Distributed Splay Suffix Arrays:A New Structure for Distributed String Search

Transcription

1 World Academy of Scence, Engneerng and Technology Internatonal Journal of Computer and Informaton Engneerng Vol:, No:0, 2007 Dstrbuted Splay Suffx Arrays:A New Structure for Dstrbuted Strng Search Tu Kun, Gu Na-je, B Kun, Lu Gang and Dong Wan-l Internatonal Scence Index, Computer and Informaton Engneerng Vol:, No:0, 2007 wasetorg/publcaton/9022 Abstract As a structure for processng strng problem, suffx array s certanly wdely-known and extensvely-studed But f the strng access pattern follows the 90/0 rule, suffx array can not take advantage of the fact that we often fnd somethng that we have just found Although the splay tree s an effcent data structure for small documents when the access pattern follows the 90/0 rule, t requres many structures and an excessve amount of ponter manpulatons for effcently processng and searchng large documents In ths paper, we propose a new and conceptually powerful data structure, called splay suffx arrays (SSA), for strng search Ths data structure combnes the features of splay tree and suffx arrays nto a new approach whch s sutable to mplementaton on both conventonal and clustered computers Keywords suffx arrays, splay tree, strng search, dstrbuted algorthm T I INTRODUCTION ODAY large databases become avalable, such as full text of newspapers of Web pages, and Genome sequences, therefore t s mportant to effcently store them on memory for quck queres In order to acheve ths goal, some ndexng data structures and searchng tools have been ntroduced The man approaches are: word ndexes, character n-gram ndexes, and suffx ndexes Word ndexes have the advantage of supportng very fast word queres, whle they have dffculty wth ndexng unstructured texts- lke DNA-sequences or some Asan language texts [] Whle character n-gram ndexes enable us to ndex unstructured texts, the search for a lengthy query or regular expresson s complcated and neffcent [2]Suffx ndexes have been desgned to overcome the above lmtatons by dealng wth arbtrary texts, but ths ncreases the cost due to the addtonal space occuped by the underlyng ndexng data Manuscrpt receved November 4, 2005 TU Kun s wth the Department of Computer scence and technology, Unversty of Scence and Technology of Chna, Hefe, Anhu, PRChna, (correspondng author to provde phone: ; e-mal: tukun@ustcedu) GU Na-je s wth the Department of Computer scence and technology, Unversty of Scence and Technology of Chna, (e-mal: gunj@ustceducn) BI Kun s wth the Department of Computer scence and technology, Unversty of Scence and Technology of Chna, (e-mal: bkun@malustceducn) LIU Gang s wth the Department of Computer scence and technology, Unversty of Scence and Technology of Chna,(e-mal: lugang@malustceducn) DONG Wan-l s wth the Department of Computer scence and technology, Unversty of Scence and Technology of Chna, (e-mal: danl@malustceducn) structure [3] [4] [5] Examples of such ndexes are: suffx tree [6], suffx array [7] Smlar concepts were ndependently proposed n Oxford Englsh Dctonary project [8] and n corpus-based natural language processng [9] Suffx ndexes can be used n many applcatons [0] A suffx tree for a text of length n over an alphabet s of sze ON ( ) (where N s the text sze) and can be bult n ON ( log ) Suffx tree enables us to fnd the longest substrng of a text that matches the query strng n OM ( ) tme, where M s the length of the pattern strng A problem of the suffx tree s ts sze and can be qute large for many applcatons The suffx array s the most compact and smple among the suffx ndexes mentoned above The constructon and searchng tme of the suffx array does not depend on the sze of alphabet The searchng tme s compettve wth the suffx tree s n practce There are a number of dfferent dstrbuted multlevel data structures that have been