AN INDEXING METHOD FOR SUPPORTING SPATIAL QUERIES IN STRUCTURED PEER-TO-PEER SYSTEMS

Transcription

1 AN INDEXING METHOD FOR SUPPORTING SPATIAL QUERIES IN STRUCTURED PEER-TO-PEER SYSTEMS Lngku Meng a, Wenun Xe a, *, Dan Lu a, b a School of Remote Sensng and Informaton Engneerng, Wuhan Unversty, Wuhan 4379, Chna b Department of Informaton Technology, Huazhong Normal Unversty, Wuhan 4379, Chna (lkmeng, xewenun)@whu.edu.cn, rock_fall@tom.com Commsson VI, WG VI/4 KEY WORDS: Peer-to-Peer, Spatal Queres, Overlap Mnmzaton, Spatal Index, Tree Structure, Dstrbuted Optmzaton ABSTRACT: To provde the effcent supportng spatal data ueres n peer-to-peer systems has recently receved much attenton. Most proposed methods tred to use hop count to represent the transmsson delay, and the total message count to estmate the cost of uery processng. For the gnorance of the dfferences between DHT lookups and spatal ueres, and dstncton between physcal networks and overlay networks, the effcency and cost of ther uery processng can t be ndcated properly. In addton, ther expermental results are acheved by usng pont data sets, whle the fact that the overlap of spatal obects usually exsts n real applcatons s not consdered, and t may cause mult path uery processng and then results n plenty of peers vstng and routng messages. In ths paper, we propose an ndexng method whch effcently supports spatal ueres n structured peer-to-peer systems. It adopts an overlap mnmzaton algorthm whch takes the uery rate of data nto account to reasonably reduce the holstc cost of ueres. We also ntroduce a dynamcally adaptve dstrbuted optmzaton scheme that dynamcally adaptng to the tme-varyng overlay archtecture and data usage concerns. Theoretcal analyss and smulaton results both ndcate that our method s effcent and effectve.. INTRODUCTION Wth the rapd growth and ncreased mportance of dstrbuted spatal data, there s an ncreasng need for massve spatal data sharng n large-scale dstrbuted systems. Whle the P2P (Peerto-Peer) systems have become a powerful means for data sharng n Internet communty. For ther potental uses n spatal data sharng, to provde the effcent supportng spatal data ueres n peer-to-peer envronments has recently receved much attenton. There are two types of P2P overlay networks: unstructured ones n whch reuests are broadcasted or routed through floodng or random walks, such as Gnutella (Rpeanu, 2), KaZaA (Lebowtz, 23), edonkey (Tutschku, 24), Freenet (Clarke, 2), and structured ones based on DHTs (Dstrbuted Hash Tables) n whch reuests are routed usng routng tables, such as Chord (Stoca, 23), CAN (Ratnasamy, 2), Tapestry (Zhao, 24), Pastry (Rowstron, 2). Snce they use floodng or random walks based methods for processng ueres reuest, whch results n a large number of messages and a traffc overhead, so lead to the poor effcency of ueres and the bad scalablty for the systems, and these restrcted ther rapd development. On the other hand, t s very easy to process a uery by usng the assgned key n structured P2P networks, and ther scalablty s very good. So the structured P2P systems are more approprate for handlng data sharng. But for the reason of usng DHTs, whch destroy the semantcs of the data obects, only exact key match ueres can be supported effcently, and t sn t an easy task to support spatal ueres n structured P2P systems. The tradtonal ndexng methods for spatal databases can be brefly classfed nto two approaches: spatal sortng-based, such as Hlbert space fllng curve (Bally, 969), Z-orderng curve (Orensten, 986), and spatal contans relatonshp-based, such as R-tree (Guttman, 984), R+-tree (Sells, 987), R*-tree (Beckmann, 99). Smlarly, the methods that support spatal data ueres n structured P2P systems also have two knds of mplements: the one that maps multdmensonal spatal data nto one-dmensonal by usng order-preservng hash functon, and the other one that dstrbute tree data structure n P2P envronment. The man problem of former approach s the spatal relatonshps between spatal obects often may be destroyed, so leads to the neffcency of the ueres. Ths s because there are no any functons can always preserve the spatal propertes. As for the