On the usage of Sorting Networks to Big Data

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1 On he sage of Soring Neorks o Big Daa Blanca López and Nareli Crz-Corés Arificial Inelligence Laboraory, Cenro de Invesigación en Compación, Insio Poliécnico Nacional (CIC-IPN), México DF, México Conry Absrac Soring daa in a comper is maybe he mos poplar classical ask in Comper Science For he majoriy of applicaions he main goal is o minimize he nmber of comparisons and execion ime ha he soring algorihm consmes Soring Neorks are algorihms ha perform exacly he same nmber of comparisons o order any inp permaion for a given inp daa size Tha is, each sep does no depend on he resl of a previos comparisons Ths, designing Soring Neorks ih a minimal nmber of comparisons becomes a very imporan ask Hoever, i is an NP-hard problem Acally, he opimal Soring Neorks ih a minimal nmber comparisons (or a leas close o he opimal) for small inp daa sizes from o 6 are pblished in he specialized lierare Of corse, hese inp daa sizes are very small o be sed in real orld problems In his ork e propose a ne sraegy o improve he QickSor performance by copling i ih some Soring Neorks o large inp daa The resls demonsrae i helps redcing he soring execion ime Keyords: Soring Neorks, QickSor Inrodcion Soring Algorihms are maybe one of he mos sdied problems in Comper Science, from he heoreical and pracical poins of vie Applicaions of hem can be fond in Daa Processing Sysems, Neork Commnicaion Sysems, Image Processing, Arificial Inelligence, Crypography, Comper Secriy, Informaion Sysems, among many ohers A large se of Soring Algorihms can be fond in he specialized lierare, sch as: qicksor, bbble sor, merge sor, shell sor, heapsor, inserion, inrosor, shear soring, ec Choosing he mos efficien algorihm sally depends on he ype of applicaion a hand In general, he Soring Algorihms can be classified ino o grops: he adapive and non-adapive An adapive algorihm execes is compare-inerchange operaions depending on he inp daa On he oher hand, he non-adapive algorihms have fixed operaions hich are execed no maer he configraion of he inp daa (e g all he possible permaions) They alays exece he same compare-inerchange operaions Soring Neorks (SN) are an example of he non-adapive algorihms Taking advanage of he divide-and-conqer sraegy ilized by he QickSor, i is designed a sraegy here some SN are copled o i in order o redce he comparisons performed by he QickSor The remaining of his paper is organized as follos In Secion some basic conceps abo Qicksor and Soring Neorks are presened In Secion he proposal is explained Secion 4 presens he experimens and resls Finally in Secion 5 some conclsions are dran Basic Conceps Qicksor Algorihm Qicksor (also knon as Pariion-Exchange Sor) as firs presened in 960 by Tony Hoare [4] I ses a divideand-conqer sraegy by dividing a large lis ino o smaller sbliss A sblis ih he smalles vales and anoher ih he greaes Then, each sblis is recrsively ordered The algorihm is as follos: ) Choose an elemen from he lis ha ill be called pivo ) Order he lis in sch a ay ha all he vales hich are less han he pivo ill be locaed o is lef (before he pivo) Frher, all he vales greaer han he pivo ill be locaed o is righ (afer he pivo) This ay, he vale in he pivo is on is final posiion ) For each sblis, repea he previos seps in a recrsively manner nil he sbliss size is zero or one This idea is illsraed in Figre QickSor is a very efficien algorihm ha on he average and bes cases makes O(n log n) comparisons for soring n elemens In he ors case i makes O(n ) Some varians o his algorihm have been presened in [6][] here heir ahors proposed some modificaions o redce he execion ime Soring Neorks SN are algorihms ih he main feare of being oblivios, i means ha heir crren operaions (comparisons) do no depend on he inp daa or he previos comparisons [5][7] Unlike oher ell knon soring algorihms (bbble

