Counting Stars and Other Small Subgraphs in Sublinear Time

Transcription

1 Coutig Star ad Other Small Subgraph i Subliear Time Mira Goe Departmet of Mathematic Bar-Ila Uiverity Ramat Ga, ISRAEL goem@math.biu.ac.il Yuval Shavitt School of Electrical Egieerig Tel-Aviv Uiverity Ramat Aviv, ISRAEL havitt@eg.tau.ac.il April 4, 0 Daa Ro School of Electrical Egieerig Tel-Aviv Uiverity Ramat Aviv, ISRAEL daar@eg.tau.ac.il Abtract Detectig ad coutig the umber of copie of certai ubgraph alo kow a etwork motif or graphlet, i motivated by applicatio i a variety of area ragig from biology to the tudy of the World-Wide-Web. Several polyomial-time algorithm have bee uggeted for coutig or detectig the umber of occurrece of certai etwork motif. However, a eed for more efficiet algorithm arie whe the iput graph i very large, a i ideed the cae i may applicatio of motif coutig. I thi paper we deig ubliear-time algorithm for approximatig the umber of copie of certai cotat-ize ubgraph i a graph G. That i, our algorithm do ot read the whole graph, but rather query part of the graph. Specifically, we coider algorithm that may query the degree of ay vertex of their choice ad may ak for ay eighbor of ay vertex of their choice. The mai focu of thi work i o the baic problem of coutig the umber of legth- path ad more geerally o coutig the umber of tar of a certai ize. Specifically, we deig a algorithm that, give a approximatio parameter 0 < ǫ < ad query acce to a graph G, output a etimate ˆν uch that with high cotat probability, ǫν G ˆν + ǫν G, where ν G deote the umber of tar of ize + i the graph. The expected query complexity ad ruig time of the algorithm are O ν G + + mi {, ν G } polylog, /ǫ. We alo prove lower boud howig that thi algorithm i tight up to polylogarithmic factor i ad the depedece o ǫ. Our work exted the work of Feige SIAM Joural o Computig, 006 ad Goldreich ad Ro Radom Structure ad Algorithm, 008 o approximatig the umber of edge or average degree i a graph. Combied with thee reult, our reult ca be ued to obtai a etimate o the variace of the degree i the graph ad correpodig higher momet. I additio, we give ome egative reult o approximatig the umber of triagle ad o approximatig the umber of legth-3-path i ubliear time. Reearch upported by the Irael Sciece Foudatio grat No. 46/08

2 Itroductio Thi work i cocered with approximatig the umber of copie of certai cotat-ize ubgraph i a graph G. Detectig ad coutig ubgraph alo kow a etwork motif [MSOI + 0] or graphlet [PCJ04], i motivated by applicatio i a variety of area ragig from biology to the tudy of the World- Wide-Web ee e.g., [MSOI + 0, KIMA04, SIKS06, PCJ04, Wer06, SSR06, GK07, DSG + 08, HBPS07, ADH + 08, HA08, GS09], a well a by the baic quet to udertad imple tructural propertie of graph. Our work differ from previou work o coutig ubgraph with the exceptio of coutig the umber of edge [Fei06, GR08] i that we deig ubliear algorithm. That i, our algorithm do ot read the whole graph, but rather query part of the graph where we hall pecify the type of querie we allow whe we tate our precie reult. The eed for uch algorithm arie whe the iput graph i very large a i ideed the cae i may of the applicatio of motif coutig. The mai focu of thi work i o the problem of coutig the umber of legth- path ad more geerally o coutig the umber of tar of a certai ize. We emphaize that we cout o-iduced ubgraph. We hall ue the term -tar for a ubgraph over + vertice i which oe igle vertex the tar ceter i adjacet to all other vertice ad there are o edge betwee the other vertice. Oberve that a legth- path i a -tar. We alo give ome egative reult o approximatig the umber of triagle ad o approximatig the umber of legth-3-path. A we how i detail below, we obtai almot matchig upper ad lower boud o the query complexity ad ruig time of approximatig the umber of -tar. Thee boud are a fuctio of the umber,, of graph vertice ad the actual umber of -tar i the graph, ad have a o-trivial form. Our reult exted the work [Fei06] ad [GR08] o ubliear-time approximatio of the average degree i a graph, or equivaletly, approximatig the umber of edge where a edge i the implet o-empty ubgraph. Note that if we have a etimate for the umber of legth- path ad for the average degree, the we ca obtai a etimate for the variace of the degree i the graph, ad the umber of larger tar correpod to higher momet. Thu, the tudy of the frequecie of thee particular ubgraph i a graph hed light o baic tructural propertie of graph. Our Reult. We aume graph are repreeted by the icidece lit of the vertice or, more preciely, icidece array, where each lit i accompaied by it legth. Thu, the algorithm ca query the degree, dv, of ay vertex v of it choice ad for ay vertex v ad idex i dv it ca query who i the i th eighbor of v. Let ν G deote the umber of -tar i a graph G. Our mai poitive reult i a algorithm that, give a approximatio parameter 0 < ǫ < ad query acce to a graph G, output a etimate ˆν uch that with high cotat probability over the coi flip of the algorithm, ǫν G ˆν + ǫν G. The expected query complexity ad ruig time of the algorithm are: { } O + mi ν G, polylog,/ǫ. + ν G The depedece o i expoetial, ad i ot tated explicitly a we aume i a cotat. The complexity of our algorithm a tated i Equatio i bet udertood by viewig Table, i which we ee that there are three regio whe coiderig ν G a a fuctio of, ad i each the complexity i govered by a differet term. Oberve that i the firt rage ν G + the complexity of the algorithm which i at it maximum whe ν G i very mall decreae a ν G icreae; i the ecod rage + <

3 ν G ν G + + < ν G ν G > Query ad Time Complexity O polylog,/ǫ ν G + O polylog,/ǫ O polylog,/ǫ ν G Table : The query complexity ad ruig time of our algorithm for approximatig the umber of -tar. ν G the complexity doe ot deped o ν G; ad i the lat rage ν G > it agai decreae a ν G icreae where i the extreme cae, whe ν G = Ω + the complexity i jut polylog,/ǫ. For example, for = 3, if ν 3 G = 4/3 the the query complexity ad ruig time of the algorihm i O /3, which i le tha the umber of tar. If ν 3 G = the the query complexity ad ruig time of the algorihm i O /3, ad if ν 3 G = 4 the the query complexity ad ruig time of the algorihm i O, which i defiitely le tha the umber of tar. We ote that a light adaptatio of the algorithm give a additive error, that i, the requiremet i that the etimate ˆν atify with high probability ν G α ˆν ν G + α for α that i larger tha ǫν G for 0 < ǫ <. The expected query complexity ad ruig time of the modified algorithm are: { } O α + + mi, α polylog. The expreio i Equatio might eem uatural ad hece merely a artifact of our algorithm. However, we prove that it i tight up to polylogarithmic factor i ad the depedece o ǫ. Namely, we how that: Ay multiplicative approximatio algorithm for the umber of -tar mut perform Ω querie. ν G + Ay cotat-factor approximatio algorithm for the umber of -tar mut perform Ω querie whe the umber of -tar i O. Ay cotat-factor approximatio algorithm for the umber of -tar mut perform Ω querie whe the umber of -tar i Ω. ν G We metio that aother type of querie, which are atural i the cotext of dee graph, are vertex-pair querie. That i, the algorithm may query about the exitece of a edge betwee ay pair of vertice. We ote that our lower boud imply that allowig uch querie caot reduce the complexity for coutig the umber of tar except poibly by polylogarithmic factor i. We alo coider other mall graph that exted legth- path: triagle, ad legth-3 path. We how that if a algorithm ue a umber of querie that i ubliear i the umber of edge, the for triagle it i hard to ditiguih betwee the cae that a graph cotai Θ triagle ad the cae that it cotai o triagle, ad for legth-3 path it i hard to ditiguih betwee the cae that there are Θ legth-3 path ad the cae that there are o uch path. Thee lower boud hold whe the umber of edge i Θ. We metio that we are curretly tudyig thee problem i a exteded query model that alo allow vertex-pair querie. For legth-3 path, eve if we allow uch querie the there i a lower boud that i liear i the umber of edge whe the umber of edge i Θ, ad for triagle there i a lower boud that i liear i whe the umber of edge i Θ.

