Boosting Spatial Pruning: On Optimal Pruning of MBRs

Transcription

1 Boostng Spatal Prunng: On Optmal Prunng of MBRs Tobas Emrch, Hans-Peter Kregel, Peer Kröger, Matthas Renz, Andreas Züfle Insttute for Informatcs, Ludwg-Maxmlans-Unverstät München Oettngenstr. 67, D-8538 München, Germany ABSTRACT Fast uery processng of complex objects, e.g. spatal or uncertan objects, depends on effcent spatal prunng of the objects approxmatons, whch are typcally mnmum boundng rectangles (MBRs). In ths paper, we propose a novel effectve and effcent crteron to determne the spatal topology between mult-dmensonal rectangles. Gven three rectangles R, A, and B n a mult-dmensonal space, the task s to determne whether A s defntely closer to R than B. Ths domnaton relaton s used n many applcatons to perform spatal prunng. Tradtonal technues apply spatal prunng based on mnmal and maxmal dstance. These technues however show sgnfcant defcences n terms of effectvty. We prove that our decson crteron s correct, complete, and effcent to compute even for hgh dmensonal databases. In addton, we tackle the problem of computng the number of objects domnatng an object o. The challenge here s to ncorporate objects that only partally domnate o. In ths work we wll show how to detect such partal domnaton topology by usng a modfed verson of our decson crteron. We propose strateges for conservatvely and progressvely estmatng the total number of objects domnatng an object. Our experments show that the new prunng crteron, albet very general and wdely applcable, sgnfcantly outperforms current state-of-the-art prunng crtera. Categores and Subject Descrptors H.3.3 [Informaton Search and Retreval]: Informaton Search and Retreval General Terms Performance. INTRODUCTION Speedng-up ueres usng mnmal boundng rectangles (MBRs) as object approxmatons s a common technue used n many dfferent ways. For example, rectangles are Permsson to make dgtal or hard copes of all or part of ths work for personal or classroom use s granted wthout fee provded that copes are not made or dstrbuted for proft or commercal advantage and that copes bear ths notce and the full ctaton on the frst page. To copy otherwse, to republsh, to post on servers or to redstrbute to lsts, reures pror specfc permsson and/or a fee. SIGMOD, June 6, 2, Indanapols, Indana, USA. Copyrght 2 ACM //6...$.. R A Fgure : Spatal prunng on MBRs. used for data sets wth spatally extended objects such as polygons or CAD models because operatons on the exact object representaton are usually much more expensve than on the object approxmatons. Furthermore, MBRs are used as spatal key for spatal access methods, e.g. the most promnent ones ncludng the R-Tree [5], R*-Tree [3], X- Tree [4] as well as specalzed adaptatons lke the TPR tree [25] and the U-Tree [22] among many others. In the last decade, MBR approxmatons have also become very popular for uncertan databases [5, 7, 8, 9, 2] n order to approxmate all possble locatons of an uncertan vector object such as a GPS sgnal. Rectangular approxmatons are commonly ntegrated nto spatal prunng methods n order to speed-up spatal ueres such as dstance-range (ε-range) ueres and k-nearest neghbor ueres. Generally, current spatal prunng methods utlze the boundares of regons, n partcular of axs-algned rectangles, n order to facltate the prunng,.e. to flter out true drops that do not match the uery predcate. In ths context, spatal prunng technues are used for numerous applcaton felds ncludng searchng n mult-dmensonal vector spaces [4, 8, 2], smlarty search n tme seres databases [9], uery processng on spato-temporal data [6, 24] and probablstc uery processng on uncertan data [5,, 22]. Most types of spatal/smlarty ueres used for the above mentoned applcatons, ncludng k-nearest neghbor (knn) ueres, reverse k-nearest neghbor (RkNN) ueres, and rankng ueres, commonly reure the followng nformaton. Gven three pont sets A, B, and R n a mult-dmensonal space R d, e.g. representng MBRs, the task s to determne whether object A s defntely closer to R than B w.r.t. a dstance functon defned on the objects n R d. If ths s the case, we say A domnates B w.r.t. R. An example of such a stuaton s depcted n Fgure. In fact, we wll focus on pont sets that represent rectangles, e.g. mnmum boundng rectangles (MBRs), because rectangles are the most preva- B

2 lent form of approxmatons for sets of ponts representng more complex objects as mentoned above. However, t should be noted that the concepts presented here can be easly extended to general pont sets representng e.g. pxels of pctures, mult-represented objects, spatal objects [6], etc. The concept of domnaton s a central problem for most types of smlarty ueres ncludng the ones mentoned above n order to dentfy true hts and true drops (prunng). For example, n case of a NN uery around R, we can prune B f t s domnated by A w.r.t. R and for an RNN uery around R, we can prune B f A domnates R w.r.t. B. The domnaton problem s trval for pont objects. However, appled to rectangles, the domnaton problem s much more dffcult to solve. The problem s that the dstance between two objects approxmated by rectangles s no longer a sngle value but s represented by an nterval. If two such dstance ntervals overlap, we cannot defntely detect whether one dstance s smaller than the other. Tradtonally, the mnmal dstance and maxmal dstance between rectangles are used to decde whch object s closer to another object. For example, n a nearest-neghbor uery we can prune all objects whose mnmal dstance to the uery object exceeds the maxmal dstance between at least one other object and the uery object. In fact, the tradtonal dstance approxmatons based on mnmal and maxmal dstance are not always sutable to determne the dstance relatonshp between objects. An example s depcted n Fgure showng three objects A, B and R each approxmated by rectangles. In order to decde whether object A s closer to R than object B, we cannot apply the mnmal/maxmal dstances because the mnmal dstance between B and R s smaller than the maxmal dstance between A and R. Here, the problem s that when comparng the maxmal dstance between A and R wth the mnmal dstance between B and R we take two dfferent postons of the object R nto account. For the maxmal dstance between