Efficient Multidimensional Searching Routines for Music Information Retrieval

Transcription

1 Effcent Multdmensonal Searchng Routnes for Musc Informaton Retreval Josh Ress Jean-Julen ucouturer Mark Sandler Department of Electrcal Engneerng Kng s College London Strand. London WC2 2LR UK Phone: {josh.ress jean-julen.aucouturer mark.sandler}@kcl.ac.uk BSTRCT The problem of Musc Informaton Retreval can often be formalzed as searchng for multdmensonal trajectores. It s well known that strng-matchng technques provde robust and effectve theoretc solutons to ths problem. However for low dmensonal searches especally queres concernng a sngle vector as opposed to a seres of vectors there are a wde varety of other methods avalable. In ths work we examne and benchmark those methods and attempt to determne f they may be useful n the feld of nformaton retreval. Notably we propose the use of KD-Trees for multdmensonal nearneghbor searchng. We show that a KD-Tree s optmzed for multdmensonal data and s preferred over other methods that have been suggested such as the K-Tree the box-asssted sort and the multdmensonal quck-sort. 1. MULTIDIMENSIONL SERCHING IN MUSIC IR The generc task n Musc IR s to search for a query pattern ether a few seconds of raw acoustc data or some type of symbolc fle (such as MIDI) n a database of the same format. Fgure 1- Feature extracton 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 perform ths task we have to encode the fles n a convenent way. If the fles are raw acoustc data we often resort to a feature extracton (fg. 1). The fles are cut nto M tme frames and for each frame we apply a sgnal-processng transform that outputs a vector of N features (e.g. psychoacoustcs parameters such as ptch loudness brghtness etc ). If the data s symbolc we smlarly encode each symbol (e.g. each note suppose there are M of them) wth a N- dmensonal vector (e.g. ptch duraton). In both cases the fles n the database are turned nto a trajectory of M vectors of dmenson N. Wthn ths framework two search strateges can be consdered: - Strng-matchng technques try to algn the two sequences of r r r r r r vectors x x } and y y } usng a set of { 1 2 x3 { 1 2 y3 elementary operatons (substtutons nsertons ). They have receved much coverage n the Musc IR communty (see for example [1]) snce they allow a context-dependent measure of smlarty and thus can account for many of the hgh-level specfctes of a muscal query (.e. replacng a note by ts octave shouldn t be a msmatch). They are robust and relatvely fast. - The other approach would be to fold the trajectores of M vectors of dmenson N nto embedded vectors of hgher dmenson M*N. For example wth M=3 and N=2: {( x y) 1 ( x y) 2( x y) 3} = ( x1 y1 x2 y2 x3 y3) The search problem now conssts of dentfyng the nearest vector n a multdmensonal data set (.e. the database) to some specfed vector (.e. the query). Ths approach may seem awkward as - We loose structure n the data that could be used to help the search routnes (e.g. the knowledge that x1 and x2 are coordnates of the same knd ). - We ncrease the dmensonalty of the search. However there has been a consderable amount of work n devsng very effcent searchng and sortng routnes for such multdmensonal data. complete revew of the multdmensonal data structures that mght be requred s descrbed by Samet et al. [23]. Non-herarchcal methods such as the use of grd fles [4] and extendable hashng [5] have been appled to multdmensonal searchng and analyzed extensvely. In many areas of research the KD-Tree has become accepted as

2 one of the most effcent and versatle methods of searchng. Ths and other technques have been studed n great detal throughout the feld of computatonal geometry [67]. Therefore we feel that Musc IR should captalze on these well-establshed technques. It s our hope that we can shed some lght on the benefcal uses of KD-Trees n ths feld and how the mult-dmensonal framework can be adapted to the peculartes of musc data. The paper s organzed as follows. In the next four sectons we revew four multdmensonal searchng routnes: The KD-Tree the K-Tree the Multdmensonal Quck-sort whch s an orgnal algorthm proposed by the authors and the Box-sssted Method. We then benchmark and compare these routnes wth an emphass on the very effcent KD-Tree algorthm. Fnally we examne some propertes of these algorthms as regards a multdmensonal approach to Musc IR. 2. THE KD-TREE 2.1 Descrpton Fg. 2 depcts the partton n 2 dmensons for ths data set. t each node the cut dmenson and the cut value (the medan of the correspondng data) are stored. The bucket sze has been chosen to be one. Fgure 2- The KD-Tree created usng the sample data. The correspondng parttonng of the plane s gven n Fg. 3. We note that ths example comes from a larger data set and thus does not appear properly balanced. Ths data set wll be used as an example n the dscusson of other methods. The K-dmensonal bnary search tree (or KD-Tree) s a hghly adaptable well-researched method for searchng multdmensonal data. Ths tree was frst ntroduced n Bentley et al.