A Cost Model for Query Processing in High-Dimensional Data Spaces

Transcription

1 A Cost Moel for Query Processing in High-Dimensional Data Spaces Christian Böhm Luwig Maximilians Universität München This is a preliminary release of an article accepte by ACM Transactions on Database Systems The efinitive version is currently in prouction at ACM an, when release, will supersee this version Name: Christian Böhm Affiliation: Luwig Maximilians Universität München Aress: Institut für Informatik, Oettingenstr 67, München, Germany, boehm@informatikuni-muenchene Copyright 2000 by the Association for Computing Machinery, Inc Permission to make igital or har copies of part or all of this work for personal or classroom use is grante without fee provie that copies are not mae or istribute for profit or commercial avantage an that copies bear this notice an the full citation on the first page Copyrights for components of this work owne by others than ACM must be honore Abstracting with creit is permitte To copy otherwise, to republish, to Post on servers, or to reistribute to lists, requires prior specific permission an/or a fee Request permissions from Publications Dept, ACM Inc, fax + (22) , or permissions@acmorg During the last ecae, multimeia atabases have become increasingly important in many application areas such as meicine, CAD, geography or molecular biology An important research issue in the fiel of multimeia atabases is similarity search in large ata sets Most current approaches aressing similarity search use the so-calle feature approach which transforms important properties of the store objects into points of a high-imensional space (feature vectors) Thus, the similarity search is transforme into a neighborhoo search in the feature space For the management of the feature vectors, multiimensional inex structures are usually applie The performance of query processing can be substantially improve by optimization techniques such as the blocksize optimization, ata space quantization or imension reuction To etermine optimal parameters an accurate estimation of the performance of inex-base query processing is crucial In this paper, we evelop a cost moel for inex structures for point atabases such as the R * -tree or the X-tree It provies accurate estimations of the number of ata page accesses for range queries an nearest neighbor queries uner Eucliean metric an maximum metric The problems spe-

2 2 Christian Böhm cific to high-imensional ata spaces, calle bounary effects, are consiere The concept of the fractal imension is use to take the effects of correlate ata into account Categories an Subject Descriptors: H3 [Information Storage an Retrieval]: Content Analysis an Inexing H28 [Database Management]: Database Applications General Terms: Performance, Theory Aitional Key Wors an Phrases: Cost moel, multiimensional inex INTRODUCTION Motivation Inexing high-imensional ata spaces is an emerging research omain It gains increasing importance by the nee to support moern applications with powerful search tools In the socalle non-stanar applications of atabase systems such as multimeia [Faloutsos et al 994a, Shawney an Hafner 994, Seil an Kriegel 997], CAD [Berchtol 997, Berchtol an Kriegel 997, Berchtol et al 997c, Jagaish 99, Gary an Mehrotra 993, Mehrotra an Gary 993, Mehrotra an Gary 995], molecular biology [Altschul et al 990, Kastenmüller et al 998, Kriegel et al 997, Kriegel an Seil 998, Seil 997], meical imaging [Keim 997, Korn et al 996], time series analysis [Agrawal et al 995, Agrawal et al 993, Faloutsos et al 994b], an many others, similarity search in large ata sets is require as a basic functionality A technique wiely applie for similarity search is the so-calle feature transformation, where important properties of the objects in the atabase are mappe into points of a multiimensional vector space, the so-calle feature vectors Thus, similarity queries are naturally translate into neighborhoo queries in the feature space In orer to achieve a high performance in query processing, multiimensional inex structures [Gaee an Günther 998] are applie for the management of the feature vectors Unfortunately, multiimensional inex structures eteriorate in performance when the imension of the ata space increases, because they are primarily esigne for low imensional ata spaces (2D an 3D) prevalent in spatial atabase systems whereas feature vectors are usually high-imensional Therefore, a number of specialize inex structures for highimensional ata spaces have been propose Most high-imensional inex structures are variants of the R-tree family [Guttman 984, Beckmann et al 990, Sellis et al 987] using either rectangular or spherical page regions Minimum bouning rectangles are use by the X-tree [Berchtol et al 996] The SS-tree [White an Jain 996] an the TV-tree [Lin et al 995] use spheres to escribe the regions of the space covere by ata pages an irectory pages The SR-tree [Katayama an Satoh 997] uses a combination of a sphere an an MBR, the intersection soli, as page region Other high-imensional inex structures which are not irectly relate to R-trees are the LSD h -tree [Henrich 998] an the Pyrami-tree [Berchtol et al 998b] The Hybri Tree combines the properties of ata partitioning (eg R-tree) an space partitioning (eg k--b trees) inex structures [Chakrabarti an Mehrotra 999] An inex structure applicable in a parallel computer is the parallel X-tree [Berchtol et al 997a] In spite of these efforts, there are still high-imensional inexing problems uner which even specialize inex structures eteriorate in performance Therefore, various optimization techniques for inex structures have been propose The most common optimization parameter in inex structures is the logical page size, ie the basic unit of transfer between