A and R, we assume that the object R s located at the upper left corner of ts rectangle approxmaton. For the mnmum dstance between B and R we assume that the object R s located at the lower rght corner of ts rectangle approxmaton. However, snce an object approxmated by a rectangle cannot be located at dfferent postons at the same tme, the two dstances between A and R and between B and R depend on each other. To the best of our knowledge, none of the exstng work, except approaches for reverse k-nn ueres [23, 2], take ths dependency nto account. In our example, n fact, t can be detected that object A s closer to R than to object B when takng the above mentoned condtons nto account. For many spatal uery applcatons, t even does not suffce to determne whether an object s domnated by another object. For example, n order to detect whether an object A approxmated by a rectangle belongs to the knns of a uery object Q (k > ), we have to determne the number of objects that domnate A w.r.t. Q. In such a case, we have to tackle the general problem of computng the number of objects domnatng a gven object A w.r.t. a gven object R whch we call domnaton count of A w.r.t. R. The challenge here s to ncorporate also sets of objects where each of the elements does not domnate A f consdered separately, but the entre set domnates A f consdered as a compound object. We say that the sngle objects of such a set domnate A only partally, whle the set domnates A n the same sense dscussed above for sngle objects. In ths paper, we propose a new decson crteron for the domnaton problem that can be used n all of the above sketched applcatons and n any algorthm desgned for the above mentoned uery types. In addton, t s the bass for our novel method to determne conservatve and progressve bounds for the domnaton count of an object effcently. In partcular, we clam the followng contrbutons. We dscuss current state-of-the-art decson crtera for the domnaton problem among rectangles focussng on ther correctness, completeness, and effcency. We propose a novel decson crteron for the domnaton problem among rectangles that s correct, complete, and can be effcently computed. We propose a number of heurstcs that can be used to estmate the domnaton count n consderaton of partal domnaton. We show how our domnaton decson crteron and our heurstcs to determne the domnaton count can be used to mprove spatal prunng strateges for a varety of spatal uery processng methods. We present extensve experments to evaluate our new prunng crtera n comparson to state-of-the-art approaches. The remander of ths paper s organzed as follows: Secton 2 ntroduces our novel domnaton decson crteron. An effectve estmaton of the domnaton count s proposed n Secton 3. The applcablty of our concepts for spatal uery problems are dscussed n Secton 4. Secton 5 presents expermental results and Secton 6 fnally concludes the paper. 2. DETERMINING DOMINATION 2. The Problem of Domnaton Decson Let D R d be a database of d-dmensonal ponts and dst be a dstance functon on objects n R d. In ths paper we wll focus on the L p norms as the most commonly used famly of dstance functons n the area of smlarty search. Intutvely, our problem s the followng. Gven the pont sets A, B, R D, we want to decde f A s defntely closer to R than B to R w.r.t. the dstance functon dst. If ths s the case, we say A domnates B w.r.t. R. In fact, we wll focus on ponts sets that represent rectangles, e.g. mnmum boundng rectangles (MBRs) because rectangles are the most prevalent form of approxmatons for sets of ponts representng more complex objects lke page regons of drectory nodes n spatal ndex structures, polygons, tme seres, uncertan objects, etc. (see above). Defnton (Domnaton). Let A, B, R R d be rectangles. The rectangle A domnates B w.r.t. R ff for all ponts r R t holds that every pont a A s closer to r than any pont b B,.e. Dom(A, B, R) r R, a A, b B : dst(a, r) < dst(b, r) ()

3 R a H ab b Fgure 2: MBR prunng example B H AB A R To determne Dom(A, B, R), Euaton s not very helpful because a rectangle contans an nfnte number of ponts n R d and t s smply not computable to test all trples a A, b B and r R. Rather, a domnaton decson crteron DDC(A, B, R) for the sngle domnaton relaton s reured, whch should fulfll the followng propertes: Correctness: f DDC(A, B, R) returns true then A domnates B w.r.t. R,.e. DDC(A, B, R) Dom(A, B, R). Completeness: f DDC(A, B, R) returns false then A does not domnate B w.r.t. R,.e. DDC(A, B, R) Dom(A, B, R). Effcency: DDC(A, B, R) can be evaluated effcently. 2.2 Exstng Domnaton Decson Crtera In the followng, X = [X mn, X max ] represents the nterval of the rectangle X n dmenson, X md = /2 (X mn + X max ) s the mean of nterval X, and x denotes the value of pont x n dmenson ( d). The Mn-/MaxDst decson crteron. Probably the most well-known decson crteron for the domnaton problem among rectangles used n many database applcatons s based on two well known metrcs defned on rectangles [2]. The mnmum dstance MnDst(A, B) between two rectangles A and B always underestmates the dstance of pont pars (a, b) A B and s defned as v 8 u < A mn B max p, f A mn > B max MnDst(A, B) = t p B mn A max p, f B mn > A max :, else (2) The maxmum dstance MaxDst(A, B) between two rectangles A and B always overestmates the dstances of all pont pars (a, b) A B and s defned as: v ux MaxDst(A, B) = t d j A max p B mn p, fa md B md B max A mn p, fb md > A md Defnton 2 (Mn-/MaxDst crteron). Let A, B, R R d be rectangles. The Mn-/MaxDst domnaton decson crteron s defned as DDC MnMax(A, B, R) MaxDst(A, R) < MnDst(B, R). Lemma. The Mn-/MaxDst decson crteron s correct,.e. DDC MnMax(A, B, R) Dom(A, B, R). (3) H BA Fgure 3: Vorono-based decson crteron on MBRs Proof. The followng holds due to the conservatve propertes of MnDst and MaxDst: DDC MnMax(A, B, R) MaxDst(A, R) < MnDst(B, R) a A, r R, b B : dst(a, r) MaxDst(A, R) < MnDst(B, R) dst(b, r) Dom(A, B, R). Lemma 2. The Mn-/MaxDst decson crteron s not complete,.e. DDC MnMax(A, B, R) Dom(A, B, R). Proof. Fgure 2 shows an example for the 2D space where DDC MnMax(A, B, R) s false although Dom(A, B, R) holds. In the examples, A = a and B = b are rectangles wth zero extenson,.e. ponts. Clearly, MaxDst(a, R) < MnDst(b, R) s not satsfed,.e. DDC