TMIX: Temporal Model for Indexing XML Documents

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1 TMIX: Temporal Model for Indexing XML Douments Rasha Bin-Thalab Department of Information System Faulty of omputers and Information Cairo University, Egypt Neamat El-Tazi Department of Information System Faulty of omputers and Information Cairo University, Egypt Mohamed E.El-Sharkawi Department of Information System Faulty of omputers and Information Cairo University, Egypt Abstrat Different models have been proposed reently for representing temporal data, traking historial information and retrieving temporal queries results effiiently. We onsider the problem of indexing temporal XML douments. In partiular, we propose an indexing sheme that uses a summary struture and a matrix that aptures the strutural relationships as well as time intervals inside a temporal XML doument. We introdue an algorithm to effiiently proess all types of temporal queries with any depth using our newly proposed index. We show that our proposed index out-performs the state of the art indies in terms of both query proessing time and support for different temporal query types. Keywords: Temporal XML, Indexing, Query Proessing, Semi- Strutured Data, Summary Shema I. INTRODUCTION Temporal data has reeived a lot of interest in different areas inluding XML; temporal XML. Temporal XML doument reords the evolution of data as new elements are being inserted, existing elements are edited or deleted. Time stamps are assoiated with XML elements to keep trak of history. Storing temporal data in XML opened interesting researh areas suh as data modeling [], [2], querying [3], indexing [4], [5], and reently keyword searh [6]. A temporal XML doument an be modeled as a tree. Eah edge in the tree is attahed with a time stamp interval denoting time validity of the inoming node. Figure presents a temporal XML tree representing a fragment of a ompany database. Employees an move from one department to another during their life work period. Our ontributions an be summarized as follows: Summarize the temporal XML doument struture into a matrix using an edge-based approah. This matrix effiiently detets if the query is overed without aessing the physial data layer. A B+ tree is built on top of the matrix entries to failitate fast probing of temporal elements. A new temporal query proessing algorithm is presented by utilizing the oalesed matrix to enhane temporal queries evaluation performane. A Temporal update algorithm is proposed and its effet on the proposed index is analyzed. Extensive experiments are implemented to evaluate the performane of our proposed index TMIX. The rest of this paper is organized as follows: Setion II presents related work in indexing temporal XML. Setion III defines preliminaries and bakground. The proposed indexing tehnique, TMIX, and its onstrution is presented in Setion IV. Query proessing and evaluation algorithms are introdued in Setion V. The effet of updates on the index is addressed in Setion VI. Experimental evaluation is shown in Setion VII. And we onlude in Setion VIII. II. RELATED WORK Several approahes address temporal XML indexing by extending regular XML indexing methods suh as TempIndex [5] whih extends the regular summary struture of XML douments to index transation time dimension. Later, TempIndex was extended by using a graph model taking referene edges and temporal onsisteny into aount [7]. However, TempIndex suffered from omplexity and usage of large spae. Later, Gao [4] proposed IB-tree struture to index valid time of nodes. Eah node is represented by (node_ode, Interval, depth). Two main issues were raised upon using this struture. The first was the spae overhead ost sine the elements were stored twie. The seond issue was the additional ost inurred to maintain all lists during updates. Zhang [8] made use of the suffix linear algorithm [9] to onstrut temporal XML tree. Another approah is TFIX [0] whih is based on FIX []. It onverts XML douments into bisimulation graph and builds asymmetri matrix for this graph. TFIX took the root interval into onsideration but it did not onsider updates. In this paper, we are providing an XML index that handles time dimension while avoiding omplexity and overhead ost of index update propagation that exist in urrent works." III. PRELIMINARIES AND BACKGROUND In this setion, we define the temporal XML data model that we use in our work as well as the shema oding number used in our temporal model. A. Data Model There are two main dimensions of time; valid time and transation time. We onsider only transation time as the main temporal dimension in this paper. Transation time has an interval duration between two points [t s, t e ] where t s represents the time when the data is inserted and t e represents the time when the data is modified or deleted. Eah element /3/$ IEEE