3 A Cost Moel For Query Processing in High-Dimensional Data Spaces 3 main memory an seconary storage [Böhm 998] an [Böhm an Kriegel 2000] propose a ynamic an inepenent page size optimization It turns out that large page sizes are optimal in high-imensional ata spaces while in meium-imensional cases smaller page sizes are optimal The optimum epens not only on the imension but also on the number of objects currently store in the atabase an on the unerlying ata istribution Weber, Schek an Blott [Weber et al 998] give evience that there always exists a imension which is high enough to make any thinkable inex structure inefficient an propose the VAfile, a metho which is base on the sequential scan of the ata set (combine with a ata compression metho) [Böhm 998] consiers the sequential scan as a special case of an optimize inex having an infinite page size A secon optimization technique is the ata space quantization which is essentially a ata compression technique It is applie in the VA-file [Weber et al 998] an in the IQ-tree [Berchtol et al 2000] The basic iea is to reuce the I/O time by representing the attributes not by their full 32 bit values but only with between 4 an 8 bits As the representation of the vectors is inexact in this concept, a refinement step is require for the results, which causes aitional I/O cost A further price to pay is a greater CPU overhea for the ecompression If query processing is I/O-boun, the cost balance of ata space quantization can be positive A thir optimization technique for inexes is imension reuction The TV-tree [Lin et al 995] uses imension reuction in the irectory Tree striping [Böhm 998] is base on the iea of ecomposing the feature vectors an storing the sub-vectors in separate inexes with moerate imensionality The results of query processing in the separate inexes must be joine Seil [Seil 997] an Faloutsos et al [Faloutsos et al 994a] apply imension reuction in the filter step of a multi-step query processing architecture For both the choice of a suitable inex structure or query processing technique as well as for the etermination of optimal parameters in optimization techniques it is crucial to provie accurate estimations of the cost of query processing In this paper we present a methoology for cost moeling of high-imensional inex structures 2 Influence Factors on the Performance of Query Processing There are various factors which have an influence on the performance of inex-base query processing First of all the ata set The efficiency of inex-base query processing epens on the imension of the ata space, the number of points in the atabase an on the ata istribution from which the points are taken Especially the correlation of imensions is of high importance for the efficiency Correlation means that the attributes of some imensions are statistically not inepenent from each other The value of one attribute is more or less etermine by the values of one or more other attributes In most cases, this epenency is not strict, but rather observable by the means of statistics From a geometric point of view, correlation means that the ata points are not sprea over the complete ata space Instea, they are locate on a lower-imensional subset of the ata space which is not necessarily a single linear subspace of the ata space For most inex structures the performance of query processing improves if this intrinsic imension of the ata set is lower than the imension of the ata space Our cost moel takes these properties of the ata set into account The impact of the imension an the number of ata points is consiere separately for low-imensional ata spaces (sections 3-4) an for high-imensional ata spaces (section 5) Section 5 evelops also a criterion for istinguishing between the low-imensional an the high-imensional case Correlation effects are hanle by the concept of the fractal imension in section 6 The metric for measuring the istance between two ata points (Eucliean metric, Manhattan metric, maximum metric) has an important influence on the query performance, too

4 4 Christian Böhm The cost formulas are thus presente for the two most important metrics, the Eucliean an the maximum metric throughout the whole paper While the Manhattan metric is relatively straightforwar, moeling more complex kins of metrics such as quaratic form istance functions [Seil 997, Seil an Kriegel 997] is future work A secon set of influence factors is connecte with the inex structure Most important is the shape of the page regions: It can be a rectangle, a sphere or a compose page region If it is a rectangle, it can be a minimum bouning rectangle or it can be part of a complete ecomposition of the ata space such as in the k--b-tree or in the LSD h -tree Most ifficult to capture in a moel are the various heuristics which are applie uring insert processing an inex construction The impact of the heuristic on the volume an extension of page regions is very har to quantify A further influence factor is the layout of pages on the seconary storage If the layout is clustere, ie pairs of ajacent pages are likely to be near by each other on isk, the performance can be improve if the query processing algorithm is conscious of this clustering effect Our cost estimations are presente for rectangular page regions Consiering spherical page regions or cyliners is straightforwar, whereas more complex shapes such as compose or intersecte solis are ifficult The impact of insertion heuristics is consiere neither in this paper nor in any cost moel for multiimensional query processing known to the author In fact, these cost moels assume iealize inex structures which o not yiel eteriorations such as heavy overlap among the page regions Such parameters coul eventually be measure for a specific inex an be integrate into the formulas, but this is future work A thir set of influence factors is ue to the choice of the query processing algorithm The HS algorithm propose by Hjaltason an Samet [Hjaltason an Samet 995] yiels a better performance in terms of page accesses than the RKV algorithm by Roussopoulos, Kelley an Vincent [Roussopoulos et al 995] Disk clustering effects can be exploite by algorithms consiering the relative positions of pages on the backgroun storage We will assume a epth-first search strategy for range queries an the HS algorithm for nearest neighbor queries 3 Paper Outline After reviewing some relate work on cost moels, we start with the introuction of moeling range queries assuming an inepenent an uniform istribution of the ata points Moreover, we assume in the beginning that queries o not touch the bounary of the ata space Range queries are transforme into equivalent point queries by accoringly aapting the page regions The central concept for this compensating aaptation is calle Minkowski sum or Minkowski enlargement We etermine from the Minkowski sum the access probability of pages an use this access probability to evelop an expectation for the number of page accesses In the next section, nearest neighbor queries evaluate by the HS algorithm are moele This is conceptually one by a reuction step which transforms nearest neighbor queries into an equivalent range query The corresponing range r can be estimate by a probability ensity function p(r) using r as variable The simplifying assumptions of a uniform an inepenent ata istribution an the ignorance of ata space bounaries will be roppe step by step in the subsequent sections First, the so-calle bounary effects are introuce an shown to be important in highimensional ata spaces Our moel is moifie to take bounary effects into account Then, we consier non-uniform ata istributions which are inepenent in all imensions Last, we formalize correlations by the means of the fractal imension an integrate this concept into our cost moels for range queries an nearest neighbor queries Experimental evalua-