4 Techique. Our tartig poit i imilar to the oe of [GR08]. Coider a partitio of the graph vertice ito Olog /ǫ bucket where i each bucket all vertice have the ame degree with repect to the etire graph up to a multiplicative factor of ± Oǫ. For a precie defiitio of thi part ee Sectio 3.. If we could get a good etimate of the ize of each bucket by amplig, the we would have a good etimate of the umber of -tar ice the vertice i each bucket are the ceter of approximately the ame umber of tar. The difficulty i that ome bucket may be very mall ad we might ot eve hit them whe amplig vertice. The approach take i [GR08] to get a multiplicative etimate of ± ǫ i to etimate the umber of edge betwee large bucket ad mall bucket, ad icorporate thi etimate i the fial approximatio. Here we firt oberve that we eed a more refied procedure. I particular, we eed a eparate etimate for the umber of edge betwee each large bucket ad each mall bucket. Note that if we have a etimate ê of the umber of edge icidet to vertice i a certai bucket, ad all vertice i that bucket have degree roughly d, the the umber of -tar whoe ceter belog to thi bucket i approximately ê d. To ee why thi i true, coider a edge u,v that i icidet to a vertex u that ha degree roughly d. The the umber of tar that iclude thi edge ad are cetered at u i roughly d. If we um thi expreio over all ê edge that are icidet to vertice i the bucket of u, the each tar that i cetered at a vertex i the bucket i couted time, ad hece we divide the expreio ê d by. A a firt attempt for obtaiig uch a etimate o the umber of edge icidet to vertice i a bucket, coider uiformly amplig edge icidet to vertice that belog to large bucket. We ca the etimate the umber of edge betwee the large bucket ad each mall bucket by queryig the degree of the other ed poit of each ampled edge. It i poible to how that for a ufficietly large ample of edge we ca ideed obtai a good etimate for the umber of -tar uig thi procedure. However, the complexity of the reultig procedure, which i domiated by the umber of edge that eed to be ampled, i far from optimal. The reao for thi ha to do with the variace betwee the umber of edge that differet vertice i the ame large bucket have to the variou mall bucket. To overcome thi ad get a almot optimal algorithm, we further refie the amplig proce. Specifically, we firt defie the otio of igificat mall bucket. Such bucket have a oegligible cotributio to the total umber of -tar where each vertex accout for the umber of tar that it i a ceter of. Now, for each large bucket B i ad igificat mall bucket B j we further coider partitioig the vertice i B i accordig to the umber of eighbor they have i B j. The difficulty i that i order to determie exactly to which ub-bucket a vertex i B i belog to, we would eed to query all it eighbor, which may be too cotly. Moreover, eve if a etimate o thi umber uffice, if a vertex i B i ha relatively few eighbor i B j the we would eed a relatively large ample of it eighbor i order to obtai uch a etimate. Fortuately, we ecouter a tradeoff betwee the umber of vertice i B i that eed to be ampled i order to get ufficietly may vertice that belog to a particular ub-bucket ad the umber of eighbor that hould be ampled o a to detect approximately to which ub-bucket a vertex belog to. We exemplify thi by a extreme cae: coider the ub-bucket of vertice for which at leat half of their eighbor belog to B j. Thi ub-bucket may be relatively mall ad till cotribute igificatly to the total umber of edge betwee B i ad B j but if we ample a vertex from thi ub-bucket the we ca eaily detect thi by takig oly cotat ample of it eighbor. For more detail ee Subectio 3.4. We ote that i the cae of the average degree umber of edge, if we igore the mall bucket for a appropriate defiitio of mall the we ca already get roughly a factor- approximatio i O time [Fei06, GR08]. However, thi i ot the cae for -tar eve whe =. To verify thi coider the cae that the graph G i a tar. There are two bucket: oe cotaiig oly the tar ceter, ad aother cotaiig all other vertice. If we igore the very mall bucket that cotai the tar ceter the we get a etimate of 0 while the graph cotai Θ legth- path -tar. 3

5 Related Work. A oted previouly, our work exted the work [Fei06, GR08] o approximatig the average degree of a graph i ubliear time. I particular, our work i mot cloely related to [GR08] where it i how how to get a etimate of the average degree of a graph G that i withi ± ǫ of the correct value dg. The expected ruig time ad query complexity of the algorithm i [GR08] are O/ dg / polylog,/ǫ. There are quite a few work that deal with fidig ubgraph of a certai kid ad of coutig their umber i polyomial time. Oe of the mot elegat techique devied i color-codig, itroduced i [AYZ95], ad further applied i [AYZ97, AR0, AG07, ADH + 08, AG09]. I particular, i [AR0] the author ue color-codig ad a techique from [KL83] to deig a radomized algorithm for approximately coutig the umber of ubgraph i a give graph G which are iomorphic to a bouded treewidth graph H. The ruig time of the algorithm i k Ok b+o, where ad k are the umber of vertice i G ad H, repectively, ad b i the treewidth of H. I [AG07] the author ue color-codig ad balaced familie of perfect hah fuctio to obtai a determiitic algorithm for approximately cout imple path or cycle of ize k i time Ok log log k O. I [ADH + 08] thee reult are improved i term of the depedece o k. We ote that amplig i alo applied i [KIMA04, Wer06], where the author are itereted i uiformly amplig iduced ubgraph of a give ize k. Other related work i thi category iclude [DLR95, GK07, BBCG08, Kou08, Wil09, GS09, BHKK09, AFS09, KW09, VW09]. I [FG04] the author coclude that mot likely there i o fk c - algorithm for exactly coutig cycle or path of legth k i a graph of ize for ay computable fuctio f : N N ad cotat c. Aother related lie of work deal with approximatig other graph meaure uch a the weight of a miimum paig tree i ubliear time ad iclude [CRT05, CS09, CEF + 05, PR07, NO08]. Orgaizatio. For the ake of the expoitio we firt decribe the algorithm ad the aalyi, a well a the lower boud, for the cae =, that i, legth- path. Thi i doe i Sectio 3 ad 4, repectively. I Sectio 5 we explai how to adapt the algorithm for legth- path i order to get a algorithm for -tar, ad i Sectio 6 we explai how to adapt the lower boud. Fially, i Sectio 7 we hortly dicu triagle ad legth-3 path. Prelimiarie Let G = V, E be a udirected graph with V = vertice ad E = m edge where G i imple o that it cotai o multiple edge. We deote the et of eighbor of a vertex v by Γv ad it degree by dv. For two ot ecearily dijoit et of vertice V,V V we let EV,V def = {v,v E : v V, v V }. Sice we hall ue the multiplicative Cheroff boud very exteively, we quote it ext. Let χ,...,χ m be m idepedet 0/ valued radom variable where Pr[χ i = ] = p for every i. The, for every η 0,], the followig boud hold: [ ] m Pr m χ i > + ηp < exp η pm/3 i= ad [ ] m Pr m χ i < ηp i= < exp η pm/. 4