[9] studed extensvely n[10] and a hghly effcent and versatle mplementaton was descrbed n [11]. It s ths second mplementaton and varatons upon t that we wll be dealng wth here. There are two types of nodes n a KD-Tree the termnal nodes and the nternal nodes. The nternal nodes have two chldren a left and a rght son. These chldren represent a k- dmensonal partton of the hyperplane. Records on one sde of the partton are stored n the left sub-tree and on the other sde are records stored n the rght sub-tree. The termnal nodes are buckets whch contan up to a set amount of ponts. onedmensonal KD-Tree would n effect be a smple quck-sort. 2.2 Method The buldng of the KD-Tree works by frst determnng whch dmenson of the data has the largest spread.e. dfference between the maxmum and the mnmum. The sortng at the frst node s then performed along that dmenson. quckselect algorthm whch runs n order n tme fnds the mdpont of ths data. The data s then sorted along a branch dependng on whether t s larger or smaller than the mdpont. Ths succeeds n dvdng the data set nto two smaller data sets of equal sze. The same procedure s used at each node to determne the branchng of the smaller data sets resdng at each node. When the number of data ponts contaned at a node s smaller than or equal to a specfed sze then that node becomes a bucket and the data contaned wthn s no longer sorted. Consder the followng data: (7-3); B (42); C (-67); D (2-1); E (80); F (1-8); G (5-6); H (-89); I (98); J (-3-4); Fgure 3- The sample data as parttoned usng the KD-Tree method. nearest neghbor search may then be performed as a topdown recursve traversal of a porton of the tree. t each node the query pont s compared wth the cut value along the specfed cut dmenson. If along the cut dmenson the query pont s less than the cut value then the left branch s descended. Otherwse the rght branch s descended. When a bucket s reached all ponts n the bucket are compared to see f any of them s closer than the dstance to the nearest neghbor found so far. fter the descent s completed at any node encountered f the dstance to the closest neghbor found s greater than the dstance to the cut value then the other branch at that node needs to be descended as well. Searchng stops when no more branches need to be descended. Bentley recommends the use of parent nodes for each node n a tree structure. search may then be performed usng a bottom-up approach startng wth the bucket contanng the search pont and searchng through a small number of buckets

3 untl the approprate neghbors have been found. For nearest neghbor searches ths reduces computatonal tme from O(log m) to O(1). Ths however does not mmedately mprove on search tme for fndng near neghbors of ponts not n the database. Tmed trals ndcated that the ncreased speed due to bottom-up (as opposed to top-down) searches was neglgble. Ths s because most of the computatonal tme s spent n dstance calculatons and the reduced number of comparsons s neglgble. 3. THE K-TREE 3.1 Descrpton K-trees are a generalzaton of the sngle-dmensonal M- ary search tree. s a data comparatve search tree a K-tree stores data objects n both nternal and leaf nodes. herarchcal recursve subdvson of the k-dmensonal search space s nduced wth the space parttons followng the localty of the data. Each node n a K-tree contans K=2 k chld ponters. The root node of the tree represents the entre search space and each chld of the root represents a K-ant of the parent space. One of the dsadvantages of the K-tree s ts storage space requrements. In a standard mplementaton as descrbed here a tree of N k-dmensonal vectors requres a mnmum of k (2 + k) N felds. Only N-1 of the 2 k branches actually pont to a node. The rest pont to NULL data. For large k ths waste becomes prohbtve. 