latter method, a crtcal performance ssue s the tree structure has to be uered n a top-down manner from the root node, so the communcaton bottlenecks are more lkely to happen on the peers that take charge of the tree nodes at hgher level, especally for root node, and t s also a sngle pont of falure. For ther good effcency n centralzed envronment and the bad performance of the former approach, usng herarchcal tree structure s a better choce. VBI-Tree (Jagadsh, 26) solved the above problem n latter approach by ntroduce a new routng table desgn usng sdeway ndex lnks, and DPTree (L 26) handled t by propose tree branch orented dstrbuton. * Correspondng author. 439

2 Although some proposed methods have acheved good results. There stll exst some ssues that are not consdered carefully. ) The dfferences between DHT lookups and spatal ueres. In a DHT lookup, the uery reuest s forwarded along a sngle path, whle t may be routed over multple paths for spatal uery. The multple peers vstng sometmes can t be avoded. However, we can decrease the number of multple peers vstng. 2) The dstncton between physcal networks and overlay networks. So the hop count can not really reflect the transmsson delay. To address the above two ssues, we propose a sute of effcent solutons, whch can be used to support any knd of herarchcal tree archtecture overlay. Our paper makes the followng two maor contrbutons. ) An nnovatve defnton of overlap s frst ntroduced. Here we take the followng propertes of the systems nto consderaton: the non-unform and tme-varyng propertes of spatal data dstrbuton and ther popularty, and peer nterests also are dfferent and tme-varyng. Then we present an overlap mnmzaton algorthm to mnmze the number of peers need to vst for process a uery. 2) We propose an effcent dstrbuted optmzaton algorthm to guarantee each peer has neghbors that are physcally close to t n the underlyng network, and t can contnuously and effcently optmze the overlay structure under dynamc network condtons. The rest of the paper s organzed as follows. Secton 2 surveys prevous work, focusng on spatal data ueres n structured P2P systems. In secton 3 we propose the defnton of overlap and the overlap mnmzaton algorthm. The dstrbuted optmzaton algorthm s explaned n secton 4. The expermental results of the above desgn, usng many metrcs, such as routng hop count, number of messages, dstrbuton percentage of delay tme of peers, are presented n secton 5. Fnally, we dscuss the conclusons and future work n secton 6. tree branch orented dstrbuton method to dstrbute the tree structure among peers n a way preservng the good propertes of balanced tree structures yet avodng sngle ponts of falure and performance bottlenecks. It s known that tree structures are very dffcult to dstrbute n P2P systems, because searchng the tree by followng paths nduces an uneven load on tree nodes at hgher level. The above two methods solve the problem by ntroduce some novel desgns. However, the efforts only am to support zero dmensonal data ueres n mult-dmensonal data space, and none of them consder the stuatons of mult-dmensonal data n multdmensonal data space. More specfcally, the ent methods can not effcently handle the ueres about lne data or polygon data, snce they omtted the overlap between adacent tree nodes, whch results n the multple peers vstng n dstrbuted systems. In addton, we should also try to keep each peer has neghbors that are physcally close to t n the underlyng network. 3. OVERLAP MINIMIZATION For the exstence of overlap n tree nodes, there wll be multple paths need to be followed even for pont data ueres, whch results n a large number of messages and plenty of peers vstng. We therefore frst gve our defnton of overlap n P2P systems, and then present an overlap mnmzaton algorthm. 