nvestgated for multkey searchng and sortng on both conventonal and dstrbuted computers [] [2] But they have shortcomngs and drawbacks dealng wth strng problems MSA (Multdmensonal Suffx Arrays) [3] s a new data structure for strng search whch combnes the features of suffx arrays and B-Trees, and s very amenable to mplementaton on both conventonal and clustered computers But MSA does not consder about the stuaton that access pattern follows the 90/0 rule That means we should pay the same cost when we fnd somethng that has been just found Ths paper focuses on the performance of strng search usng suffx array when the access pattern follows the 90/0 rule In ths paper, we propose a new structure, the splay suffx arrays, for nvestgatng and refnng hgh-performance algorthms for searchng strngs on both conventonal computers and clusters The paper s organzed as follows Secton II ntroduces the related work of strng search usng suffx array Secton III gves the problem statement Then we descrbe the splay suffx array (SSA) and dstrbuted splay suffx array (DSSA) and ther algorthms n secton IV Secton V shows the expermental results, and secton VI s the concluson II RELATED WORK A suffx array s a lnear structure composed of ponters to every suffx n the text (snce the user normally bases hs queres upon words and phrases, t s customary, n documents, to ndex only word begnnngs) These ndex ponters are sorted accordng to a lexcographcal orderng of ther respectve suffxes To fnd patterns n the text, bnary search s performed Internatonal Scholarly and Scentfc Research & Innovaton (0) scholarwasetorg/ /9022

2 World Academy of Scence, Engneerng and Technology Internatonal Journal of Computer and Informaton Engneerng Vol:, No:0, 2007 Internatonal Scence Index, Computer and Informaton Engneerng Vol:, No:0, 2007 wasetorg/publcaton/9022 on the array at O(log N) cost Suffx array of strng S, denoted SA, s an array of sze n storng the sorted order of suffxes of text, e, SA[] s j ff suff j s the th lexcographcally smallest suffx A closely related concept s the longest common prefx (lcp) whch s an array of some length often coupled wth suffx arrays One mportant nformaton about lcp s the length of lcp of adjacent suffxes n the suffx arrays The smplest approach to buld the suffx array sequentally s to perform a tradtonal sort of the ponters, such as mergesort of qucksort However, there exst specalzed algorthms for sequental constructon of suffx arrays, such as the orgnal one of Manber and Myers [7] and, more recently, those of Sadakane [4] MSD radx sort [5] and Multkey Qucksort [6] are known as the fastest algorthms for sortng strngs lexcographcally Many dstrbuted algorthms for ths problem are generalzatons of general purpose sortng algorthms adapted to suffx arrays: mergesort(msort [7]) qucksort (Qsort [8] and G-Qsort [9]) and of MMsort [20] have been used All above algorthms pad man attenton to the constructon of suffx array Dfferent from them, MSA (Multdmensonal Suffx Arrays) [3] proposed a new data structure for nvestgatng and refnng hgh-performance algorthms for searchng strngs on both conventonal computers and clusters The man drawback of MSA s that we should pay the same cost when we fnd somethng that has been just found, especally when the access pattern follows the 90/0 rule We combne the features of splay trees and suffx arrays nto a new data structure The purpose of ths s to mprove the performance of suffx arrays when the access pattern follows the 90/0 rule and reduce the amount of ponter manpulatons for effcently processng at the same tme III PROBLEM STATEMENT Let S=a 0, a,,a N- be a text of length N Denote the substrng a, a +,, a j by SA[,j] Each a s a member of the fnte alphabet Denote the suffx that starts at poston n the text S by S =SA[,N]= a, a +,, a N- The number s called ndex of the suffx S The suffx array SA