5 A Cost Moel For Query Processing in High-Dimensional Data Spaces 5 tions showing the practical applicability of the cost moels an their superiority over relate approaches are provie in each section separately 4 Terminology Due to the high variety of cost formulas we will evelop in this paper the notation an the ientifiers we have to use are complex There are a few general ientifiers for basic cost measures which will be use throughout this paper: V for some volume R for the expecte istance between a query point an its nearest neighbor P for some probability istribution function X for the access probability of an iniviual page A for the expectation of the number of page accesses The bounary conitions for the corresponing cost measure such as the istance metric an basic assumptions such as uniformity or inepenence in the istribution of ata points are enote by subscripte inices of the general ientifiers For instance, X nn,em,l,ui means the access probability for a nearest neighbor query (nn) using the Eucliean metric (em) on a low-imensional ata space (l) uner uniformity an inepenence assumption (ui) We istinguish: - the query type: range query (r), nearest neighbor (nn) an k-nearest neighbor (knn) - the applie metric: Eucliean metric (em) an maximum metric (mm) - the assume imensionality: low-imensional (l) an high-imensional (h) - the ata istribution assumption: uniform/inepenent (ui), correlate (cor) Especially the metric ientifier (em or mm) is sometimes left out if an equation is vali for all applie metrics In this case, all terms lacking the metric specification are unerstoo to be substitute by the same metric, of course The volume V is specifie by a few inices inicating the geometric shape of the volume Basic shapes are the hyper-sphere (S), the hyper-cube (C) an the hyper-rectangle (R) The Minkowski sum (cf section 3) of two solis o an o 2 is marke by a plus symbol ( o o 2 ) The clipping operation (cf section 5) is marke by the intersection symbol ( o o 2 ) Often, inices enote only the type of some geometrical object: V R S stans for the Volume of a Minkowski sum of a rectangle an a sphere We use the following notational conventions throughout this paper: Our atabase consists of a set of N points in a -imensional ata space DS R Usually, we will restrict ourselves to the ata spaces normalize to the unit hypercube [0] It is unclear to what extent the normalization of the ata space affects the correctness of the result set This epens on the mapping between the object space an the feature space in the feature transformation which is out of the scope of this paper Generally, the multi-step architecture of query processing requires that in the feature space more objects must be retrieve than the user requests as similar objects The ratio is calle the filter selectivity The mapping into a normalize ata space may affect this filter selectivity positively or negatively The problem of moeling the filter selectivity for various mappings an similarity measures is subject to our future work We will use the Eucliean metric δ em an the maximum metric δ mm for istance calculations between two points P an Q: δ em ( P, Q) = ( Q i P i ) 2 an δ mm ( P, Q) = max { } 2 0 i < 0 i < Q i P i

6 6 Christian Böhm The average number of points store in a ata page will be enote as the effective ata page capacity,ata In our cost estimations, we o not consier irectory page accesses for two reasons: First, irectory pages are often assume to be resient in the atabase cache after their first access [Berchtol et al 996] But even if the cache is too small to store the complete irectory, the number of accesse ata pages is often even larger than the number of available irectory pages Even if the ratio of accesse pages ecreases when moving from the root level to the leaf level, the ata page accesses are the preominant factor in the query processing cost For concreteness, we restrict ourselves to point access methos with rectilinear page regions minimizing the overlap in the inex construction The moel can be extene for structures with spherical regions in a straightforwar way 5 Algorithms for Query Processing in High-Dimensional Spaces In this paper, we will consier two kins of queries: range queries an nearest neighbor queries A range query is efine as follows: We are given a atabase DB of N points in a - imensional ata space, a query point Q, a raius r calle query range an a istance metric δ Retrieve all points in the atabase which have a δ -istance from Q not greater than r: RangeQuery(DB,Q,r, δ ) = { P DB δ( P, Q) r} By Q an r, a high-imensional hypersphere is efine, the query sphere An inex base algorithm for range queries traverses the tree in a epth-first search All pages intersecting the query sphere are accesse In a k-nearest neighbor search, the region of the ata space where ata points are searche is not previously known Given is also the atabase DB, the query point Q, the istance metric δ an a number k of objects to be retrieve from the atabase The task is to select the k points from the atabase with minimum istance from Q: knnquery( DB, Q, k, M) = { P 0 P k DB P DB\ { P 0 P k } i, 0 i < k:δ( P i, Q) > δ( P, Q) } This can be etermine in a epth-first search, as in the RKV algorithm by Roussopoulos, Kelley an Vincent [Roussopoulos et al 995] This algorithm keeps a list of the k closest points which is upate whenever a new, closer point is foun The istance of the last point in the list is use for pruning branches of the tree from being processe Although several further information is use for pruning, the algorithm cannot guarantee a minimum number of page accesses The HS algorithm propose by Hjaltason an Samet [Hjaltason an Samet 995] an in a similar form earlier by Henrich [Henrich 994] performs neither a epth-first search nor a breath-first search, but accesses pages in an orer by increasing istance from the query point This can be implemente using a priority queue The algorithm stops when all pages with a istance less than the current pruning istance are processe The etails of this algorithm can be taken from the original literature [Henrich 994, Hjaltason an Samet 995] or one of the seconary publications [Berchtol et al 997b, Böhm 998] In contrast to the RKV algorithm, the optimality of the HS algorithm is provable (cf [Berchtol et al 997b]), because it accesses exactly the pages in the knn-sphere, ie a sphere aroun the query point which touches the k-th nearest neighbor In contrast, RKV accesses a number of aitional pages which epens on the effectivity of the applie heuristics Therefore, the performance of the RKV algorithm is ifficult to estimate, an we will restrict ourselves to the HS algorithm in our cost moel