3.2 Method Consder the case of two-dmensonal data (k=2 K=4). Ths K-tree s known as a quad-tree and s a 4-ary tree wth each node possessng 4 chld ponters. The search space s a plane and the parttonng nduced by the structure s a herarchcal subdvson of the plane nto dsjont quadrants. If the data conssts of the 10 vectors descrbed n Secton 2.2 then the correspondng tree s depcted n Fg. 4 and the parttonng of the plane n Fg. 5. Fgure 5- The sample data as parttoned usng the KTree method. Searchng the tree s a recursve two-step process. cube that corresponds to the boundng extent of the search sphere s ntersected wth the tree at each node encountered. The bounds of ths cube are mantaned n a k-dmensonal range array. Ths array s ntalzed based on the search vector. t each node the drecton of search s determned based on ths ntersecton. search on a chld s dscontnued f the regon represented by the chld does not ntersect the search cube. Ths same general method may be appled to weghted radal and nearest neghbor searches. For radal searches the radus of the search sphere s fxed. For nearest neghbor searches t s doubled f the nearest neghbor has not been found and for weghted searches t s doubled f enough neghbors have not been found. 4. MULTIDIMENSIONL QUICKSORT 4.1 Descrpton For many analyses one wshes to search only select dmensons of the data. problem frequently encountered s that a dfferent sort would need to be performed for each search based on a dfferent dmenson or subset of all the dmensons. We propose here a multdmensonal generalzaton of the quck-sort routne. 4.2 Method Fgure 4- The KDTree created usng the sample data. Note that much of the tree s consumed by null ponters. The data s descrbed as a seres of N vectors where each 1 2 D vector s D dmensonal = ( X... X ). quck-sort s k k k k performed on each of the D dmensons. The orgnal array s not modfed. Instead two new arrays are created for each quck-sort. The frst s the quck-sort array an nteger array where the value at poston k n ths array s the poston n the data array of the k th smallest value n ths dmenson. The second array s the nverted quck-sort. It s an nteger array where the value at poston k n the array s the poston n the quck-sort array of the value k. Keepng both arrays allows one to dentfy both the locaton of a sorted value n the orgnal array and the locaton of a value n the sorted array. Thus f 1 X has the second smallest value n the thrd dmenson then t may be

4 X represented as 23. The value stored at the second ndex n the quck-sort array for the thrd dmenson wll be 1 and the value stored at the frst ndex n the nverted quck-sort array for the thrd dmenson wll be 2. Note that the addtonal memory overhead need not be large. For each floatng-pont value n the orgnal data two addtonal nteger values are stored- one from the quck-sort array and one from the nverted quck-sort array. We begn by lookng at a smple case and showng how the method can easly be generalzed. We consder the case of two-dmensonal data wth coordnates x and y where we make no assumptons about delay coordnate embeddngs or unformty of data. Suppose we wsh to fnd the nearest neghbor of the vector = ( x y). If ths vector s poston n the x-axs quck-sort s and ts poston n the y-axs quck-sort s j ( and j are found usng the nverted quck-sorts) then t may also be represented = = ( x y ) = = ( x y ). as x x x j y j y j y Usng the quck-sorts we search outward from the search vector elmnatng search drectons as we go. Reasonable canddates for nearest neghbor are the nearest neghbors on ether sde n the x-axs quck-sort and the nearest neghbors on ether sde n the y-axs quck-sort. The vector = ( x y ) correspondng to poston -1 n the x- 1 x 1 x 1 x axs quck-sort s the vector wth the closest x-coordnate such that x 1 x < x. Smlarly the vector + 1 x = ( x + 1 x y+ 1 x ) correspondng to +1 n the x-axs quck-sort s the vector wth the closest x-coordnate such that x1 > x. nd from the y-axs quck-sort we have the vectors j 1 y = ( x j1 y y j1 y ) and = ( x y ). These are the four vectors adjacent to j+ 1 y j+ 1 y j+ 1 y the search vector n the two quck-sorts. Each vector's dstance to the search vector s calculated and we store the mnmal dstance and the correspondng mnmal vector. If x 1 x x s greater than the mnmal dstance then we know that all vectors 1 x... 2 x must also be further away than the 1x mnmal vector. In that case we wll no longer search n decreasng values on the x-axs quck-sort. We would also no longer search n decreasng values on the x-axs quck-sort f has been reached. Lkewse f 1x x+ 1 x x s greater than the mnmal dstance then we know that all vectors + 1 x x must also be further away than the N x mnmal vector. If ether that s the case or has been N x reached then we would no longer search n ncreasng values on the x-axs quck-sort. The same rule apples to y j 1 y y and y y j+ 1 y. We