University of Crete Computer Science Department QUERY ORDERING BASED TOP-K ALGORITHMS FOR QUALITATIVELY SPECIFIED PREFERENCES IOANNIS KAPANTAIDAKIS

Transcription

1 Unversty of Crete Computer Scence Department QUERY ORDERING BASED TOP-K ALGORITHMS FOR QUALITATIVELY SPECIFIED PREFERENCES by IOANNIS KAPANTAIDAKIS Master s Thess Heraklon, January 27

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3 Unversty of Crete Computer Scence Department QUERY ORDERING BASED TOP-K ALGORITHMS FOR QUALITATIVELY SPECIFIED PREFERENCES by Ioanns Kapantadaks A thess submtted n partal fulfllment of the requrements for the degree of MASTER OF SCIENCE Author: Supervsory Commttee: Ioanns Kapantadaks, Computer Scence Department Vassls Chrstophdes, Assocate Professor, Supervsor Georgos Georgakopoulos, Assstant Professor, Member Ioanns Tztzkas, Assstant Professor, Member Approved by: Trahanas Panos, Professor Charman of the Graduate Studes Commttee Heraklon, January 27

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5 QUERY ORDERING BASED TOP-K ALGORITHMS FOR QUALITATIVELY SPECIFIED PREFERENCES Ioanns Kapantadaks Master Thess Unversty of Crete Computer Scence Department Abstract Preference modellng and management has attracted consderable attenton n the areas of Databases, Knowledge Bases and Informaton Retreval Systems n recent years. Ths nterest stems from the fact that a rapdly growng class of untraned lay users confront vast data collectons, usually through the Internet, typcally lackng a clear vew of ether content or structure, moreover, not even havng a partcular object n mnd. Rather, they are attemptng to dscover potentally useful objects, n other words, objects that best sut ther preferences. A modern nformaton system, consequently, should enable users to quckly focus on the k best object accordng to ther preferences. In ths thess, modellng preferences as bnary relatons, we ntroduce effcent algorthms for the evaluaton of the top-k objects. Prevous related work treated preference expressons as black boxes and dealt wth the dea of exhaustvely applyng domnance tests among database objects n order to determne the best ones, resultng n quadratc costs. On the contrary, we advocate a query orderng based approach. Our key dea s to explot the semantcs of the nput preference expresson tself, n terms of both the operators and the preferences nvolved, to defne an orderng over those queres, whose evaluaton s necessary for the retreval of the top-k objects. We ntroduce two novel algorthms, LBA and TBA. LBA defnes an orderng over queres whch are essentally conjunctons of atomc selecton condtons, contanng all attrbutes that the user preferences nvolve. The algorthm ensures that the way and order n whch objects are fetched respect user preferences, avodng any domnance testng, and accessng only the top-k objects,

6 each of them only once. From a dfferent angle, TBA defnes an order of queres whch are dsjunctons of atomc selecton condtons over sngle attrbutes, and uses approprate threshold values to sgnal object fetchng termnaton, ensurng that all remanng objects are worse than those fetched. Domnance tests are performed only for already retreved objects. Analytcal study and expermental evaluaton show that our algorthms outperform exstng ones under all problem nstances. Supervsor: Vassls Chrstophdes Assocate Professor

7 ΑΛΓΟΡΙΘΜΟΙ ΚΟΡΥΦΑΙΩΝ-Κ ΑΠΑΝΤΗΣΕΩΝ ΒΑΣΙΣΜΕΝΟΙ ΣΕ ΔΙΑΤΑΞΕΙΣ ΕΠΕΡΩΤΗΣΕΩΝ ΓΙΑ ΠΟΙΟΤΙΚΩΣ ΚΑΘΟΡΙΣΜΕΝΕΣ ΠΡΟΤΙΜΗΣΕΙΣ Ιωάννης Καπανταϊδάκης Μεταπτυχιακή Εργασία Τμήμα Επιστήμης Υπολογιστών Πανεπιστήμιο Κρήτης Περίληψη Τα τελευταία χρόνια, η μοντελοποίηση και η διαχείριση των προτιμήσεων έχουν προσελκύσει ιδιαίτερη προσοχή στους τομείς των Βάσεων Δεδομένων, των Βάσεων Γνώσης και των Συστημάτων Ανάκτησης Πληροφοριών. Αυτό το ενδιαφέρον πηγάζει από το γεγονός ότι ολοένα και περισσότεροι μη ειδικευμένοι κοινοί χρήστες έρχονται σε επαφή με τεράστιες συλλογές δεδομένων, συνήθως μέσω του Διαδικτύου, χωρίς, κατά κανόνα, να έχουν μια σαφή άποψη ούτε για το περιεχόμενο ούτε και για τη δομή της πληροφορίας, χωρίς καν να έχουν ένα συγκεκριμένο αντικείμενο υπ όψει τους. Πιο πολύ προσπαθούν να ανακαλύψουν αντικείμενα που ενδεχομένως θα τους είναι χρήσιμα, αντικείμενα, με άλλα λόγια, που ταιριάζουν καλύτερα στις προτιμήσεις τους. Συνεπώς, ένα σύγχρονο πληροφοριακό σύστημα θα πρέπει να διευκολύνει τους χρήστες στο γρήγορο εντοπισμό των k βέλτιστων αντικειμένων βάσει των προτιμήσεων τους. Στην εργασία αυτή, μοντελοποιώντας τις προτιμήσεις ως δυαδικές σχέσεις, εισάγουμε αποδοτικούς αλγορίθμους αποτίμησης των k βέλτιστων αντικειμένων. Η έως τώρα σχετική έρευνα, αντιμετώπιζε τις εκφράσεις επί προτιμήσεων ως «μαύρα κουτιά» και εφάρμοζε την ιδέα των εξαντλητικών διαδοχικών ελέγχων υπεροχής μεταξύ των αντικειμένων μιας βάσης δεδομένων για τον προσδιορισμό των καλύτερων εξ αυτών, γεγονός που οδηγούσε σε τετραγωνικά ως προς τον αριθμό των αντικειμένων κόστη. Αντιθέτως, εμείς υποστηρίζουμε μια προσέγγιση που