7 A Cost Moel For Query Processing in High-Dimensional Data Spaces 7 2 REVIEW OF RELATED COST MODELS Due to the high practical relevance of multiimensional inexing, cost moels for estimating the number of necessary page accesses have alreay been propose several years ago The first approach is the well-known cost moel propose by Frieman, Bentley an Finkel [Frieman et al 977] for nearest neighbor query processing using maximum metric The original moel estimates leaf accesses in a k-tree, but can be easily extene to estimate ata page accesses of R-trees an relate inex structures This extension was publishe in 987 by Faloutsos, Sellis an Roussopoulos [Faloutsos et al 987] an with slightly ifferent aspects by Aref an Samet in 99 [Aref an Samet 99], by Pagel, Six, Toben an Wimayer in 993 [Pagel et al 993] an by Theooriis an Sellis in 996 [Theooriis an Sellis 996] The expecte number of ata page accesses in an R-tree is This formula is motivate as follows: The query evaluation algorithm is assume to access an area of the ata space which is a hyper cube of the volume V = /N, where N is the number of objects store in the atabase Analogously, the page region is approximate by a hypercube with the volume V 2 = /N In each imension, the chance that the projection of V an V 2 intersect each other, correspons to V + V 2 if n To obtain a probability that V an V 2 intersect in all imensions, this term must be taken to the power of Multiplying this result with the number of ata pages N, yiels the expecte number of page accesses A nn,mm,fbf The assumptions of the moel, however, are unrealistic for nearest neighbor queries on high-imensional ata for several reasons First, the number N of objects in the atabase is assume to approach to the infinity Secon, effects of high-imensional ata spaces an correlations are not consiere by the moel Cleary [Cleary 979] extens the moel of Frieman, Bentley an Finkel [Frieman et al 977] by allowing non-rectangular page regions, but still oes not consier bounary effects an correlations Eastman [Eastman 98] uses the existing moels for optimizing the bucket size of the k-tree Sproull [Sproull 99] shows that the number of ata points must be exponential in the number of imensions for the moels to provie accurate estimations Accoring to Sproull, bounary effects significantly contribute to the costs unless the following conition hols: where V S ( r) A nn,mm,fbf = C eff N >> C eff V S -- 2 is the volume of a hypersphere with raius r which can be compute as V S ( r) with the gamma-function Γ( x) which is the extension of the factorial operator x! = Γ( x + ) into the omain of real numbers: Γ( x + ) = x Γ( x), Γ( ) = an Γ( 2 -- ) = π For example, in a 20-imensional ata space with = 20, Sproull s formula evaluates to N >> 0 We will see later, how ba the cost estimations of the FBF moel are if substantially fewer than a hunre billion points are store in the atabase Unfortunately, Sproull still assumes for his analysis uniformity an inepenence in the istribution of ata π = Γ ( 2 + ) r

8 8 Christian Böhm Figure : Evaluation of the moel of Frieman, Bentley an Finkel points an queries, ie both ata points an the center points of the queries are chosen from a uniform ata istribution, whereas the selectivity of the queries (/N) is consiere fix The above formulas are also generalize to k-nearest neighbor queries, where k is also a user-given parameter The assumptions mae in the existing moels o not hol in the high-imensional case The main reason for the problems of the existing moels is that they o not consier bounary effects Bounary effects stans for an exceptional performance behavior, when the query reaches the bounary of the ata space As we show later, bounary effects occur frequently in high-imensional ata spaces an lea to pruning of major amounts of empty search space which is not consiere by the existing moels To examine these effects, we performe experiments to compare the necessary page accesses with the moel estimations Figure shows the actual page accesses for uniformly istribute point ata versus the estimations of the moel of Frieman, Bentley an Finkel For high-imensional ata, the moel completely fails to estimate the number of page accesses The basic moel of Frieman, Bentley an Finkel has been extene in two ifferent irections The first is to take correlation effects into account by using the concept of the fractal imension [Manelbrot 977, Schröer 99] There are various efinitions of the fractal imension which all capture the relevant aspect (the correlation), but are ifferent in the etails, how the correlation is measure We will not istinguish between these approaches in our subsequent work Faloutsos an Kamel [Faloutsos an Kamel 994] use the box-counting fractal imension (also known as the Hausorff fractal imension) for moeling the performance of R- trees when processing range queries using maximum metric In their moel they assume to have a correlation in the points store in the atabase For the queries, they still assume a uniform an inepenent istribution The analysis oes not take into account effects of high-imensional spaces an the evaluation is limite to ata spaces with imensions less or equal to 3 Belussi an Faloutsos [Belussi an Faloutsos 995] use in a subsequent paper the fractal imension with a ifferent efinition (the correlation fractal imension) for the selectivity estimation of spatial queries In this paper, range queries in low-imensional ata spaces using Manhattan metric, Eucliean metric an maximum metric were moele Unfortunately, the moel only allows the estimation of selectivities It is not possible to exten the moel in a straightforwar way to etermine expectations of page accesses Papaopoulos an Manolopoulos use the results of Faloutsos an Kamel an the results of Belussi an Faloutsos for a new moel publishe in a recent paper [Papaopoulos an