2 Pivo Pivo Pivo Ieraion x 0 =4 x = x = x = c 0 c c c c 4 y 0 = y = y = y =4 Fig Ieraion RECURSIVE PARTITION OPERATION OF THE QUICKSORT ALGORITHM sor, qicksor, ec), he seqence and nmber of comparisons are exacly he same no maer he inp configraion (permaion) The SN exhibis o main feares: The comparisons (called comparaors) are fixed before he SN execion, Some comparisons can be execed in a parallel manner A SN is composed by a se of comparaors, here each of hem execes an acion compare-inerchange beeen o elemens (a, b) The elemen a ms be no graer han b, if so, he vales ms be inerchanged o (b, a) So, for a given inp lis ih size n, he se of comparaors conforming he SN are applied o i, hen he op is he lis monoonically non decreasing ordered Typically, he SN are graphically represened by n horizonal lines represening he n inp daa Frher, some verical lines ha represen comparisons beeen he vale a is op exreme and he vale a is boom If he vale a he op is graer han he vale a he boom, hese vales ms be sapped The inp daa are placed a he lef, hen, afer hey have raveled across he horizonal lines and execed he comparisons fond, he op is obained a he righ The daa ms be ascendan sored from op o boom See for example a SN for n = 4 inps illsraed in Figre Each inp daa is se on he horizonal lines labeled as x 0, x, x, x The verical lines are he comparaors c 0, c, c, c, c 4, each receiving o vales, i e, he comparaor c 0 receives he vales x 0 and x, and so on All he daa vales go from lef o righ execing a Fig SORTING NETWORK FOR n = 4 INPUTS compare-inerchange each ime a comparaor is fond So, he comparaors c 0 and c are execed firs, hen c and c, and finally c 4 c 0 evalaes 4 >, hs he vales of x 0 and x are sapped c evalaes <, so he vales of x and x remain iho change This process conines nil all he comparaors are applied, so he final sored lis y 0, y, y, y a he righ accomplishes y 0 y y y As a maer of fac, if an opimal SN for inp size n can be designed (i e ih minimal nmber of comparaors), hen i means ha is he bes manner o sor n daa Designing SN ih minimal nmber of comparaors and/or high parallelism is a classical ineresing problem in Comper Science Acally, noadays i is an open research area I is imporan o noice ha he opimal SN for inp size greaer han n = 6 are no kno Acally, only loer bonds regarding he nmber of comparaors are heoreically knon [5] The mos sdied SN is he one ih inp size n = 6, hich is a relaively small vale, considering he hge qaniy of informaion ha he modern sysems ms handle The bes knon SN n = 6 has only 60 comparaors, for example, he one designed by Green [5] is illsraed in Figre In [] K E Bacher proposed an ineresing algorihm called Merge Odd-Even o merge o SN ino one Tha is, if e have a SN ih inp size n, hen, i is possible o obain a SN ih inp size n by merging o copies of he original SN size n each By folloing his algorihm i is possible o obain SN ih larger inp sizes An example, o increase he size of inp daa in n from SN for n = 4 A se of operaions o order and o op liss g and h are considered In he Figre 4 are shon o liss o re-arrange The lis has he nmbers {,, g } in ordered A he same ime, second lis called are composed by {,,, h } The g + h is he op of he merging neork, he nmbers of he merged liss in ascending order are {,, g+h, g+h } ie, a firs, a lis g + h can be bild by merging neork ih he odd-indexed nmbers of he o inp liss and he even- Usally SN for inp sizes greaer han n = 6 are considered as large