2 Company [990, now] [990, now] [995, now] [2000, now] Dept. 2 Dept.2 3 Emp.3 4 [99, 994] [996, now] name Employees name.2. 7 Employees name.3. 9 salary Prodution [993, 994] [99, 993] Sales [2000, now] [996, 2002] George 8000 Emp..2. Emp Emp Emp name name 5 salary 6 name 7 salary [992, 993] [99, 992] salary name 2 salary Jane 3000 John Mary 4000 Mihel 4000 Fig. : Temporal XML for Company of Employees has two attributes that stores start and end time to be used as transation time. We use the word "now" to indiate urrent time. Handling urrent time was overed in detail by Cliford et al [2]. We use the temporal XML model defined in [7] whih is based on transation time model. B. Doument Order and Numbering Shemes In temporal XML, there is no unique order between all nodes, sine eah node has its own validity interval. At eah instant of time there is a different order for nodes as long as eah node an hange or modify. We use a disrete time domain to impose linear order through intervals. Eah interval ontains an ordered pair of time points [t s, t e ]. The relation order t s < t e means that the interval validity ordered from t s up to t e. For any instant of time t there is a total order for eah snapshot of doument D(t). There are ases for any pair of nodes n and n 2, n < t n 2 and n 2 < t2 n for any two instants t and t 2 respetively. Although, an interval I might exist where the order of two nodes is not hanged, where I = n.i n 2.I 2. The doument has a partial order at a given instant of time t. Several shema numbering for nodes were introdued to enfore an ordering in a doument. We use dynami dewey [3] ordering sheme in our work sine it an effiiently apture the relationship between elements as well as doument hierarhy. In addition, it allows insertions without relabeling sibling nodes. C. Summary Graph Summary struture methods provide an abstrat view of the doument to be navigated rather than navigating the physial nodes. We use the summary struture defined in [4] sine it obtains a minimal summary for different shemas; regular or omplex ones. Figure 2 presents a summary graph of the temporal XML tree represented in Figure. The summary graph has only six summary nodes with distint labels: Company, Dept, Employees, Emp, name, and salary. Eah data node in the original doument is mapped to its orresponding summary vertex as shown Table I. The summary graph does not onsider time intervals into aount. This is due to the large diversity in interval ranges whih makes it diffiult to summarize these ranges into a unique lass whih might lead to an inrease in the summary size. Dept Company 0 2 Employees 4 name 3 Emp 5 salary Fig. 2: Summary Struture of Company DB TABLE I: Nodes Enoding Index Label 0 Company Dept 2 Employees 3 Emp 4 name 5 salary IV. TEMPORAL INDEXING (TMIX) In this setion, we propose a new index TMIX for temporal XML douments. TMIX uses a summary graph to preserve the hierarhial relationships between nodes of a temporal

3 TABLE II: Temporal Matrix TM [990,now] 0 [2000,now] [99,994],[996,now] 0 [990,now] [99,994],[996,now] [99,994],[996,now] [99,994],[996,now] XML doument. Additional strutures are needed to over the temporal relationships between these nodes. We reate a temporal matrix TM to integrate the temporal and strutural nodes relationships. Moreover, a temporal equivalene lass TEC is reated to get an effiient semanti temporal retrieval. Temporal Query Label -> Signatures TXML Construt Index Temporal Matrix Index Bloks B-tree file A. Storage Strutures Given the summary graph SG of a temporal XML doument D. For eah edge e = (s i, s j ) in SG, there is a orresponding matrix ell in a temporal matrix TM. Definition (TM). Temporal matrix is an (n m) 2- dimensional struture that holds SG relationships between labels. where n is the number of distint labels of parent nodes (not onsidering leaf labels), m is the number of distint labels of hildren nodes (not onsidering the root label) and T M ( i, j) entry stores the oalesed intervals of the orresponding time intervals of the data nodes. From Definition, a TM ell stores the oalesed intervals in the orresponding nodes plaed in the parent hild intersetion. oalesing funtion is a temporal operation [5] whih finds intervals that are neither adjaent nor overlap. Table I