8 βασίζεται στη διάταξη επερωτήσεων. Η κύρια ιδέα μας βασίζεται στην εκμετάλλευση της σημασιολογίας μιας έκφρασης από προτιμήσεις, σε ό,τι αφορά τόσο τους εμπλεκόμενους τελεστές όσο και τις εμπλεκόμενες προτιμήσεις, με σκοπό τον ορισμό μιας διάταξης μεταξύ εκείνων των επερωτήσεων, των οποίων η αποτίμηση είναι αναγκαία, ώστε να ανακτηθούν τα k βέλτιστα αντικείμενα. Παρουσιάζουμε δύο πρωτότυπους αλγόριθμους, τους LBA και TBA. Ο LBA ορίζει μια διάταξη επερωτήσεων, οι οποίες ουσιαστικά αποτελούν συζεύξεις ατομικών συνθηκών επιλογής, συμπεριλαμβάνοντας όλα τα γνωρίσματα που εμπλέκονται στις προτιμήσεις του χρήστη. Ο αλγόριθμος εξασφαλίζει ότι ο τρόπος και η σειρά ανάκτησης των αντικειμένων σέβεται τις προτιμήσεις του χρήστη, αποφεύγοντας τους ελέγχους υπεροχής, και προσπελάζοντας μόνο τα k βέλτιστα αντικείμενα, μία φορά το καθένα. Από διαφορετική οπτική, ο TBA ορίζει μια διάταξη επερωτήσεων που αποτελούν διαζεύξεις ατομικών συνθηκών επιλογής πάνω σε μοναδικά γνωρίσματα, και χρησιμοποιεί κατάλληλα κατώφλια τιμών για να σημάνει τη διακοπή της ανάκτησης των αντικειμένων, εξασφαλίζοντας ότι όλα τα εναπομείναντα αντικείμενα είναι χειρότερα των ανακτηθέντων. Εν προκειμένω, πραγματοποιούνται έλεγχοι υπεροχής μόνο για τα ήδη ανακτηθέντα αντικείμενα. Τόσο η αναλυτική μελέτη όσο και η πειραματική αποτίμηση καταδεικνύουν την υπεροχή των αλγορίθμων που παρουσιάζουμε έναντι των υφισταμένων σε όλες τις περιπτώσεις του προβλήματος. Επόπτης: Βασίλης Χριστοφίδης Αναπληρωτής Καθηγητής

9 Στους γονείς μου Κώστα και Μαρία και στον αδερφό μου Μάνο

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11 Ευχαριστίες Καταρχήν θα ήθελα να ευχαριστήσω τον επόπτη μου κ. Βασίλη Χριστοφίδη για τα όσα μου προσέφερε στα δυόμισι και πλέον χρόνια της συνεργασίας μας. Χωρίς την ουσιαστική του καθοδήγηση, τη διαρκή στήριξη και τη βοήθεια που πάντα ήταν διατεθειμένος να προσφέρει, η παρούσα εργασία δεν θα μπορούσε να ολοκληρωθεί. Επίσης ευχαριστώ τον καθηγητή κ. Ιωάννη Τζίτζικα με τον οποίο είχα την ευκαιρία να συνεργαστώ κατά την εκπόνηση της εργασίας αυτής και να αποκομίσω σημαντικές γνώσεις. Παράλληλα θα ήθελα να ευχαριστήσω θερμά τον κ. Γεώργιο Γεωργακόπουλο για τις πολύτιμες παρατηρήσεις του οι οποίες βελτίωσαν την παρούσα εργασία. Ένα μεγάλο ευχαριστώ αξίζει στον Περικλή Γεωργιάδη, με τον οποίο μοιραστήκαμε πολλές ώρες συζητήσεων. Η συμβολή του στην κατανόηση πολύπλοκων προβλημάτων ήταν καθοριστική ενώ η προθυμία του για βοήθεια είναι άξια αναφοράς. Επίσης, θα ήθελα να ευχαριστήσω το Πανεπιστήμιο Κρήτης και την ομάδα Πληροφοριακών Συστημάτων του Ινστιτούτου Πληροφορικής για όσα μου προσέφεραν αυτά τα χρόνια και για τις γνώσεις που απέκτησα κατά τις σπουδές μου. Ένα μεγάλο ευχαριστώ επίσης ανήκει σε όλους τους φίλους/ες μου με τους οποίους συνεργάστηκα καθ όλη την διάρκεια των σπουδών μου. Τους εύχομαι ότι καλύτερο στη ζωή τους. Τελευταίο αλλά μεγαλύτερο ευχαριστώ ανήκει στους γονείς μου Κώστα και Μαρία και στον αδερφό μου Μάνο για την αμέριστη συμπαράστασή τους σε όλες τις δυσκολίες. Για το λόγο αυτή η εργασία αυτή είναι αφιερωμένη σε αυτούς και ελπίζω να αποτελέσει μια μικρή ανταμοιβή για τις θυσίες και τις προσπάθειές τους όλον αυτό τον καιρό. Καπανταϊδάκης Γιάννης

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13 Contents Chapter 1 : Introducton...1 Chapter 2 : Orders and Preferences Introducton to Order Theory Bnary Relatons Orders Graphcal Representaton of Partal Orders Notons on Posets From Partal Order of Elements to Lnear Order of Blocks Qualtatve Preference Model User Preferences From Tuple to Object Orderng...22 Chapter 3 : Top-k Algorthms Object-based orderng Block Nested Loop (BNL) Best Query Based Orderng Lattce Based Algorthm (LBA) Threshold Based Algorthm (TBA)...51 Chapter 4 : Expermental Evaluaton Expermental Envronment Preference and Testbed Generator Metrcs Expermental parameters Performance parameters Query Patterns and Evaluaton Plans The effect of database sze Unform Testbed Correlated Testbed Ant-Correlated Testbed...74

14 4.6 The effect of atomc preferences sze The effect of preference dmensons Effect of the < orderng Effect of the Number of Objects Requested k Conclusons...85 Chapter 5 : Related Work Related Frameworks for Preference Modellng Kesslng s Framework Chomck s Framework Top-k Algorthms Skylne Algorthms...94 Chapter 6 : Concluson and Future Work...97 Bblography...11

15 Lst of Fgures Fgure 1: Evaluatng the top- k objects accordng to qualtatvely specfed preferences...2 Fgure 2: The Hasse dagram of over X /...13 Fgure 3: The <, < orderng of blocks B, B1, B 2 of a partally ordered set O...17 Fgure 4: Hasse dagram for three atomc preferences...21 Fgure 5: A relaton R...23 Fgure 6: The Hasse dagram of ƒ P over R...23 Fgure 7: Actve and nactve objects...24 Fgure 8: Block Nested Loop Algorthm...3 Fgure 9: Best Algorthm...34 Fgure 1: Query Orderng Framework...37 Fgure 11: The QB array of P= P1& P2...4 Fgure 12: LBA Algorthm...41 Fgure 13: ConstructQueryBlocks functon...42 Fgure 14: ParetoComp functon...43 Fgure 15: ProrComp functon...43 Fgure 16: Evaluate functon...44 Fgure 17: Evaluate functon for the < varaton of LBA...46 Fgure 18: A Query Orderng framework example...51 Fgure 19: Threshold Based Algorthm...55 Fgure 2: OrderObjects functon...56 Fgure 21: GetNextBlock functon...57 Fgure 22: Hasse dagram for our default atomc preferences...63 Fgure 23: Total tme...71 Fgure 24: # domnance tests...71 Fgure 25: Total Tme Analyss...71 Fgure 26: LBA scalablty over database sze...71

