Partial Restreaming Approach for Massive Graph Partitioning.

Transcription

1 Partal Restreamng Approach for Massve Graph Parttonng. Ghzlane Echbarth, Hamamache Kheddouc To cte ths verson: Ghzlane Echbarth, Hamamache Kheddouc. Partal Restreamng Approach for Massve Graph Parttonng.. SITIS The 10th Internatonal Conference on SIGNAL IMAGE TECHNOLOGY INTERNET BASED SYSTEMS, 2014, Marrakech, Morocco <hal > HAL Id: hal Submtted on 4 May 2017 HAL s a mult-dscplnary open access archve for the depost and dssemnaton of scentfc research documents, whether they are publshed or not. The documents may come from teachng and research nsttutons n France or abroad, or from publc or prvate research centers. L archve ouverte plurdscplnare HAL, est destnée au dépôt et à la dffuson de documents scentfques de nveau recherche, publés ou non, émanant des établssements d ensegnement et de recherche franças ou étrangers, des laboratores publcs ou prvés. Dstrbuted under a Creatve Commons Attrbuton - NonCommercal - NoDervatves 4.0 Internatonal Lcense

2 Partal Restreamng Approach For Massve Graph Parttonng Ghzlane ECHBARTHI Unversty Lyon1 Lyon, France Emal: Hamamache KHEDDOUCI Unversty Lyon1 Lyon, France Emal: Abstract Graph parttonng s a challengng and hghly mportant problem when performng computaton tasks over large dstrbuted graphs; the reason s that a good parttonng leads to faster computatons. In ths work, we ntroduce the partal restreamng parttonng whch s a hybrd streamng model allowng only several portons of the graph to be restreamed whle the rest s to be parttoned on a sngle pass of the data stream. We show that our method yelds parttons of smlar qualty than those provded by methods restreamng the whole graph (e.g ReLDG, ReFENNEL) [1], whle ncurrng lower cost n runnng tme and memory snce only several portons of the graph wll be restreamed. I. INTRODUCTION Actual graph datasets are massve. The World Wde Web conssts of trlons of unque lnks [2], Facebook contans over one bllon of actve users [3] and Twtter conssts of mllons of users, bologcal networks are also of smlar szes, lke proten nteracton networks, to name but a few. However, the growng sze of graph datasets makes t challengng when performng usual computatons over graphs, such as communty detecton, countng trangles, dentfyng proten assocatons as well as many other computaton tasks. One soluton to ths problem s to dvde the graph over several clusters and then run parallel algorthms to perform computatons. Graph parttonng s an NP-hard problem amng to dvde a graph dataset nto dstnct sets under the constrants of equlbratng the cluster szes and mnmzng the number of edges crossng the clusters, ths s the balanced graph parttonng problem. As seen n prevous works [4], [5], a good parttonng leads to fast computatons, the reason s that the balanced szes of clusters make sure that each processor s assgned the same amount of data and the mnmzed crossng edges mnmze the overhead network. In the settng of dynamc graphs, ths problem s called streamng graph parttonng where the graph s serally processed, each vertex loaded s assgned to a cluster. Our contrbuton. In ths paper, we ntroduce the partal restreamng parttonng; t s a hybrd model of data streamng where a porton of sze C s restreamed several tmes, and the rest of the graph dataset s streamed once, lke n the streamng model used n [4], [5]. The man strength of our proposed method avods restreamng the whole graph dataset to mnmze runtme and memory cost. We show that by restreamng at most half of the graph dataset we can obtan a good partton as n the settng of full restreamng [1]. As parttonng heurstcs, we use LDG and Fennel. As an extenson of partal restreamng, we ntroduce the selectve partal restreamng parttonng, n whch several portons of the graph dataset are selected for restreamng accordng to ther average degree and densty. We show that the selectve method