bult on S s an array of length N storng the sequence of ndexes p 0,p,,p N- such that S p 0 < S p < < S p N-, where < denotes the lexcographc order Now assume that we wll perform a long seres of T > N operatons on a suffx array We concern about the performance of suffx array when the access pattern follows the 90/0 rule That means we often fnd somethng that has been just found IV DISTRIBUTED SPLAY SUFFIX ARRAYS (DSSA) A Splay tree A splay tree [2] s a varety of self-adjustng bnary search tree Statc bnary search trees ether do not adjust n response to changng balance or access patterns, or f they do (as n AVL trees), nformaton s copously mantaned at each node to mantan balance Sleator and Tarjan [2] devsed splay trees to allow a bnary tree to self-adjust n response to varyng access patterns and yet reman approxmately balanced wthout storng addtonal balance nformaton Then, nstead of provdng a frm O(log N ) tme guarantee for each operaton, the amortzed tme s O(log N) whle some ndvdual operatons may be more expensve The fundamental heurstc used n splay trees to accomplsh ths task s called SPLAYING SPLAYING moves the currently accessed node to the root of the tree through a seres of rotatons whle keepng the tree roughly balanced durng ths move Thus, SPLAYING can re-balance a tree as well as reduce the amortzed cost of accessng nodes by keepng frequently accessed nodes near the root of the tree B SSA and SSA algorthms Dfferent from standard splay trees, each node n splay suffx array contans a value to pont out the start poston of the correspond suffxes n suffx arrays So we can apply Manber and Myer s strng searchng method [7] frstly to the suffxes whch are often accessed We show the splay suffx arrays of text BANANA n Fg Fg Splay Suffx Arrays (SSA) ) The constructon of SSA Compare wth suffx array, splay suffx array takes advantage of splay tree to mprove the performance of strng searchng The cost s the operaton of SPLAYING and the space to store the splay tree The followng lemma captures the essence of how we construct the splay suffx array Lemma : Let P(x, M, N) be the performance of strng searchng usng data structure x, then P( SSA, M, N ) P( suffxarray, M, N ), f T log( ) T = N R = log N Where M s the sze of pattern strng, and N s the sze of text, R s the number of nodes of splay suffx array, and N s the number of the suffxes that belong to the node R of splay suffx arrays Proof: From [2] and [7], we know that the max complexty of T operatons of splay tree whch has R nodes s OT ( log R ),and the complexty of strng searchng of suffx array s OM ( + log N) Then t s easy to see: P( SSA, M, N ) = T log R + T * M + T = log N = T * M + T = log( N* R) P( suffxarray, M, N ) = T * M + T log N = T * M + T = logn Internatonal Scholarly and Scentfc Research & Innovaton (0) scholarwasetorg/ /9022

3 World Academy of Scence, Engneerng and Technology Internatonal Journal of Computer and Informaton Engneerng Vol:, No:0, 2007 Internatonal Scence Index, Computer and Informaton Engneerng Vol:, No:0, 2007 wasetorg/publcaton/9022 Q T log( N * R) T = = log N P( SSA, M, N ) P( suffxarray, M, N ) Now we are ready to descrbe the algorthm of the constructon of SSA The key pont here s T log( ) T = N R = log N That means to keep N [N/R ]when the correspond node n SSA s deep, and let the nodes that have more than[n/r]suffxes to be near the root So when we construct the SSA, we perform the followng steps: a) Dvde all suffxes nto dfferent buckets accordng to ther frst three characters We frst propose that strngs be processed on a character by character for the frst three characters The reason for ths splttng after the frst three characters s that many common words n the Englsh dctonary contan the same frst three characters [3] For example, then, theory, theatre So we can dvde the text nto dfferent buckets, suffxes n the same