16 Fgure 27: TBA scalablty over database sze...71 Fgure 28: Total tme...73 Fgure 29: # domnance tests...73 Fgure 3: Total Tme Analyss...73 Fgure 31: LBA scalablty over database sze...73 Fgure 32: TBA scalablty over database sze...73 Fgure 33: Total tme...75 Fgure 34: # domnance tests...75 Fgure 35: Total Tme Analyss...75 Fgure 36: LBA scalablty over database sze...75 Fgure 37: TBA scalablty over database sze...75 Fgure 38: Total tme...77 Fgure 39: # domnance tests...77 Fgure 4: Total Tme Analyss...77 Fgure 41: Total tme, unform testbed (1 MB)...8 Fgure 42: Total tme, unform testbed (1 MB)...8 Fgure 43:# queres evaluated - pareto composton...8 Fgure 44: # queres evaluated-prortzed composton...8 Fgure 45: Total tme, unform testbed (1 MB)...83 Fgure 46: Total tme, unform testbed (1 MB)...83 Fgure 47: : # queres evaluated-pareto composton...83 Fgure 48: # queres evaluated-prortzed composton...83 Fgure 49: Total tme, unform testbed (1 MB)...83 Fgure 5: :# queres evaluated - pareto composton...83 Fgure 51: Total tme, 1MB unform testbed...84 Fgure 52: #Domnance tests, 1MB unform testbed...84 Fgure 53: Total tme, 1MB unform testbed...85 Fgure 54: #queres evaluated, 1MB unform testbed...85 v

17 Lst of Tables Table 1: Metrc values for the Unform Testbed...68 Table 2: Metrc values for the Correlated Testbed...68 Table 3: Metrc values for the Ant-correlated Testbed...69 Table 4: Metrc values (ncreasng atomc preference sze)...76 Table 5: Metrc values (ncreasng dmensonalty-pareto composton)...78 Table 6: Metrc values (ncreasng dmensonalty-prortzed composton)...78 Table 7: Metrc values (ncreasng k )...84 Table 8: Proposed algorthms n varous cases...85 v

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19 Chapter 1: Introducton Preference modellng and management has attracted consderable attenton n the areas of Databases, Knowledge Bases and Informaton Retreval Systems n recent years. Ths nterest stems from the fact that a rapdly growng class of untraned lay users confront vast data collectons, usually through the Internet, typcally lackng a clear vew of ether content or structure, moreover, not even havng a partcular object n mnd. Rather, they are attemptng to dscover potentally useful objects, n other words, objects that best sut ther preferences. A modern nformaton system, consequently, should enable users to quckly focus on the k best object accordng to ther preferences. In recent years, a lot of research effort has been made for the representaton of user preferences. Manly there are two dfferent approaches of such type of personalzaton, the qualtatve ([7], [8], [9], [13], [15], [24], [3]) and the quanttatve ([1], [14], [22], [23]). In the qualtatve approach, the preferences between objects are specfed drectly, typcally usng bnary relatons. In the quanttatve approach, preferences are specfed ndrectly usng scorng functons that assocate a numerc score wth every object. An object o s preferred to an object o f the score of o s hgher than the score of o. The qualtatve approach s more powerful (n terms of expressve power) than the quanttatve one, because we can model quanttatvely specfed preferences usng preference relatons, whle not every (ntutvely plausble) preference relaton can be captured by scorng functons [8]. Moreover, there s no obvous method that the users could follow for specfyng and combnng scores. In ths work we confne ourselves to the qualtatve approach for the representaton of user preferences. More precsely, we advocate a qualtatve preference framework n whch users can defne atomc and complex preferences as well. An atomc preference s defned as a reflexve and transtve bnary relaton (.e., non-antsymmetrc preorder) over the doman of an attrbute. On the other hand, a complex preference s an expesson that mposes 1

20 prortes over the atomc preferences by usng avalable preference constructors (e.g. Pareto, Prorzaton). Object Relaton od Make Color o1 bmw red o2 bmw blue o3 vw black o4 vw red o5 aud black o6 fat red o7 aud green Object Based Orderng o1 o2 o4 o5 o3 B B 1 B 2 Top-k Objects o1 o2 Query Based Orderng q1=bmw red q2=bmw blue o4 o5 o3 Q P = P M & P C P C q3=aud red q4=bmw black q5=aud blue Q 1 [red] P M [blue] [bmw] q6=vw red q7=aud black q8=vw blue Q 2 [black] [aud] q9=vw black Q 3 [vw] Fgure 1: Evaluatng the top- k objects accordng to qualtatvely specfed preferences Assume for example an object relaton R( Make, Color ) descrbng cars, as depcted n Fgure 1 where for smplcty objects are dentfed by a od. A user wshng to purchase a car may state that he prefers red and blue cars to black ones. Furthermore, he also prefers a bmw to an aud, and the latter to vw. Fnally, he states that preferences on (M)ake ( P M ) are as mportant as on (C)olor ( P C ) ( P M & P C ). Snce preferences P M and ndvdual attrbutes are consdered as atomc preferences whle P C are defned over P M & P s a preference expresson. Let us frst consder the atomc preference P M stated on the doman of the attrbute Make. The doman values appearng n P M (.e.,bmw, aud, vw ) mply that only objects featurng the correspondng terms are of nterest to the user. Furthermore, snce n our example the user s not wshng to further restrct hs car selecton (.e., no addtonal selectons were made), preference P M wll partton objects of R nto objects that match the C 2

21 dsjuncton of the nvolved terms ( M = bmw) ( M = aud) ( M = vw) and nto objects that do not (e.g., o 6 ). However, a preference lke P M not only parttons the matchng objects accordng to the preference terms, but also orders the resultng partton (e.g., n decreasng order of preference) under the form of a block sequence (.e., a lnear order of blocks or sets). Accordng to the database of Fgure 1 and preference P M we wll have the followng block sequence: { o, o } { o, o } { o, o } gven that bmw ( o 1, o 2 ) precedes aud ( o 5, o 7 ) and vw ( o 3, o 4 ), and thus should be placed on the top block of objects returned to the user (note that object o 6 s fltered out). When a user preference spans more than one attrbutes, such as P M & P, we need to flter out objects by consderng a dsjuncton of term C conjunctons rather than atomc terms. The result of the preference P M & P wll thus consst of blocks of objects matchng the dsjuncton of the Cartesan Product of the terms nvolved n P M and ( M = bmw C = red) ( M = bmw C = blue) P C ( o 6 and o 7 are fltered out): ( M = aud C = red) ( M = aud C = blue) ( M = bmw C = black) ( M = vw C = red) ( M = vw C = blue) ( M = aud C = black) ( M = vw C = black) Then, to order ths partton, we need to examne the relatonshp of the user preferences stated per attrbute: n our example preferences P M and P C are consdered to be of equal mportance ( PM & P C ). Gven that red ( o 1 ) or blue bmw s ( o 2 ) are the most preferred ones (top block), whle black vw ( o) 3 are the least preferred ones (bottom block), we obtan the followng sequence of blocks (note that blocks that te n terms of preferences are merged): { o} { o } { o } { o } { o } We can easly observe that not all conjunctons of preference terms wll yeld non empty results. It s worth notcng that the resultng lnear order of blocks essentally lnearzes the order of objects nduced by the preference P M C & P C 3