mproves the partton qualty. The rest of the paper s organzed as follows. In Secton 2, we dscuss the related work. In Secton 3 we present our partal restreamng approach. In Secton 4 we present our evaluaton set up. Secton 5 presents and dscusses our man results. In Secton 6 we conclude. II. RELATED WORK Graph parttonng s a well studed problem. Many of methods and heurstcs have been proposed n the ltterature, each one presentng strengths and weaknesses. But n the settng of streamng parttonng, only few heurtcs are proposed, and were frst ntroduced by Stanton et al. n 2012 [4]. Stanton et al. proposed ten dfferent heurstcs and found that the best performant heurstc was Lnear Determnstc Greedy LDG. Snce then, streamng parttonng methods become the trend n parttonng graphs especally when dealng wth huge datasets, the reason s that these methods are well suted for bg graphs snce they process nodes on the fly and do not ncur as hgh computaton cost as n the classc parttonng methods. After Stanton et al., Tsourakaks et al. came up wth a new streamng parttonng heurstc called FENNEL [5]. FENNEL relaxes the balance constrant unlke LDG, producng parttons wth lower edges cut but hgher balance slackness. In 2013, Nshmura et al. [1] proposed the restreamng parttonng, whch mproves parttons qualty whle preservng the balance constrant. Though classcal methods are not suted anymore for processng huge graphs, we cte METIS [6], a multlevel method for graph parttonng yeldng hgh qualty parttons. METIS have been usually used n order to compare onlne and offlne methods. 1) The streamng model: Recent work n streamng parttonng [1], [4], [5] adopts almost the same streamng graph model n whch vertces arrve, each wth ts adjacency lst. The vertces arrve n a certan order: Random, Breath Frst

3 Search or Depth Frst Searh [7], [5], and once the vertex s assgned to a shard, t s never replaced afterwards. The streamng graph model conssts n a cluster of k machnes each one of capacty C such that the total capacty of the k machnes can hold the whole graph. When a vertex s loaded, a parttoner must decde n whch one of the k machnes the vertex must be placed. III. PROPOSED APPROACH In ths secton, we ntroduce the partal restreamng parttonng. In Secton 3.1 we frst present the partal restreamng model used. In Secton 3.2 we present the nstance of the partal restreamng parttonng model usng Lnear Determnstc Greedy LDG [4] and Fennel [5] as heurstcs for parttonng. In Secton 3.3, we gve another varant of partal restreamng, called Selectve Partal Restreamng parttonng where portons to be restreamed are selected dependng on ther degree and densty. Notatons. We wll be usng the followng notaton throughout the paper. We consder a smple undrected graph G = (V, E), let V = n be the number of vertces n G and E = m be the number of edges of G. Let the current vertex loaded be v, and N(v) represents hs neghbors. K s the number of clusters or parts we wsh to dvde the graph to. s s the number of the streamng teratons. Let P t = (S1, t, Sk t ) be a partton of the graph. S,, S k s called clusters such that S V and S S j = for every j. P t 1 = (S1 t 1, S t 1 k ) represents the partton obtaned from the precedent stream. Let e(s, S V ) be the set of edges wth ends belongng to dfferent clusters. We defne e(s,s V ) λ = m as the fracton of edge cut. We note that λ should be mnmzed durng parttonng. Each cluster S s of sze C. In our work we set C = n k. C represents also the sze of the porton to be restreamed. We can choose that several portons of the graph should be restreamed, then β s the number of portons to be restreamed (each porton s of sze C). A. Partal Restreamng Model We conssder a smple streamng model as descrbed n Secton 2, where vertces arrve n a random order wth ther adjacency lsts, the heurstc used for parttonng must make a decson about the cluster to place the current vertex. In the partal