9 A Cost Moel For Query Processing in High-Dimensional Data Spaces 9 Figure 2: The Minkowski Sum Manolopoulos 997] Their moel is capable of estimating ata page accesses of R-trees when processing nearest neighbor queries in a Eucliean space They estimate the istance of the nearest neighbor by using the selectivity estimation of Belussi an Faloutsos [Belussi an Faloutsos 995] in the reverse way We will point out in section 4 that this approach is problematic from a statistical point of view As it is ifficult to etermine accesses to pages with rectangular regions for spherical queries, they approximate query spheres by minimum bouning an maximum enclose cubes an etermine upper an lower bouns of the number of page accesses in this way This approach makes the moel inoperative for highimensional ata spaces, because the approximation error grows exponentially with increasing imension Note that in a 20-imensional ata space, the volume of the minimum bouning cube of a sphere is by a factor of V S ( 2) = larger than the volume of the sphere The sphere volume, in turn, is by V S ( 2) = 27, 000 times larger than the greatest enclose cube An asset of the moel of Papaopoulos an Manolopoulos is that queries are no longer assume to be taken from a uniform an inepenent istribution Instea, the authors assume that the query istribution follows the ata istribution The concept of fractal imension is also wiely use in the omain of spatial atabases, where the complexity of store polygons is moele [Gaee 995, Faloutsos an Gaee 996] These approaches are of minor importance for point atabases The secon irection, where the basic moel of Frieman, Bentley an Finkel nees extension, are the bounary effects occurring when inexing ata spaces of higher imensionality Arya, Mount an Narayan [Arya et al 995, Arya 995] presente a new cost moel for processing nearest neighbor queries in the context of the application omain of vector quantization Arya, Mount an Narayan restricte their moel to the maximum metric an neglecte correlation effects Unfortunately, they still assume that the number of points is exponential with the imension of the ata space This assumption is justifie in their application omain, but it is unrealistic for atabase applications Berchtol, Böhm, Keim an Kriegel [Berchtol et al 997b] presente in 997 a cost moel for query processing in high-imensional ata spaces, in the following calle BBKK moel It provies accurate estimations for nearest neighbor queries an range queries using the Eucliean metric an consiers bounary effects To cope with correlation, the authors propose to use the fractal imension without presenting the etails The main limitation of the moel are () that no estimation for the maximum metric is presente, (2) that the number of ata pages is assume to be a power of two an (3) that a complete, overlap-free coverage of the ata space with ata pages is assume Weber, Schek an Blott [Weber et al 998] use the cost moel by Berchtol et al without the extension for correlate ata to

10 0 Christian Böhm show the superiority of the sequential scan in sufficiently high imensions They present the VA-file, an improvement of the sequential scan Ciaccia, Patella an Zezula [Ciaccia et al 998] aapt the cost moel [Berchtol et al 997b] to estimate the page accesses of the M- tree, an inex structure for ata spaces which are metric spaces but not vector spaces (ie only the istances between the objects are known, but no explicit positions) Papaopoulos an Manolopoulos [Papaopoulos an Manolopoulos 998] apply the cost moel for eclustering of ata in a isk array Two papers [Rieel et al 998, Agrawal et al 998] present applications in the ata mining omain This paper is base on the BBKK cost moel which is presente in a comprehensive way an extene in many aspects The extensions not yet covere by the BBKK moel inclue all estimations for the maximum metric, which are evelope aitionally throughout the whole paper The restriction of the BBKK moel to numbers of ata pages which are a power of two is overcome (cf sections 52 an 53) A further extension of the moel regars k-nearest neighbor queries The corresponing cost formulas are presente in section 44 The numerical methos for integral approximation an for the estimation of the bounary effects were to the largest extent out of the scope of [Berchtol et al 997b] Improve evaluation methos which are not containe in the BBKK moel are escribe in sections 43, 52 an 53 These evaluation methos base on precompute volume functions for intersection solis have a high practical importance, because costly numerical methos such as a Montecarlo integration can be completely performe uring the compile time Finally, the concept of the fractal imension, which was also use in the BBKK moel in a simplifie way (the ata space imension is simply replace by the fractal imension) is in this paper well establishe by the consequent application of the fractal power laws 3 RANGE QUERY In this section, we assume uniformity an inepenence in the istribution of both, ata an query points Moreover, we ignore the existence of a bounary of the ata space or assume at least that page regions an queries are istant enough from the space bounary that the bounary is not touche We start with a given page region an a given query range r an etermine the probability with which the page is accesse uner the uniformity an inepenence assumption for the query point The uniformity an inepenence assumption means that the query point is a -imensional stochastic variable which is istribute such that each position of the ata space has the same probability 3 The Minkowski Sum The corresponing page is accesse, whenever the query sphere intersects with the page region To illustrate this, cf figure 2 In all figures throughout this paper, we will symbolize queries by spheres an page regions by rectangles Also our terminology ( query sphere, for example) will often reflect this symbolization We shoul note that queries using the maximum metric rather correspon to hypercubes than hyperspheres We shoul further note that not all known inex structures use hyperrectangles as page regions Our concepts presente here are also applicable if the shape of the query is cubical or the shape of the page region is spherical For estimating the access probability of a page, we make conceptually the following transformation: We enlarge the page region so that if the original page touche any point of the query sphere, then the enlarge page touches the center point of the query It is obvious from figure 2 that the page region becomes enlarge by a sphere of the same raius r whose center point is rawn over the surface of the page region All positions insie the grey