22 as depcted n Fgure 1. However, users usually do not wsh to obtan the entre lnear order of blocks but only the top- k objects that best sut ther preferences. In ths thess, we devse effcent top- k evaluaton algorthms. Specfcally, for the gven user preferences our objectve s to compute and delver a lnear order of n blocks (.e., sets) of objects, where n s the smallest nteger that satsfes the nequalty n 1 = B k. In such a lnear order, each block would correspond to a screen of objects that s shown to the user, satsfyng the followng propertes wth respect to the user preferences: a) Each block conssts of non comparable objects. b) The frst block contans the most preferred objects. c) For each block B other than the frst and for each object n B there s a more preferred object n the prevous block (alternatvely but not equvalently, for each block B other than the last and for each object n B there s a less preferred object n the next block). The objects n are called the top- k objects. Exstng algorthms ([8], [29], [3]) for the evaluaton of the top- k objects accordng to qualtatvely specfed preferences, follow an object-based orderng approach (Fgure 1). The key dea of ths approach s to sequentally apply domnance - tests (.e., compare two objects to determne whether one s better than the other wth respect to user preferences) for every possble par of objects. The results of these tests actually specfy a preorder (.e., a reflexve and transtve bnary relaton) over the objects of a relaton. Subsequently, the algorthms of ths approach lnearze the preorder.e., they turn the preorder to a lnear order of blocks n a reasonable manner that respects the preorder and fnally pck and delver to the user the top- k objects. The man characterstc of the object-based orderng approach s that the flow of control of the algorthms of ths famly s ndependent of the user preferences. n 1 = B Despte the wde applcablty of the object-based orderng approach (snce t can be used for any number of atomc preferences wthout ndexng or sortng of 4

23 the database objects), the algorthms of ths famly are not approprate for large database systems and real scale Web applcatons snce they have serous drawbacks. An algorthm that follows the object-based orderng approach wll access all objects of a relaton R at least once and wll perform at least one domnance test for every object n R. The total number of domnance tests that such an algorthm performs s 2 On ( ) where n s the number of objects n R. Ths makes them napproprate for large databases. Moreover, exstng algorthms are nadequate for on-lne (.e., ncremental) processng snce the entre preorder over the objects of the relaton R needs to be specfed n order to return the top- k objects progressvely (.e., top-1, top- 2,..., top- k ). We advocate a query-based orderng approach for the evaluaton of the top- k objects and we ntroduce two novel algorthms called LBA (Lattce Based Algorthm) and TBA (Threshold Based Algorthm) that follow ths approach. Contrary to the object-based orderng approach, the flow of control of our algorthms, takes nto account the preference expresson gven as nput, as well as, the value orderng of the nvolved atomc preferences. The man dea of the query-based orderng approach s to use the specfed user preferences for defnng an orderng over queres that need to be evaluated n order to retreve the top- k objects (see Fgure 1). In partcular, LBA defnes an orderng over queres whch are essentally a unon of conjunctons of atomc selecton condtons, contanng all attrbutes that the user preference nvolves. A query answer of Q precedes Q f the objects n the Q (denoted by ans( Q ) ) are more preferred than the objects n ans( Q ). The evaluaton of such a query answer Q returns the next block of the B.e., B = ans( Q ). In Fgure 1, accordng to the specfed user preferences the frst query that LBA wll construct s the followng: = where 1 : Q : { q, q} 1 2 q = M = bmw C = redand q : = 2 M = bmw C = blue snce the objects n ans( Q ) are clearly the top objects of R (there cannot be another object better than the objects n ans( Q ) ). The evaluaton of Q wll 5

24 return the frst block of the answer B. Nevertheless, such a query based algorthm should be also able to dynamcally reformulate the queresq, capturng each block when some of the partal queres q j of Q return empty anwsers. To make ths clear, recall agan our example of Fgure 1 and assume that o ( bmw, blue ) was replaced by an object o (, ) 2 2 aud blue. Now, the evaluaton of Q wll return only o 1 (.e., ans( q 2) = ). However, for o 2 there wll not be a more preferred object n the prevous block (.e., n B ) snce o 1 s not better that o 2. Thus, the (c) property does not hold. Therefore LBA wll replace query q 2 (whch results n no objects) wth query q : ' ' ' ' 2 = color = blue make = aud for whch t holds that ans( q 2 ) contans the best objects of relaton R that are not worse than objects n ans( q 1). Q wll be reformed as follows: Q: = { q1, q 2}. The evaluaton of Q wll return o 1 and o 2. Now for each of the remanng objects n R there s a more preferred object n the prevous block (.e., n B = { o1, o 2} ). Notce that LBA wll never perform a domnance testng over objects. The algorthm explots user preferences and retreves objects such that t s ensured that the objects are fetched n a way that respects user preferences. Moreover, LBA wll only access the top- k objects and only once (assumng that avalable ndexes exst). LBA s also sutable for on-lne processng snce t returns the next block of the answer B wthout havng to compute prevously the followng blocks of B n the lnear order. However, consder a scenaro where the total number of objects of a relaton R s relatvely very small compared to the number of dstnct values that each doman contans (.e., the selectvty of each doman value s small) or/and the number of attrbutes that the user preference nvolves s qute large. Snce LBA constructs queres that are actually a combnaton of atomc selecton condtons that contan all attrbutes that user preferences nvolve, n such a scenaro LBA 6

