Extracting Partition Statistics from Semistructured Data

Transcription

1 Extrating Partition Statistis from Semistrutured Data John N. Wilson Rihard Gourlay Robert Japp Mathias Neumüller Department of Computer and Information Sienes University of Strathlyde, Glasgow, UK Abstrat The effetive grouping, or partitioning, of semistrutured data is of fundamental importane when providing support for queries. Partitions allow items within the data set that share ommon strutural properties to be identified effiiently. This allows queries that make use of these properties, suh as branhing path expressions, to be aelerated. Here, we evaluate the effetiveness of several partitioning tehniques by establishing the number of partitions that eah shemean identifyover a given data set. Inpartiular, we explore the use of parameterised indexes, based upon the notion of forward and bakward bisimilarity, as a means of partitioning semistrutured data; demonstrating that even restrited instanes of suh indexes an be used to identify the majority of relevant partitions in the data. Introdution Effiient resolution of branhing path expressions over XML data requires that data items having a struture that mathes thatspeified by the query an be identified without traversing the omplete data set. This an be ahieved with a suitable form of index graph. Suh graphs an provide a strutural summary of the omplete data set and allow data items with similar struture (and semantis) to be olleted together and aessed effiiently. Consequently, index graphs impliitly define a partitioning of the data, with the number, and nature, of partitions identified varying with hoie of indexing tehnique. In this paper, we explore the effetiveness of a number of forms of index graph through an exploration of the number of partitions that eah index an identify over a given data set. The different forms of index that an be used with semistrutured data eah define a set of partitions over the data. Eah vertex on the index graph orresponds to exatly one partition, with all data therein sharing some notion of ommon strutural properties. The size of the index graph, and therefore the number of partitions identified, varies dramatially with hoie of indexing tehnique. There is a tradeoff between the size of the index graph and auray onsequently the hoie of index and the number of partitions identified, will vary onsiderably depending on whih form of index has been seleted. Construting a suitable index graph, and identifying a set of partitions, is a fundamental step in the effiient proessing of semistrutured data. In addition to the benefits of effiient aess provided by the index graph, partitions provide a valuable opportunity for optimisation. Firstly, it is likely that data within a given partition will be aessed together. Seondly, partitions apture regular aspets of the data set, presenting an opportunity to exploit tehniques aimed at suh data (whih have been extensively studied for the relational model). Index graphs, and the partitions that they identify, are entral to the effiient proessing of XML and an be used as the basis of query proessing models in the absene of globally valid shemata or when ost-based optimisers are unavailable. The remainder of this paper is strutured as follows. Setion 2 summarises the neessary onepts and terminology, while Setion 3 desribes a number of tehniques for partitioning semistrutured data. Setion presents the main ontribution of this paper: an evaluation of how effetive different indexing tehniques are at identifying partitions of the data. Setion 5 onludes the paper. 2 Bakground The data graph representation of semistrutured data is a direted node-labelled graph. The verties of this graph represent data items, with the edges representing strutural relationships. Diretly reating a data graph from the flatfile representation of XML gives a tree rather than a graph:

2 it is only when additional relationships, suh as those enoded using ID:IDREF referenes, are inluded that a graph ontaining yles an be obtained. Here, we refer to a tree view of the data that is both a spanning tree and is rooted at the root vertex as a distinguished spanning tree. The forms of index graph of interest here are all targeted at path expressions (i.e. they do not index the values of the atomi data) and provide a graph derived from the data graph that an assoiate multiple data items with a single vertex. Branhing path expressions are a form of query that an be performed over a data graph. These queries are omprised of a sequene of labels (a label path), and an ontain both forward and bakward separators. Evaluating suh a query onsists of finding all verties on the data graph that have a path leading to them mathing the path given by the labels with forward separators (this is referred to as the primary path) that also satisfy the onditions given by the bakward separators. If a given index graph is smaller than the data graph it summarises, then it is possible to evaluate a query more effiiently using the index than it would be to traverse the omplete data set. The use of bisimilarity as a means of partitioning semistrutured data was first demonstrated by Buneman et al. []. Suh indexes group verties on the data graph aording to the equality of the omplete set of outgoing and inoming edges to a given vertex. This produes a refined set of partitions over a given data set that an be used as an aid to the effiient proessing of branhing path expressions. However, suh index graphs an beome too large to be useful. Kaushik et al. [5] introdue a parameterised index based upon bisimilarity, with a view to using suh parameters to limit the size of the index graph. More reently, He and Yang [] proposed a multi-resolution index upon bisimilarity: a form of index with the degree of bisimilarity varied aording to the needs of eah node. 3 Partitioning Tehniques Partitions an be ategorised in one of three ways (depending on the type of verties ontained therein). Firstly, a partition omprised solely of verties representing atomi data is said to be an atomi partition. Seondly, a partition that ontains only omplex verties, those verties that ombine the information stored in other verties by means of a set of outgoing edges, is said to be aomplexpartition. Finally, a partition that ontains both atomi and omplex verties is said to be a mixed partition. Suh distintions will be onsidered when ontrasting different partitioning shemes. Tehnique (Label Partitions) This is the simplest partitioning tehnique disussed here. For a given data set (Figure ), all verties arrying an idential label are grouped together. Using this approah, there will be exatly one &5 lass &6 & &25 &2 &3 & is &2 teahing &3 &6 assistant lass leturer&5 "Databases" &26 &7 &7 leturer staff &8 &27"Sotirios" & & &9 &8 &29 "S.A.D." &28 "3839" student &2 &2 "Mathias"&3 &3 "359" &2 &9 &22 & researher staff& researh &23 projet "John" &33"Domains" &32 "358" Figure. An example data graph staff student Tag Map T6 T7 T8 T9 Struture Container T7 T8 C / T9 C2 / Data Container S.A.D. Sotirios Mathias Data Container Figure 2. XMill ontainers. atomi partition and one omplex partition for eah sourespeifi tag present in the data graph (inluding the root vertex). Tehnique 2 (Parent Partitions) This tehnique is based on the equality of the label of a node s parent node in the distint spanning tree system [7]. A vertex on a data graph may have more than one inoming edge (parent node), so it beomes possible for a node to be present in more than one partition. The total number of partitions that an be identified is bounded by the size of the label alphabet. Sine the root vertex has no inoming edge it does not belong to any partition. Inversely, there is no partition orresponding to the tag label, as atomi verties have no outgoing edges. Figure 2 illustrates the transformation of our example data graph as used with XMill. This approah auses all data items representing a to be merged into one atomi partition, regardless of whether they are the s of, projets or ourses. This effet of ombining unrelated information is referred to as partitionmixing. Tehnique 3 (Path Partitions) This partitioning tehnique attempts to avoid partition mixing by partitioning the vertex set based upon the entire path from the root node to a given vertex. This may result in a given vertex being part of more than one partition. In fat, yli graphs will result in an unbounded number of partitions being identified. However, the equivalent tehnique based upon the distinguished spanning tree is bounded by the size of the node set giving a usable definition of partitioning. This is equivalent to the strong DataGuide [3] and an result in partition splitting (Figure 3). Tehnique (Depth Partitions) The notion of partitioning that we are using here allows for the haraterisation of partitions based upon node level. Here, all nodes that our at the same depth in the tree view of the data are grouped