then look at the four vectors 2 x + 2 x j and 2 y j +. If any of these s closer than 2 y the mnmal vector then we replace the mnmal vector wth ths one and the mnmal dstance wth ths dstance. If x 2 x x s greater than the mnmal dstance then we no longer need to contnue searchng n ths drecton. smlar comparson s made for x + 2 x x y j2 y y and y j+ 2 y y. Ths procedure s repeated for 3 x + 3 x j and 3 y j + and so on untl all search 3 y drectons have been elmnated. We fnd whch of these four vectors s closest to the search vector. We then search the next four closest ponts. If any of these ponts s further away n ts drecton of search than the mnmal dstance then we can elmnate that drecton from further search. lso f we reach the end of the data set n any drecton then we can elmnate that drecton. We fnd the dstance of the four ponts from our pont of nterest and f possble replace the mnmal dstance. We then proceed to the next four ponts and proceed ths way untl all drectons of search have been elmnated. The mnmal vector must be the nearest neghbor snce all other neghbor dstances have ether been calculated and found to be greater than the mnmal dstance or have been shown that they must be greater than the mnmal dstance. Extenson of ths algorthm to hgher dmensons s straghtforward. In n dmensons there are 2n possble drectons. Thus 2n mmedate neghbors are checked. mnmal dstance s found and then the next 2n neghbors are checked. Ths s contnued untl t can be shown that none of the 2n drectons can contan a nearer neghbor. It s easy to construct data sets for whch ths s a very neffcent search. For nstance f one s lookng for the closest pont to (00) and one were to fnd a large quantty of ponts resdng outsde the crcle of radus 1 but nsde the square of sde length 1 then all these ponts would need to be measured before the closer pont at (10) s consdered. However smlar stuatons can be constructed for most multdmensonal sort and search methods and preventatve measures can be taken. 5. THE BOX-SSISTED METHOD The box-asssted search method was descrbed by Schreber et al. [11] as a smple multdmensonal search method for nonlnear tme seres analyss. grd s created and all the vectors are sorted nto boxes n the grd. Fg. 6 demonstrates a two-dmensonal grd that would be created for the sample data. Searchng then nvolves fndng the box that a pont s n then searchng that box and all adjacent boxes. If the nearest neghbor has not been found then the search s expanded to the next adjacent boxes. The search s contnued untl all requred neghbors have been found. One of the dffcultes wth ths method s the determnaton of the approprate box sze. The sort s frequently talored to the type of search that s requred snce a box sze s requred and the preferred box sze s dependent on the type of search to be done. However one usually has only lmted a pror knowledge of the searches that may be performed. Thus the approprate box sze for one search may not be approprate for another. If the box sze s too small then many boxes are left unflled and many boxes wll need to be searched. Ths results n both excessve use of memory and excessve computaton.

5 Ths s true even for hgher dmensonal data although the convergence s much slower. Fgure 6- The sample data set as grdded nto 16 boxes n two dmensons usng the box-asssted method. The choce of box dmensonalty may also be problematc. Schreber et al.[11] suggest 2 dmensonal boxes. However ths may lead to neffcent searches for hgh dmensonal data. Hgher dmensonal data may stll be searched although many more boxes are often needed n order to fnd a nearest neghbor. On the other hand usng hgher dmensonal boxes wll exacerbate the memory neffcency. In the benchmarkng secton we wll consder both two and threedmensonal boxes. Fgure 7- The dependence of average search tme on data set sze. In Fgure 8 the KD-Tree s shown to have an exponental dependence on the dmensonalty of the data. Ths s an mportant result not mentoned n other work provdng dagnostc tests of the KD-Tree. [10 12] It mples that KD-Trees become neffcent for hgh dmensonal data. It s not yet known what search method s most preferable for neghbor searchng n a hgh dmenson (greater than 8). 