MnMax(a, b, R) s false. The Vorono lne H ab between a and b,.e. the lne contanng all ponts that have eual dstance to a and b, whch s the dashed lne n Fgure 2 dvdes the 2D space nto two half spaces. It s obvous that all ponts above that lne (located n the half space contanng a) have a dstance to a that s smaller than the dstance to b. Thus, accordng to Defnton 2., a domnates b w.r.t. all objects whch le completely above H ab. As a conseuence, Dom(a, b, R) holds. Let us note that the Mn-/MaxDst domnaton decson crteron s complete for two arbtrary rectangles A and B f R s a pont,.e. R has no extenson n all dmensons. In addton, the Mn-/MaxDst domnaton decson crteron can be computed effcently n O(d) tme snce the calculaton of MnDst and MaxDst s lnear n d. Vorono-based decson crteron. The Vorono plane H ab between two ponts a and b that has been used n the proof of Lemma 2 s used n [23] as a dfferent decson crteron for ponts. In a d-dmensonal space H ab = {x R d dst(a, x) = dst(b, x)} s a (d )-dmensonal hyperplane contanng all ponts havng eual dstance to a and to b. It dvdes the space nto two half-spaces H ab (a) contanng a and H ab (b) contanng b. If a rectangle R les completely wthn one of these half-spaces, then R s closer to the respectve pont n the same half-space. In the example of Fgure 2, R s n the half-space H ab (a), thus all r R are closer to a than to b. A Vorono hyperplane between a pont and a rectangle has been proposed n [2]. For the general case of two rectangles, we need to construct the Vorono plane H AB between two rectangles A and B whch s the ntersecton of all Vorono half-spaces between all pars of ponts of the correspondng rectangles and can be defned as H AB = {x R d MnDst(x, B) = MaxDst(x, A)}, see []. An example of a Vorono plane between two rectangles A and B s H AB, depcted n Fgure 3. Ths Vorono

4 plane s pecewse lnear and curvlnear (cf. [] for more detals on the Vorono plane between two rectangles). If a rectangle les completely wthn the half-space H AB(A), then R s defntely closer to A. However, to determne the half-space contanng all ponts that are defntely closer to B than to A, H AB(B) cannot be used and the Vorono plane H BA has to be computed. The reason s that unlke n the case of ponts, there exst ponts p for whch nether Dom(A, p, R) nor Dom(B, p, R) s true. Intutvely, the Vorono-based domnaton decson crteron states that A domnates B w.r.t. R f R s completely contaned n the half-space H AB(A). Defnton 3 (Vorono-based crteron). Let A, B, R R d be rectangles. The Vorono-based decson crteron s defned as DDC Vorono (A, B, R) R H AB(A). Lemma 3. The Vorono-based decson crteron s correct and complete,.e. DDC Vorono (A, B, R) Dom(A, B, R). Proof. By defnton of H AB the statement holds: DDC Vorono (A, B, R) R H AB(A) a A, b B: R H ab (a). a A, b B, r R : dst(a, r) < dst(b, r) Dom(A, B, R). Computng any Vorono plane between any a A and b B to obtan the curvlnear plane as depcted n Fgure 3 s rather complex. To the best of our knowledge, there exsts no effcent soluton for ths problem. However, t s clear that any such algorthm must scale exponentally n the dmensons, snce even for the smple case where b s a pont, the number of dfferent peces of the plane s eual to the number of corners of A whch s n O(2 d ) (cf. [2] for a dscusson on the computaton of such Vorono planes). Corner-based decson crteron. The corner-based decson has recently been proposed as a prunng crteron for RkNN search of spatal objects n R 2 []. Ths approach explots the property that the sde H AB(A) of H AB that s responsble for prunng s convex for RkNN ueres. Thus, f a rectangle R s not fully contaned n H AB(A) (.e. R cannot be pruned), then at least one corner of R must be contaned n H AB(B). Therefore, t s suffcent to consder only corners of MBRs. The Mn-/MaxDst decson crteron, that s correct and complete n the case where only ponts are consdered, s then appled to the corners. For more detals on ths decson crteron, refer to []. However, snce ths crteron reures to consder all 2 d corners of MBRs, the complexty must scale n O(2 d ). Summary. Table summarzes the dscusson of exstng decson crtera for the domnaton problem. It can be observed, that none of these approaches meets all the desred propertes,.e. ether s not complete or suffers from exponental runtme. The fourth approach n Table called Optmal s our new decson crteron whch s descrbed n the next secton. 2.3 A Correct, Complete, and Lnear-Tme Domnaton Decson Crteron We wll derve a new decson crteron that s correct, complete, and can be computed n O(d) tme. Our novel domnaton decson crteron can be derved from the orgnal defnton of domnaton n Defnton by applyng the followng sx euvalences. Table : Overvew decson crtera Crteron Correct Complete Effcent DDC MnMax YES NO YES: O(d) DDC Vorono YES YES NO: O(2 d ) DDC Corner YES YES NO: O(2 d ) DDC Optmal YES YES YES: O(d) Euvalence. a A, b B, r R : dst(a, r) < dst(b, r) r R : MaxDst(A, r) < MnDst(B, r) Proof. () If the left-hand sde holds for each r R then t also holds for that a A and b B that maxmze and mnmze the dstance to r, respectvely. These ponts a A and b B obvously determne the values of MaxDst and MnDst, respectvely. (2) If the rght-hand sde holds for each r R as well as for that a A and b B that maxmzes and mnmzes the dstance to r,.e. determnes the value of MaxDst and MnDst, respectvely, then t also holds for any a A and any b B. Euvalence 2. r R : MaxDst(A, r) < MnDst(B, r) r R : p Pd MaxDst(A, Pd r)p < p Proof. Follows drectly from the defnton of MaxDst and MnDst for L p norms (see above). Euvalence 3. p Pd r R : MaxDst(A, Pd r)p < p r R : P d (MaxDst(A, r)p MnDst(B, r ) p ) < Proof. p Pd r R : MaxDst(A, Pd r)p < p r R : P d MaxDst(A, r)p < P d r R : P d MaxDst(A, r)p P d < r R : P d (MaxDst(A, r)p MnDst(B, r ) p ) < Euvalence 4. r R : P d (MaxDst(A, r)p MnDst(B, r ) p ) < max P d r R( (MaxDst(A, r)p MnDst(B, r ) p )) < Proof. Instead of consderng all possble r R, t s suffcent to consder only that pont r R whch maxmzes the left-hand sde of the neualty. If the neualty holds for ths pont r, then t obvously holds for all possble r R and vce versa. The next euvalence reures the followng lemma: Lemma 4. Let F : R d R be a functon that s summed by treatng each dmenson ndependently,.e. there exsts a functon f : R R such that F (o) = f(o ) Also, let A R d be a rectangle and σ := argmax a A (F (a))