3. Defnton of Overlap In the X-tree method (Berchtold, 996), the followng two defntons of overlap are gven. If a tree node contans n hyper rectangles {R,, R n }, the overlap can formally be defned as 2. RELATED WORK There has been a substantal amount of research on spatal data ueres n P2P systems. As mentoned, most ent proposed methods generally can be brefly classfed nto two categores: one s the reducton of multdmensonal spatal data to one dmenson, then the ent DHTs can be drectly used to support spatal ueres, and the other s dstrbuton of spatal contans relatonshp-based ndexng structure. The former category ncludes SCRAP (Ganesan, 24), MAAN (Ca 24), PRoBe (Sahn, 25). SCRAP uses a two-step soluton to partton the data space. In the frst step, t map mult-dmensonal spatal data down to one-dmensonal by usng a space-fllng curve, then the one-dmensonal data could be range parttoned across the dynamc avalable peers. MAAN supports mult-attrbute range ueres through multple sngleattrbute resoluton by usng localty preserved hashng to map a range of data space to Chord, and the effcency may be very poor. PRoBe uses a mult-dmensonal logcal space and maps data tems onto the space based on ther attrbute values, and the space s dvded nto hyper-rectangles, wth each mantaned by a peer wthn the system. In the latter category, VBI-Tree s an abstract data structure buld based on a vrtual bnary balanced tree structure. It was nspred by BATON (Jagadsh, 25) structure each peer corresponds an nternal node and a leaf node. DPTree uses a Overlap { = () {... ( R R ) A denotes the volume covered by A. For the dstrbuton of spatal data s nonunform, the modfed defnton took ths nto account. That s { p p ( R R )} { WeghtedOverlap = (2) { p p R } R {... A denotes the number of data elements contaned n A. Obvously, the popularty between data elements s dfferent. A more accurate defnton of overlap needs to take the uery rate 44

3 of data elements nto account. So we propose the followng new defnton. p { p p ( R R )} { NewOverlap = (3) { p p R } all {... p s a data element p denotes the uery rate of p, whch s the number of receved ueres about p durng the hstory perod. all denotes the uery rate of all data elements, whch s the number of receved ueres about all data elements stored n the local peer durng the hstory perod. As data element popularty and peer nterest s tme-varyng, we use exponental movng average technue to calculate the value of uery rate, whch gve more weght to the observatons n most recent perods wthout dscardng other values, rather than drectly usng the observaton values durng the entre hstory perod. So the new formula for calculatng uery rate s as follows. 4. DISTRIBUTED OPTIMIZATION ALGORITHM We defne the delay tme of a peer p as D(, whch s the sum of latences to all ts neghbours. Its defnton can formally be defned as neghbors( D ( = L( p, n) (6) n neghbors( p s a peer neghbours ( s the set of p s neghbours n s one of ts neghbours L(p, n) s the latency of peer p to n. It s easy for us to know that the optmum stuaton s to keep the mnmum value of total delay tme D sum ( of all peers S nvolved n the system. S sum( = D( p (7) p S D ) = w + w) prev ( (4) S s the set of all peers nvolved n the system s ent vald value for uery rate prev s the prevous vald one s the ent observed one w [,] s a constant value whch represents a weght factor value for new observaton. 3.2 Overlap Mnmzaton Algorthm If an overlap wll appear on one peer, we should choose to allow ts exstence or adust the poston of some nfluenced peers to mnmze the overlap. The latter acton wll be taken only when the benefts brought about by the exstence of overlap s less than ts cost. If the overlap oced n a peer, and t wll affect n data elements {d, d 2,, d n }, wth the fractons of area of the overlap to them are {f, f2,, f n }, and the average cost to process uery about these data elements are {c, c 2,, c n }. We also assume t wll cost some extra system resources SR to allow the exstence of the overlap. If the benefts of the overlap less than ts cost, then we keep the overlap, or adust some peers. That s Snce n the P2P systems t s mpractcal to calculate the above value, we propose a dstrbuted optmzaton algorthm that s an teratve algorthm that each peer executes perodcally and uses nteger lnear optmzaton to mnmze the sum of the delay tme of a peer and one of ts neghbours. The algorthm s executed perodcally on two adacent peers n the system, whch we call them seeds. Frstly, they mutually exchange the routng table of ther neghbours, and