6. BENCHMRKING ND COMPRISON OF METHODS In ths secton we compare the suggested sortng and searchng methods namely the box asssted method the KD- Tree the K-tree and the multdmensonal quck-sort. ll of these methods are preferable to a brute force search (where no sortng s done and all data vectors are examned each tme we do the searchng). However computatonal speed s not the only relevant factor. Complexty memory use and versatlty of each method wll also be dscussed. The versatlty of the method comes n two flavors- how well the method works on unusual data and how adaptable the method s to unusual searches. The multdmensonal bnary representaton and the unform K-Tree descrbed n the prevous two sectons are not compared wth the others because they are specalzed sorts used only for exceptonal crcumstances. 6.1 Benchmarkng of the KDTree One beneft of the KD-Tree s ts rough ndependence of search tme on data set sze. Fgure 7 compares the average search tme to fnd a nearest neghbor wth the data set sze. For large data set sze the search tme has a roughly logarthmc dependence on the number of data ponts. Ths s due to the tme t takes to determne the search pont s locaton n the tree. If the search pont were already n the tree then the nearest neghbor search tme s reduced from O(log n) to O(1). Ths can be accomplshed wth the mplementaton of Bentley's suggested use of parent ponters for each node n the tree structure. [10]. Fgure 8- log plot of search tme vs dmenson. Fgure 9 shows the relatonshp between the average search tme to fnd n neghbors of a data pont and the value n. In ths plot 10 data sets were generated wth dfferent seed values and search tmes were computed for each data set. The fgure shows that the average search tme s almost nearly lnearly dependent on the number of neghbors n. Thus a varety of searches (weghted radal wth or wthout excluson) may be performed wth only a lnear loss n speed. The drawbacks of the KD-Tree whle few are transparent. Frst f searchng s to be done n many dfferent dmensons ether a hghly neffcent search s used or addtonal search trees must be bult. lso the method s somewhat memory ntensve. In even the smplest KD-Tree a number ndcatng the cuttng value s requred at each node as well as an ordered array of data (smlar to the quck-sort). If ponters to the parent node or prncpal cuts are used then the tree must contan even more nformaton at each node. lthough ths ncrease may at

6 frst seem unmportant one should note that expermental data typcally conssts of floatng ponts or more. In fact some data analyss routnes requre a mnmum of ths many ponts. If the database s also on that order then memory may prove unmanageable for many workstaton computers. Fgure 11- Comparson of search tmes for dfferent methods usng 2 dmensonal random data. Fgure 9- plot of the average search tme to fnd n neghbors of a data pont as a functon of n. We have mplemented the KD-Tree as a dynamc lnked lbrary consstng of a set of fully functonal object orented routnes. In short they consst of the followng functons Create(*phTree ncoords ndms nbucketsze*aponts); FndNearestNeghbor(Tree*pSearchPont *pfoundpont); FndMultpleNeghbors(Tree *psearchpont *pnneghbors *aponts); FndRadalNeghbors(Tree *psearchpont radus **paponts *pnneghbors); ReleaseRadalNeghborLst(*aPonts); Release(Tree); 6.2 Comparson of methods The KD-Tree mplementaton was tested n tmed trals aganst the multdmensonal quck-sort and the box-asssted method. In Fgure 10 through Fgure 13 we depct the dependence of search tme on data set sze for one through four dmensonal data respectvely. In Fgure 10 the multdmensonal quck-sort reduces to a one-dmensonal sort and the box asssted method as descrbed by [12] s not feasble snce t requres that the data be at least two-dmensonal. We note from the slopes of these plots that the box-asssted method the KDTree and the KTree all have an O(n log n) dependence on data set sze whereas the quck-sort based methods have approxmately O(n 1.5 ) dependence on data set sze for 2 dmensonal data and O(n 1.8 ) dependence on data set sze for 3 or 4 dmensonal data. s expected the brute force method has O(n 2 ) dependence. Despte ts theoretcal O(n log n) performance the KTree stll performs far worse than the box-asssted and KDTree methods. Ths s because of a large constant factor worse performance that s stll sgnfcant for large data sets (64000 ponts). Ths constant worse performance relates to the poor balancng of the KTree. Whereas for the KDTree the data may be permuted so that cut values are always chosen at medans n the data the KTree does not offer ths opton because there s no clear multdmensonal medan. In addton many more branches n the tree may need to be searched n the KTree because at each cut there are 2 k nstead of 2 branches. Fgure 10- Comparson of search tmes for dfferent methods usng 1 dmensonal random data. Fgure 12- Comparson of search tmes for dfferent methods usng 3 dmensonal random data.