5 be the object n A that maxmzes F. Then, the followng holds: Proof. max ( f(a )) a A Def F (a) = max ( f(a )) = a A Def F (a) max (f(a )) a A Def = max(f (a)) a A f(σ ) Def σ = σ = F (σ) max (f(a )) a A Euvalence 5. max (P d r R MaxDst(A, r)p MnDst(B, r ) p ) < P d max (MaxDst(A, r ) p MnDst(B, r ) p ) < r R Proof. Ths follows from lemma 4 by substtutng F (r) = MaxDst(A, r) MnDst(B, r) The fnal euvalence (euvalence 6) makes the euaton computable. It s based on the assumpton that for fndng the maxmum r n dmenson, t s suffcent to consder the boundary ponts (R mn and R max ) of the nterval R. Ths assumpton s proven n the followng two lemmas. Lemma 5. Let A and B be ntervals. The functon f : R R defned as f(x) = MaxDst(A, x) p MnDst(B, x) p has no local maxmum. Proof. A formal proof for ths lemma can be found n the appendx of the extended verson of ths paper [3]. Lemma 6. Let f : R R be a functon that has no local maxmum and I = [I mn, I max] R be an arbtrary fnte nterval. The value that maxmzes f n the nterval I must be ether I mn or I max,.e. argmax(f()) {I mn, I max} I Proof. Let p [I start, I end ] be the value that maxmzes f n I,.e. p = argmax I (f(a)). Then, I : f() f(p), n partcular, f(i mn) f(p) and f(i max) f(p). Note that f(i mn) < p and f(i max) < p cannot both be true, because ths would be a contradcton to the assumpton that f(x) has no local maxmum. Thus t must ether hold that f(i start) = f(p) or f(i end ) = f(p),.e. I mn = argmax I (f(x)) or I max = argmax I (f(x)). Now we can derve the fnal euvalence. Euvalence 6. P d maxr R (MaxDst(A, r ) p MnDst(B, r ) p ) < dp max r {R mn,r max } (MaxDst(A, r ) p MnDst(B, r ) p ) < Proof. Follows from lemma 5 and 6. Defnton 4 (optmal decson crteron). Our novel optmal domnaton decson crteron s defned as DDC Optmal (A, B, R) max r {R mn,r max } (MaxDst(A, r ) p MnDst(B, r ) p ) < Lemma 7. The novel optmal domnaton decson crteron s correct and complete. Proof. Correctness and completeness follow drectly from euvalences to 6. Obvously, the novel optmal domnaton decson crteron can be computed n O(d) tme and, thus, fulflls all three desred propertes mentoned n Secton DOMINATION COUNT COMPUTING In most applcatons, testng the sngle domnaton relaton of only two rectangles (w.r.t. a reference rectangle) s too basc. Rather, n the context of a set of rectangles O R d, the number of rectangles A O that domnate a gven rectangle B w.r.t. R (referred to as domnaton count) s reured. For example, a knn uery algorthm can use the nformaton that at least k rectangles of O domnate rectangle B O w.r.t. a uery rectangle R to dentfy B as true drop that can be pruned. The number of rectangles that domnate a gven rectangle can analogously be used e.g. for RkNN ueres and nverse rankng ueres. Defnton 5 (domnaton count). Let B, R R d be rectangles and O be a set of rectangles. The domnaton count of B w.r.t. R s defned by: DC(O, B, R) = mn{ A O :MaxDst(A, r)<mndst(b, r) } r R Intutvely, f the domnaton count of B w.r.t. R s k, then for each pont r R there exst at least k rectangles A O whch are closer to r than B. Let us note that the domnaton count of B w.r.t. R cannot be computed by smply countng the number of rectangles that domnate B w.r.t. R by means of Defnton because ths does not nvolve groups of rectangles that domnate R collectvely, but not ndvdually. An example of such a group of rectangles s shown n Fgure 4. Nether rectangle A nor rectangle A 2 domnates B w.r.t. R. However, B s domnated partally by A and partally by A 2, respectvely,.e. t s domnated by A and A 2 w.r.t. specfc subregons of R. However, when consderng any pont r R, rectangle B s domnated by at least one of the two rectangles A, A 2 w.r.t. r and, thus, B s domnated by the group A = {A, A 2} accordng to Defnton. In general, the problem of fndng the subregon wth the mnmal domnaton count s hard. Frst, the computaton of the ntersecton of a half-space and a hyper-polyhedron becomes ncreasngly complex [23] for ncreasng dmensonalty. Secondly, the number of subregons grows very fast. To gve a bref ntuton of the possble number of subregons generated by a total of n objects, consder the case of axs parallel prunng regons. If n d, then each object may splt R n a dfferent dmenson, resultng n a total of 2 n subregons. For n > d, balanced splttng of dmensons results n at least ( + n d d ) subregons. If d s assumed

6 to be constant, then ( + n d d ) O(n d ). Thrdly, the resultng subregons can be complex d-dmensonal polygons, partcularly the subregons could have not only straght sdes but also parabolc sdes whch makes computatons nvolvng these polygons very complex. Though we are not able to compute the exact domnaton count of a gven rectangle effcently, we can try to fnd effcent solutons for approxmatng the domnaton count of a rectangle. In prncpal, n order to determne the domnaton count of B w.r.t. R we need to take the two consttutng types of domnatons nto account: The frst part s to count all objects A for whch Dom(A, B, R) holds. Ths number s called basc domnaton count. Ths can be done usng e.g. DDC Optmal. The second and more challengng part s to detect all mnmal groups A that domnate B as a group but do not contan an element that already domnates B separately,.e. each A A only partally domnate B. The consderaton of ths type of domnaton reures the concept of partal domnaton whch wll be ntroduced later on. A smple lower bound of the domnaton count can be acheved by computng the basc domnaton count. Intutvely, the basc domnaton count smply counts the number of rectangles that (completely) domnate the rectangle B w.r.t. rectangle R,.e. neglects groups of rectangles that only partally domnate B separately but completely domnate B as a group. Defnton 6 (Basc Domnaton Count). Let O = {A,..., A N } be a set of d-dmensonal rectangles and let B, R R d be two rectangles. The basc domnaton count of B w.r.t. R s the number of objects n O that domnate B w.r.t. R, formally: DC basc (O, B, R) = {A O Dom(A, B, R)}. Usng our novel domnaton decson crteron DDC Optmal, the basc domnaton count DC basc can be computed n O(N d). Ths s worth notng snce exstng decson crtera only allow ether to compute the exact DC basc value n exponental tme or to compute an approxmaton of the DC basc value n lnear tme. In the latter case, we would obtan a lower bound of DC basc whch makes the lower boundng estmaton of the domnaton count even more loose. As dscussed above, the domnaton count also takes nto account all sets of rectangles that ncrease the domnaton count of a rectangle as a group and that do not contan any element that does so separately. Therefore, we need the concept of partal domnaton. In the remander of ths secton, we wll frst formalze the concept of partal domnaton. In partcular we wll dscuss how our novel domnaton decson crteron DDC Optmal can be used for () detectng partal domnaton and () dervng a conservatve approxmaton of the domnaton count. 