25 wll have many frutless fetchng attempts (.e., resultng n no object) because R does not necessarly contan objects for every query that LBA wll construct. Therefore n a scenaro lke the one descrbed above LBA s performance s expected to drop. For ths reason we dense a second algorthm called TBA (Threshold Based Algorthm). Lke LBA, TBA defnes an order of queres however these queres are dsjunctons of atomc selecton condtons over just one attrbute. As a result, TBA s expected to have less frutless fetchng attempts. Moreover TBA uses approprate threshold values n order to determne when the fetchng of objects should stop. These values work as a guarantee ensurng that objects that were not fetched are worst than the ones that were already fetched (.e., work as an upper bound of the unseen objects). For defnng the orderng of queres, TBA takes nto account the selectvtes of the atomc selecton condtons so that to avod fetchng more objects than those actually requred. However, TBA needs to perform domnance tests but only for the already retreved objects. Thus, unlke object-based orderng algorthms, TBA avods exhaustve domnance testng among all objects whch leads to quadratc costs. In a nutshell, the contrbutons of ths thess are: We advocate a smple, yet expressve, framework for specfyng qualtatvely specfed preferences as preorders. We ntroduce a query based orderng approach for the evaluaton of the top- k objects. Unlke object-based orderng approaches, the key dea of ths approach s to explot the partcular user preference semantcs to defne an orderng over those queres, whose evaluaton s necessary for the retreval of the top-k objects. Inspred by the query orderng based approach, we desgned and mplemented two progressve algorthms (LBA, TBA) for qualtatvely specfed preferences and we study ther performances. We report the results of an extensve expermental evaluaton on large datasets that shows that the algorthms that we propose outperform the exstng ones under all problem nstances that we tested. 7

26 The rest of ths thess s organzed as follows. In Chapter 2, we ntroduce some prelemnary materal n Order Theory and present our proposed qualtatve preference model. Chapter 3, fully descrbes the most common algorthms that have emerged so far and our novel top- k algorthms. In Chapter 4, we analyze expermentally the performance of the varous top- k algorthms presented earler. Chapter 5 dscusses related work and fnally, Chapter 6 summarzes our contrbutons and dentfes ssues for further research. 8

27 Chapter 2: Orders and Preferences 2.1 Introducton to Order Theory Bnary Relatons A bnary relaton R s an arbtrary assocaton of elements of one set wth elements of another (or perhaps the same) set. More specfcally, a bnary relaton R from X to Y s a subset of the Cartesan Product X Y (.e., R X Y ). The statement ( x, y) R s read x s R-related to y, and s denoted by xry or R( xy., ) If X = Y then we smply say that the bnary relaton s over X. There are several categorzatons of bnary relatons over a set X, based on whch axoms they satsfy. Common axoms (or relaton propertes) defned for bnary relatons are the followng: reflexvty: x X t holds that xrx. rreflxvty: x X t holds that ( xrx) symmetry: x, y X t holds that f xry then yrx antsymmetry: x, y X t holds that f xry and yrx then x = y asymmetry: x, y X t holds that f xry then ( yrx) transtvty : x, yz, X t holds that f xry and yrz then xrz completeness : x, y X t holds that xry or yrx (or both) 9

28 2.1.2 Orders Certan mportant types of bnary relatons can be characterzed by the axoms they satsfy. These types of relatons are called orders. Below we present the most mportant orders whch we ntend to use n the followng chapters of our work n order to formally defne the model of preferences that we use 1 : Defnton 2.1: A bnary relaton s a preorder, denoted by, f t s reflexve and transtve. A set that s equpped wth a preorder s called a preordered set. Defnton 2.2: A bnary relaton s an equvalence relaton, denoted by, f t s reflexve, symmetrc and transtve. For an equvalence relaton on a set X, the set of the elements of X that are related to an element, say x X, s called the equvalence class of element x, often denoted as [ x ]. Defnton 2.3: A bnary relaton whch s reflexve, antsymmetrc and transtve s called a partal order and t s denoted by. A set wth a partal order s called a partally ordered set or poset. Defnton 2.4: A bnary relaton s a strct partal order, denoted by <, f t s rreflexve and transtve, and therefore asymmetrc. Note that f a preorder s also antsymmetrc, t becomes a partal order, whereas f t s also symmetrc t becomes an equvalence relaton. Let ƒ be a nonantsymmetrc preorder (.e., a reflexve and transtve relaton) over X. The asymmetrc part of ƒ s the bnary relaton over X, defned as ( x, y) X X, x y xƒ y ( yƒ x). The symmetrc part of ƒ s the bnary relaton over X defned as ( x, y) X X, x y xƒ y yƒ x. It s easy to see that the asymmetrc part comprses a strct partal order (.e., an rreflexve, asymmetrc, transtve) relaton, whereas the symmetrc one, an equvalence relaton (.e., a reflexve, symmetrc, transtve relaton). A partal order (.e., a reflexve, antsymmetrc, transtve) relaton derves from ƒ 1 The preference model that we use s an extended verson of [28] 1

29 among the equvalence classes of the quotent set X / [ x] [ y] xƒ y and [ x] = [ y] x y. as follows: For ƒ, beng a non-antsymmetrc preorder over X, t holds that: s transtve s transtve x y y z x z x y y z x z For ƒ, beng a non-antsymmetrc preorder over X, ts asymmetrc and symmetrc parts are dsjont and ther unon equals ƒ (.e., symmetry parttons ƒ ). For any two elements x and y of a partally ordered set, f x y and x y, due to antsymmetry we can wrte x < y. Smlarly, for any two elements x and y of a preordered set, f x y and ( y x), we can wrte x y. In ether case, f x < y (respectvely, x y ) and there s no z such that x < z and z < y, (respectvely, there s no z such that x z and z y wll say that y s a cover of x, and denote t as x y. ) we A partal order whch s complete s called a total (or lnear) order or a chan. A preorder whch s complete s called a weak order or a complete preorder. Elements x and y of a set X, for whch t holds that xry or yrx are sad to be comparable; otherwse, x and y are ncomparable. More formally, we defne: Defnton 2.5: Gven a relaton R over a set X, the ncomparablty relaton (usually denoted as when R s some order), s defned as the complement relaton c R over the same set X ;.e., c xr y, ff ( xry) ( yrx). For example, x y means that elements x and y are ncomparable to each other (.e., none of the relatons xry and yrx hold). Note that the above termnology may be msleadng when R s a strct partal order, as ts 11