6 We hall ay that a evet hold with high cotat probability if it hold with probability at leat δ for a mall cotat δ. Let µ be a meaure defied over graph ad let G be a ukow graph over vertice. A algorithm for etimatig µg i give a approximatio parameter ǫ, the umber of vertice,, ad query acce to the graph G. Here we coider two type of querie. The firt are degree querie. Namely, for ay vertex v, the algorithm may ak for the value of dv. The ecod are eighbor querie. Namely, for ay vertex v ad for ay i dv, the algorithm may ak for the i th eighbor of v. 3 We do ot make ay aumptio o the order of the eighbor of a vertex. Baed o the querie it perform we ak that the algorithm output a etimate ˆµ of µg uch that with high cotat probability over the radom coi flip of the algorithm, ˆµ = ± ǫ µg, where for γ 0, we ue the otatio a = ± γb to mea that γb a + γb. 3 A Algorithm for Approximatig the Number of Legth- Path I thi ectio we decribe ad aalyze a algorithm for etimatig the umber of legth- path -tar i a graph G, where we deote thi umber by lg. I all that follow we coider udirected imple graph. We tart by givig the high-level idea behid the algorithm. 3. A High-Level Decriptio of the Algorithm Let β = ǫ/c where c > i a cotat that will be et ubequetly, ad let t = t = Olog /ǫ. For i = 0,...,t, let log +β o that B i def = {v : dv + β i, + β i] }. 3 We refer to the B i a degree bucket. Note that ice degree are iteger, the iterval of degree i each bucket i actually + β i, + β i ], ad ome bucket are empty. For implicity we do ot ue floor ule it ha a ifluece o our aalyi, ad whe we write a b for a that i ot ecearily a iteger e.g., +β i the we iterpret it a a b. We alo have that a b = 0 for a < b ad i particular whe a 0 < b. Note that if i i the umber of ode with degree i i the graph the lg = i=0 i i. Suppoe that for each bucket B i we could obtai a etimate, ˆb i, uch that β B i ˆb i + β B i. If we let ˆl = t + β i ˆbi i=0 the β lg ˆl + β 4 lg, 5 where we have ued the fact that +β i +β i + β3 for + β i. If we et β ǫ/8, the we get a etimate that i withi ± ǫ of the correct value lg. The difficulty i that i order to obtai uch a etimate ˆb i of B i i ubliear time, that i, by amplig, the ize of the ample eed to grow with / B i. Our algorithm ideed take a ample of vertice, but it ue the ample oly to etimate 3 Oberve that a degree query ca be emulated by log eighbor querie, but for the ake of the expoitio we allow degree querie. 4 5

7 the ize of the large bucket, for a appropriate threhold of largee. Uig the etimated ize of the large bucket it ca obtai a etimate o the umber of legth- path whoe mid-poit belog to the large bucket. A oted i the itroductio, it i poible that oly a mall or eve zero fractio of the legth- path have a mid-poit that belog to a large bucket. Thi implie that we mut fid a way to etimate the umber of legth- path whoe mid-poit i i mall bucket for thoe mall bucket that have a oegligible cotributio to the total umber of legth- path. def To thi ed we do the followig. Let E i,j = EB i,b j. For each large bucket B i ad mall bucket B j uch that the umber of legth- path whoe mid-poit i i B j i oegligible, we obtai a etimate ê i,j to the umber, E i,j, of edge betwee the two bucket. The etimate i uch that if E i,j i above ome threhold, the ê i,j = ± β E i,j, ad otherwie, ê i,j i mall. Our etimate for the umber of legth- path whoe midpoit i i a mall bucket i ê i,j + β j, 6 j / L where L deote the et of idice of the large bucket. For a illutratio, ee Figure. Thi etimate doe B i, i L u B j, j / L w v degv + β j E i,j ê i,j E i,j Figure : A illutratio for the legth- path whoe midpoit i i a mall bucket. ot take ito accout legth- path i which o vertice o the path belog to L. However, we hall et our threhold of largee o that the umber of uch path i egligible. I additio, thi etimate take ito accout oly half of the legth- path i which two vertice o the path do ot belog to L, oe of them i the midpoit. However, we hall et our threhold of largee o that the umber of uch path i alo egligible. Oe way to etimate ê i,j for i L ad j / L i to uiformly elect radom eighbor of vertice ampled i B i ad check what bucket they belog to. Thi will ideed give u a good etimate with high probability for a ufficietly large ample. However, the variace i the umber of eighbor i B j that differet vertice i B i have implie that the ample ize ued by thi cheme i igificatly larger tha eceary. I order to obtai a etimate with a maller ample, we do the followig. For each i L ad j / L we coider partitioig the vertice i B i that have eighbor i B j ito ub-bucket. Namely, for r = 0,...,i, def B = {v B i : + β r < Γv B j + β r }. 7 Figure illutrate the defiitio of B. By the defiitio of B, 6

8 B i v B j Γv B j + β r B Figure : A illutratio for the defiitio of B. i B + β r = ± β E i,j. 8 r=0 Now, if we ca obtai good etimate of the ize of the ubet B, the we get a good etimate for E i,j. The difficulty i that i order to determie to which ub-bucket B a vertex v belog we eed to etimate the umber of eighbor that it ha i B j. Thi i ulike the cae we eed to determie for a vertex v to which bucket B i it belog, where we oly eed to perform a igle degree query. I particular, if v B i,j,0, that i, v ha a igle eighbor i B j, we mut query all the eighbor of v i order to determie that it belog to B i,j,0. What work i our favor i the followig tradeoff. Whe r i large the B may be relatively mall eve if EB,B j i oegligible o that we eed to take a relatively large ample of vertice i order to hit B. However, i order to determie whether a vertex i B i belog to B for large r, it uffice to take a mall ample of it eighbor. O the other had, whe r i relatively mall the B mut be relatively big if EB,B j i oegligible. Therefore, it uffice to take a relatively mall ample o a to hit B ad the we ca afford performig may eighbor querie from the elected vertice. We ext preet our algorithm i detail ad the aalyze it. 3. The Algorithm I what follow we aume that we have a rough etimate l uch that lg l lg. We later remove thi aumptio. Recall that for ay two bucket B i ad B j we ue the horthad E i,j for EB i,b j. 7