3. Partal Domnaton The concept of partal domnaton (cf. Fgure 4) was frst ntroduced n [2] (the authors used the term partal prunng ) for boostng RkNN ueres n the 2D space. It can be appled to any other smlarty uery type analogously. Defnton 7 (partal domnaton). Let A, B, R R d be rectangles. A domnates B partally w.r.t. R, denoted by P Dom(A, B, R) f A domnates B for some, but not all r R,.e. B H A2B H AB Fgure 4: Partal Domnaton example for an RNNuery P Dom(A, B, R) A A 2 ( a A, b B, r R : dst(b, r) > dst(a, r)) (4) V r R : a A, b B : dst(b, r) > dst(a, r) (5) Ineualty 4 holds f A does not domnate B w.r.t. all ponts r R. Note that Ineualty 4 s smply the negaton of Dom(A, B, R) and can also be computed n O(d) usng our novel decson crteron DCC Optmal. Ineualty 5 s only satsfed f there exsts an r R for whch B s domnated by A. Obvously, the sets of objects that domnate B as a group can only contan rectangles A that partally domnate B,.e. for whch P Dom(A, B, R) holds. In other words, for the computaton of the second part of the domnaton count of a rectangle B, we could use the detecton of partal domnaton as a frst step because only those rectangles A for whch P Dom(A, B, R) holds could be the elements of those set of rectangles that domnate B as a group. Partal domnaton can effcently be detected by applyng the followng sx euvalences analogously to Secton 2.3. We start wth neualty 5. Euvalence 7. r R : a A, b B : dst(b, r) > dst(a, r) r R : MaxDst(A, r) < MnDst(B, r) Proof. Ths proof s analogous to the proof of Euvalence,.e. t explots that the DDC MnMax, decson crteron s optmal n the case where R s a pont. Euvalence 8. r R : MaxDst(A, r) < MnDst(B, r) r R : p Pd R MaxDst(A, Pd r)p < p Proof. Follows drectly from the defnton of MaxDst and MnDst for L p norms. Euvalence 9. p Pd r R : MaxDst(A, Pd r)p < p r R : P d MaxDst(A, r)p MnDst(B, r ) p < Proof. p Pd r R : MaxDst(A, Pd r)p < p r R : P d MaxDst(A, r)p < P d r R: P d MaxDst(A, r)p P d < r R : P d MaxDst(A, r)p MnDst(B, r ) p <

7 Euvalence. r R : P d MaxDst(A, r)p MnDst(B, r ) p < MIN P d r R( MaxDst(A, r)p MnDst(B, r ) p ) < Proof. The ratonale for euvalence s that f there exsts an r R for whch the left-hand sde returns less than, then ths also holds for the r whch mnmzes the term on the rght-hand sde and vce versa. B H AB A R Euvalence. mn P d r R( MaxDst(A, r)p MnDst(B, r ) p ) < P d mnr R (MaxDst(A, r ) p MnDst(B, r ) p ) < Proof. Ths proof s analogous to the proof of Euaton 5 usng mnmzaton nstead of maxmzaton. Analogously to Euvalence 6, the last euvalence below makes the euaton computable. Agan, we need two lemmas. Lemma 8. Let D be a one dmensonal vector database usng L p-norm. Let A and B be ntervals. The functon f : R R: f(x) = maxdst(a, x) p mndst(b, x) p may have a local mnmum only at A.mean. Proof. Ths proof s contaned n the formal proof of lemma 5 n the appendx of the extended verson [3]. Lemma 9. Let f : R R be a functon that has at most one local mnmum at x. For any fnte nterval I R = [I start, I end ] the followng holds: argmn(f()) {I start, I end, x} I That s, the pont of the nterval I that mnmzes f(x) must be ether the lower or the upper bound of I, or the local mnmum x. Proof. The proof s smlar to the proof of Lemma 6 and thus omtted here. In consderaton of the above lemmas we now derve the fnal euvalence: Euvalence 2. P d mnr R (MaxDst(A, r ) p MnDst(B, r ) p ) < P d mnr (MaxDst(A, r)p MnDst(B, r ) p ) <, where r {R mn, R max, A md } Proof. Drectly follows from Lemma 8 and Lemma 9. Thus, usng the formula n Euvalence 2 we can effcently detect all partal domnatons. However, as ndcated above, ths s only the frst step towards computng the domnaton count. In fact, we need to determne that subregon of the reference rectangle R, for whch the domnaton count s mnmal. Snce we cannot test all possble ponts r R (see also the dscusson above), we propose three heurstcs to conservatvely approxmate the domnaton count of a rectangle. Fgure 5: Partal domnaton usng grd parttonng 3.2 Domnaton Count Estmaton Usng the technues proposed n Sectons 2 and 3. we can check f an MBR A domnates B completely or partally w.r.t. R. These tests are generally applcable as long as the nvolved objects are MBRs. For calculatng the domnaton count of B t s therefore possble to splt R nto smaller MBRs and then calculate the domnaton count for each cell ndvdually. The followng three heurstcs use dfferent approaches for splttng R to estmate the domnaton count Domnaton Count Estmaton based on grd parttonng A straght forward approach for splttng R s performed by usng a grd wth a fxed number m of parttons n each dmenson. Consderng the example n Fgure 5, we can (usng the decson crtera for domnaton and partal domnaton) assert that A domnates B w.r.t. all dark gray cells and partally domnates B w.r.t. all lght gray cells of R. For the rest of the cells (whte) A does not domnate B. Usng ths grd parttonng, the domnaton count (DC(O, B, R)) can be estmated by the mnmum domnaton count of all cells c R, that s: DC grd (O, B, R) = mn (DC basc (O, B, c )) Ths estmaton s vald as we know that B s domnated by at least ths amount of A O w.r.t. each cell c R. An example for the grd based partal prunng s gven n Fgure 6. Here an MBR R s parttoned nto 6 cells. In addton two Vorono hyperplanes H A B and H A2 B are shown. The objects O = {A, A 2} and B generatng the hyperplanes are ommted here. For the area on the rghthand sde of H A B, object B s domnated by object A and for the left-hand sde of H A2 B, B s domnated by A 2. It s clear that nether A nor A 2 (fully-) domnate B wth respect to the whole MBR R. For each cell the conservatve domnaton count DC basc (O, B, c ) s shown. Wth respect to dark cells, A and A 2 domnate B and thus the cells have a value of 2. Wth respect to lght cells, only one of the two objects domnates B, therefore they get marked wth a value of. By takng the mnmum value of all cells c R we obtan DC grd (O, B, R) =. The advantage of ths approach s, that t returns a very accurate estmaton of the domnaton count whle avodng expensve materalzaton of the Vorono hyperplanes. The accuracy can be boosted by ncreasng the number of splts per dmenson. In return ncreasng m wll hghly ncrease the runtme of the algorthm, as the number of cells c R s m d. Ths mples that ths approach s not applcable for hgh dmensons. For