lists the labels used in the summary struture with their odes as shown in Figure 2. The orresponding temporal matrix is shown in Table II. TABLE III: Temporal Equivalene Class Summary Edge Coding Interval Equivalene Node Company Dept 0 [990, now]. [995, now].2 Dept name 4 [990, now].. [995, now].2. Dept Employees 2 [99, 994]..2 [996, now].2.2 Company Emp 0 3 [2000, now].3 [99, 993]..2.2 Employees Emp 2 3 [993, 994]..2. [996, 2002] [2000, now].2.2. [99, 993] Emp name 3 4 [993, 994]..2.. [996, 2002] [2000, now].2.2..,.3. [99, 993] Emp salary 3 5 [993, 994] [996, 2002] [2000, now] ,.3.2 Definition 2 (TEC). Consider a summary graph SG of temporal XML doument D, a temporal equivalene lass TEC is defined by a triple < E s, IClass, Enode > where E s is the set of parent-hild pairs for all edges in SG, I Class is the set Temporal Query Proess Node Results Fig. 3: TMIX Framework of intervals in D and E node is the set of all nodes in D having the same interval. TEC maps eah summary edge with an interval to the orresponding nodes representing the edge end. TMIX uses a B-tree index for failitating query proessing by having both strutural and temporal information available in key retrieval. A key in the B-tree is omposed of the first two attributes in TEC. The arhiteture of TMIX is presented in Figure 3. B. Index Constrution The index onstrution algorithm is presented in Algorithm. The algorithm parses the input XML using a SAX parser that passes over the XML doument only one in depth first order. While onstruting TMIX strutures, a value index is also built to support value prediates by storing values separately. Eah value is assoiated with its node parent id. The separate index provides a diret aess to values to prune searh spae. The omplexity of the index onstrution algorithm is O(n m β) where n is the number of nodes in the XML doument, m is the number of distint labels and β is the number of oalesed intervals. The spae Requirements of TMIX is omposed of matrix and B-tree size. The matrix TM size is equal to O(k l) where k is the number of distint labels and l is less than k by the number of leaf nodes. The B-tree index size equals to O(k + n), k is the number of keys and n is the number of nodes in XML doument. V. QUERY PROCESSING This setion illustrates query evaluation algorithms to evaluate queries using our proposed index TMIX. The main idea is based on taking advantage of indexing edges (e) with their time intervals (I). Edges are enoded by hashing its start and end labels enoding as shown in Figure 2 with time interval to form the key pair < e, I>, as shown in Table III. We use TXPath [7] whih is a temporal extension of XPath 2.0 as a query language to represent temporal queries. Algorithm 2 illustrates query proessing using TMIX index. The input TXPath is deomposed into a sequene expressions. We

4 Algorithm Index Constrution Input: Temporal XML Doument D Output: T B, M, T D // Interval: ontains start and end time // N ode: stores an element; label and Interval // M: stores the temporal matrix T M // P ath Stak : a stak that stores a sequene of Node from the root up to a leaf node // Key Sig : an array that holds an id for parent- hild nodes and an Interval struture // Eq Key : a ditionary that ontains Key signatures and Node values // T B: a B tree file // T D: a ditionary of the labels with their odes : M 2: Eq Key 3: P ath Stak 4: T D 5: while not end of D do 6: if start event then 7: reate N ode // Node struture store label and interval 8: N ode.label element.name 9: N ode.interval.start element.start 0: N ode.interval.end element.end : P ath Stak.push(Node) 2: else 3: if losed event then 4: N ode P athstak.pop 5: if P ath Stak then 6: Node P P ath Stak.top 7: parent MEntry(T D, Node P.label) 8: else 9: parent 0 // root has no parent 20: end if 2: hild MEntry(T D, N ode.label) 22: if M[parent, hild] = then 23: M[parent, hild] N ode.interval 24: else 25: AppCovInt(M, parent, hild, N ode) 26: end if 27: Key Sig.p Hash(parent, hild) 28: Key Sig.Interval Node.Intravl 29: if Eq Key ontains Key Sig then 30: append Node to Eq Key [Key Sig ] 3: else 32: reate new Key Sig in Eq Key 33: add Node to Eq Key [Key Sig ] 34: end if 35: if Stak P ath is then 36: insert Eq Key into T B 37: end if 38: end if 39: end if 40: end while Proedure AppCovInt(M, p,, N) begin if N.Interval M[p, ] then oalese N.Interval to M[p, ] end if end Proedure Funtion MEntry (T D, L) begin searh for L in T D if not exist then add L to T D end if return index of L in T D end Funtion distinguish between four expressions; parent-hild, anestordesendant, time