9 Algorithm Etimatig the umber of legth- path for G = V, E Iput: ǫ ad l.. Let β def = ǫ 3, t def = log +β, ad. Uiformly ad idepedetly elect Θ log t θ elected vertice that i, we allow repetitio. def θ = ǫ/3 l/3 3t 4/3. ǫ vertice from V, ad let S deote the multiet of 3. For i = 0,...,t determie S i = S B i by performig a degree query o every vertex i S. { S 4. Let L = i : i S θ }. { +β i } If max θ > 4 l the termiate ad retur For each i L ru Algorithm to get etimate {ê i,j } j / L for { E i,j } j / L. 6. Output ˆl = S i S + β i + j / L ê i,j + β j. Algorithm Etimatig { E i,j } for a give i L ad all j / L Iput: L, i L, ǫ ad l. def ǫ3/ l / =. For each 0 p i let θ p, where c c t 5/ +β p/ i a cotat that will be et i the aalyi, ad where t = log +β for β = ǫ/3. Let p 0 be the mallet value of p atifyig 4 θ p +.. For p = i dow to p 0 iitialize Ŝp i,j,p =. 3. For p = i dow to p 0 do: a Let p = Θ θ p t β log t, ad let g p = Θ +β i p logt β. b Uiformly, idepedetly at radom elect p vertice from S p+ where S i+ = V ad let S p be the multiet of vertice elected. c Determie S p i = S p B i by performig a degree query o every vertex i S p. If S p i < p 4+β θ p, the go to Step 4. Ele, if S p i > p 4 l the termiate ad retur +βi 0. d For each v S p i elect uiformly, idepedetly at radom g p eighbor of v, ad for each j / L let γ p j v be the umber of thee eighbor that belog to B j. If g p dv the coider all eighbor of v. 8

10 l l 3/ 3/ < l l > Query ad Time Complexity O/ l /3 polylog,/ǫ O / polylog,/ǫ O 3/ / l / polylog,/ǫ Table : The query ad time complexity of Algorithm. e For each j / L ad for each v S p i \ p >p Ŝp i,j,p, if + β p dv < γp j v g p v + βp dv the add v to Ŝp i,j,p. 4. For each j / L let ê i,j = i p=p 0 p Ŝp i,j,p + βp. 5. Retur {ê i,j } j / L. Theorem If lg l lg the with probability at leat /3, the output, ˆl, of Algorithm atifie { ˆl = } ± ǫ lg. The query complexity ad ruig time of the algorithm are O + mi l /, 3/ polylog, /ǫ. /3 l / Table give the domiat term i the complexity of the algorithm i three differet regio of the value of lg a a fuctio of. We firt prove the ecod part of Theorem, cocerig the complexity of the algorithm ad the tur to provig the firt part, cocerig the quality of the output of the algorithm. We later how how to remove the aumptio that the algorithm ha a etimate l for lg. 3.3 Proof of the ecod part of Theorem The ruig time of Algorithm i liear i it query complexity, ad hece it uffice to boud the latter. Note that recoutig i doe here. Namely, eve though S p S p+, the querie doe o S p are recouted at tep p. The query complexity of Algorithm i the um of Θ log t = O θ ǫ log l the /3 ǫ 4 log/ǫ ize of the ample elected i Step of the algorithm ad the umber of querie performed i the executio of Algorithm. I order to boud the latter we firt oberve that if Algorithm did ot termiate i Step 4, the l/3 i L : + β i t /3 = O. 9 Similarly, if Algorithm did ot termiate i ay of it executio i Step 3c, the, ice β = Θǫ, i L ad p 0 p i : S p p i = O l + β i. 0 I additio, it trivially alway hold that S p i mallet value of p atifyig 4 θ p+ where θ p def ǫ /3 p. Recall that p ru from i dow to p 0 where p 0 i the ǫ3/ l / = log +β 9 c t 5/ +β p/. That i, p 0 = ǫ 3 l c t5

11 for a certai cotat c. Thi implie that if if l, the p ǫ 3 0 = 0, ad otherwie it may be larger. Therefore, the total umber of querie performed i the executio of Algorithm i upper bouded by: { } i p + mi p, p 4 l +β i g p p=p 0 i i + i p=p 0 mi c t5 { p, p 4 l +β i } g p. For the firt ummad we apply Equatio 9, the defiitio of θ i, the fact that β = Θǫ, ad that i t, ad get: i i t 9/ log t + β i/ t O ǫ7/ l / / t 3/ log t l/3 t /3 ǫ = O /3 Turig to the ecod ummad, { i p, p 4 l mi p=p 0 = = O i p=p 0 O mi mi +β i ǫ7/ l / = O l t7 log t /3 ǫ 4 } g p { + β p/ t3/ log t l / ǫ 7/, } { + β i l /, l / + β i. l / + β i p/ t3/ log t ǫ 7/ t3/ log t logt ǫ / } i p 0 + β p 0/ + βi p logt β k=0 + β k/.3 I order to boud the expreio i Equatio 3 we firt ote that if + β i l / the +βi / l /, / while if + β i l / l /, the / a well. Sice + β p0/ = ad i p 0 / +β i k=0 + β k/ = O/β, if p 0 = 0 the the right-had-ide of Equatio 3 i upper bouded by O / t5/ log t logt ǫ 3/. 4 If p 0 > 0 the the boud i Equatio 4 hould be multiplied by + β p0/. By defiitio of p 0 we have that + β p0/ = O t 5/ ǫ 3/ l, ad o we get the tighter boud: / O / t5/ log t logt ǫ 3/ + β p0/ 3/ = O l t0 log t logt / ǫ

12 The total umber of querie performed i the executio of Algorithm i hece upper bouded by { } O + mi l /, 3/ polylog,/ǫ. 6 /3 l / 3.4 Proof of the firt part of Theorem I what follow we claim that certai evet occur with high cotat probability, ad i ome cae we claim that they hold with larger probability e.g., polyt. I all cae the tatemet hold for ufficietly large cotat i the Θ otatio for the ample ize ued by the algorithm. The ext lemma follow by applyig the multiplicative Cheroff boud ad the uio boud ad recallig that the ize of the ample S i Θ log t θ ǫ, where θ i a defied i Step of Algorithm. Lemma With high cotat probability, for every i uch that B i θ it hold that S i S = ± ǫ Bi 8, ad for every i uch that B i < θ it hold that S i S < θ. Proof: The proof follow from the multiplicative Cheroff boud. Sice Exp[S i ] = B i S it hold that if B i θ the [ Si Pr S > + ǫ ] [ Bi = Pr S i > + ǫ ] Bi 8 8 S [ I the ame maer we prove that Pr Si S < ǫ ] Bi 8 Exp[S i ] < θ S, o it hold that < e ǫ B i S 9 < e Θ ǫ θ log t θ ǫ = polyt. 7 < polyt. I additio, if B i < θ the [ ] Si Pr S > θ Pr[ S i > Exp[S i ]] < e Exp[Si]/3 < e θ S 3 = e Θ ǫ θ log t θ ǫ = polyt, ad the lemma follow. A a direct corollary of the Lemma ad the defiitio of L i Algorithm we get: Corollary With high cotat probability, for every i L we have that S i S = ± ǫ Bi 8, ad for every i / L we have that B i < 4θ. The firt part of Corollary implie that with high cotat probability the etimate S i +β i S i cloe to the actual umber of legth- path whoe mid-poit belog to a bucket B i uch that i L. It alo implie that Algorithm doe ot termiate i Step 4 with high cotat probability. To verify thi, firt oberve that ice l lg, for every i t we have that B +β i i lg l. By the defiitio of L, for every i L we have that S i S θ. If the termiatio coditio hold, that i, there +β i > 4 l, the S i +β i exit a idex i L for which θ Corollary, with high cotat probability, for every i L we have that S i S = ± ǫ 8 +β i that B i > l lg, ad we reach a cotradictio. S > 4 l for that idex i. But by Bi, which implie The remaider of the aalyi deal with the quality of the etimate for the umber of legth- path i G whoe mid-poit i ot i L.

