex grah, and two addtonal arameters ; > 0. The algorthm runs n tme 0:5+ =oly(), and accets any grah havng (normalzed) second egenvalue at most. We beleve that the algorthm rejects any grah that s -far from havng second egenvalue at most =O(1), and rove the valdty of ths belef under an aealng combnatoral conjecture. Suorted by MIERVA Foundaton, Germany. 0

2 1 Bref Introducton Ths memo reorts artal results regardng the task of testng whether a gven bounded-degree grah s an exander. The model s of testng grah roertes as formulated n [1]: The (randomzed) algorthm s gven ntegers d and, a dstance arameter (as well as some roblemsecc arameters), and oracle access to a -vertex grah G wth degree bound d; that s, query (v; ) [] [d] s answered by the th neghbor of v n G (or by a secal symbol n case v has less than neghbors). For a redetermned roerty P, the algorthm s requred to accet (wth robablty at least /3) any grah havng roerty P, and reject (wth robablty at least /3) any grah that s -far from havng roerty P, where dstance between grahs s dened as the fracton of edges (over d) on whch the grahs der. Loosely seakng, the secc roerty consdered here s beng an exander. More recsely, for a gven bound < 1, we consder the roerty, denoted E, of havng a normalzed by d adjacency matrx wth second egenvalue at most. Actually, we further relax the roerty testng formulaton (as n []): Usng an addtonal arameter 0, we only requre that the algorthm must accet (wth robablty at least /3) any grah havng roerty E (.e., havng second egenvalue at most ); and the algorthm must reject (wth robablty at least /3) any grah that s -far from havng roerty E 0 (.e., from any grah that has second egenvalue at most 0 ). Settng 0 = we regan the more strct formulaton of testng whether a grah has second egenvalue at most. We menton that the ( ) lower bound on \testng exanson" (resented n [1]) contnues to hold for the relaxed formulaton above, rovded that 0 < 1. Ths s the case snce the lower bound s establshed by showng that any o( )-query algorthm fals to dstngush between a very good exander and an unconnected grah wth several huge connected comonents. 1 In vew of the above, we shall be content wth any sub-lnear tme algorthm for testng exanson. Below, we resent a arameterzed famly of algorthms. For any > 0, the algorthm has runnng-tme n 0:5+ =oly() and s suosed to satsfy the above requrement wth 0 = =7. Unfortunately, we can only rove that ths s ndeed the case rovded that a certan combnatoral conjecture (resented n Secton.) holds. Conventons and otatons We consder -vertex grahs of degree bound d, whch should be thought of as xed. We consder the stochastc matrx reresentng a canoncal random walk on ths grah, where canoncal s anythng reasonable (e.g., go to each neghbor wth robablty 1=d). The egenvalues below refer to ths matrx. By we denote the clamed second egenvalue (.e., we need to accet grahs havng second egenvalue at most ). By we denote the dstance arameter: we need to reject grahs that are -far from havng second egenvalue at most 0, where 0 > s related to. The algorthm resented below s arameterzed by a small constant > 0 that determnes both ts comlexty (.e., O( 0:5+ =oly())) and ts erformance (.e., 0 = =O(1) ). To be of nterest, the algorthm must use < 0:5. 1 In the latter case, the grah has (normalzed) second egenvalue equal 1. 1