11 A Cost Moel For Query Processing in High-Dimensional Data Spaces marke region of figure 2 are the positions of the query point, where the page is accesse, an all positions outsie the grey area are the positions of the query point, where the page is not accesse Therefore, the marke volume ivie by the ata space volume irectly correspons to the access probability of the page As we assume for simplicity that the unit hypercube [0] is the ata space, the ata space volume correspons to The concept of enlarging a given geometrical object by another object in the escribe way is calle Minkowski sum [Kaul et al 99] The Minkowski sum of two objects A an B is efine as the vector sum of all points of the objects: A B = { a + b a A, b B} The main application of the Minkowski sum is robot motion planning In the context of cost moeling, only the volume V A B of the Minkowski sum is neee Therefore, we will not strictly istinguish between the Minkowski sum as a spatial object (soli) an its volume For the etermination of the volume, we have to consier various cases The simplest case is that both solis are hyperrectangles with sie lengths a i an b i, for 0 i < In this case, the volume V R R of the Minkowski sum of two rectangles is the rectangle with the sie lengths c i, where each c i correspons to the sum of a i an b i : V R R ( ( a 0,, a ) b, (,, b 0 )) = ( a i + b i ) 0 i < If both solis, query sphere an page region are hyperspheres with raius r q an r p, the Minkowski enlargement correspons to a hypersphere with the raius r q + r p The corresponing volume of the hypersphere can be evaluate by the following formula: V S S ( r q, r p ) The evaluation of the volume becomes more complex if query an page region are shape ifferently This is illustrate in a 3-imensional example in figure 3, which epicts some of the solis by which the page region (gray) becomes enlarge: At each corner of the gray soli, we have a part (/8) of a sphere (rawn is only one part s ) The compose parts woul form one complete sphere of raius r (the query raius) At each ege of the region, we have a part (/4) of a cyliner The parts form three complete cyliners where the raius is the query raius The length of c, for instance, is the with of the page region Last, at each surface of the region, we enlarge the region by a cuboi where two sie lengths are efine by the surface of the region an the remaining sie length is again r In the general - imensional case, our page region has not only corners, eges an 2-imensional surfaces, but surface segments with imensions up to Each surface with imensionality k is enlarge by a part of a hypercyliner which is spherical in k imensions In the remaining k imensions, the hypercyliner has the shape of the surface segment to which it is connecte Before we etermine the volume of the Minkowski enlargement in the most complex case of a hypersphere an a hyperrectangle, let us solve it for the simpler case that the page region is a hypercube with sie length a In this case, all k-imensional surface segments have the volume a k Still open is the question, how many such surface segments exist an what the volume of the assigne part of the hypercyliner is The number of surface segments can be etermine by a combinatorial consieration All surface segments (incluing the corners an the page region itself) can be represente by a -imensional vector over the π = Γ ( r q + r p )

12 2 Christian Böhm c 3 = LU* *U* s = LUL c = *UL page region =*** y z x c 2 = L*L **L Figure 3: Example of a 3-imensional Minkowski sum symbols L, U an * Here, the symbol L stans for the lower bounary, U for the upper bounary an the star stans for the complete interval between the lower an upper bounary Each position of a symbol L, U or * correspons to one imension of the original ata space Using this notation, the corners have no star, the eges have one star, the 2- imensional surfaces have two stars in the vector, an so on The hyperrectangle itself has stars, no L an no U in its escription vector In our example in figure 3, s is assigne to the vector (LUL), an c is assigne to the vector (*UL), because it covers the entire range in the x-irection an it is fixe at the upper en of the y-irection an at the lower en of the z-irection The number of k-imensional surface segments correspons to the number of ifferent vectors having k stars The positions of the stars can be arbitrarily selecte from positions in the vector, yieling a binomial number of possibilities The remaining k positions are fille with the symbols L an U Therefore, the number of surface segments SSEGM(k) of imension k is equal to: SSEGM( k) = k 2 k The fraction of the hypercyliner at each surface segment is /2 -k, because the cyliner is halve for each of the k imensions Therefore, we get the following formula for the Minkowski sum of a hypersphere with raius r an a hypercube with sie length a: V S C ( r, a) = a + SSEGM( k) a k k Γ k r k 0 k < + 2 a k π k = k Γ k r k 0 k + 2 π k In the most complex case of non-cubical hyperrectangles, the k-imensional surface segments must be summe up explicitly, which is a costly operation Instea of the binomial