13 We itroduce the followig otatio. For j / L ad σ {,,3}, let l σ j G, L deote the umber of legth- path i G whoe mid-poit belog to B j ad uch that the umber of vertice o the path that belog to B k for k / L icludig j i σ. For σ {,,3} let l σ G,L = j / L lσ j G, L ad for every j / L let l j G,L = 3 σ= lσ j G, L. We firt oberve that with high cotat probability both l 3 G,L ad l G,L are relatively mall. Lemma 3 With high cotat probability, l 3 G,L ǫ 4 lg ad l G,L ǫ 4 lg. Proof: Firt oberve that by the ecod part of Corollary ad the defiitio of θ we have that with high cotat probability, B j < 8t /3 ǫ/3 l/3. 8 j / L By our aumptio that l lg, l 3 G,L j / L B j 3 ǫ /3 l/3 /8t /3 3 < ǫ 8 l ǫ lg. 9 4 I order to boud l G,L we oberve that ice the total umber of legth- path i lg, for every bucket B j we have that +β j + lg/ Bj, ad o + β j l/ G. 0 B j / Therefore, l G,L j / L B j + β j B k ǫ/3 l/3 4t /3 k/ L l / G B j / j / L ad the proof i completed. ǫ/3 l/3 4t /3 l / G t ǫ/3 l/6 t /3 < ǫ lg, 4 Lemma 3 implie that i order to obtai a good etimate o the umber of legth- path whoe midpoit belog to mall bucket it uffice to get a good etimate o the umber of uch path that have at leat oe ed-poit i a large bucket. 4 We ext defie the otio of igificat bucket for bucket B j uch that j / L. Roughly peakig, o-igificat mall bucket are bucket that we ca igore, or, more preciely, we ca udercout the umber of edge betwee vertice i them ad vertice i large bucket. 4 The aertio follow from the firt part of Lemma 3, which boud l 3 G, L. The reao we alo eed a boud o l G, L will be made clear ubequetly.

14 Defiitio Sigificat mall bucket For every j / L we ay that j i igificat if + β j B j ǫ c 3 t l, where c 3 i a cotat that will be et i the aalyi. We deote the et of idice of igificat bucket B j where j / L by SIG. Note that by the defiitio of SIG, Let ad recall that θ r def ǫ3/ l / = j / L,j / SIG l j G,L < ǫ c 3 l ǫ c 3 lg. E j def = t E j,k, 3 k=0 c t 5/ +β r/. We have the followig lemma cocerig igificat bucket. Lemma 4 If j SIG, the for every r uch that B > 0 for ome i we have that E j c /c / 3 t θ r + β r. ǫ The implicatio of Lemma 4 i roughly the followig. Coider ay j SIG ad a o-empty ub-bucket B. Recall that by the defiitio of B the umber of edge betwee B ad B j i approximately B +β r. Suppoe that B i mall, ad i particular, that it i maller tha θ r. The the umber of edge betwee B ad B j a a fractio of all the edge icidet to B j, that i, E j, i Oǫ/t, which i egligible. Thi mea that we may uderetimate the ize of uch mall ub-bucket without icurrig a large error. Proof: Sice j i igificat, + β j > ǫ l c 3 t B j. 4 Sice the graph cotai o multiple edge, B j + β r for each r uch that B i ot empty. Therefore, by the defiitio of E j ad B j, ad Equatio 4, E j B j + β j ǫ l B j + β c 3 t c / 3 t / ǫ/ l/ + β r/ c /c / 3 t ǫ 5 6 θ r + β r, 7 ǫ3/ l / where the lat iequality follow from the defiitio of θ r = ad the proof i completed. c t 5/ +β r/ Armed with Lemma 3 ad 4 we ow tur to aalyzig Algorithm. We tart with a high-level dicuio ad the tur to the precie detail. 3

15 The high level tructure of the aalyi of Algorithm. Recall that the algorithm work iteratively a follow. It firt take uiformly, idepedetly, at radom a ample S i from V, ad i further iteratio 0 p < i the ample S p i elected uiformly, idepedetly, at radom from S p+. Sice the ame vertex may be elected more tha oce, the S p may actually be multi-et. For each p the algorithm trie to etimate B i,j,p by decidig for each vertex v S p B i whether it belog to B i,j,p. Thi i doe by amplig from the eighbor of the vertex ad checkig what fractio of it eighbor belog to B j. If the fractio i withi ome iterval, the the vertex i aumed to belog to B i,j,p ad i put i a correpodig ubet Ŝp i,j,p. The difficulty i that thi etimate of the fractio of eighbor i B j may deviate omewhat from it expected value. A a reult, vertice that belog to B i,j,p may ot be deemed o becaue the umber of eighbor they have i B j i cloe to the lower boud of + β p or the upper boud + β p, ad i the ample they fall outide of the required iterval. Similarly, vertice that do ot belog to B i,j,p but have a umber of eighbor i B j that i cloe to +β p or +β p, that i, that belog to B i,j,p or B i,j,p+ may fall iide the required iterval ad are the added to Ŝp i,j,p. If the ize of the ample S p wa the ame for all p the the above would t really be a difficulty: we could take a igle ample S = S i ad work iteratively from p = i dow to p = 0. For each p we would coider oly thoe vertice v that were ot yet added to Ŝp i,j,p for p > p ad decide whether to add v to Ŝp i,j,p. By the above dicuio, for every r ad every v B the vertex v would be put either i or i Ŝr or i Ŝr. The algorithm would the output, a a etimate for E i,j, the um over Ŝ r+ + all 0 p i, of S Ŝp i,j,p + βr. If S B i cloe to it expected ize for each r the the deviatio of the fial etimate from E i,j ca be eaily bouded. However, a p decreae from i to 0 we eed to ue a maller ample S p. Recall that a maller ample uffice ice θ p icreae whe p decreae, ad it i eceary to ue a maller ample becaue the cot of etimatig the umber of eighbor i B j icreae a p decreae. Thu, i each iteratio p, the ew, maller ample, S p, i elected from the ample S p+ of the previou iteratio. What we would like to eure i that: The ize of each ubet S p fractio of S p+ wa added to Ŝp+ def = S p B i cloe to it expectatio; If ome i,j,p+ for r = p + or r = p, the i the ew ample Sp, the ize of S p S p+ \ Ŝp+ i,j,p+ i cloe to it expectatio. Here, whe we ay cloe to it expectatio we mea up to a multiplicative factor of ± Oǫ. Thi hould be the cae ule the expected value i below ome threhold which i determied by θ r. If the expected value i below the threhold the it uffice that we do ot get a igificat overetimate. To udertad the idea for why thi uffice, ee the dicuio followig Lemma 4. Further detail follow. Recall that p deote the ize of the ample S p, where p = Θ θ p t β log t. The ext lemma etablihed that by our choice of p, if a fixed ubet of S p+ i ufficietly large, the the umber of it vertice that are elected i S p i cloe to the expected value, ad if it i mall the few of it vertice will appear i S p. Lemma 5 follow directly by applyig a multiplicative Cheroff boud ad will be applied to variou ubet of the ample S p. Lemma 5 For ay fixed choice of Sp+ S p+, if S p+ p+ 3t, 4 + β i+ S p+ p+ Sp S p+ p+ 4 + θ p 8 the, with probability at leat β S p+ i + p+,