3 sep sep sep sep4 sep5 sep6 g h Fig SN WITH INPUT SIZE n = 6 DESIGNED BY GREEN IT IS THE BEST KNOW WITH 60 COMPARATORS Fig 5 ODD-EVEN MERGESORT SCHEME TWO SN FOR n = 8 INPUTS IS CONSTRUCTED BY TWO SN FOR n = 4 Co C Co C C C C C C4 C4 g h M E R G E g+h O d d E v e n x x x y y y Fig 4 ODD-EVEN MERGESORT SCHEME FOR TO ORDER TWO SN FOR n = 8 INPUTS IS CONSTRUCTED BY TWO SN FOR n = 4 Fig 6 ODD-EVEN MERGING SCHEME indexed nmbers of he o inp liss The loes op of he odd merge is lef alone and becomes he loes nmber of final lis The seps o ordered o liss are : ) Merge he keys of by odd-indexed {,, 5, } ih he odd-indexed {,, 5, } o form he seqences {x, x, x, } of he keys in order ) Merge he keys of by even-indexed {, 4, 6, } ih he even-indexed {, 4, 6, } o form he seqences {y, y, y, } of he keys in order ) Then se = x and se parallel comparaors o se i = min(x i+, y i ) and i+ = max(x i+, y i ) i = {,,, } The sep and sep can be apply in parallel and each of he involves abo (g + )/ keys A more deailed informaion abo he Theorem is [][5][][8] In regard o bild a SN o n = 8 from o SN for n = 4 ih he Odd-Even merge mehod, he Figre 5 exhibi he nmber of comparaors ha i has fll I has 9 comparaors in 6 seps or layers The sep arranged o lis o n = 4, boh in non decreasing order In sep 4 as sed for comparaors o re-arrange (merge) he 8 elemens and hen, o comparaors o merge he for ordered liss (sep 5) and hen, in sep 6 as sed hree comparaors o merge he ordered seqences o form one ordered lis conaining all 8 elemens Neverheless, his mehod orks beer for small inp sizes, and decreases is performance as he inp sizes increase, ie for large inp sizes he resling SN old have more comparaors han he opimal Noice ha for a given inp daa size n he corresponding SN has fixed is comparaors, hich means ha a ne SN ms be designed if he inp daa size is differen han n

4 In his sense he SN are less flexible han he convenional adapive soring algorihms Proposal Considering ha he QickSor is a very efficien algorihm based on a divide-and-conqer sraegy, and he opimal SN are only knon for small inp daa sizes, his proposal consiss on copling a SN ih he QickSor algorihm o order big inp daa in an efficien manner The algorihm ill be named Qick+SN The general idea is o apply QickSor in convenional ay o he inp daa as many imes as necessary nil he sbliss are small enogh o be sored by a given SN of inp size n This idea is illsraed in Figre 7 Wih his small change in he QickSor i is possible o improve is execion ime hile mainaining is flexibiliy, in he sense ha he algorihm is able o order any inp daa size Le s sppose ha e have seleced a deermined SN o ork ih an inp size eqal o n = 6 (hich is he greaes near-opimal knon) For a given inp daa A o be ordered ih size Z (here Z is a large nmber), he algorihm Qick+SN is defined as follos: ) Divide an array A ino o pars (a l and a r ) by selecing he elemen in he middle posiion as pivo denoed by m ) Compare each elemen of a l and a r agains he pivo and move all he elemens less han m o he lef array a l, and all he elemens greaer han m o he righ array a r ) Verify if he resling liss are size n: If so, hen he sblis is sored by he SN If no, hen go o sep recrsively The op of his algorihm is he lis A compleely ordered Noice ha Qick+SN orks by applying a specific SN for a deermined inp size n Therefore, only if a resling lis has exacly n elemens hen he SN can be applied o i Each ime ha he QickSor splis he lis ino o, i is no possible o kno he resling sizes Hence, i is no possible o kno in advance he nmber of imes ha he SN ill be applied 4 Experimenal design In order o assess he proposed algorihm efficiency, a se of experimens ere condced The Qick+SN as applied o daa liss (nmbers) ih differen inp sizes, inp permaions Addiionally, i as experimened by sing differen SN The configraion for each experimen is a combinaion of he folloing opions: Inp daa size: To liss conformed by and nmbers o be sored ere sed as inp Inp permaions: Three differen configraions of he inp liss ere sed: ly generaed nmbers, SN Pivo SN Pivo Fig 7 Pivo SN SCHEME OF THE PROPOSED QUICK+SN ALGORITHM Ieraion Ieraion nmbers hich are already ordered, and inverse ordered nmbers To differen SN ere ilized for n = 6 and n = 56 Therefore, e have elve experimen ses ih differen configraions Each se as execed 0 imes The ime and comparisons performed as aken by he