7 dmensonalty guaranteed that an excessve number of boxes needed to be searched and for the Lorenz data where the hghly non-unform dstrbuton ensured that many boxes went unflled. The K-tree also performed poorly for hgh dmensonal data (four and fve dmensonal) due to the exponental ncrease n the number of searched boxes wth respect to dmenson. summary of the comparson of the four routnes can be found n Table 2. The adaptve and flexble crtera refer to the next secton. Table 2- Comparson of some features of the four routnes. Ratng from 1=best to 4=worst. Fgure13- Comparson of search tmes for dfferent methods usng 4 dmensonal random data. However all of the above trals were performed usng unform random nose. They say nothng of how these methods perform wth other types of data. In order to compare the sortng and searchng methods performance on other types of data we compared ther tmes for nearest neghbor searches on a varety of data sets. Table 1 depcts the estmated tme n mllseconds to fnd all nearest neghbors n dfferent pont data sets for each of the benchmarked search methods. The unform nose data was smlar to that dscussed n the prevous secton. Each Gaussan nose data set had a mean of 0 and standard devaton of 1 n each dmenson. The dentcal dmensons and one vald dmenson data sets were desgned to test performance under unusual crcumstances. For the dentcal dmensons data unform random data was used and each coordnate of a vector was set equal e.g X = ( X ) = ( X ). For the data wth only one vald dmenson unform random data was used n only the frst dmenson e.g X = ( X ) = ( X 00). In all cases the KD-Tree proved an effectve method of sortng and searchng the data. Only for the last two data sets dd the multdmensonal quck-sort method prove faster and these data sets were constructed so that they were n effect onedmensonal. In addton the box method proved partcularly neffectve for hgh dmensonal Gaussan data where the lgorthm Memory Buld Search dapt. Flexble KDTree 2 3/4 1 yes yes KTree 3 3/4 3 no no Quck-sort yes yes Boxsssted no yes 7. INTERESTING PROPERTIES FOR MUSIC IR The mult-dmensonal search approach to Musc IR and the correspondng algorthms presented above have a number of nterestng propertes and conceptual advantages. 7.1 daptve to the dstrbuton truly mult-dmensonal approach enables an adaptaton to the dstrbuton of the data set. The KD-Tree algorthm for example focuses ts dscrmnatng power n a non-unform way to best ft the densty of the data. Ths could be effcent for say search tasks n a database where part of the features reman quas constant e.g. a database of samples whch are all pure tones of a gven nstrument wth quas constant ptch and a varyng brghtness. It s nterestng to compare ths adaptve behavor wth a strng-matchng algorthm that would have to Table 1- Nearest neghbor search tmes for data sets consstng of ponts. The brute force method multdm. quck-sort the box asssted method n 2 and 3 dmensons the KDTree and the KTree were compared. n X ndcates that t wasn t possble to use ths search method on ths type of data. The fastest method s gven n bold and the second fastest method s gven n talcs. Data set Dmenson Brute Qucksort Box (2) method Box (3) method KDTree Unform nose Gaussan X KTree Gaussan Identcal dmensons One vald dmenson

8 compare sequences that all begn wth aaa. The latter can t adapt and systematcally tests the frst three dgts whch s an obvous waste of tme. 7.2 Independent of the metrc and of the alphabet ll the methods presented here are blnd to the metrc that s used. Ths s especally useful f the set of features s composte and requres a dfferent metrc for each coordnate e.g. ptches can be measured modulo 12. The routnes are also ndependent of the alphabet and work for ntegers as well as for floatng-ponts. Ths makes them very general as they can deal wth a varety of queres on mxed low-level features and hghlevel meta-data such as: Nearest neghbor ( ptch 1 ptch2 ptch3" BCH" ) 7.3 Flexblty There are a varety of searches that are often performed on multdmensonal data. [7] Perhaps the most common type of search and one of the smplest s the nearest neghbor search. Ths search nvolves the dentfcaton of the nearest vector n the data set to some specfed vector known as the search vector. The search vector may or may not also be n the data set. Expansons on ths type of search nclude the radal search where one wshes to fnd all vectors wthn a gven dstance of the search vector and the weghted search where one wshes to fnd the nearest N vectors to the search vector. Each of these searches (weghted radal and nearest neghbor) may come wth further