16 ad if S p+ < θ p p+ 8 Let S p i the with probability at leat S p S p+ p < + 3t 4, β i + θp 8. def = S p B i ad let S p def = S p B. Note that Si i+ = B i ad S i+ = B. Sice θ p i mootoically decreaig with p o that p i mootoically icreaig with r, ad becaue + β i+ i+ + β, Lemma 5 implie the ext corollary. Corollary 6 With high cotat probability, for every i L ad j / L, ad for every r uch that B 4 θ r, we have that for every r p i, + β i+ i p+ B O the other had, if B < 4 θ r the for every p. Sp p S p p < + β θr 4 β i p+ + B. i + Lemma 5 alo implie that with high cotat probability, Algorithm doe ot termiate i Step 3c. Recall that the algorithm termiate i Step 3c if Sp i p 4+β θ p ad Sp i +β i p > 4 l. By Lemma 5, with probability at leat, for every i ad p, if B 3t i < 6 θ p, the Sp i p +β 6 θ p, ad if B i 6 θ p, the Sp i + β B p i. Aumig thi i i fact the cae, if B i < 6 θ p the Sp i < p 4+β θ p, o that the algorithm will ot termiate. O the other had, if B i 6 θ p, the Sp i p + β i + β i + β B i + βlg < 4 l, 8 o that the algorithm will ot termiate i thi cae a well. The ext Lemma deal with the etimate we get for the umber of eighbor that a vertex i B i ha i B j, ad it too follow from the multiplicative Cheroff boud. I the lemma ad what follow we hall ue the otatio Γ j v def = Γv B j ad d j v def = Γ j v. Lemma 7 Let i L, j / L ad for each 0 p i, let g p = Θ +β i p logt β. For ay r p ad for ay fixed choice of a vertex v S p, if we take a ample of ize gp of eighbor of v ad let γ p j v be the umber of eighbor i the ample that belog to Γ j v, the with probability at leat 6 t, + β djv dv γp j v g p + β djv dv. I additio, for each r p ad v S p, with probability at leat 6 t, γ p j v g p < + βp dv. 5

17 The ext lemma i cetral to our aalyi. Ideally we would have liked each vertex i the ample to be added to it correct ubet. That i, if v S r = Sr B the ideally it hould be added to Ŝr. However, ice the deciio cocerig whether to add a vertex to a particular ubet deped o amplig it eighbor ad etimatig the umber of eighbor that it ha i B j, we caot eure that it will be added to preciely the right ubet. However, we ca eure with high probability that it will ot be added to a ubet Ŝ p i,j,p for p that differ igificatly from r. Lemma 8 With high cotat probability, for every i L, j / L, 0 r i ad v B uch that v i elected i the iitial ample S i, the vertex v may belog to either Ŝr+ + or to Ŝr or to Ŝr, but ot to ay other Ŝr. I other word, Ŝr B + B B. Proof: By the defiitio of B, if v B the +β r < d j v +β r. By Lemma 7, for each p r + with probability at leat 6 t, + βr + β dv < γp j v g p + β That i, for each p r + ad i particular for r p r +, + βr+ dv. 9 + β r dv < γp j v g p + βr+. 30 dv O the other had, for p r +, with probability at leat 6 t, γ p j v g p < + βp dv. 3 By takig a uio boud over all vertice v, ad for each v B over all 0 p i, thi implie that:. for r + p i, o vertex i S p i added to Ŝp ;. for r p r + the followig hold: If a vertex v belog to S r+, the it may be added to Ŝr+ +, ad if ot, the it may be added to Ŝr aumig v Sr. If it wa added to either of the two ubet ad it i elected i S r, the it i added to Ŝr ad it will ot be added to Ŝ p for ay p < r. We are ow ready to prove that the etimate ê i,j computed by Algorithm are eetially cloe to the correpodig value of E i,j. Recall that SIG deote the et of all igificat idice a defied i def Defiitio ad that E j = t k=0 E j,k. Lemma 9 For a appropriate choice of c i the defiitio of θ i Step of Algorithm ad of c 3 i Defiitio, with high cotat probability, for all j / L, if j SIG, the ǫ E i,j ǫ 8 6 E j ê i,j + ǫ E j, 4 6

18 ad if j / SIG the Proof: Recall that êi,j + β j ǫ 4t lg. ê i,j = i p=p 0 Ŝp p i,j,p + βp. 3 By Lemma 8, with high cotat probability, for every uch that r p 0 +, the cotributio of vertice i B to thi um i Ŝ r+ + B r+ + β r+ + Ŝ r B r + β r + Ŝ r B r + β r. 33 Aume from ow o that thi i i fact true ad deote thi um by ê. Coider firt the cae that B < 4 θ r. By Corollary 6, with high cotat probability, for every, if B < 4 θ r, the S p < + β θr p 4 for every p. Aumig thi i i fact the cae, we have that If j SIG, the by Lemma 4 we have that E j c /c / ê 3 4 θ r + β r+. 34 ê 3 t ǫ θ r + β r. Therefore, ǫ c 4 t E j, 35 for c 4 = c /c / 3 uig β /3. If j / SIG the + β j ǫ l /. c / t B j 3 Uig the fact that + β r B j becaue there are o multiple edge ad by the defiitio of θ r we get that ê + β j / c t 5/ ǫ3/ l / B j / ǫ l c / t B j 3 ǫ c 5 t l 3 ǫ c 5 t3lg, 36 for c 5 = c c / 3 /. We ow tur to the cae that B 4 θ r. By Corollary 6, with high cotat probability, for every, if B 4 θ r, the for every r p i, i p+ + β B i+ Sp p β i p+ + B. 37 i + Aume from thi poit o that thi i i fact the cae. Fixig uch a choice of, let S r+ def = S r+ \ Ŝr+ + ad S r def = S r \ Ŝr+ + Ŝr. 38 7

19 That i, Sr+ S r i the ubet of vertice i S r+ = S r+ B that were ot added to Ŝr+ + ad i the ubet of vertice i Sr = Sr B that were added to either Ŝr+ + or to Ŝr. Let α def = S r+ S r+ ad α def = S r S r+ S r. 39 Sice S r+ = Ŝr+ B S r+ accordig to the defiitio of α we have that Ŝr+ ad imilarly Ŝr+ + B r+ = α S r+ Ŝr+ + B r+ where the two ubet o the right-had ide are dijoit, B = α S r+. By Equatio 37, r+ + β α B β α B, I order to obtai boud o the ecod ad third term i Equatio 33, aume firt that both That i, α S r+ r+ r+ S r+ θ r 4 θ r 4 ad ad S r r α r+ S S r r Uder thi aumptio, by Lemma 5, with probability at leat 3t 4, ad r+ S S r r r+ S S r r Similarly, with probability at leat 3t 4, ad r S Sr r r S Sr r + β Sr+ i r+ = + β Sr i r = θ r 4. 4 θ r β α S r+ i r+, 44 β α S r+ i r β r+ α S S r i r, 46 β r+ α S S r i r. 47 Aume that Equatio ideed hold. Oberve that Sr+ S r = Ŝr B where the two ubet o the right-had ide are dijoit, o that by the defiitio of α we have that S r 8