3 3 The algorthm 1:5 ln We set L =. Ths guarantees that a grah wth second egenvalue at most mxes well n ln(1=) L stes (.e., the devaton n max-norm of the end robablty from the unform dstrbuton s at most?1:5 ). The followng algorthm evaluates the dstance of the end robablty (of an L-ste random walk startng at a xed vertex) from the unform robablty dstrbuton. It s based on the fact that the unform dstrbuton over a set has the smallest ossble collson robablty, among all dstrbutons over ths set. Reeat t def = (1=) tmes (1) Select unformly a start vertex, denoted s. () Perform m def = ( 0:5+ =) random walks of length L, startng from vertex s. (3) Count the number of arwse collsons? between the endonts of these m walks. 1+0:5?= m () If the count s greater than then reject If all reettons were comleted wthout rejecton then accet. Comment: Random walks were used before n the context of testng grah roertes (n the bounded-degree model): Seccally, O( e =oly()) such walks were used by the barttness tester of []. Random walks seem much more natural here. Analyss Fxng any start vertex s, we denote by s;v the robablty that a random walk of length L startng at s ends n v. The collson robablty of L-walks startng at s s gven by v s;v 1 By our choce of L, f the grah has egenvalue at most then (for any startng vertex s) the collson robablty of L-walks startng at s s very close to 1= (.e., s smaller than (1=) + (1= ))..1 Aroxmaton of the collson robabltes The rst ssue to address s the aroxmaton to Eq. (1) rovded by Stes (){(3) of the algorthm. Lemma 1 Wth robablty at least 1? (1=3t), the aroxmaton rovded by Stes (){(3) s wthn a factor of 1 1?= of Eq. (1). Thus, wth robablty at least =3, all aroxmatons rovded by the algorthms are wthn a factor of 1 1?= of the correct value. Proof: For every < j, dene a 0-1 random varable ;j so that ;j = 1 f the endont of the th ath s equal to the endont of the j th ath. Clearly, def = E[ ;j ] = P v s;v, for every < j. Usng Chebyshev's nequalty we bound the robablty that the count rovded by Stes (){(3) devates from ts (correct) exected value. Let P def = f(; j) : 1 < j mg and = 1?=. Pr ;j? jp j > jp j 3 Var[ 5 P ;j] (1) ( jp j ) ()

4 def Denote ;j = ;j?. The rest of the roof needs to deal wth the fact that the random varables assocated wth P are not arwse ndeendent. Seccally, for four dstnct ; j; 0 ; j 0, ndeed ;j and 0 ;j 0 are ndeendent, and so E[ ;j 0 ;j 0] = E[ ;j] E[ 0 ;j0] = 0; but for < j 6= k the random varables ;j and ;k are not ndeendent (snce they both deend on the same th walk). Stll Var 3 0 ;j 5 = E 6@ = E ;j 1 A h ;j + h E ;j + jp j + m3 6 v (;j);(;k)p & j6=k [m] 3 s;v j6=kf+1;:::;mg E h ;j ;k E [ ;j ;k ] snce ;j ;k = 1 f and only f all three random walks end at the same vertex. Usng ( P v 3 s;v) 1=3 ( P v s;v) 1=, and m < 3 jp j, we obtan 3 ;j 5 jp j + jp j 3= 3= < (jp j ) 3= (3) Var Combnng Eq. () and (3), we obtan Pr ;j? jp j > jp j 3 5 < (jp j ) 1= Usng 1= and jp j > m 1+ = ( ), the denomnator s at least ( ). Recallng that = 1?= and t = O(1=), the lemma follows. As an mmedate corollary we get Corollary If the grah has second egenvalue at most then the above algorthm accets t wth robablty at least =3. Another mmedate corollary s the followng Corollary 3 Suose that for at least a =O(1) fracton of the vertces s n G the collson robablty of L-walks startng at s s greater than. Then the algorthm rejects wth robablty?= 1+0:8 at least =3. Thus, f a grah asses the test (wth robablty greater than 1=3) then t must have less than (=O(1)) excetonal vertces; that s, vertces s for whch the collson robablty of L-walks startng at s s greater than 1+0:8?=. Comment: ote that by changng arameters n the algorthm (.e., t = ( =) and m = ( 0:5+ =)), we can make the fracton of excetonal vertces smaller than?. Ths may hel n closng the ga (below), and only ncreases the comlexty from 0:5+ =oly() to 0:5+3 =oly(). 3