8 H AB H A2B H AB H A2B H AB H A2B Fgure 6: Domnaton Count estmaton usng grd parttonng. each cell c, DC basc (O, B, c ) can be computed n a sngle scan of the objects for whch P Dom(A, B, R) holds usng the DDC Optmal (c.f. Defnton 4). Thus the total tme complexty s n O(d O m d ) Domnaton Count Estmaton based on slces In order to reduce the runtme of the domnaton count estmaton, we propose a second algorthm, whch s not based on a grd parttonng. Instead of cells, ths approach consders slces. Therefore an MBR R s splt nto m slces s dm n each of the d-dmensons ( dm d). Ths results n d m overlappng slces. The domnaton count DC(O, B, R) can then be approxmated by computng, for each dmenson, the mnmal domnaton count of all slces and usng the result of the dmenson maxmzng ths estmaton. DC slce (O, B, R) = max dm (mn (DC Basc(O, B, s dm ))) For example, the domnaton count DC(O, B, s ) for each slce s (.e. each row and each column) and each cell c s shown n Fgure 6 for a 2 dmensonal MBR R. The mnmal domnaton count consderng all rows s, whle the mnmal domnaton count w.r.t. all columns s. Thus DC slce (O, B, s ) = n ths example. The complexty of ths algorthm s n O(m d). However, ths approach yelds much worse results than the grd-based approach for an dentcal m parameter. Detals can be found n our experments (Secton 5) Domnaton Count Estmaton based on bsectons We next propose a bsecton based approach that yelds much better effcacy, whle stll beng lnear n d. Ths approach works teratvely. Durng each teraton, one secton of R s chosen to be splt evenly (mean splt) n one dmenson. After m splts, ths results n m + sectons s s... s m = R and t holds that: DC bsect (O, B, R) = mn (DC basc (O, B, s )) The challenge here s to wsely choose the splt secton of R and the dmenson to splt n each teraton. We propose to splt the secton s R wth the lowest domnaton count estmaton. Ths decson s optmal, because the estmaton of DC(O, B, R) s determned by the secton whch results n the lowest domnaton count. Thus, n order to ncrease the domnaton count approxmaton, s must be splt. If the decson for s s ambguous, then one of the canddates of s s chosen arbtrarly. To determne the (a) ntal settng H AB H A2B (c) after 2nd splt (b) after st splt H AB H A2B (d) after 3rd splt Fgure 7: Example for computng DC bsect splt axs, the heurstc tests each dmenson, and greedly uses the dmenson that yelds the hghest domnaton count DC bsect (O, B, R) consderng the two resultng bsectons of s. In the case of tes the axs s chosen whch maxmzes the sum P m = DC basc(o, B, s ). An example s shown n Fgure 7. Consderng Fgure 7(a) t s clear that none of the two objects A, A 2 O that are responsble for the Vorono hyperplanes H A B and H A2 B domnates B w.r.t. R. Begnnng wth the y-axs as splt axs would result n two eu-szed MBRs both of whch result n a domnaton count DC bsect (O, B, R) of and therefore the approxmaton of DC(O, B, R) does not ncrease. Choosng the x-axs as splt axs would result n two eu-szed MBRs shown n Fgure 7(b) yeldng the same domnaton count approxmaton but a hgher sum ( P m = DC basc(o, B, s ) = ). In the next teraton, the rght MBR s chosen to be splt, snce t s responsble for the lowest domnaton count approxmaton. Both possble splt axes are eual accordng to our heurstc. In the example, the y-axs s chosen arbtrarly (c.f. Fgure 7(c)). The thrd (see Fgure 7(d)) splt of the lower-rght MBR ncreases DC bsect (O, B, R) to. The bsecton-based Domnaton Count Estmaton algorthm uses m teratons. In each teraton there exst exactly sectons of whch the secton wth the lowest conservatve domnaton count has to be found. Ths yelds a complexty of O(m 2 ) but can be reduced to O(m log(m))) by usng a Prorty Queue to fnd the secton wth the lowest conservatve domnaton count. For the greedy heurstc, n each teraton, each dmenson has to be tested to determne the best splt axs n O(m d). Thus we get a total complexty of O(m log(m) + m d) = O(m max(log(m), d)), where m s the number of teratons. 4. BOOSTING SIMILARITY QUERIES In ths secton, we wll show how the concepts of domnaton and domnaton count can be used to boost the prunng power of smlarty search algorthms. Nearest-Neghbor Search. For a knn uery wth uery object Q, any object O D can be pruned f DC(D, O, Q) k. Note, that for a knn uery, the uery object corresponds

9 DDC MnMax B,9 DDC Optmal,8,7 R A Rat to,6 5,5,4,3,2, sde length =. sde length =.2 sde length =.3 sde length =.4 sde length = Dmenson Fgure 8: Refnement areas for fxed R and A to the reference object R n Defnton 4. Thus, DDC Optmal has an advantage over DDC MnMax n the general case but s euvalent n the specal case where Q s a pont, because then DDC MnMax s optmal. However, as dscussed above, there are many applcatons n whch the uery object s a rectangle. Reverse Nearest Neghbor Search. For a general RkNN uery wth uery object Q, any object O DB can be certanly pruned f DC(D, Q, O) k. For RkNN ueres, the uery object corresponds to the object B n Defnton 4. Thus DDC Optmal s superor to DDC MnMax also n the specal case where the uery object s gven as a pont. True ht detecton. Our decson crteron DDC Optmal can be used to prune potental result canddates by beng able to decde that they must not be part of the result set. A problem very smlar to prunng s the detecton of true hts,.e. to uckly decde that a potental result canddate must be part of the result set. For example, n the case of knn ueres, an object B s a true ht, f there may be at most k objects that can be closer to R than B. In other words, B s a true ht, f t domnates at least D k objects. Thus, for a knn uery, an object B s a true ht f {A D dom(b, A, Q)} > D k. For a RkNN uery, an object B s a true ht f {A D dom(q, A, B)} D k. The concept of partal domnaton can be appled to true ht detecton as well. Inverse Smlarty Rankng. The problem of nverse rankng s to determne for a gven uery object Q the number of objects that are closer to a gven reference object R. Such ueres are useful e.g. to determne the fnancal standng of bank customers n relaton to exstng customers. In ths scenaro, the attrbutes of customers are often uncertan (e.g. ncome of 4k 5k) and thus modeled by uncertan regons,.e. rectangles. Lower and upper bounds for the rank of Q are DC(D, Q, R) + and D {A D dom(q, A, R)} +, respectvely. 