20 Ŝr B r+ = α S S r. By Equatio 37 ad Equatio 44, Ŝr B r ad imilarly by Equatio 37 ad Equatio 45, = α S r+ S r Ŝr B r r + β α α Sr+ i r+ + β α α B, 48 β α α B, 49 Fially, by our aumptio which hold with high probability, that ampled vertice i B are added either to Ŝr+ r + or to Ŝr or to Ŝr, all vertice i S Sr are added to Ŝr. Therefore, by Equatio 37, 44 ad 46 ad the defiitio of α ad α, Similarly by Equatio 37, 45 ad 47, r Ŝr B r = S Sr r + β Sr i = = Ŝr B r + β i + β i + β i r r+ S S α r r α S r+ r+ α α S r+ r+ + βα α B. 50 βα α B. 5 The boud i Equatio 48 5 were obtaied for the cae that both α ad α are above certai threhold. If α S r+ /r+ < θ r r+ 4, that i, S / r+ < θ r 4, the by Lemma 5, with probability at leat 6t, 4 S r+ S r r + β θ r 4 ad r S Sr r + β θ r 4 9

21 a well. Similarly, if α S r+ leat 3t 4,. S r / r < θ r r 4, that i, S r S Sr r + β θ r 4 By combiig all the boud above we get that for B 4 θ r, ad /r < θ r 4 ê + β α B + β r+ + α α B + β r + α α B + β r + θ r + β r, the with probability at B + β r+ + θ r + β r, 5 ê B + β r θ r + β r+. 53 Similarly to what we have how for the cae that B < 4 θ r ee Equatio 34 36, if we let def E = EB,B j the we get that for j SIG, + β 3 E ǫ c 4 t E j ê + β 3 E + ǫ c 4 t E j, 54 ad for j / SIG, ê + β j + β 3 E + β j + ǫ c 5 t3 l, 55 for c 4 = c /c / 3 ad for c 5 = c c / 3 /4. Let LARGEi,j deote the ubet of idice r for which B 4 θ r. By Equatio 35 ad 36 for the cae that B < 4 θ r, ad Equatio 54 ad 55 for the cae that B 4 θ r, ad by takig a uio boud over all i,j ad r we get that the followig boud hold with high cotat probability. Firt, for every j SIG, ê i,j = i Ŝp p i,j,p + βp p=p 0 + r LARGEi,jê + ǫ 8 + ǫ 8 + ǫ 4 r LARGEi,j r/ LARGEi,j E i,j + ǫ c E j + ǫ E j 4 c 4 ê E + ǫ c E j + ǫ E j 4 c 4 E j, 56 where the lat iequality hold coditioed o c 4 ad c 4 which are fuctio of c ad c 3 beig ufficietly large ad i particular hold for ay c 3 ad c 3 c / 3. Recall that p 0 i the mallet value 0

22 of p atifyig 4 θ p +. Sice B for every while B 4 θ r for every r LARGEi,j, we have that r p 0 + for every r LARGEi,j. Therefore, ê ê i,j r LARGEi,j ǫ E ǫ 8 c E j r LARGEi,j 4 ǫ i E E ǫ E j 8 r=0 ǫ E i,j ǫ + 8 c 4 c 4 ǫ 8 r/ LARGEi,j E j E i,j ǫ 6 E j, 57 where the lat iequality hold for ufficietly large c 4 ad c 4 ad i particular wheever c 3 ad c 64 c / 3. O the other had, for j / SIG, + β j êi,j = + ǫ 8 + ǫ 8 < ǫ 4t lg, r LARGEi,jê + β j + r LARGEi,j ǫ c 3 t l + ǫ t c 5 r/ LARGEi,j E + β j + ǫ c 5 tlg + c 4 ê + β j ǫ c 5 t lg + c 5 lg 58 where i Equatio 58 we built o the defiitio of igificat bucket ad the lat equatio hold for ufficietly large c 3, c 5 ad c 5 ad i particular for ay choice of c 3 3 ad c 64/c / 3. By takig c 3 3 ad c 64 c / 3, the proof of Lemma 9 i completed. Puttig it all together: provig the firt part of Theorem. Recall that ˆl = S i + β i S + ê i,j + β j. 60 j / L Let lg,l deote the umber of legth- path i G whoe mid-poit belog to a bucket B i uch that i L, ad let lg,l deote the umber of legth- path whoe mid-poit belog to a bucket B j uch that j / L o that lg,l + lg,l = lg. By the firt part of Corollary ad the ettig of β we have that with high cotat probability: S i S + β i = 59 ± ǫ lg,l. 6 4

23 Turig to the ecod ummad i Equatio 60, by Lemma 9, j / L ê i,j + β j = j / L,j SIG + j / L,j SIG ê i,j + β j j / L,j/ SIG + ǫ 4 ê i,j + β j E j + β j + ǫ 4 lg + ǫ 4 E j + β j + ǫ 4 lg j / L + ǫ lg,l + ǫ lg. 6 4 I the other directio, recall that l σ G,L = j / L lσ j G,L where for j / L ad σ {,,3}, we let l σ j G,L deote the umber of legth- path whoe mid-poit belog to B j ad uch that the umber of vertice o the path that belog to B k for k / L icludig j i σ, j / L ê i,j + β j j / L,j SIG j / L ǫ 8 j / L,j / SIG ǫ E i,j ǫ 8 6 E j E i,j + β j + β j + βǫ lg 6 ǫ E i,j + β j 8 8 ǫ l G,L + l G,L ǫ lg 63 4 ǫ l G,L + l G,L + l 3 G,L 8 l G,L l 3 G,L ǫ 4 lg ǫ lg,l 3ǫ lg, where i Equatio 63 we ued Equatio baed o the defiitio of SIG ad takig c 3 3 a it wa et previouly, ad i the lat iequality we applied Lemma 3. By combiig Equatio 6, 6 ad 64 we get that ˆl = ± ǫlg with high cotat probability. 3.5 Removig the aumptio o l Our aalyi build o the aumptio that lg l lg. I order to get rid of thi aumptio we oberve that if we ru Algorithm with l > lg the our aalyi implie that with high cotat probability ˆl + ǫ lg + ǫ l. 8 Thi i true becaue: the algorithm till obtai with high probability a etimate of lg,l that doe ot overetimate lg,l by more tha a factor of + 4 ǫ ; For the umber of legth- path whoe mid-poit i i a bucket B j where j / L ad j SIG the approximatio factor