30 complement c R may capture two very dfferent stuatons: ether ncomparablty ndeed, or comparable equalty; only when R s a preorder or a partal order the term ncomparablty have ts lteral meanng Graphcal Representaton of Partal Orders Any relaton R over a (fnte) set X may be vsually represented by a drected graph ( V, E ), wth a bjectve mappng of the elements of X onto the vertces of V and a bjectve mappng of the pars of R onto the edges of E. A graph of a partal order (or a preorder, accordngly) would be very busy, carryng a lot of redundant nformaton: self-loops (, vv ) for every node, dervng from reflexvty, as well as transtve edges ( v1, v 3), wth both ( v1, v 2) and ( v2, v 3) beng present. Furthermore, one may make two more observatons: antsymmetry ensures that n such a graph there could not be any two vertces v 1 and v 2 wth both edges ( v 1, v 2 ) and ( v 2, v 1 ) present; n conjuncton wth transtvty, antsymmetry also forbds any longer loop, meanng that the graph, wth the excepton of self-loops, has one and only drecton. Explotng the above, a partal order may be graphcally represented by a Hasse dagram. Before we formally defne a Hasse dagram we need to ntroduce the followng auxlary defntons: Defnton 2.6: The transtve closure of a bnary relaton R on a set X s the mnmal transtve relaton R on X that contans R. Defnton 2.7: The reflexve closure of a bnary relaton R on a set X s the mnmal reflexve relaton R on X that contans R. Defnton 2.8: The transtve reducton of a bnary relaton R on a set X s the mnmum relaton R on X wth the same transtve closure as R. Defnton 2.9: The reflexve reducton of a bnary relaton R on a set X s the mnmum relaton R on X wth the same reflexve closure as R. Now we can proceed to the formal defnton of a Hasse dagram: 12

31 Defnton 2.1: A Hasse dagram of a partal order s a drected acyclc graph of ts reflexve and transtve reducton, where drecton s omtted, as t s mpled by the dagram s upward orentaton. A Hasse dagram may also depct a nether symmetrc, nor antsymmetrc preorder. In ths case t essentally represents the partal order of the equvalence classes of the quotent set X /, rather than the preorder tself. The Hasse dagram of an equvalence relaton s smply a set of non-connected nodes, each of whch s a representatve of an equvalence class. So, n all cases above, the Hasse dagram obeys conventons of what each nodes stands for (class representatves n some cases) and the only case where t drectly depcts a relaton s the case of a strct partal order. Note that, n all cases, all lnes n a Hasse dagram correspond to the cover relaton (.e., the transtve reflexve reducton of a partal order), reflectng on the strct part < of the partal order. Example 2.1: Let us assume the Hasse dagram for a set X = { abcde,,,,, f}, a preorder over X, wth d b, b a, d a, c a, d f and f d. Such a dagram cannot represent the preorder tself drectly, so, as dscussed above, t wll depct the partal order of the equvalence classes of the quotent set X /. There are fve equvalence classes [ a] = { a}, [ b] = { b}, [] c = {} c, [ d] = { d, f}, [] e = {} e, and let s choose a sngle representatve from each one to use n the dagram. The resultng Hasse dagram s llustrated n Fgure 2. [a] [b] [c] [e] [d] Fgure 2: The Hasse dagram of over X / 13

32 2.1.4 Notons on Posets In a partally ordered set there are some elements that play a specal role. The most basc examples are gven by the maxmal and the mnmal elements of a poset. Defnton 2.11: Let a partal order over a set of elements X. An element x X s a maxmal element of, f x X such that x x. Defnton 2.12: Let a partal order over a set of elements X. An element x X s a mnmal element of, f x X such that x x. We may partton the elements of a partal order relaton X nto nonoverlappng parts called blocks (or layers or buckets) that cover all of X usng varous topologcal crtera. To defne our approach formally we need some auxlary defntons that we adapt from [28]: Defnton 2.13: Let us call path from an element x to an element x of a partal order, any sequence of pars of the form ( x, x1),( x1, x2),,( xn 1, xn),( xn, x ) such that x x, 1 xn x and x 1 x for = 2 n. The nteger n + 1 s called the length of the path, and t s clear that there may be zero, one or more paths from x to x. Now, assume that B contans all elements that are maxmal (or mnmal) wth respect to. The defnton of each other block B reles on the noton of dstance of an element from B. Defnton 2.14: The dstance of an element x from B s defned to be the length of the longest path from an element x to the element x, when x ranges over all elements of B. Block B s defned to be the set of all elements that are at dstance from B. Note that f x belongs to B, then ts dstance s defned to be equal to. It s easy to see that elements of the same block are ncomparable to each other (otherwse they wouldn t have the same dstance from B ). 14

33 2.1.5 From Partal Order of Elements to Lnear Order of Blocks Let be a bnary order relaton (e.g., a preorder, a partal order, or a strct partal order) on a set O and let 2 O be the powerset of O mnus. In ths secton, we defne relatons over subsets of O,.e., over 2 O, whch derve from the ntal order over O. Defnton 2.15: Assumng that XY, 2 O for whch does not necesserly hold X Y =, we defne the followng relatons over 2 O : Let 2 X Y, ff x X, y Y such that x y Let X Y, ff y Y, x X such that x y Note, that apart from transtvty, whch s trval to prove, whether other order axoms hold n each of these relatons over sets, depends on the nature of the ntal poset and probably other assumptons, and need to be proved, thus the use of the term set order, nstead of set relaton may be abusve. Let B, B1,, Bn sequence of blocks of O that were produced as descrbed n prevous secton. 3 a Theorem 2.1: If B contans the maxmal elements of, a relaton s defned between blocks B, B1,, Bn (.e., Bn Bn 1 B). Proof 2.1: For every element x n B there s a longest path p from some element of B to x. Let x be the predecessor of x n p (.e., x x ). Clearly, the sub-path of p endng n x s the longest path from B to x (otherwse, p s not the longest path to x thus a contradcton). It follows that x s n 1 B and that x x. 2 As a rule of thumb, the frst quantfer runs on the left operand set, the second on the rght, and the outer quantfer s denoted by the lne above t. 3 For each of these blocks B t holds B 2 O. 15

34 Theorem 2.2: If B contans the mnmal elements of, a relaton s defned between blocks B, B1,, Bn (.e., B B1 Bn ). Proof 2.2: For every element x n B there s a longest path p from some element of B to x. Let x be the predecessor of x n p. The sub-path of p endng n x s the longest path from B to x (otherwse, p s not the longest path to x thus a contradcton). It follows that x s n B 1 and that x x. Theorem 2.3: relaton defnes a lnear order of blocks. Proof 2.3: Clearly s reflexve and transtve (snce s reflexve and transtve). Moreover, snce each block conssts of mutually ncomparable elements t s also antsymmetrc and thus s a partal order. Due to the defnton B B j ff x B, x j B j such that x x j, we have n B B B n 1 between blocks.. As a result, actually defnes a lnear order Smlarly, we can prove that also defnes a lnear order of blocks. Therefore snce and relatons defne lnear orders of blocks from now on we could wrte < and < to denote those lnear orders. Example 2.2: Fgure 3 llustrates the two ndvdual orderngs < and < between the blocks B, B1, B 2 of a partally ordered set O. < orderng occurs f we use as basc block B the block that contans the maxmal elements of O and < occurs f we use as B the block that contans the mnmal elements. 16