5. EXPERIMENTAL EVALUATION Ths secton evaluates the effectveness and effcency of our novel domnaton decson crteron n comparson to the prevalent DDC MnMax decson crteron. After that we evaluate the performance of our domnaton-count-detecton approach whch s based on the concept of partal domnaton. Fnally, we exemplarly wll show how our new methods nfluences the performance of exstng smlarty search Fgure 9: Rato of the refnement areas of DDC Optmal and DDC MnMax w.r.t. dmenson and sze of MBRs methods desgned for knn and RkNN ueres. For all experments the underlyng dstance functon s the eucldan norm. 5. Sngle Object Domnaton We frst evaluate the effectveness gan of DDC Optmal compared to DDC MnMax n consderaton of the decson power. In order to measure the decson power, we take for a gven par of rectangles R and A the regon nto account contanng all ponts that cannot be detected to be domnated by A w.r.t. R. In the remnder we call ths regon refnement area, snce all objects ntersectng ths area mght be refned n order to detect the domnaton relaton. It should be clear that the smaller ths area, the hgher the correspondng domnaton power. Fgure 8 exemplarly shows the refnement areas for the 2-dmensonal MBRs A and R w.r.t. both crtera DDC MnMax and DDC Optmal, respectvely. In ths example, object B s detected to be domnated by A only f we apply DDC Optmal nstead of DDC MnMax. The refnement areas depend on several condtons such as poston, shape, dstance and extenson of the MBRs specfyng the refnement area as well as the dmensonalty of the space. For our experment evaluatng the domnaton power, pars of MBRs R and A are postoned n [, ] d wth a fxed MnDst of.5 and eual dstances n each dmenson. The length of each sde of the two MBRs was scaled from. to.5 and dmenson screened from to. The gan of the domnaton power s measured by the rato of the volumes of the refnement area w.r.t. DDC Optmal and the refnement area w.r.t. DDC MnMax by means of Monte-Carlo-Samplng. The results n Fgure 9 show that DDC Optmal leads to a much hgher decson power. The effect becomes more evdent as the number of dmensons and the extenson of the MBRs ncrease. As expected, ncreasng the extenson of the MBRs leads to dmnshng completeness of the DDC MnMax decson crteron. It s notable, that the DDC MnMax crteron suffers consderably from an ncreasng dmensonalty. Note that we used MBRs of eual sde length as we observed that ths settng favors the decsons power based on DDC MnMax n order to make a far comparson. In fact, the advantage of the gan of the decson power based on DDC Optmal wll ncrease even further for non-uadratc rectangles. In addton to the above experment whch s more from a theoretcal pont of vew, we compared the number of domnaton relatons detected by applyng DDC Optmal and DDC MnMax. Therefore we randomly generated one ml-

10 postve decsons MnMax Optmal agg regated domnat ton coun nt GRID 62 SLICE 6 58,,,2,3,4,5,6,7,8,9, extent decsons per MBR (a) Postve decsons made (a) Accuracy vs. effcency 4,4 2 dmenson=2 dmenson=3 dmenson=4 dmenson=5,2 GRID SLICE gan fa actor 8 6 n factor ga,8,6 4,4 2,2,,,2,3,4,5,6,7,8,9,,,2,3,4,5 extent,,2,3,4,5,6 extent (b) Factor of postve decsons made more by the optmal crteron Fgure : Comparson of MnMax- and optmalcrteron on synthetc data lon trples of rectangles (A, B, R) wth a fxed extent (.e. the sum of sde lengths) n the [, ] 2 space. For each trple we tested f the decson crteron s able to determne whether Dom(A, B, R) holds. Fnally we aggregated the number of postve decsons for dfferent extents of the MBRs. The results are llustrated n Fgure (a). Note that an extent of zero yelds ponts nstead of rectangles such that both crtera perform eual. However, we can observe that wth ncreasng extent, the percentage of postve decsons of DDC Optmal compared to DDC MnMax ncreases consderably. The gan of the decson power based on DDC Optmal over DDC MnMax s llustrated n Fgure (b) showng the factor of postve domnaton decsons usng DDC Optmal n comparson of that usng DDC MnMax. We vared the dmensonalty of the rectangle space up to 5 dmensons. Here we can observe that the gan ncreases wth ncreasng extent. In contrast, when ncreasng the dmensonalty, the gan of the decson power decreases. The reason s that n ths settng, the extent of the MBRs s fxed for all dmensonalty settngs such that the average sde length per dmenson decreases and MBRs converge to ponts for hgh dmensonalty. 5.2 Domnaton Count Estmaton The next experments evaluate the accuracy of the domnaton count estmaton of a rectangle B w.r.t. a rectangle R for the approaches proposed n Secton 3.2: Basc Domnaton Count Estmaton (DC basc ), grd parttonng (DC grd ), slce parttonng (DC slce ) and bsecton based parttonng (DC bsect ). For these experments, we gener- (b) Performance w.r.t. MBR sze Fgure : Heurstcs for partal domnaton ated one thousand three-dmensonal MBRs wth random poston. One MBR R was postoned n the center of the data space. Then we computed the conservatve domnaton count w.r.t. R for each MBR usng the four approaches mentoned above. We performed several runs for dfferent parametrc settngs and averaged the results. Fgure shows the performance of all four approaches n terms of estmated domnaton count. Frst, we want to get a grasp of the relatonshp between accuracy of the domnaton count estmaton and the cost reured for the domnaton count