ondition, and value prediates. When parenthild expression "/" is enountered between two sequene labels, hashing ode is applied to the labels enoding to get the first part of searhing key. Otherwise, if anestor-desendant relationship "//" is enountered, the diret parents of a given hild is alulated by alling SearhAD algorithm. When a temporal prediate (asking for a start or end of interval) is enountered, the omputed key is attahed by the given interval. If the interval is inluded within the TM, key is used to prune searh and retrieves a list of <node id, I> pairs that satisfy the ondition. Otherwise if no temporal prediate is deteted, only the first part of the key is used in the retrieval proess. Value prediate is evaluated using value index whih retrieves a list of nodes ids whih math the given value. The intermediate lists resulted from eah expression are stored in M edresult. At the end, a strutural join proess is performed on the intermediate result based on their Dewey enoding to get the answer of the query. Algorithm 3 presents SearhAD funtion whih is based on a temporal matrix TM. The algorithm is a dynami stak that searhes for the diret parents of the speifi node in the shema. In the worst ase, omplexity of SearhAD is O(l 2 ), where l is number of distint labels. Query proessing omplexity is O(m + α l 2 ), where m is query length, α is the number of anestor-desendant relationships in the query and l is the number of distint labels. Sine m and α are onstants, we an say that the query proessing omplexity is O(l 2 ) whih is totally dependent on the matrix size. I/O omplexity is based on the number retrieved nodes from the B tree whih is O(log d n) where d is the size of page used in B tree file and n is the number of elements in the temporal XML doument. There are several types of temporal queries [6] suh as projetion, snapshot, sliing, join and aggregate whih are all supported by TMIX. Consider the following example for aggregation and join query whih asks for employees names who were in sales department when "Mary" joined the department for the first time. Using TXPath, the query is expressed as follows: let $m=min(//dept[name="sales"]/emp[name="mary"]/start) return //dept[name="sales"]/emp[$m start and $m end]/name. Minimum aggregation funtion is applied to the list of start times that is retrieved from let statement. The returned start time (2000 as in Figure ) is used to evaluate the time ondition. Value prediate dept[name="sales"] retrieves

5 Algorithm 2 Query Proessing Input: Temporal query: query, Ditionary T D, Temporal Matrix: T M, B-tree File: T B, V F Output: Result: set of retrieved nodes : j 0 // ounter for intermediate results 2: Query Parse query into expressions 3: repeat 4: L p 0 5: Get parent and hild enoding from T D 6: if parent-hild relationship is enountered then 7: if TM(parent, hild) then 8: L p parent 9: end if 0: Query.next : end if 2: if An_Des relationship is enountered then 3: L p SearhAD(parent, hild) 4: if L p.length == 0 then 5: print : Error no relationship is deteted 6: end if 7: Query.next 8: end if 9: if value_prediate then 20: MedResult j retrieve(value from V F file) 2: Query.next 22: end if 23: if time_ond then 24: for parent in L p do 25: if T M[parent, hild] inlude time_ondition then 26: key Hash(parent, hild) 27: time_interval extrat interval from time_ondition 28: MedResult j retrieve(key, time_interval) from T B file 29: Query.next 30: else 3: exit no available intervals 32: end if 33: end for 34: else 35: for parent in L p do 36: key Hash(parent, hild) 37: MedResult j retrieve(key ) 38: end for 39: Query.next 40: end if 4: j j + 42: until end of Query 43: Result j i=0 MedResult i Algorithm 3 SearhAD Input: T M, parent, hild Output: new list of parents List parent : ColumnStak // stak stores olumns number 2: List parent // list of deteted parents 3: if TM[parent,hild] 0 then 4: Append parent to List parent 5: return List parent 6: end if 7: while true do 8: for i = 0 to T M.olumn.length do 9: if T M[parent, i] 0 then 0: ColumnStak.push(i) : end if 2: end for 3: if ColumnStak then 4: parent ColumnStak.pop 5: while parent > T M.rows.length do 6: if ColumnStak then 7: parent ColumnStak.pop 8: end if 9: end while 20: if TM[parent,hild] 0 then 2: Append parent to List parent 22: end if 23: else 24: if List parent.length > 0 then 25: return List parent 26: else 27: print "Error: No parent-hild relationship exist" 28: exit 29: end if 30: end if 3: end while department node.2, and time prediate searhes for employees in department sales who have