24 i at mot + ǫ ; 3 The additioal error caued by overetimatig the umber of legth- path whoe mid-poit i i a bucket B j where j / L ad j / SIG i at mot ǫ l. 8 Suppoe we ru Algorithm with l > lg. The with high cotat probability ˆl < + ǫ l. O the other had, if we ru Algorithm with lg l < lg, the with high cotat probability, ˆl ǫlg > ǫ l, which i greater tha + ǫ l for every ǫ < /3. Therefore, we do the followig. Startig from l =, we repeatedly call a light variat of Algorithm with our curret etimate l. The variat i that we icreae all ample ize by a factor of Θlog log o a to reduce the failure probability of each executio to O/log ad we ru the algorithm with ǫ = mi{ǫ,/4}. I each executio we reduce the previou value of l by a factor of, ad top oce ˆl > ǫ l, at which poit we output ˆl. By the above dicuio, with high cotat probability we do ot top before l goe below lg, ad coditioed o thi, with high probability O/log we do top oce lg l < lg or poibly, oe iteratio earlier, whe lg l < lg with ˆl = ± ǫlg. Sice there i a o-zero probability that the algorithm doe ot top whe lg l < lg, we ext boud the expected ruig time { of the algorithm. } The total ruig time of all executio util lg l < lg i O + mi /, 3/ polylog,/ǫ. Oce l < lg /3 lg / lg, the algorithm may termiate i Step 4 of Algorithm or i Step 3c of Algorithm, but if it doe ot, the the probability that ˆl ǫ l i ay particular executio i upper bouded { by O/log }. Sice the executio are idepedet, the expected ruig time i O + mi /, 3/ polylog, /ǫ. lg /3 lg / We thu have the followig theorem. Theorem With probability at leat /3, the aforemetioed algorithm, which ue Algorithm a a ubroutie, retur a etimate ˆl that atifie ˆl = { ± ǫ lg. } The expected query complexity ad ruig time of the algorithm are O + mi /, 3/ polylog, /ǫ. lg /3 lg / 4 Lower Boud for Approximatig the Number of Legth- Path I the ext theorem we tate three lower boud that together match our upper boud i term of the depedece o ad lg. I what follow, whe we refer to a multiplicative approximatio algorithm for the umber of legth- path, we mea a algorithm that output a etimate ˆl that with high probability atifie lg/c ˆl ClG for ome predetermied approximatio factor C where C may deped o the ize of the graph. If C i a cotat the the algorithm i a cotat factor approximatio algorithm. Theorem 3. Ay multiplicative approximatio algorithm for lg mut perform Ω querie. l /3 G. Ay cotat-factor approximatio algorithm for lg mut perform Ω querie whe the umber of legth- path i O. 3. Ay cotat-factor approximatio algorithm for lg mut perform Ω 3/ querie whe the umber of legth- path i Ω. 4. Proof of Item i Theorem 3 l / G To etablih the firt item i Theorem 3 we how that every ad for every value of l, there exit a family of -vertex graph for which the followig hold. For each graph G i the family we have that lg = Θl, 3

25 but it i ot poible to ditiguih, makig o/l /3, with high cotat probability betwee a radom graph i the family ad the empty graph for which lg = 0. Each graph i the family imply coit of a clique of ize b = l /3 ad a idepedet et of ize b. The umber of legth- path i the graph i b b = Θl. However, i order to ditiguih betwee a radom graph i the family ad the empty graph it i eceary to perform a query o a vertex i the clique. The probability of hittig uch a vertex i o l /3 G querie i o. Γ V \S v S S = c l d = + l/ d = u Γ S w Figure 3: A illutratio for the proof of Item i Theorem 3. O the left-had-ide i a graph i G, ad o the right-had-ide are the correpodig eighborhood table, Γ V \S ad Γ S. Each row i Γ V \S correpod to a vertex i V \S ad each row i Γ S correpod to a vertex i S. A coectig lie betwee a pair of etrie i the two table idicate that there i a edge betwee the two correpodig vertice. 4. Proof of Item i Theorem 3 Sice we have already etablihed i Item i Theorem 3 that there i a lower boud of Ω l /3 G, ad ice for lg 3/ we have that l /3 G /, we may coider the cae that lg > 3/ >. To etablih Item i Theorem 3 we how that every, every cotat c ad every < l < /c there exit two familie of -vertex graph for which the followig hold. I both familie the umber of legth- path i Θl, but i oe family thi umber i a factor c larger tha i the other family. However, it i ot poible to ditiguih with high cotat probability betwee a graph elected radomly i oe family ad a graph elected radomly i the other family uig o querie. We firt preet two familie that iclude ome graph with multiple edge ad elf-loop, ad the modify the cotructio to obtai imple graph. l/ The graph familie. I the firt family, deoted G, each graph i a uio of d = matchig. l/ Thu, each vertex ha degree d = ad lg = d < l. 65 A radom graph i G i determied by imply electig d radom matchig. I the ecod family, deoted G, each graph i determied a follow. There i a mall ubet, S, of c vertice, where each vertex i S ha 4

26 l l/ degree d = +, ad each vertex i V \S ha degree d = like all vertice i the graph i G. If we view each vertex i S a havig d port oe for each icidet edge ad each vertex i V \ S a havig d port, the a graph i the family G i defied by a perfect matchig betwee the c d + c d port we aume thi umber i eve, otherwie, d ad d ca be lightly modified. For a illutratio, ee the left-had-ide of Figure 3. Thu, d l + lg > c = c > cl. 66 Procee that cotruct graph i the familie. I order to how that it i hard to ditiguih betwee graph elected radomly from the two familie i o querie, we follow [GR0, KKR04] ad defie two procee, P ad P, that iteract with a approximatio algorithm A. The proce P awer the querie of A while cotructig a radom graph i G, ad the proce P awer the querie of A while cotructig a radom graph i G. We coider the ditributio over the repective query-awer hitorie, q,a,...,q t,a t, ad how that for hitorie of legth o the ditributio are very cloe, implyig that A mut have a high failure probability if it perform oly o querie. Detail follow. For implicity we aume that for every vertex that appear i either a eighbor query or a awer to uch a query, both proceed give the degree of the vertex for free, o there i o eed for degree querie. We alo aume that a awer u to a eighbor query v,i come with the label i of the edge from u ide of the edge. Clearly ay lower boud uder thee aumptio give a lower boud without the aumptio. The Proce P. The proce P maitai a d table Γ. A graph i G correpod to a perfect matchig betwee the table etrie. That i, if there i a edge v,u i the graph, ad the edge i labeled i from v ide ad i from u ide, the Γv,i = u,i ad Γu,i = v,i. Thu, a radom electio of a graph i G correpod to a radom electio of a perfect matchig betwee the etrie of Γ. Such a matchig ca be cotructed iteratively, where i each iteratio a umatched etry i the table i elected arbitrarily ad matched to a uiformly elected yet-umatched etry. The proce P fill the etrie of Γ i the coure of awerig the querie of the algorithm A: Give a query q t+ = v,i, the proce P awer with a uiformly elected yet-umatched etry, u,i The Proce P. The proce P maitai two table: oe c d table, Γ V \S, ad oe c d table, Γ S. The row of Γ V \S correpod to vertice i V \ S, ad the row of Γ S correpod to vertice i S. A radom graph i G ca be determied i the followig iterative maer. I each tep, a pair v,i i elected arbitrarily amog all pair uch that:. either there i already a row labeled by v i oe of the two table but the etry v,i i yet-umatched, or. there i o row labeled by v. I the latter cae, we firt elect, uiformly at radom, a ot yet labeled row i oe of the two table, ad label it by v. We the elect, uiformly at radom, a ot yet matched etry i oe of the two table. If the row of the elected etry i ot yet labeled, the we give it a radom label amog all ot yet uued label i {,...,}. 5