restreamng model, two major phases exst: the restreamng phase and the one pass streamng phase. Namely, a frst loaded porton of the graph dataset of sze beta C s gong to be restreamed and the rest s gong to be processed n the smple streamng phase. In other words, mult-passes of the stream are allowed only for a graph subset. Let P t be the partton obtaned at tme t, P t 1 represents the partton obtaned at the precedent teraton of the restream. When we attan the number of restreamng teratons allowed for the porton concerned, we contnue streamng the rest of the graph dataset n a sngle pass of data stream. In the secton below we descrbe the parttonng heurstcs used n our model. B. Partal Restreamng Parttonng As a parttonng strategy, we use the state-of-the-art heurstcs [4], [5]. In ths secton we descrbe how we adapt these heurstcs to our partal restreamng model. 1) Lnear Determnstc Greedy : LnearDetermnstcGreedyLDG s the best performng heurstc n [4]. It greedly assgns the vertces to clusters whle addng a penalzaton for bg clusters to emphasze balance. LDG assgns vertces to clusters that maxmze: LDG =argmax (P t N(v)) (1 P t C ) (1) In our streamng model, we consder 2 phases: the restreamng phase and the smple streamng phase whch conssts n one pass. In the frst phase, the LDG functon wll use nformaton about the last parttonng to decde about the placement of the current vertex such that: P artldg =argmax (P t 1 N(v)) (1 P t C ) (2) Where P artldg makes a reference to partal LDG for partally restreamng LDG. In the second phase, the one pass streamng phase, the vertces are assgned followng the LDG functon as follows: P artldg =argmax (P t N(v)) (1 P t C ) (3) 2) Fennel: Lke done for LDG, we adapt Fennel functon to the two phases of our streamng model. In the frst restreamng phase, F ennel s as follows: P artf ennel =argmax (P t 1 N(v)) αγ( P t ) γ 1 (4) Where P artldg makes a reference to partal F ennel for partally restreamng F ennel. Whle n the second phase, P artf ennel s as follows: P artf ennel =argmax C. Selectve Partal Restreamng Parttonng (P t N(v)) αγ( P t ) γ 1 Instead of restreamng the frst loaded porton of the graph dataset, we select relevant portons of sze C that would lead to a good qualty parttonng. We set degree and densty parameters as crtera for selectng portons to be restreamed n the frst phase of the model. In other words, when a porton s loaded, we check ts average degree and ts average densty, f t s hgher than the average degree (respectvely average densty) of the whole graph, we select ths porton for restreamng, otherwse t wll be processed n the second phase of one pass streamng. selecton crterons. In order to select a porton for restreamng, two crtera are consdered: 1) Average degree : The average degree of a porton s the average of vertces degrees nsde the porton. Notce that the degree (5)

4 of a vertex s the number of ts neghbors no matter they are nsde or outsde the porton concerned. We take the average degree as a crteron to make sure that the porton whch s gong to decde for the parttonng of the graph must nfluence the parttonng decson of a large number of vertces. 2) Average densty : The average densty of a porton represents the number of edges nsde t. It s mportant to have edges nsde the porton to make better parttonng decson as the objectve functons used make decson dependng on edges, otherwse the parttonng of the porton wll be done at random and t wll defntly lead to lower qualty parttonng. The average degree and densty of the whole graph aren t always avalable, we can substtute ths by progressvely addng degree and densty nformaton as the data s loaded. A porton wth hgh densty and hgh degree vertces should act lke a kernel to yeld parttons of good qualty. In fact, portons wth vertces havng hgh degree would nfluence the parttonng of a large number of vertces, and the densty crteron nsde the porton makes sure to take nto consderaton the edges to make better decson for the parttonng. In Secton 5, we show that by selectng portons of hgh degree average and hgh densty average we obtan parttons wth better qualty than those obtaned by smply restreamng the frst loaded porton. IV. EVALUATION SET UP 1) Evaluaton datasets: Two types of graph datasets were used: web and socal. We test our methods n ten graph datasets lsted n Table I, all obtaned from the SNAP repostory [8]. Vertces wth 0 degree and self-loops were removed. All the graphs were made undrected by recprocatng the edges. 