lifespan intervals that inlude year This retrieves employee node A strutural join is performed afterwards to retrieve name node VI. TEMPORAL UPDATES There are four basi temporal update operations: insertion, deletion, modifiation, and subtree reloation. Sine modifiations and sub tree reloation are a sequene of insertion and deletions, we desribe the effets of only insertion and deletion operations on our index and how to propagate that effet. The insertion of a new node, in a temporal XML doument, requires determining the loation of the newly inserted node. As in [7], a path expression is used to speify that loation. A parent label of the new node is also speified paired with the label of the new node and both are oded using label signature (see Table I). Furthermore, two parts must be provided with the new node; node id and interval. Node id is omputed by using the numbering sheme dynami dewey [3] based on the urrent entries nodes. Sine we use transation time model, the start time of interval of new node is assigned to urrent time (t ) of insertion statement and its end is assigned to "now" expression. Algorithm 4 represents the insertion algorithm where the result nodes from evaluating the path expression

6 Giga Byte Giga Bytes Giga Bytes are given as inputs. During insertion, there is no need to reonstrut the index even if the new node hanges the struture, only a new parent-hild is added to the matrix entries. In addition, there is no hange made to the temporal matrix unless there is a new time entry with no overage interval. Algorithm 4 Insertion Pseudo Code Input: matrix: T M, urrent time: t, node: n, parent id: p, hild id, urrent nodes: CN, index file: T F, V F Output: updated index strutures T M, T F, V F : reate interval I = [t, now] 2: if <p,> T M then 3: oalese I in TM(p,) 4: else 5: reate a new <p,> entry in T M 6: T M(p, ) I. 7: end if 8: for eah urrent node in CN do 9: assign the new node a new Dewey ode id [8] 0: T F.insert(< p,, I >, id) : V F.insert(value, id) 2: end for Physial deletion is not allowed in temporal XML. A node is logially deleted by hanging its end time validity from "now" value to the deletion time t. A path expression is used to determine the nodes to be logially deleted. Algorithm 5 presents the deletion algorithm. T M matrix is updated by updating only the intervals related to the deleted nodes entries. Update is also needed in the B tree index based on the node entry key. Algorithm 5 Deletion Pseudo Code Input: matrix: T M, urrent time: t, node: n, parent id: p, hild id, urrent nodes: CN, index file: T F Output: updated index strutures T M, T F : for eah node n in CN do 2: n.i.end = t - 3: update n.i in TM(p,) 4: T F.modify(< p,, I >, n.id) 5: end for VII. EXPERIMENTAL EVALUATION In this setion we evaluate the performane of TMIX. All experiments were implemented in # with a Berkeley DB implementation of B+tree. We ompare TMIX against TSuffix [8] and TFIX [0]. All tests were onduted on a PC with Pentium dual ore CPU and 6 GB RAM running Windows 7. A. Storage Utilization To evaluate the performane of TMIX, we use both syntheti and real data sets. We use Employees data set whih was generated by Wang [2]. Employee data set is omposed of multiple douments versions. The text values of a single element is modified over 7 years. In addition, we use Xbenh [9], TCSD (Text-Centri Single Doument) and Xmark [20] with sale fator. Both Xbenh and Xmark were onverted to temporal sets by attahing validity attributes to eah element with normal inrease of intervals over 2 years. Storage utilization for the three algorithms is presented graphially in Figures 4(a), 4(b) and 4(). It is lear that TSuffix has the smallest index size as it is built based on paths only. Although TMIX indexes both struture and value, it is lose in size to TSuffix speially in Xmark and Xbenh data sets. On the other hand, TFIX is the least effiient index in terms of storage utilization sine it depends on the whole number of elements inside the XML doument TMIX TSuffix TFIX (a) Employees Index TMIX TSuffix TFIX () Xbenh Index 0 TMIX TSuffix TFIX (b) Xmark Index Fig. 4: Indexes Sizes (Douments size in MB) B. Query Performane To evaluate our index, we use the temporal queries lassified by Bertino [2] whih are used to benhmark the effiieny of any temporal index. Temporal queries for the different data sets are listed in Table IV. We first investigate slie range query type using the three queries Q, Q2 and Q3. The evaluation time for these queries is presented in Figures 5(a), 5(b) and 5() respetively. It is lear from the