3 & & is is is &2 teahing &3 & researh 2 2 teahing 3 researh 2 teahing 3 researh &5 lass &6 lass &7 staff &8 student &9 staff & projet 3 5 researher 2 assistant lass lass staff student projet lass 6 staff 7 student projet leturer 6 assistant researher a 9 a 2 a 22 a 2 a 2 a 23 &5 leturer &3 &9 &2 researher & &7 &2 &23 &2 leturer &6 assistant & &8 &22 &2 & &3 359 &25 Databases &26 S.A.D. &27 Sotirios &29 Mathias &3 John & &33 Domains a 5 leturer 3 a 6 Figure 3. Strong DataGuide. Figure. Depth partitions. Figure 7. The skeleton representation. Soure Size (MB) Note Xmark PubMed Ensembl Rat MAGE Syntheti data, sale fator Figure 8. Data strutures investigated. is teahing teahing 2 3 researh 2 3 researh assistant 2 researher 5 leturer assistant lass 6 lass 7 staff 8 student 9 staff projet 3 researher 2 lass lass staff student staff projet a leturer 5 leturer a23 a2 2 a22 a 2 Figure 5. The A() index graph. is Figure 6. The (,) F+B Index graph. together. Although suh a partitioning may initially seem to be of limited use, ertain queries an be aelerated with an index based upon suh a partitioning. For example, the query/*/*/*, an be resolved effiiently using this tehnique. If this tehnique is used on the graph view of the data then, as was the ase with Path Partitions, an unbounded number of partitions an be identified. Given that the partitions identified do not take aount of node labelling, mixed partitions an be identified. Figure shows suh a partitioning imposed on the data graph. Tehnique 5 (Skeleton Partitions) This tehnique is based on the onept of forward bisimilarity of a node [2], i.e. the equality of the olletion of outgoing paths. This tehnique states that two nodes in the tree are said to be bisimilar if they have the same label and the same ordered sequene of bisimilar hild nodes. This allows ommon substrutures in a doument to be identified, for example, the two staff nodes shown in Figure ; however, this tehnique splits the two lass nodes beause of the differenes in their respetive sub-trees. Given that only outgoing edges are onsidered when omputing bisimilarity, all atomi data is grouped in a single atomi partition. Tehnique 6 (Loal Bakward Bisimilarity Partitions) Here the entire set of inoming edges is used to determine the partition of eah node [6]. This gives an index that is similar to that of path partitioning, whilst avoiding nodes being present in multiple partitions based on eah inoming edge. Sine, in pratie, path expressions are often of limited length, the lengths of the paths used when alu- lating the bisimilarity an also be limited. The definition of bisimilarity then beomes reursive, stating that a vertex is k-bisimilar to another if they have equal labels and have the same set of inoming edges from verties that are (k )- bisimilar. Thus, the parameter k an be used to ontrol the balane between index size and index overage. Figure 5 shows the index graph of the example soure based on - bisimilarity. Again, every vertex defines a partition of the data set. The twostaff verties that were ontained in a single partition when using skeleton partitions are now split As seen previously, all s are merged into a single partition regardless of whether they refer to, projets or ourses. Inreasing the value of k would overome this limitation. Tehnique 7 (Forward and Bakward Bisimilarity Partitions) This method ombines the strutural properties identified by tehniques suh as skeleton partitions, with the ontextual properties that an be identified using bakward bisimilarity [5]. The resulting index graph is overing for general branhing path expressions without value prediates. As with loal bakward bisimilarity, the lengths of paths onsidered an be limited. Here, two parameters, k b and k f, that restrit the lengths of inoming and outgoing paths respetively, an be used to redue the omplexity of the final index graph. Figure 6 shows the index graph based on (,)-F+B-bisimilarity, The and leturer verties ourring below lass verties have been split, although their inoming paths of length one are idential. This is beause they an be distinguished based on their siblings by using a branhing path expression whose length does not exeed one. For example, the branhing path expression lass[/]/ selets vertex & but not vertex &, whih were ombined in partition in Figure 5 but are split into the partitions and. Partition Statistis Our analysis using various data soures (Figure 8) explores both the expressive power of a given partitioning