13 A Cost Moel For Query Processing in High-Dimensional Data Spaces 3 multiplie with a k, we have to summarize over all k combinations of the sie lengths of the hyperrectangle, ie the power set 2 { 0 } : V S R ( r, a) a π k = ij { i i k } 2 { 0 } j = Γ k ( r) k 0 k + 2 We shoul note that in most cases the etermination of the Minkowski sum of a hypersphere an a hyperrectangle is an operation which is too costly, because it involves a number of basic operations (multiplications), which is exponential in the imension The explicit etermination of the Minkowski sum of a real hyperrectangle an a hypersphere of high imensionality must be avoie, even if exactness is sacrifice A usual work-aroun is to transform the page region into a hypercube with equivalent volume However, exactness is sacrifice in this approach, because the Minkowski sum of a non-cubical hyperrectangle is larger than the Minkowski sum of a volume-equivalent hypercube Our experiments later will show that this approximation oes not cause serious errors in the cost estimations for the X-tree, which strives for square-like page regions For other inex structures, this effect coul play a more important role Estimating access probabilities in such cases is subject to future work 32 Estimating Rectangular Page Regions To make moeling manageable, we assume page regions which have the form of a hypercube Observe that structures as the X-tree or the R * -tree try to prouce such page regions an that our preiction matches the experimental observation We exploit that the number of points enclose in an arbitrary hypercube of sie length a nonboun is proportional to the volume of this hypercube By the inex nonboun we inicate that this hypercube is not the minimum bouning rectangle of the enclose points, ie a nonboun is the expecte sie length of the page regions before taking the MBR effect (minimum bouning rectangle) into account Similarly, V R,nonboun is the volume of the page region without consiering the MBR effect Due to the proportionality, the number of points store in a page ivie by the number N of all points in the atabase is equal to the volume of the page ivie by the volume V DS of the ata space: C eff N V R,nonboun V DS a nonboun C = = a eff nonboun = N In this formula for the sie length of a typical page region, we assume a complete coverage of the ata space with page regions This assumption is not meaningful for minimum bouning rectangles Usually, there is a small gap between two neighboring page regions An expectation for the with of this gap uner uniformity an inepenence assumption can be etermine by projecting all points of a page onto the coorinate axis which is perpenicular to the gap The average istance between two neighboring projections is times the sie length of the region, because the projections of the points are uniformly istribute This is also the expecte value for the with of the gap by which the sie length of the page region is ecrease compare with a nonboun Therefore, the sie length of a minimum bouning rectangle can be estimate as: a = a nonboun = N k

14 4 Christian Böhm Compensation Factor Compensation Factor Capacity Dimension Figure 4: The Compensation Factor for Consiering Gaps The consieration of gaps between page regions is particularly important if the effective page capacity is low Figure 4 shows the compensation factor for varying page capacities (left sie) an for varying imensions (right sie) It shows the factor which ecreases the volume of a page region when gaps are consiere If the compensation factor is, no compensation is neee The smaller the compensation factor is, the higher is the impact of the MBR effect The left iagram shows the compensation factor for a fixe imension = 6 with varying capacity The strongest MBR effect occurs for low capacities For a typical capacity between 20 an 40 points per ata page (which is the most frequently applie capacity accoring to our experience), the compensation factor ranges between 40% an 70% The right iagram shows the compensation factor for a fixe effective page capacity (30 points) an varying imension Most compensation is necessary for large imensions 33 Expecte Number of Page Accesses By inserting the expecte sie length a into the formulas for the Minkowski enlargement, it is possible to etermine the access probabilities of typical ata pages uner uniformity an inepenence assumption This is for the maximum metric: X r,mm,ui ( r) ( 2r + a) = = 2r C eff N Note again that the volume correspons to the access probability as the ata space is normalize to the unit hypercube For Eucliean metric, the access probability for range queries with raius r evaluates to: X r,em,ui ( r) = 0 k k a k π k Γ k r k + 2 = k C k eff π k N Γ k r k 0 k + 2

15 A Cost Moel For Query Processing in High-Dimensional Data Spaces 5 From these access probabilities, the expecte number of page accesses can be etermine by multiplying the access probability with the number of ata pages N : N A r,mm,ui ( r) = 2r For Eucliean metric, the corresponing result is: A r,em,ui ( r) = N k C k eff π k N Γ k r k 0 k NEAREST NEIGHBOR QUERY In [Berchtol et al 997c] an [Böhm 998], the optimality of the HS algorithm [Hjaltason an Samet 995] for nearest neighbor search was proven The HS algorithm yiels exactly the same page accesses as an equivalent range query, ie a range query using the istance to the nearest neighbor as query range This enables us to reuce the problem of moeling nearest neighbor queries to the problem of moeling range queries, which was solve in section 3 Therefore, we have to estimate the nearest neighbor istance an apply this istance as the raius in the range query moel Like in section 3 we start with the assumptions of an inepenent, uniform ata istribution an we will ignore bounary effects These effects will be investigate in epth in section 5 an section 6 4 Coarse Estimation of the Nearest Neighbor Distance A simple way to estimate the nearest neighbor istance is to choose a sphere in the ata space such that an expecte value of one ata point is containe in it accoring to the current point ensity an to use the raius of this sphere as an approximation of the actual nearest neighbor istance In the case of the maximum metric, we get the following formula: --- = V N q = ( 2r) r For Eucliean metric, the corresponing formula is: π --- = V N q = Γ ( 2 + ) r r = N Γ( 2 + ) N π Unfortunately, this approach is not correct from the point of view of stochastics, because the operation of builing an expectation is not invertible, ie the expectation of the raius cannot be etermine from the expectation of the number of points in the corresponing sphere The approximation etermine by this formula is rather coarse an can be use only if a fast an simple evaluation is of higher importance than the accuracy of the moel The general problem is that even uner uniformity an inepenence assumption the nearest neighbor istance yiels a certain variance, when several range queries are execute 42 Exact Estimation of the Nearest Neighbor Distance A stochastically correct approach is to etermine a istribution function for the nearest neighbor istance, an to erive an expectation of the nearest neighbor istance from the =