bucket have same frst three characters We can put one or more buckets nto a node On the other hand, f there are too many suffxes n one bucket, we can dvde t nto several nodes Then we can apply Manber and Myer s strng sortng method [7] to each node to fnsh the constructon of splay suffx array The complexty wll be ON ( ) b) Let every bucket be a node of SSAThe complexty wll be OR ( ) c) Use Manber and Myer s strng sortng method [7] to fnsh the constructon of suffx array n each node The T complexty wll be O( ) = N 2) Strng searchng Let f ( s) denote the frst three characters of strng s, g( R ) denote the frst suffx of the suffxes that belong to the node R of SSA, where R 0 denotes the root node Let m denote the pattern strng Then we can perform the followng steps: a) If f ( m) = f ( g( R )),then use Manber and Myer s strng 0 searchng method, the complexty wll be O(log N + M) ;else do b) 0 b) If f ( m) < f ( g( R )),then compare f(m) and f ( g( R lchld )) ;else compare f(m) and f ( g( R rchld)) c) Do ths recursvely tll we fnd the node R that f ( m) = f ( g( R )) and then do d), or f there s no such node then we can say that no match poston The complexty wll be O(log R ) d) Use Manber and Myer s strng searchng method, the complexty wll be O(log N M) + Adjust the structure; the complexty wll be O(log R ) C DSSA and DSSA algorthms Our dstrbuted model s that of a cluster Assume that we have a number p of computers, we call each one computng node A problem n the mplementaton of DSSA s how to reduce the amount of ponter manpulatons and communcaton complexty for effcently processng In order to resolve ths we change the structure of SSA a lttle If there are too many nodes n SSA, then the cost wll be hgh when there s only one computng node to mantan all of SSA nodes So we choose some of computng nodes to mantan the nodes of SSA We just keep 3~5 SSA nodes n one computng node, and then the SPLAYING can be operated effcently n t We call these computng nodes SSA-keepng nodes Each leaf node n SSA-keepng node keeps a value to pont another SSA-keepng node The relatonshps between these SSA-keepng nodes are same as the relatonshps between the nodes of splay tree We choose one computng node, called Home node, to mantan the splay tree that represents the relatonshp between the SSA-keepng nodes We show the DSSA of a cluster whch has 0 nodes n Fg 2a and Fg 2b Fg 2(a) The real structure of the cluster Home node Fg 2 (b) The correspond logcal structure of DSSA of the cluster n Fg 2 (a) We keep the structure of SSA n the SSA keepng nodes (node, 4, 7) nstead of keepng the structure n all of the computng nodes And we mantan the splay tree that represents the relatonshp of SSA keepng nodes n the Home node (node 0) Now only Home node and SSA-keepng nodes need to communcate when the pattern strng s not found, and each node needs to communcate no more than two SSA-keepng nodes ) The constructon of DSSA Let R be the number of SSA keepng nodes and P denote the Internatonal Scholarly and Scentfc Research & Innovaton (0) scholarwasetorg/ /9022

4 World Academy of Scence, Engneerng and Technology Internatonal Journal of Computer and Informaton Engneerng Vol:, No:0, 2007 Internatonal Scence Index, Computer and Informaton Engneerng Vol:, No:0, 2007 wasetorg/publcaton/9022 number of computng nodes It s clear that R P Then we can perform the followng steps: a) Dvde all suffxes nto dfferent buckets accordng to ther frst three characters The complexty wll be ON ( ) Send each bucket to the correspond computng node The communcaton complexty wll be OP ( ) b) Let every bucket be a node of SSA and choose some of the computng nodes to be SSA keepng node Construct the splay tree that represents the relatonshp between the SSA-keepng nodes n Home node The complexty wll be OR ( ) c) Use Manber and Myer s strng sortng method [7] to fnsh the