35 a b a b B b B 2 c d c d B 1 a d B 1 e e A partally ordered set O B2 < B1 < B B 2 e c B < B < B 1 2 B Fgure 3: The <, < orderng of blocks B, B1, B 2 of a partally ordered set O 2.2 Qualtatve Preference Model Let R( ) denote a relaton scheme, where R s the name of the relaton and A1 A2 A n = {,,, } s a set of attrbute names wth assocated domans dom( A ). Wthout loss of generalty, we assume the attrbute domans par-wse dsjont,.e., dom( A ) dom( A ) = for every j [1 n]. As null values j are possble, n order to keep notatons smple, we use dom( A ) to denote dom( A ) { }, where " " stands for the null value. We shall also use the notaton dom( A) = dom( A1 ) dom( A m ) and dom A = dom A 1 dom A m ( ) ( ) ( ) to denote the Cartesan Product and the unon of domans, of a non empty set of attrbutes A. An object over a scheme R( ) assocates wth each A a value taken from ts doman. As usual oa [ ] denotes the projecton of an object o onto a non empty set of attrbutes A. A relaton R over the scheme R( ) (also called an nstance) s a fnte set of objects o such that o[ ] dom( ) User Preferences In order to proceed to the general preference defnton we should take nto consderaton two mportant factors: 17

36 The user, most tmes, s not n poston to know the objects that a database contans. Thus, the preferences should be defned on structures of nformaton that are not nfluenced by the avalable objects. Such structures are the attrbute domans. The number and the nature of attrbutes that are nvolved n a preference expresson vary. Therefore, the defnton of a preference should be based on the attrbutes that the preference nvolves. Defnton 2.16: Let us assume a relaton scheme R( ). A preference P A over a non empty set of attrbutes A= { A1 A m } 2 \ s a non-antsymmetrc partal preorder over dom( A) = dom( A1 ) dom( A m ) 4 denoted as P = ( dom( A), ƒ ), where ƒ P dom( A) dom( A). For m -tuples A P A A vv, doma ( ), vƒ P A v s nterpreted as v s at most as preferable as v (or equvalently, v s at least as preferable as v ). We shall pronounce those v, v for whch both vƒ P A v and v ƒ v hold, as P A equally preferred or ndstngushable w.r.t. preference P A. As symmetry holds by defnton and reflexvty wth transtvty are nherted from ƒ P A, the preference equalty relaton s an equvalence relaton,.e., equally preferred tuples v, v are equvalent, or belong to the same equvalence class, thus we wll denote ths relaton as v v. P A If vƒ P A v but ( v ƒ P A v), we can wrte v P A v whch s nterpreted as v s (strctly) more preferable than v. As asymmetry and rreflexvty holds by defnton and transtvty s nherted from ƒ P A, the asymmetrc part P A of ƒ P A comprses a strct partal order relaton. If nether vƒ P A v nor v ƒ P A v hold, then we wll say that v, v are ncomparable and we wll wrte v P A v. The ncomparablty relaton carres symmetry, by defnton, but apart from t, t satsfes no other order axoms n general. 4 The order of factors wthn the Cartesan Product s consdered of no partcular sgnfcance. 18

37 When A s a trval sngle-factor Cartesan Product (.e., A= { A }) we wll call P = ( dom( A ), ƒ ) an atomc preference over A, where dom( A ) s the doman P of A. For a preference P = ( dom( A), ƒ ), the values of doman dom( A ) can be A P A separated n two concrete categores, proportonally whether they take actve part or no n the preorder ƒ P A. Defnton 2.17: Gven a preference P = ( dom( A), ƒ ), a value v dom( A) A P A that s not nvolved n the partal preorder relaton n any other way except though reflexvty (.e., v dom( A), v v, such that vƒ P A v or v ƒ v) wll be called nactve (otherwse, t wll be called actve) and clearly t s ncomparable to all other values of dom( A ). P A We denote V( PA, A ) the set of actve values of dom( A ) accordng to P A and c V ( P, A ) the set of nactve values, respectvely. Notce that: A V( P, A) = V( P, A) where V( P, A ) denotes the set of actve m A = 1 A values from the doman of A appearng n an atomc preference P A A c V ( P, A) V ( P, A) = dom( A) A A c V( P, A) V ( P, A) = A A We make ths seperaton snce nactve values actually don t take part n the orderng ƒ P A, creatng, thus, a sense of ndfference of the user to each nactve value. In essence, there s no need to take nactve values nto account snce only actve values have nterest to a partcular user and need to be specfed (regardless of whether they are actually nstantated n R ). Next we dscuss preference expressons, whch capture the semantcs of combnng or syntheszng user preferences over more than one ndvdual attrbutes. 19

38 Defnton 2.18: Let P be an atomc preference over any ndvdual attrbute A of a relaton R. A preference expresson P s defned as: P= P ( P& P) ( P& P) ( P P) ( P P) It should be stressed out that an atomc preference appears only once n a preference expresson. Semantcally, ths nterprets a current lmtaton n ths work; we are able to capture the relatve mportance among dfferent preferences, where each s provded and mentoned only once. On the other hand, ths comples wth the fact that values n the actve preference domans also appear only once n the respectve preferences. A preference expresson P A, as defned above, spans over the set A of attrbutes t nvolves, so we are able to use V( P, A ) as defned earler. The bnary preference operators & and A defne the pareto and prortzed composton operatons on two preferences as follows. Defnton 2.19: Preference P = ( dom( A), ƒ ) s the pareto preference A P A (denoted by P = P & & 1 P ) of m atomc preferences P = ( dom( A ), ƒ ) A m P defned over A when v= ( v1 v m ), v = ( v 1 v m ) dom( A) we have: v v: ff [1 m], such that & v v and j [1 m] { } t P holds v ƒ v j Pj j v v: ff [1 m] t holds & v v P v v n all other cases & Defnton 2.2: Preference P = ( dom( A), ƒ ) s the prortzed preference, A P A (denoted by PA = P1 Pm) of m atomc preferences P = ( dom( A ), ƒ ) P defned over A when v= ( v1 v m ), v = ( v 1 v m ) dom( A) we have: v v: ff [1 m] such that v v and j [1 1] t holds P v v j Pj j 2