restrctons. For nstance ponts or collectons of ponts may be excluded from the search. ddtonal functonalty may also be requred. The returned data may be ordered from closest to furthest from the search vector and the sortng and searchng may be requred to handle the nserton and deleton of ponts. That s f ponts are deleted from or added to the data these addtonal ponts should be added or deleted to the sort so that they can be removed or ncluded n the search. Such a feature s essental f searchng s to performed wth real-tme analyss. Most sortng and searchng routne presented above are able to perform all the common types of searches and are adaptable enough so that they may be made to perform any search. 7.4 note on dmensonalty One of the restrctons shared by the multdmensonal search routnes presented on ths paper s ther dependence on the dmensonalty of the data-set (not ts sze). Ths s detrmental to the sheer foldng of the trajectory search as presented n the ntroducton especally when t nvolves long M-sequences of hgh-n-dmenson features (dmenson M*N may be too hgh). However as we mentoned n the course of ths paper there are stll a varety of searches that can ft nto the multdmensonal framework. We notably wsh to suggest: - Searches for combnatons of hgh-level metadata (M=1) - It s possble to reduce N wth classc dmensonalty reducton technques such as Prncpal Component nalyss or Vector Quantzaton - It s possble to reduce M by computng only 1 vector of features per audo pece. It s the approach taken n the Muscle Fsh technology [13] where the mean varance and correlaton of the features are ncluded n the feature vector. - It s possble to reduce M by computng the features not on a frame-to-frame bass but only when a sgnfcant change occurs ( event-based feature extracton see for example [14]) - For fnte alphabets t s always possble to reduce the dmenson of a search by ncreasng the sze of the alphabet. For example searchng for a set of 9 notes out of a 12 sem-tone alphabet can be reduced to a 3D search over an alphabet of 3 12 symbols. 8. CONCLUSION We ve presented and dscussed four algorthms for a multdmensonal approach to Musc IR. The KD search tree s a hghly adaptable well-researched method for searchng multdmensonal data. s such t s very fast but also can be memory ntensve and requres care n buldng the bnary search tree. The k tree s a smlar method less versatle more memory ntensve but easer to mplement. The box-asssted method on the other hand s used n a form desgned for nonlnear tme seres analyss. It falls nto many of the same traps that the other methods do. Fnally the multdmensonal quck-sort s an orgnal method desgned so that only one search tree s used regardless of how many dmensons are used. These routnes share a number of conceptual advantages over the approaches taken so far n the Musc IR communty whch -we beleve- can be useful for a varety of muscal searches. The am of the paper s to be only a revew and the startng pont of a reflecton about search algorthms for musc. In partcular we stll have to mplement specfc musc retreval systems that use the results presented here. 9. REFERENCES [1] K. Lemstrom Strng Matchng Technques for Musc Retreval. Report Unversty of Helsnk Press. [2] H. Samet pplcatons of Spatal Data Structures: ddson-wesley [2] H. Samet The desgn and analyss of spatal data structures: ddson-wesley [3] H. Hnterberger K. C. Sevck and J. Nevergelt CM Trans. On Database Systems vol. 9 pp [4] N. Pppenger R. Fagn J. Nevergelt and H. R. Strong CM Trans. On Database Systems vol. 4 pp

9 [5] K. Mehlhorn Data Structures and lgorthms 3: Multdmensonal Searchng and Computatonal Geometry: Sprnger-Verlag [6] F. P. Preparata and M. I. Shamos Computatonal geometry: n ntroducton. New York: Sprnger- Verlag [7] J. Orensten Informaton Processng Letters vol. 14 pp [8] J. H. Bentley Multdmensonal Bnary Search Trees Used for ssocatve Searchng Communcatons of the CM vol. 18 pp [9] J. H. Fredman J. L. Bentley and R.. Fnkel n algorthm for fndng best matches n logarthmc expected tme CM Trans. Math. Software vol. 3 pp [10] J. L. Bentley K-d trees for semdynamc pont sets n Sxth nnual CM Symposum on Computatonal Geometry vol. 91. San Francsco [11] T. Schreber Effcent neghbor searchng n nonlnear tme seres analyss Int. J. of Bfurcaton and Chaos vol. 5 pp [12] R. F. Sproull Refnement to nearest-neghbour searchng n k-d trees lgorthmca vol. 6 pp [13] E. Wold T. Blum et al. Content Based Classfcaton Search and Retreval of udo n IEEE Multmeda Vol.3 No. 3 Fall 1996 p [14] F. Kurth M. Clausen Full Text Indexng of Very Large udo Databases n Proc. 110 th ES Conventon msterdam May 2001.