constructon of suffx array n each node Because every node can do ths parallel, the complexty wll be O(logN max ), where N max =max (N, P) 2) Strng searchng a) Fnd the SSA keepng node R n Home node R0 that f ( m) f ( g( R )), and adjust the structure of splay tree that R0 keeps The complexty wll be OT ( log R ), f we do ths operaton T tmes b) Let R lssa denote the left SSA keepng node of R and R rssa denote the rght SSA keepng node of R If f ( m) < f ( g( R )),then send message to R lssa, and compare f ( m) and f ( g( R lssa)) ;else send message to R rssa,compare f ( m) and f ( g( R rssa)) Do ths recursvely tll we fnd the node R that f ( m) = f ( g( R )) and then do c), or f there s no such node then we can say that no match poston The complexty wll be O(log R ) And the communcaton complexty wll be O(log R ) c) Use Manber and Myer s strng searchng method, the complexty wll be O(log N M) + Adjust the structure; the complexty wll be O(log R ) We show the process of strng searchng n Fg 3 Assume that the strng we fnd s n node 5 Frstly, we fnd the rght SSA keepng node n Home node In ths case, the word rght means that we can fnd the clue of node 5 n t And then adjust the structure of splay tree n Home node Secondly, we search the SSA keepng node contnuously tll we fnd the node 5 Fnally, use Manber and Myer s strng searchng method to fnd the strng And then we adjust the structure V EXPERIMENTAL RESULTS In ths secton, we show the performance of DSSA We mplemented DSSA usng MPI The programs were executed on HP RX2600 cluster Fg 5 shows the relatonshp between the SSA and the access patterns whch are shown n Fg 4 The sze of text strng s 6kB, and the pattern strng s s 8kB From Fg4 we know that when the access pattern follows the 90/0 rule, we access the strngs that begn wth the more often than we access the strngs that begn wth acc It s clear n Fg5 that the DSSA s neffcent when we fnd somethng randomly The reason s that we should change the structure of DSSA when we access t But t s amenable to mprovement when the access pattern follows the 90/0 rule Because t takes advantage of the fact that we often fnd somethng that we have just found We compare the strng searchng tme between the DSSA and one of the dstrbuted versons of suffx array [20] The results presented n Fg 6 show the fact that DSSA s very sutable for the stuaton that we often fnd somethng that we have just found The frst three characters of pattern strngs Access tme of DSSA ( s ) the sea add acc Fg 3 The process of strng searchng "90/0" pattern Random pattern Tme (s) Fg 4 Two dfferent access patterns "90/0" pattern Random pattern Tme ( s ) Fg 5 The relatonshp between DSSA and access pattern Internatonal Scholarly and Scentfc Research & Innovaton (0) scholarwasetorg/ /9022

5 World Academy of Scence, Engneerng and Technology Internatonal Journal of Computer and Informaton Engneerng Vol:, No:0, 2007 Internatonal Scence Index, Computer and Informaton Engneerng Vol:, No:0, 2007 wasetorg/publcaton/9022 VI CONCLUSION We propose a new structure for strng searchng The structure, called Splay Suffx Array (SSA), combnes the features of splay trees and suffx arrays nto a new data structure The purpose of ths s to mprove the performance of suffx arrays when the access pattern follows the 90/0 rule and reduce the amount of ponter manpulatons for effcently processng DSSA offer substantal advantages over both strng splay trees and suffx arrays n terms of memory space and tme The complexty of DSSA s very amenable to performance mprovement through several parameters, ncludng effcent mplementaton of DSSA, explotaton of parallelsm, and the hgh-performance capabltes of computer cluster archtecture Executng tme ( s ) DSSA Suffx Array Tme ( s ) Fg 6 Comparson between DSSA and Suffx Array REFERENCES [] I H Wrtten, A Moffat, and T C Bell Managng Ggabytes: Compressng