39 v v: ff [1 m] t holds v v P v v n all other cases Note that a pareto preference s a combnaton of mutually non domnatng preferences. On the other hand, a prortzed preference treats P 1 as more mportant than preference P 2, whch n turn s more mportant than P 3, etc., up to preference P m. Both operators are assocatve, allowng us to apply each of then on more than two operands, wthout usng parentheses; the Pareto operator s commutatve, but not the Prortzaton one, whereas nether of the two s dstrbutve over the other. Let us, also, assume that when omttng parentheses, the operators take prorty from left to rght. Example 2.3: Consder, for nstance, the relaton schema R( ABC,, ) where the doman of attrbutes s gven respectvely by the sets dom( A) = { a1, a2, a3}, dom( B) {,, } = b1b 2 b3, dom C c1 c2 c3 ( ) = {,, }. Also suppose that a user has defned the atomc preferences P1 = ( dom( A), ƒ ), P1 P2 = ( dom( B), ƒ ), P2 P3 = ( dom( C), ƒ ) such that P3 a ƒ a, 3 P 1 1 a ƒ a, 3 P 1 2 b ƒ b, 2 P 2 1 b ƒ b, 3 P 2 2 c ƒ c, c ƒ 2 P 3 1 c 3 P 3 2 correspondng Hasse dagrams 5 :. Notce that all values here are actve. Fgure 4 depcts ther P 1 P 2 P 3 [b 1 ] [c 1 ] [a 1 ] [a 2 ] [a 3 ] [b 2 ] [b 3 ] [c 2 ] [c 3 ] Fgure 4: Hasse dagram for three atomc preferences 5 Recall that the above Hasse dagram represents the partal order of the equvalence classes of the quotent sets. 21

40 Now, consder two elements ( a3, b1, c 1), ( a, b, c ) dom( A) dom( B) dom( C). Accordng to the above defntons we have: 1. ( a3, b1) and b ƒ P 1& P2 1 1 b 1 P 2 1 ( a, b ) due to a a (.e., snce 3 P 1 1 a ƒ a and a 3 a 1 ) 3 P ( a3, b1, c1) ( a, b, c ) because P 1& P2& P a a but c 3 P 1 1 c 3 P ( a1, b1, c3) prortzed to P 1 ( a, b, c ) as both contan b 1 and because P 3 s P 2 P3 P ( a3, b1, c1) mportant ( P1& P2) P ( a, b, c ) due to (1) and snce P 3 s the least From Tuple to Object Orderng Gven a preference P = ( dom( A), ƒ ), we can use the defntons above to nfer A P A a non-antsymmetrc partal preorder of the objects themselves n a relaton R through projecton, as follows: o, o R, o ƒ o ff o[ A] ƒ o [ A] PA PA oƒ P A o means that o s at most as preferable as o, (or equvalenty, o s at least as preferable as o ). The process of comparng two objects n order to decde whch one s more preferred than the other s referred n the lterature as domnance testng. If both oƒ P A o and o ƒ P A o hold, we shall pronounce those objects o, o as equally preferred and we wll denote ths relaton as o o (.e., belong to the same equvalence class). If P A oƒ P A o but ( o ƒ o), we can wrte P A o P A o whch s nterpreted as o s (strctly) more preferable than o or o domnates o. Fnally f nether oƒ P A o nor o ƒ P A o hold, then we wll say that o, o are ncomparable and we wll wrte o P A o. 22

41 Now let be the partal order relaton whch derves from ƒ P A among the equvalence classes of the quotent set R /.[ o] [ o ] wll mean that the equvalence class of o s at most as preferable as the equvalence class of o. However f [ o] [ o ] and [ o] [ o ] we can wrte [ o] < [ o ] meanng that the user prefers object o (or any object equvalent to o ) to object o (or any object equvalent to o ). P A Example 2.4: Assume preference P= P1& P2 where P1 = ( dom( A), ƒ ), P1 P2 = ( dom( B), ƒ ) are the atomc preferences of Fgure 4 and the followng relaton R : P2 od A B C o a b c o a b c o a b c o a b c Fgure 5: A relaton R Accordng to the defntons descrbed above we wll have the followng relatons over the objects of R : o2 P o1, o3 P o1, o4 P o1, o2 P o3, o2 P o4, o o. As a result, there are three equvalence classes [ o1] = { o1}, 3 P 4 [ o2] = { o2, o3}, [ o4] = { o4} defned. The resultng Hasse dagram s llustrated n Fgure 6. [o 1 ] [o 2 ] [o 4 ] Fgure 6: The Hasse dagram of ƒ P over R 23

42 Now, gven a preference P over A n a relaton R, we shall call actve those objects o R whch contan actve values over every attrbute n A 6 whle the rest of the objects are called nactve; Actve objects, denoted as Act( P, A ) are those that represent tems whch are nterestng to the user, as they contan the combnatons of nterestng attrbute values. Moreover, actve objects can be ordered wth respect to others, whle nactve ones are those that cannot be ordered. DB Objects Top-k Actve Objects Inactve Objects Fgure 7: Actve and nactve objects For example assume that the relaton R of Fgure 5 contans one more object o5( a4, b1, c 1). Snce a4 V( P1, A), o 5 s consdered as nactve object and clearly t cannot be ordered wth respect to the remanng objects of R. In exstng frameworks ([7], [8], [9], [13], [15], [24], [3]), nactve objects are consdered as ncomparable to the actve ones and thus returned n the frst block of the result (as undomnated). For nstance, o 5 wll be returned n the same block as object o 1, although the user defntely prefers the latter. As a consequence, our approach parttons objects of R nto actve and nactve ones and reles only on the actve objects to retreve the top- k objects of a relaton R (Fgure 7). 6 oa [ ] V( P, A) A 24

43 Chapter 3: Top-k Algorthms In the prevous chapter we have shown how from a gven preference P = ( dom( A), ƒ P ), we can nfer a partal order (whch derves from ƒ P ) among the equvalence classes of the quotent set R / P. Furthermore, we have seen how to form a lnear order of mutually dsjont blocks of classes of objects that respects and as a result the ntal user preferences. In ths lnear order each block would correspond to a screen of ncomparable equvalence classes that s shown to the user, wth the most preferred classes of objects appearng frst. Nevertheless, the presence of equvalent and of ncomparable objects leaves space for more than one dfferent orderngs of R. In partcular we have consdered two lnear orders of blocks ( <, < ) that actually satsfy the above requrements. Let us assume a < orderng between the blocks of the answer. In order to retreve the most preferred objects of the partal order, all the succedng blocks have to be prevously computed snce a < order mposes a down to top orentaton. Thus, n the general case that we assume where the number of avalable actve objects n relaton R s large and the number k s small, a < orderng s not sutable due to the expensve computaton cost and ts lack of progressveness (we actually have to order the entre relaton R ). As a result of ths observaton we manly focus on the < orderng. For any gven user preference P = ( dom( A), ƒ ) and an nteger k, our purpose s to provde effcent evaluaton algorthms for computng the top- k objects of R. P Problem Statement: Gven a relaton R our objectve s for every possble P = ( dom( A), ƒ ) and k parameter to compute and delver to the user a lnear P order of n blocks of equvalence classes of objects B, B1,, Bn 1, where n s 25