Qick+SN applied o each se as obained The saisical resls abo he ime are shon in Table and and respec o he nmber of comparisons are exhibis in he Table and 4 All of hem have as firs colmn he inp configraion, as second colmn sho he size of he inp daa, in he hird colmns are shoing he ime consmed (in seconds) sing Qick+SN and he oher case, he nmber of comparisons performed In he las colmn he original Qicksor resls are shon The Table and are presened he ime and comparison performed of a SN for n = 6 The SN ih n = 56 as designed by combining copies of Green s SN (shon in Figre ) ih he algorihm Merge Odd-Even menioned in Secion Is performance is presened in Table and 4 I can be observed for he case of random sored inp daa, ha he Qick+SN ih n = 6 operforms he original QickSor Hoever, for he cases here he inp daa as already sored or invered, he QickSor obained beer resls han Qick+SN n = 6 Ths, for all he configraions and inp daa sizes, he Qick+SN ih n = 56 obained bes resls ih he minimal execion imes

5 Table STATISTICAL RESULTS (IN SECONDS) OF QUICK+SN AND QUICKSORT SN TO n = 6 WAS UTILIZED Inp Inp +SN Original Configraion Size Z ih n = 6 QickSor Average 098 Median Bes 089 Wors 0947 Average 40 Median 07 Bes 977 Wors Table STATISTICAL RESULTS (IN COMPARISONS PERFORMED) OF QUICK+SN AND QUICKSORT SN TO n = 6 WAS UTILIZED Inp Inp +SN Original Configraion Size Z ih n = 6 QickSor Average 9587 Median 077 Bes Wors Average Median Bes Wors Average 0787 Median Bes 0097 Wors Average Median n Bes Wors Average 4 Median 9669 Bes 908 Wors 5499 Average Median Bes Wors Average 667 Median 466 Bes 4569 Wors Average Median Bes Wors Average Median Bes Wors 759 Average Median Bes Wors Table STATISTICAL RESULTS (IN SECONDS) OF QUICK+SN AND QUICKSORT SN TO n = 56 WAS UTILIZED Inp Inp +SN Original Configraion Size Z ih n = 56 QickSor Average 0046 Median Bes Wors 0485 Average Median Bes 058 Wors Table 4 STATISTICAL RESULTS (IN COMPARISONS PERFORMED) OF QUICK+SN AND QUICKSORT SN TO n = 56 WAS UTILIZED Inp Inp +SN Original Configraion Size Z ih n = 56 QickSor Average Median Bes Wors 874 Average Median Bes Wors Average Median Bes Wors Average Median n Bes Wors Average Median 059 Bes Wors Average Median Bes Wors Average Median 0098 Bes 0095 Wors Average Median Bes Wors Average Median Bes Wors Average Median Bes Wors

6 5 Conclsions Taking advanage of heir inheren feares, i as proposed a combinaion of he ell-knon algorihm QickSor ih he algorihm called Soring Neorks The experimenal resls shoed ha he proposal is compeiive by obaining beer execion imes han he original QickSor I as also noiced ha he execion imes ere improved hen SN for larger inp daa sizes ere ilized I is necessary o experimen ih larger inp daa, and also ih SN for differen sizes Frher, a formal sdy relaed o he algorihms complexiy is necessary 6 Acknoledgmens The ahors acknoledges sppor from CONACyT hrogh projecs nmber 804 and 07 The firs ahor acknoledges sppor from CONACyT hrogh a scholarship o prse gradae sdies a CIC-IPN References [] S W Baddar and K E Bacher Designing Soring Neorks: A ne Paradigm Springer, 0 [] K E Bacher Soring neorks and heir applicaions In Proceedings of he April 0 May, 968, spring join comper conference, AFIPS 68 (Spring), pages 07 4, Ne York, NY, USA, 968 ACM [] D Canone and G Cincoi Qickheapsor, an efficien mix of classical soring algorihms In Gambosi G Bongiovanni GC and Pereschi R, ediors, CIAC, volme 767 of Lecre Noes in Comper Science, pages 50 6 Springer, 000 [4] C A R Hoare Algorihm 64: Qicksor Commn ACM, 4(7):, Jly 96 [5] D E Knh The Ar of Comper Programming, Volme III: Soring and Searching, nd Ediion Addison-Wesley, 998 [6] U Kocamaz Increasing he efficiency of qicksor sing a neral neork based algorihm selecion model Inf Sci, 9:94 05, April 0 [7] D G OConnor and R J Nelson Soring sysem ih n-line soring sich Unied Saes Paen nmber,09,4, 6, April, [8] Kenneh E Bacher Sherenaz W Al-Haj Baddar