16 6 Christian Böhm corresponing probability ensity function From this probability ensity function, the expectation of the number of page accesses can also be erive Accoring to the rules efining a istribution function P(r) for the nearest neighbor istance we must etermine the probability, with which the actual nearest neighbor istance is smaller than the stochastic variable r The nearest neighbor istance is larger than r if an only if no ata point is containe in the sphere with raius r The event istance is larger than r is the opposite of the event neee in our istribution function Therefore, P(r) is as follows: P( r) = ( V( r) ) N Here, V( r) moels the probability with which a point selecte from a uniform istribution is not containe in the volume Therefore, ( V( r) ) N is the probability with which all points are outsie the volume Subtracting this from is the probability of at least one point being insie Due to the convergence of the limit k lim + -- ν ν the istribution function can be approximate for a large number of objects N by the following formula: P( r) e N V( r) This approximation yiels negligible relative errors for a large N (starting from 00) an will facilitate the evaluation later in this paper For maximum metric an Eucliean metric, the probability istribution function P(r) can be evaluate in the following way: P mm ( r) = ( ( 2r) ) N, From the istribution function P(r) a probability ensity function p(r) can be erive by ifferentiation: For maximum an Eucliean metric, this evaluates to: p mm ( r) N ( ( 2r) ) N (, r 2r ) Figure 5 shows some probability ensity functions for 00,000 uniformly istribute ata points in a two-imensional an an 8-imensional ata space, respectively To etermine the expecte value of the nearest neighbor istance, the probability ensity function multiplie with r must be integrate from 0 to infinity: ν = e k P em ( r) = Γ ( 2 + ) r p( r) P mm ( r) = = r π P( r) = r P em ( r) N π p em ( r) r r Γ r N π = = ( 2 + ) Γ r ( 2 + ) N

17 A Cost Moel For Query Processing in High-Dimensional Data Spaces 7 = 2 = 8 p mm (r) p em (r) p mm (r) p em (r) r Figure 5: Probability Density Functions r r p( r) r, R mm = N ( ( 2r) ) N ( 2r ) r, The integration variable is enote by instea of the more usual to avoi confusion with the ientifier staning for the imension of the ata space The integral is not easy to evaluate analytically 43 Numerical Evaluation R = We present two numerical methos to evaluate the integrals presente at the en of section 42 numerically The first is base on the binomial theorem Basically, the probability ensity function p(r) is a polynomial of the egree N It can be transforme into the coefficient form p( r) = a 0 r N + a r N + using the binomial theorem We emonstrate this for the maximum metric: ( ( 2r) ) N = N ( ) i ( 2r) i ; i 0 0 π R em N Γ r N π ( 2 + ) Γ r = r ( 2 + ) p mm ( r) 0 0 i N = N ( 2r) N r ( ) i ( 2r) i i 0 i N This alternating series can be approximate with low error bouns by the first few summans ( 0 i i max ) if the absolute value of the summans is monotonically ecreasing

18 8 Christian Böhm This is possible if the power of r ecreases in a steeper way with increasing i than the binomial increases This is guarantee if the following conition hols: N ( 2r) r -- 2 N --- p 2 N mm ( r) ( 2r) N ( ) i ( 2r) i r i Therefore, we approximate our formula for the expecte nearest neighbor istance in the following way: R mm N ( 2r) N ( ) i ( 2r) i r i N N 0 i i max N i 0 i i max 0 i i max ( ) i ( 2r) ( i + ) r N ( ) i i N 0 i i max i The same simplification can be applie for the Eucliean metric However, an alternative way base on a histogram-approximation of the probability ensity function yiels lower approximation errors an causes even a lower effort in the evaluation To facilitate numerical integration methos such as the milebox approximation, the trapezoi approximation or the combine metho accoring to Simpson s rule [Press et al 988], we must etermine suitable bounaries, where the probability ensity function has values which are substantially greater than 0 If we consier for example figure 5, we observe that for =8, p mm (r) is very close to 0 in the two ranges 0 r 005 an 06 r Only the range between the lower boun r lb = 005 an the upper boun r ub = 06 contributes significantly The criterion for a sensible choice of lower an upper bouns is base on the istribution function which correspons to the area below the ensity function We choose the lower boun r lb such that the area in the ignore range [0r lb ] correspons to a probability less than 0% an o the same for the upper boun r ub We get the following two conitions, resulting from the approximation of the istribution function: P( r) e N V( r) 000 Integration can therefore be limite to the interval from r lb to r ub The integral can be approximate by a sum of trapezois or by a sum of rectangles: r ub r lb R = r p( r) r r ub r lb i + r lb p r ub r lb i + r lb i N P( r) e N V( r) 0999 ln0999 r r lb = V r r N ub = V ln000 N i max 0 i < i max i max i max