2) Methodology: We run our methods on the ten graph datasets, we choose to fx exprments parameters as follows: k = 40, s = 10 and β = k/2 (whch means that we are restreamng the half of the graph). Notce that the orderng of vertces s done at random. We frst compare our methods to the sngle pass methods and then to the full restreamng methods [1]. Afterwards, we compare the partal restreamng methods (P artldg, P artf ennel) and the selectve methods (P SelectLDG, P SelectF ennel) on fve graph datasets. We run 5 executons on 5 dfferent random orders and show the results obtaned. In order to see how the partton qualty reacts to the changng porton sze beng restreamed, we examne the results for dfferent values of β. We run P artldg and P artf ennel (partally restreamng LDG and partally restreamng F ennel respectvely) on WebGoogle and LveJournal for dfferent values of β. Ultmately, we show the runnng tme gan for the partal methods (P artldg, P artf ennel) over the ten graphs for k = 40 and s = 10. Runnng tme gan s calculated as follows: Gan P artldg = ReLDG P artldg ReLDG LDG Where ReLDG refers to the executon tme of the verson of LDG where the whole graph s restreamed, LDG s the one pass streamng verson. Gan P artf ennel = ReF ENNEL P artf ENNEL ReF ENNEL F ENNEL Same as Gan P artldg, ReF ENNEL refers to the executon tme of the verson of F ENNEL where the whole graph s restreamed and F ENNEL s the one pass streamng verson. V. RESULTS AND DISCUSSION In ths secton, we present and dscuss our results. Notce that the metrcs presented n tables II, III and IV represent the fracton of edge cut λ. And before we delve n the results we gve a bref summary about t. 1) Summary of our results - Results that were obtaned show that by augmentng β, or by augmentng the sze of the porton to be restreamed, we obtan a better qualty partton. - By restreamng the frst loaded half of a graph, we obtan parttons of smlar qualty than those yelded by restreamng the whole graph dataset. - The partal selectve restreamng method mproves the partton qualty. - Restreamng only a porton of a graph dataset ncurs lower run tme cost. Compared wth full restreamng methods, partal methods takes half of runnng tme n case of restreamng half of the graph dataset. 2) Performance: dscusson In Table II, we clearly see that the partal methods outperforms the sngle pass methods over all the graph datasets and t mproves the partton qualty. In table III we observe that our partal methods yelds almost the same fracton of edge cut as n full restreamng methods, wth an average dfference of 4%. In Table IV, we run PartFENNEL and PartLDG on 5 dfferent random orders and compare t to PSelectFEN- NEL and PSelectLDG (Partal Selectve FENNEL and Partal Selectve LDG). As we expected, partal selectve methods outperforms the partal methods n all the 5 graphs that we tested, and t outperforms METIS also except for Astro-ph and Webnd (0.593 vs and vs respectvely). In Fgure 1 we see that the sze of the proton restreamed nfluences the partton qualty: the bgger β s the lower fracton of edge cut. In other words, the more nformaton we have about the precedent streamng teraton the better s the edge cut. For Fennel, the smple partal methods outperforms the selectve ones because the selectve methods enhance the balance whch lead to hgher edge cut.