then each seed peer measures the latences to the neghbours of the other seed peer. Fnally, we can determne whether or not to swap ther neghbours based on the measured values. The functon that we want to mnmze durng the teraton of the algorthm s as follows. Seeds D p Seeds Seeds neghbours( ( = L( p, n) (8) p Seeds n neghbours( Seeds are two adacent peers n the system It can be proved that global convergence can be acheved but proofs are omtted due to space lmtatons. n = f c < SR s the uery rate of d. (5) 5. EXPERIMENTAL RESULTS For the evaluaton of the above two algorthms we dd compare the expermental results between our optmzed method and the orgnal one n a tree structure overlay. Snce the structure here ddn t consder the propertes of the physcal network, we use an Internet node latency measurement results (Wong, 25) from Merdan proect n Cornell Unversty. As there s only 44

4 pont data smulator component n the desgn, we also use a spatal data generator (Unversty of Praeus, 26) to create polygon data, whch are Zpf dstrbuton. We test the network wth dfferent number of nodes N from to 25. For each test, 5 pont ueres are executed, and then the average value s taken. In Fgure we present the results of number of messages to locate data to process pont ueres for pont data and polygon data, and the number can be used as a metrc of scalablty of the system. As we observe that our algorthm could reduce the uery processng cost for polygon data, but nearly the same as orgnal algorthm for pont data. Number of messages Number of nodes Orgnal for pont data Optmzaton for pont data Orgnal for polygon data Optmzaton for polygon data Fgure. Number of message comparson In Fgure 2 we demonstrate the number of routng hops comparson, whch can reflect the effcency of a system. Smlar as the above results, the fgure ndcated that our algorthm could sgnfcantly mprove the effcency for polygon data, but not for pont data Number of routng hops Number of nodes Orgnal for pont data Optmzaton for pont data Orgnal for polygon data Optmzaton for polygon data Fgure 2. Number of routng hops comparson Fgure 3 shows the results of dstrbuted optmzaton algorthm for keep adacent peers n physcal network as neghbors n overlay network, whch prove that the method has good localty propertes for tree-based overlay networks. Dstrbuton Percentage Our algorthm Delay tme Orgnal algorthm Fgure 3. Dstrbuton percentage of delay tme of peers 6. CONCLUSION AND FUTURE WORK In ths paper, we proposed our ndexng method for supportng spatal ueres n P2P systems, whch allows us to reduce the cost of routng messages and mprove the effcency of routng hops. Addtonally, by recognzng that hop count can not reflect the actual tme reured for processng ueres, we also focus on the reducton of the delay tme of each hop besdes the hop count, and hence decrease the total tme. The method s based on two newly proposed algorthms: overlap mnmzaton algorthm and dstrbuted optmzaton algorthm. For the future, we wll augment our method to nclude other effcent uery algorthms, such as range uery and knn uery. Upon completon of ths work, we also plan to run comprehensve performance evaluaton on our method. ACKNOWLEDGEMENTS The research s supported by Natural Scence Foundaton of Chna (47777). REFERENCES Rpeanu, M. 2. Peer-to-Peer Archtecture Case Study: Gnutella Network. st Internatonal Conference on Peer-to-Peer Computng (P2P 2), Lnköpngs unverstet, Sweden, pp Lebowtz, N., Rpeanu, M., Werzbck A. 23. Deconstructng the Kazaa network. 3 rd IEEE Workshop on Internet Applcatons (WIAPP 23), San Jose, Calforna, USA, pp Tutschku, K. 24. A Measurement-based Traffc Profle of the edonkey Flesharng Servce. 5 th Passve and Actve Network Measurement Workshop (PAM 24), Antbes Juan-les-Pns, France, pp Clarke, I., Sandberg, O., Wley B., et al. 2. Freenet: A Dstrbuted Anonymous Informaton Storage and Retreval System. Workshop on Desgn Issues n Anonymty and Unobservablty 2, Berkeley, Calforna, USA, pp Stoca, I., Morrs, R., et al. 23. Chord: a scalable peer-to-peer lookup protocol for Internet applcatons. IEEE/ACM Transactons on Networkng, (), pp

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