computaton. Therefore, the cost s measured n terms of number of calls of DDC Optmal. It should be clear that when ncreasng the number of MBR parttons, and thus the reured number of DDC Optmal calls, the estmaton accuracy of all approaches mproves, except for the basc approach snce t does not use any parttonng. Fgure (a) shows the results for MBRs wth an extent of.3. It can be seen that all approaches show a sgnfcant mprovement compared to DC basc when ncreasng the number of allowed DDC Optmal calls. In partcular, the accuracy ncreases very fast at the begnnng of the parttonng process but slows down later on. We can also observe that DC bsect sgnfcantly outperforms the other approaches when allowng more than 27 DDC Optmal calls per MBR, whle DC grd performs best for 8 or less DDC Optmal calls. In the next experment, as shown n Fgure (b), we fxed the number DDC Optmal calls per MBR to 64 and vared ther extent. We measured the gan of the domnaton count over DC basc. Here, agan, DC bsect outperforms the other approaches n partcular for larger MBR szes. Note that for a gven applcaton, the optmal number of parttons depends on the cost for evaluatng a canddate object. The

11 ge accesses pa accesses s page a es ccesse page ac p k (a) T AC dataset 2 k (c) Unform dataset AKKRZ AKKRZ AKKRZ k (e) F orest dataset latons calcul stance ds calculatons dstance c alculatons dstance c E,E+66 E+5,E+5 E+4,E+4,E+3 3,E+2,E+,E+,E+8,E+7,E+6,E+5,E+4 4,E+3,E+2,E+,E+ 2 k (b) T AC dataset 2 k (d) Unform dataset,e+9,e+8,e+7,e+6,e+5,e+4,e+3,e+2,e+,e+ 2 k (f) F orest dataset AKKRZ AKKRZ AKKRZ Fgure 2: AKKRZ usng dfferent decson crtera. Page accesses (left sde) and dstance calculatons (rght sde). hgher that cost, the more parttons can be used n order to reduce the total runtme. 5.3 Impact on Standard Spatal Query Processng Methods In our last experments, we evaluate the mpact of our approaches on the performance of standard uery processng methods. Here, we refer to Secton 4, descrbng how our methods can be plugged nto state-of-the-art uery processng methods. In partcular, we exemplarly consder the most promnent uery methods, the k-nearest neghbor (k- NN) search and the reverse k-nearest neghbor (Rk-NN) search. For ths evaluaton we use one synthetc dataset, contanng k unformly dstrbuted 5D ponts, and two real world datasets TAC [26] consstng of D ponts and Forest[7] contanng 582 D ponts. Frst, we evaluate the mpact of our two domnaton-count estmaton approaches DC basc and DC bsect on a reverse k-nearest neghbor search method. As a baselne, we use the algorthm proposed n [2] (n the followng referred to AKKRZ ) for Rk-NN search on the Eucldean space usng an R -Tree. The AKKRZ algorthm orgnally uses the Mn- /MaxDst decson crteron to conservatvely prune canddates. We evaluate the mpact by comparng the uery performance of the orgnal AKKRZ algorthm wth the verson where we replace the domnaton count estmaton wth our We use the R -Tree provded n the Elk Framework [] methods. Note, that wth except of the domnaton count estmaton method, both Rk-NN versons are dentcal. The results llustrated n Fgures 2(a), 2(c) and 2(e) show the uery performance of both Rk-NN versons n terms of average number of page accesses for varyng parameter k and dfferent datasets. It s notable that the enhanced algorthm reures less page access by almost a full order of magntude on all datasets. Usng DC bsect to apply the paradgm of partal prunng based on bsectons (c.f. Secton 3.2.3) wth a maxmum number of ten splts per MBR, the number of page accesses can be sgnfcantly dropped even further. The large performance ncrease compared to the orgnal verson of AKKRZ can be explaned by the fact that our domnaton decson crteron has a much hgher prunng power on large MBRs compared to the orgnal verson that s based on the Mn/MaxDst crteron. Ths allows us to prune canddates already on a hgh drectory level and, thus, to prune a large number of canddate MBRs very early. Besde the I/O cost, t s also mportant to consder the cpu cost snce the accuracy of our domnaton count estmaton methods s hghly nfluenced by the cpu cost spent for the estmaton process, as shown n the prevous secton. For ths reason, n addton to the I/O cost evaluaton we evaluate the cpu cost measured by the number of dstance calculatons reured for the competng technues as the cpu cost are manly dstance computaton bounded. We counted the total number of dstance calculatons. Calls of DDC Optmal and DDC MnMax were penalzed wth two dstance calculatons 2. The resultng numbers of total dstance calculatons are shown n Fgures 2(b), 2(d) and 2(f). It can be observed that the enhanced AKKRZ algorthm usng DC basc sgnfcantly outperforms the basc AKKRZ by close to two orders of magntude. The ratonale for ths s that the number of calculatons ncreases uadratc n the number of canddates. However, the hgh computatonal cost reured when applyng partal prunng becomes evdent here. Usng DC bsect wth a maxmum of ten splts, the number of dstance calculatons ncreases by a factor of about fve. Fnally, we evaluate the mpact of DDC Optmal and partal domnaton on k-nn ueres among objects approxmated by MBRs. These experments are based on three artfcal datasets that rely on the three datasets used n the foregong experments (TAC, Unform, Forest). Each vector n a dataset defnes the center of an MBR. For each of the resultng datasets MBRs were chosen randomly as uery MBR Q for a -NN uery on the remanng dataset. Here we dd not apply any ndex structure. The performance of the competng approaches were measured by the average number of canddates that could nether be pruned nor be reported as true hts. The results showng the performance n terms of the number of remanng canddates are depcted n Fgure 3 for varyng extent of the MBRs. It can be observed, that DC basc sgnfcantly reduces the number of canddates compared to prunng based on DDC MnMax on all datasets. The relatve performance boost ncreases for an ncreasng extent of the MBRs. We also found out n our experments, that the parameter k has no sgnfcant nfluence on the relatve performance boost. Fgure 3 also shows, that DC bsect s able to further boost the performance, especally for large MBRs. 2 n concordance wth run-tme experments omtted here due to space consderatons