5 TABLE I GRAPH DATASETS USED FOR OUR TESTS. Graph N M Avgdeg type wkvote socal enron socal Astro ph socal slashdot socal Web nd web stanford web Web google web Web berkstan web Lve journal socal orkut socal TABLE II COMPARISON OF FRACTION EDGE CUT FOR LDG AND PARTRELDG, FENNEL AND PARTFENNEL. RESULTS ARE OBTAINED FOR TEN GRAPH DATASETS WHERE k = 40 AND s = 10 AND β = k/2. Graph LDG PartLDG FENNEL PartFENNEL wkvote enron astro ph slashdot webnd stanford webgoogle web berkstan lve journal orkut TABLE III COMPARISON OF FRACTION EDGE CUT FOR RELDG AND PARTRELDG, REFENNEL AND PARTFENNEL. RESULTS ARE OBTAINED FOR TEN GRAPH DATASETS WHERE k = 40 AND s = 10 AND β = k/2. Graph ReLDG PartLDG ReFENNEL PartFENNEL wkvote enron astro ph slashdot webnd stanford webgoogle web berkstan lve journal orkut PartFennel Webgoogle graph LveJournal graph PartLDG Webgoogle graph LveJournal graph Runnng tme gan s represented n Table V, n all our graph datasets, PartFennel have a gan of 49.93% and PartLDG 48.85%. Those results are obtaned whle restreamng the half of the graph datasets, whch s normal snce the gan s equal to 50% approxmately. λ 0.25 λ VI. CONCLUSION AND FUTURE WORK K/4 K/2 3K/4 β K/4 K/2 3K/4 β Fg. 1. Varaton of λ wth the growng sze of porton beng restreamed represented by β for Webgoogle graph and LveJournal. On the left P artf ennel and on the rght P artldg. We have demonstrated that by restreamng only subsets of a graph dataset we actually obtan as good qualty parttons as those yelded by restreamng the whole graph dataset, whle fastenng the runnng tme parttonng and mnmzng the cost n memory. We also showed that by selectng portons of the graph havng hgh average degree and densty we mprove the partton qualty. Good parttonng s a result of havng a good kernel n clusters. In future work, we plan to refne our selecton crtera for selectng portons worth restreamng.

6 TABLE IV FRACTION OF EDGE CUT FOR PARTIAL METHODS P artldgandp artf ENNEL VERSUS PARTIAL SELECTIVE METHODS P SelectLDGandP SelectF ENNEL AND METIS,(1.001) INDICATES THAT THE SLACKNESS ALLOWED IS 001. RESULTS ARE OBTAINED FOR 5 GRAPH DATASETS WHERE k = 40 AND s = 10 AND β = k/2 FOR 5 DIFFERENT RANDOM STREAM ORDERS. Graphs PartLDG PSelectLDG PartFennel PSelectFennel Mets(1.001) wkvote enron Astro ph slashdot Web nd TABLE V RUNNING TIME GAIN COMPUTED FOR P artldg AND P artf ennel OVER EXECUTION ON TEN GRAPHS WITH k = 40 AND s = 10. Graphs Partal ReLDG Gan Partal ReFennel Gan Wkvote 47.2% 58.5% Enron 50% 48.3% Astro ph 44.5% 44% Slashdot 39.6% 47.5% Web nd 51.4% 49.5% Stanford 54.5% 52.4% Web google 57.5% 52.7% Web berkstan 50.6% 49.6% Lve journal 40.5% 45.6% Orkut 52.7% 51.2% REFERENCES [1] J. Nshmura and J. Ugander, Restreamng graph parttonng: Smple versatle algorthms for advanced balancng, n Proceedngs of the 19th ACM SIGKDD Internatonal Conference on Knowledge Dscovery and Data Mnng, ser. KDD 13. New York, NY, USA: ACM, 2013, pp [Onlne]. Avalable: [2] [3] bllon-monthly-actve-users-874-mllon-moble-users-728-mllon-dalyusers/. [4] I. Stanton and G. Klot, Streamng graph parttonng for large dstrbuted graphs, n Proceedngs of the 18th ACM SIGKDD Internatonal Conference on Knowledge Dscovery and Data Mnng, ser. KDD 12. New York, NY, USA: ACM, 2012, pp [Onlne]. Avalable: [5] C. Tsourakaks, C. Gkantsds, B. Radunovc, and M. Vojnovc, Fennel: Streamng graph parttonng for massve scale graphs, n Proceedngs of the 7th ACM Internatonal Conference on Web Search and Data Mnng, ser. WSDM 14. New York, NY, USA: ACM, 2014, pp [Onlne]. Avalable: [6] G. Karyps and V. Kumar, A fast and hgh qualty multlevel scheme for parttonng rregular graphs, SIAM J. Sc. Comput., vol. 20, no. 1, pp , Dec [Onlne]. Avalable: [7] Karyps and Kumar, Multlevel graph parttonng schemes, pp , [8]