and Indexng Documents and Images Van Nostrand Renhold, 994 [2] W B Cavnar Usng an n-gram based document representaton wth a vector processng retreval model In Proc Of the 3 rd Text Retreval Conference (TREC-3), pages NIST specal publcaton, 995 [3] G A Stephen Strng Searchng Algorthms World Scentfc Publshng, 994 [4] M Crochemore and W Rytter Text Algorthms Oxford Unv Press, New York, 994 [5] D Gusfeld Algorthms on Strngs, Trees, and Sequences Cambrdge Unv Press, 997 [6] E M McCreghl A Space-Economcal Suffx Tree Constructon Algorthm Journal of the ACM, 23, pp , 976 [7] U Manber and G Myers Suffx Arrays: A New Method for On-Lne Strng Searches SIAM J Comput 22(5), pp , 993 [8] G H Gonnet, R A Baeza-Yates, et al New ndces for text: PAT trees and PAT arrays In W B Frakes and R A Baeza-Yates, edtors, Informaton Retreval: Data Structure and Algorthms, pages Prentce-Hall, New Jersey, 992 [9] M Nagao and S Mor A new method of n-gram statstcs for large number of n and automatc extracton of words and phrase form large text data of Japanese In Proc of COLING 94, pages 6-65,994 [0] A Apostolco The myrad vrtues of subword trees In Combnatoral Algorthms on Words, pages Sprnger-Verlag, 985 [] A Fellah Concurrent and Dstrbuted Data Structures for Multkeys Sortng on Computer Clusters IEEE Proc Of the 6 th Intern Symp On Hgh Performance Comput Systems and Applcatons Pp 28-, Moncton, Canada, 2002 [2] A Fellah, A Maamr and M Abaza Dstrbuted Data Structures for Multkey Sortng Intern J of Parallel and Dstrbuted Systems and Networks, Vol 2(2), pp 62-68, 999 [3] Fellah, A and Mawson, R Dstrbuted multdmensonal suffx arrays for strng search Communcatons, Computers and sgnal Processng, 2003 PACRIM 2003 IEEE Pacfc Rm Conference on, Volume: 2, pp , 2003 [4] K Sadakane A fast algorthm for makng suffx arrays and for burrows-wheeler transformaton In Proc DCC 98, pages 29-38, 998 [5] D E Knuth Sortng and Searchng, volume 3 of The Art of Computer Programmng Addson-Wesley, 973 [6] J L Bentley and R Sedgewck Fast algorthms for sortng and searchng strngs In the 8 th Annual ACM-SIAM Sympo On Descrete Algorthms, pages , 997 [7] J Ktajma, G Navarro, B Rbero, and N Zvan Dstrbuted generaton of suffx arrays: a qucksort-based approach In Proc WSP 97, pages Carleton Unversty Press, 997 [8] J Ktajma, B Rbero, and N Zvan Network and memory analyss n dstrbuted parallel generaton of PAT arrays In Proc 8 th Brazlan Symp On Comp Arch Hgh-Performance Processng, pages Brazlan Comp Soc, 996 [9] G Navarro, J Ktajma, B Rbero, and N Zvan Dstrbuted generaton of suffx arrays In Proc CPM 97, LNCS 264, pages 02-5, 997 [20] Ktajma, JP, Navarro, G A fast dstrbuted suffx array generaton algorthm Strng Processng and [2] Informaton Retreval Symposum,999 and Internatonal Workshop on Groupware, Sept,999,22-24 Pages:97-04 [22] D D Sleator and R E Tarjan Self-adjustng Bnary Search Trees Journal of the ACM 32, pp , 985 Tu Kun was born n 980 He s a PhD student n the Department of Computer Scence and Technology, USTC Hs research nterests nclude parallel and dstrbuted computng Gu Na-je was born n 96 He s a Professor and Doctoral Advsor n the Department of Computer Scence and Technology, USTC Hs research nterests nclude parallel computng archtecture, nterprocessor communcaton, and hgh-performance computng B Kun was born n 98 He s a PhD student n the Department of Computer Scence and Technology, USTC Hs research nterests nclude parallel and dstrbuted computng Lu Gang was born n 978 He s a PhD student n the Department of Computer Scence and Technology, USTC Hs research nterests nclude nterprocessor communcaton and moble computng Dong Wan l was born n 98 He s a PhD student n the Department of Computer Scence and Technology, USTC Hs research nterests nclude parallel and dstrbuted computng Internatonal Scholarly and Scentfc Research & Innovaton (0) scholarwasetorg/ /9022