5 . The ga We beleve that the followng conjecture (or somethng smlar) s true. Conjecture: Let G be an -vertex grah of degree-bound d. Suose that for all but at most =O(1) fracton of the vertces s n G the collson robablty of L-walks startng at s s at most 1+0:8?=. Then G s -close to a -vertex grah (of degree-bound d) n whch the collson robablty of L-walks startng at any vertex s at most 1+?=. The conjecture s very aealng: Suosedly, you add d edges connectng at random the excetonal vertces to the rest of the grah. Ignorng for a moment the ssue of reservng the degree bounds, ths seems to work { but we cannot rove t. Indeed, one can show that the revously excetonal vertces enjoy rad mxng, but t s not clear that the added edges wll not cause harm to the mxng roertes of non-excetonal vertces..3 Fnshng t o Once the ga s closed, we have the followng stuaton: If the algorthm rejects wth robablty smaller than =3 then the nut grah s -close to a grah n whch the collson robablty of L-?= 1+ walks startng at any vertex s at most. But the excess of the collson robablty beyond 1= s nothng but the square of the dstance, n norm, of the robablty vector ( s;v ) v[ ] from the unform robablty vector (.e., ( P v s;v)? (1=) = P v( s;v? (1=)) ). Thus, for every s the dstance, n norm, of the robablty vector ( s;v ) v[ ] from the unform robablty vector s at q?= most =?(0:5+), where = =. The lan now s to \reverse" the standard egenvalue to rad-mxng connecton. That s, nfer from the rad-mxng feature that the grah has a small second egenvalue. Such a lemma has aeared n [3]: Lemma (Lemma.6 n [3]): Consder a regular connected grah on vertces, let A be ts normalzed adjacency matrx and denote the absolute value of the second egenvalue of A. Let ` be an nteger and ` denote an uer bound on the maxmum, taken over all ossble start vertces s, of the derence n orm between the dstrbuton nduced by an `-ste random walk startng at s and the unform dstrbuton. Then ( `) 1=`. ote that by the above, we have L <?(0:5+). Ths does not gve anythng useful when alyng the lemma drectly. Instead, we aly the lemma after boundng ` for ` = O(L). (The followng may be an oversght, but that's how we argue t now.) Clam 5 Let ` be dene as n Lemma. Then k` ( `) k, for every nteger k. Proof: Let B = A` be the stochastc matrx reresentng an `-ste random walk, and let ~e 1 ; :::; ~e denote robablty vectors n whch all the mass s on one vertex. Let ~ denote the unform robablty vector. Then ` (res., k`) equals the maxmum of kb~e? ~k (res., kb k ~e? ~k) taken over all the ~e 's. Consderng the bass of ~e 's, let ~z be a zero-sum vector (such as ~e? ~). That s, ~z s wrtten n the bass of ~e 's as ~z = P z ~e, and P z = 0. We obtan

6 kb~zk = = B = Snce P jz j q P z = k~zk, we get Usng B~ = ~, we get for every and the clam follows. kb~zk z ~e!? z B(~e? ~) kz B(~e? ~)k jz j kb(~e? ~)k jz j! ` ` k~zk z B~ kb k ~e? ~k = kb(b k?1 ~e? ~)k ` kb k?1 ~e? ~k k < ` Combnng Lemma and Clam 5, we obtan the followng Corollary 6 Suose that for every s the dstance, n norm, of the robablty vector ( s;v ) v[ ] from the unform robablty vector s at most?(0:5+). Then, for every constant < =3, the second egenvalue of the grah s at most. So once the ga s lled, we are done (usng = = and =3). Proof: Let 0 be the second egenvalue of the grah. Then, for every k we have Substtutng for L = 1:5 ln, we get ln(1=) 0 ( kl ) 1=kL (1? k) ln kl L k 1=kL? k 1=kL (1? k) ln = ex kl = = (1? k) ln k ((1:5 ln )= ln(1=)) 3? ln 3k > ln for sucently large k (snce < =3). We get ln 0 > ln, and the corollary follows. 5

7 Comment: We have 0 for any < =6 (e.g., = =7 wll do). One may be able to ncrease the exonent (.e., ) somewhat, but a lnear deendency (of the exonent ) on seems unavodable (under the current aroach). References [1] O. Goldrech and D. Ron. Proerty Testng n Bounded Degree Grahs. In Proc. of the 9th ACM Sym. on Theory of Comutng, ages 06{15, [] O. Goldrech and D. Ron. A sublnear Bartte Tester for Bounded Degree Grahs. Combnatorca, Vol. 19 (), ages 1{39, [3] O. Goldrech and M. Sudan. Comutatonal Indstngushablty: A Samle Herarchy. JCSS, Vol. 59, ages 53{69, [] M. Parnas and D. Ron. Testng the dameter of grahs. In Proceedngs of Random99,