Figures that TMIX has the best performane beause both TSuffix and TFIX indies traverse irrelevant physial nodes to satisfy the time prediate. Whereas TMIX traverses only relevant nodes aording to the generated key that is based on query time prediate and query struture. The seond query type asks for historial key value. Sine value prediates were not onsidered in TSuffix index, we only limit our omparison to TFIX in queries Q4, Q5 and Q6. Figures 5(d), 5(e) and 5(f) show the evaluation time for these three queries. TMIX outperforms TFIX using XMARK data set. TFIX integrates values and struture of XML doument while TMIX reates a separate index for values. As a result, TMIX depends on the retrieved nodes from value prediates. Number of value nodes in queries Q4 and Q6 are muh

7 TABLE IV: Query Benhmarks Query Type Q# Data Set Expression Q Employees //employee/firstname[ ] Temporal Slie Range Q2 Xmark //site/regions//item/shipping[ ] Q3 Xbenh //authors[ ] Q4 Employees //ompany//employee[title = "Engineer"] Key only Q5 Xmark //item[loation = "Georgia"] Q6 Xbenh //item//author[name_of_ity ="Dubai"] Q7 Employees //employee[empno="0025"][salary[ ]] Key and Range Q8 Xmark //Afria/item[shipping ="internationally"][ ] Q9 Xbenh //item[subjet = "Biographies"][ ] (a) Q - Employees (b) Q2 - XMark () Q3 - XBenh (d) Q4 - Employees (e) Q5 - XMark (f) Q6 - XBenh (g) Q7 - Employees (h) Q8 - XMark (i) Q9 - XBenh (j) U3: Update Complexities (k) U4: Deletion Operation (l) U5: Insertion Operation Fig. 5: Key Queries Evaluation Time and Update greater than that in Q5 for XMark. This explains the different performane of TMIX in the three data sets. Finally, Figures 5(g), 5(h) and 5(i) show the evaluation time for the third type of temporal queries, time and key value ranges (Q7, Q8 and Q9 respetively). Again TMIX is only ompared with TFIX and disarded TSuffix index whih do not handle value prediates. It is lear that TMIX outperforms TFIX using all data sets. TMIX an answer queries with time prediates effiiently sine keys in TMIX are grouped aording to time intervals. On the other hand, TFIX ontinues its degraded performane in evaluating time prediates sine nodes are retrieved aording to their struture only. All irrelevant retrieved nodes have to be visited and mathed with the query time prediate and this extensively inreases the query evaluation time using TFIX. C. Performane of Update Operations Using update operations, we limited our experiments on the employees data set to perform meaningful queries. The update queries are all listed in Table V. The first three update queries U, U2 and U3 range in omplexity. Figure 5(j) shows the response time for the three queries. We an see that U3 outperforms the first two updates by more than a fator of 2. The reason is that the time prediate prunes the searh spae of updated nodes whih in turn redues proessing time signifiantly. However, U effiieny is slightly better than U2. This is beause U2 needs to join lists that result from prediate values. TMIX is ompared against TSuffix in Figures 5(k) and 5(l) using update queries. U4 insertion query proessing is presented in Figure 5(k). TSuffix proessing time inreases as the number of elements inreases. TMIX performane is muh faster than TSuffix sine many nodes are pruned from the beginning using the time prediate. Deletion query U5 is presented in Figure 5(l), TMIX also outperforms TSuffix. VIII. CONCLUSION In this paper we presented TMIX, an indexing tehnique for temporal XML douments. The index is based on summa-

8 TABLE V: Update Queries U type Update DDL expression U Modify salary of employees whose fname For $m in employees[name = "Deborah"]/salary is Deborah to 5000$ Insert $m Values 5000 Modify salary for employees whose fname For $m in employees[name = "Deborah"] U2 is Deborah and title is engineer to 5000$ [title= "Engineer"]/salary Insert $m Values 5000 Modify salary for employees whose fname For $m in employees[name = "Deborah"] U3 is Deborah and title is engineer and [title= "Engineer"]/salary[T s 990 and T s 2000] start time in the range [ ] Insert $m Values 5000 U4 Delete all salaries for all employees whose start sine 990 For $m in employees/salary[t s 990] Delete $m U5 Insert a new hild alled job tax For $m in employees[t s 988 && T s 995] for all employees who start in the Insert $m/job_tax ompany within the range [ ] rizing the doument struture and elements time intervals in a matrix. A B+ tree has been built based on the TMIX matrix for effiient retrieval of physial nodes. 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