4 Soure Size V Label Parent Path Depth Skeleton Total number of partitions Xmark 322, ,662 PubMed,288, ,37 Ensembl Rat 2,556, MAGE 3,39, ,699 Total number of atomi partitions Xmark 322, PubMed,288,555 8 Ensembl Rat 2,556,2 MAGE 3,39, Total number of omplex partitions Xmark 322, ,66 PubMed,288,555 Ensembl Rat 2,556,2 MAGE 3,39, , ,689 Total number of mixed partitions Xmark 322,327 8 PubMed,288,555 Ensembl Rat 2,556,2 MAGE 3,39, bw= bw= Figure 9 shows the number of distint partitions that an be identified over our sample data when using partitioning Tehniques 5 (the partitioning tehniques that do not make use of parameters). The results shown in Figure 9 learly demonstrate that ompeting partitioning tehniques produe signifiantly different numbers of partitions for a given data. In ontrast to the regular struture of the Ensembl file (Figure 9), the XMark data exhibits few indiations of a regular struture sine its data, by design, is onvoluted. This is indiated by the results presented in Figure 9. In the XMark data set there is no orrespondene between the number of omplex partitions identified by label partitioning and path partitioning (or indeed, any of the hosen shemes). A large number of partitions are identified using the skeleton partitioning tehnique, even though the size of the data graph is signifiantly smaller than the data graph of the other sample douments. This is a onsequene of similar elements ourring in many ontexts in XMark generated files. Of the remaining two example files, PubMed and MAGE, the results shown indiate that they do have some degree of varying ontent (i.e. tags appearing in different ontexts), but that these real data soures do not have the same omplexity as the synthetially generated XMark data. When using the skeleton partitioning tehnique, the number of partitions identified for the PubMed and MAGE files is omparable to that of the XMark data, although, again, it should be noted that there is a signifiant differene in the size of the data sets. With the exeption of the regularly strutured Ensembl file, the number of partitions identifiable using the Skelefw= fw= fw=2 fw=3 fw= fw=5 Figure. XMark: Depth = bw= bw= fw= fw= fw=2 fw=3 fw= fw=5 Figure. XMark: Depth = Figure 9. Stati partitions ounts. tehnique and the omplexity of the underlying data soure. Partition Tehniques 5 are lassified asfixed-partitioning tehniques, in that they do not support the use of parameters to influene the operation of the algorithm. With parameterised-partitioning tehniques however, a range of parameter ombinations are possible.. Fixed Partitioning Statistis ton tehnique greatly exeeds that of the more simple partition tehniques. This suggests that tehniques based upon bisimulation are likely to be the most appliable when aelerating branhing path expressions or when lustering the data aording to its ontext..2 Parameterised Partitioning Statistis In ontrast to partitioning Tehniques 5, Tehniques 6 and 7 define a family of partition strutures, with one or more parameters being used to speify the exat nature of any given instane of the partitioning. Points in the parameter spae that orrespond to either stable or rapidly hanging numbers of partitions are of interest. Knowledge of suh points may be benefiial when automating parameter hoie for suh indexes. The sets of partitions that an be identified using Tehnique 6 (loal bakward bisimilarity) are a subset of the sets of partitions that an be identified using Tehnique 7 (forward and bakward bisimilarity). Thus, only the results obtained using Tehnique 7 are given here. Figures and show the numbers of partitions obtained using a parameterised index for tree depths of and 3. The results are shown as 3D-surfaes to allow the interation of the two parameters that ontrol the size of the forward and bakward bisimilarity to be seen. For any given parameter ombination (i, j), the set of partitions identifiable will inlude all partitions, or a refinement thereof, that are identifiable using a lower value of i and/or j. Thus, as the bisimilarity is extended in either diretion, the number of identifiable partitions inreases until suh a point is enountered when all the partitions identifiable using this tehnique have been enountered. The maximal number of partitions that an be identified using suh indexes is equal to the number of partitions identified by the F&B index [5]. Two trends emerge from the results presented in Figures and. Firstly, it is possible identify a maximal set of partitions using relatively low values for the three parameters that define the index. Seondly, the rate at whih we approah this maximal number of partitions inreases with tree depth. For the XMark doument, it was found that the

5 bw= bw= fw= fw= fw=2 fw=3 fw= fw=5 Figure 2. Pubmed: Depth = bw= bw= fw= fw= fw=2 fw=3 fw= fw=5 Figure 3. Pubmed: Depth = 3 maximal number of partitions that ould be identified using this tehnique was very lose to the number of verties on the data graph: 285,372 partitions ompared to 322,327 verties on the data graph. Figures 2 and 3 show the number of parameters that an be identified in the PubMed file using parameterised indexes of tree depths and 3. In eah ase, the overall shape of the surfaes is similar to that obtained when using XMark data, although the surfaes are smoother and more losely resemble that whih would be obtained over normally distributed data. These smoother surfaes indiate that the struture of the PubMed data is more regular than that of XMark. The most signifiant differene in the data gathered over PubMed is that the maximal number of partitions identified is signifiantly smaller than the size of the original data graph: 73,87 partitions ompared to a data graph of,288,555 verties. Here, the full F&B index would be an order of magnitude smaller than the original data graph. Again, it an be seen that inreasing the tree depth inreases the rate at whih the number of partitions grows. Overall, it an be seen that even restrited instanes of parameterised overing indexes have a omplexity similar to that of the full F&B index [5]. Thus, if it is intended to provide a partitioning (index graph) that is smaller than that obtained with F&B index then the parameters for the parameterised index an only be drawn from a limited range. When ompared to the size of the data graph, the maximal number of partitions identified over our two sample douments differed onsiderably. For XMark the maximal number of partitions approahes the size of the data graph, whereas for PubMed the maximal number of partitions was an order of magnitude smaller than the size of the data graph. The input data determines whether it is viable to use higher values for the index parameters. 5 Conlusions This paper summarised seven different tehniques for partitioning semistrutured data, ontrasting their effetive ness over various XML douments. As expeted, the number of identifiable partitions varied greatly with hoie of partitioning tehnique; however, it was also found the number of partitions varied greatly with the nature of the soure data. Partitioning onvoluted data soures, suh as XMark data, an result in an index graph that approahes the omplexity of the data graph. However, this was not found to be the ase for real soures of semistrutured data. Hene, index graphs that potentially have a omplexity equal to that of the data graph may well be viable over real data soures. Of the seven tehniques desribed, those based upon bisimulation (Tehniques 5, 6 and 7) identified signifiantly more partitions for data with varying struture than the more simplisti partitioning tehniques. Furthermore, employing both forward and bakward bisimulation (Tehnique 7) allowed signifiantly more partitions to be identified than is possible using bisimulation in only one diretion (e.g. Skeleton Partitions). This suggests that partitioning on forward and bakward bisimulation is the most appropriate tehnique for both aelerating branhing path expressions and for lustering the atomi data aording to ontext. Additionally, it was demonstrated that the number of partitions identified using parameterised indexes quikly approahes the maximum as the parameters ontrolling the forward and bakward bisimilarity are inreased. Hene, in situations were it is desired to limit the size of the index graph (and the number of partitions) then it will be neessary to draw values for these parameters from a restrited set of values. Referenes [] P. Buneman, S. Davidson, G. Hillebrand, and D. Suiu. A query language and optimisation tehniques for unstrutured data. In Pro. of the 996 ACM SIGMOD, pages 55 56, 996. [2] P. Buneman, M. Grohe, and C. Koh. Path queries on ompressed XML. In Pro.of 29th VLDB, pages 52, 23. [3] R. Goldman and J. Widom. DataGuides: Enabling query formulation and optimization in semistrutured databases. In Pro. of 23rd VLDB, pages Morgan Kaufmann, 997. [] H. He and J. Yang. Multiresolution indexing of XML for frequent queries. In Pro. of 2th ICDE, pages IEEE Computer Soiety, 2. [5] R. Kaushik, P. Bohannon, J. Naughton, and H. Korth. Covering indexes for branhing path queries. In Pro. of the 22 ACM SIGMOD, pages 33, 22. [6] R. Kaushik, P. Shenoy, P. Bohannon, and E. Gudes. Exploiting loal similarity for indexing paths in graph-strutured data. In Pro. of the 8th ICDE, pages 29, 22. [7] H. Liefke and D. Suiu. XMill: An effiient ompressor for XML data. In Pro. of the 2 ACM SIGMOD, pages 53 6, 2.