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Indexing transaction time databases

Toga Bozkaya, Meral Ozsoyoglu

presented by

Priyatham Pamu

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Overview

The problem addressed: Indexing time intervals of temporal objects assuming a

record based storage model with object-versioning.

Object-versioning Different versions of the same time-varying entity are kept linked

to each other (by means of time-invariant key attributes in relational model, by using pointers in object-oriented data model).

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Overview

We consider a transaction time database with two states: The past state contains the temporal data that has once been valid,

but is not valid any more (i.e. historical data). We suggest to use IB+-tree, AD* tree, R trees depending on

the application requirement. The current state contains only the data that is valid at the current

time. We suggest to use a version of B+-tree structure called

Modified B+-tree for indexing the current state. We can also use IB+-tree structure indexing valid time intervals in an

effective way.

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Existing Temporal index structures Time-index AP-tree (Append-only tree) TP-index (Time polygon index) SR-tree (Segment R-tree) Snapshot Index ‘windows’ indexing scheme

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Temporal data model A temporal data model can support two types of time

Transaction time data Bounded by the current time pt and the time when the database

was initiated. Data is never deleted from a transaction time database, and the

new data arrives in an append-only fashion. Valid time data

Validity of some temporal entity may span into future - No upper bound

Valid time databases are not append-only databases.

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A Temporal Indexing Scheme Assumptions

We assume a discrete time model in our scheme. The time points are mapped to natural numbers starting from 0

(time when database is initiated) to current time point denoted by the variable now.

In a transaction database, all object versions with their transaction time intervals of the form [a,now] (a<=now) belong to the current state of the database as they are currently valid.

All other versions that have transaction time intervals of the form [a,b] (a<=b and b<now) belong to the past state of the database as their validity had ceased at some point in the past.

Transaction time databases are append-only databases where random deletions or insertions do not take place.

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Temporal Indexing Scheme contd… We denote a temporal object version as a 2-tuple, (I,V), where I is the

transaction time interval of the object version V. Operations defined on a transaction time databases are

Deletion: If the current version v of an object o is to be deleted; ([ts,now],v) is deleted from the database and ([ts,td],v) is inserted to the past state where td is the time of deletion.

Insertion: For insertions, all we do is create the initial version v of an object o and insert to the current state. ([ts,now],v) is inserted to the current state where ts is the time of insertion.

Updates: In case of updates, we have to create a new most recent version ([tu,now],v’) of the object o. This new version is inserted into the current state and the old version ([ts,tu-1],v) is migrated to the past state.

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Figure for current state and past state.

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Indexing the current state Features

Unlike past-index, there are deletions as well as insertions to the current-index.

Deletions are done when object versions migrate from the current state to the past state of the database.

Insertions are done when new object versions are inserted into the database.

For current-index, starting points of the transaction time intervals have to be indexed, because the finish points of all such intervals are the same (i.e. now). Hence we only need a 1-D index structure for indexing the current state.

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Indexing the current state…..

Properties specific to the current-state. Insertions come ordered in starting time, and deletions are arbitrary.

We exploit the above property to come up with a B+ like structure (i.e.MB+ -tree) that supposedly has a higher storage utilization which also directly affects the height of the structure and hence the search efficiency.

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MB+ -tree MB+-tree is simply a modified version of B+-tree. MB+-tree

Insertions are done from the right end, and can handle deletions from anywhere in the tree.

The nodes along the rightmost path of the tree can have as few as one child, and the rightmost leaf can have as few as one key.

The deletion algorithm is same as it is in B+ tree. Depending on the distribution of the duration of the transaction time

intervals, this structure may provide considerable improvement over the regular B+-tree in both efficiency and storage.

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MB+-tree

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Indexing the past state Three different indexing tree structures are proposed.

Interval B+-trees AD*trees One and two-dimensional R-trees.

These index structures meet different requirements for different applications.

In past state, we don’t have any dynamic deletions other than vacuuming. It is an append-only database.

Since IB+ -trees are similar to Interval trees, we discuss Interval trees first.

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Interval-trees

Interval-tree is a binary tree (AVL or Red-black tree...) that is augmented to support operations on a dynamic set of intervals.

A node x of an Interval-tree contains an interval (int[x]), and the key of x is the starting point of that interval.

An inorder tree walk of the data structure lists the intervals in sorted order by their starting points. In addition, each node contains the maximum finish point of the intervals stored in the subtree rooted at that node.

Insertions and Deletions can be done in O(log2n).

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A balanced interval tree figure

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INTERVAL-SEARCH

INTERVAL-SEARCH(T,I)

(For a given interval I[is,if], find an interval that intersects with I in the interval-tree T.) (left[x] and right[x] stand for the left and right child of a node x)

(1)x=root(T),

(2) while x!=NIL and I does not intersect the interval int[x] do (2.1) if left[x] != NIL and max[left[x]] >=is, then x=left[x]

(2.2) else x=right[x]

(3) Return x if it is not NIL.

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Interval B+-tree structure IB+-tree is a direct generalization of the Interval-tree to a multi-way

B+-tree structure. It is basically a B+-tree on the starting points of intervals where each node is augmented with the same kind of information as binary Interval-trees.

Unlike in Interval-trees, internal nodes of IB+-trees do not keep data intervals. All data intervals are kept in the leaf nodes.

Number of children is equal to the number of keys. Refer to the figure for internal node structure.

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Internal IB+ node structure figure.

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INTERVAL-SEARCH for IB+-tree

INTERVAL-SEARCH (N,I) (For a given search interval I[is,if], find an interval that intersects with I in the

Interval B+-tree T. Here, N is node of the Interval B+-tree and the initial call is INTERVAL-SEARCH(root(T),I).)

INTERVAL-SEARCH(root(T),I)(We assume that N has k children (if an internal node), or k data items (if a leaf

node))

(1) If N is a leaf node then check if there is an intersection interval with I among the intervals in N.

(2)else if N is an internal node then (2.1) i=1; (2.2) if I intersects [ai,mi] then INTERVAL-SEARCH(ci,I)

else if i<k then i=i+1, goto 2.2.

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INTERVAL-SEARCH …… Note that INTERVAL-SEARCH algorithm returns one interval (if

there exists atleast one) that intersects with the given search interval.

To obtain all the intersecting intervals we have to use the links between the leaf nodes for a sequential search

from that point on. Since the intervals in the leaf nodes are sorted as per their starting points, so it is efficient.

Or we can follow all the child pointers that satisfy condition in step2.2 of the algorithm.

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Insertions and Deletions in IB+tree Insertion and deletion operations for IB+tree are similar to those for

B+trees with the only exception of a little overhead to maintain the augmented information.

For every merge operation done during insertion or deletion, the maximum fields for all the nodes along the ancestral path have to be updated accordingly.

Complexity of insertion and deletion operations for IB+trees is still O(logkn) (the same as B+-trees), where n is the number of leaf nodes and k is the average fanout of a node in the tree.

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IB+-tree figure.

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Storage Utilization of IB+-tree Storage utilization of IB+ tree is similar to B+tree.

Let size of a node (leaf or internal) be Mbytes

Let each pointer take p bytes while a key value is taken as k bytes.

Max number of entries in leaf node is P1=M/(2k+p) where the tuple identifier is p bytes.

Max number of entries in internal node is Pi=M/(2k+p) which same as P1.

If there are N intervals to index, there will be (N/P1 ln2) leaf nodes as each node is shown to be ln2 full on the average.

Number of internal nodes = (N/P1 ln2) (1/(P1 ln2) + 1/(P1 ln2)2+…..(h-1) terms……

=N/(P1 ln2 –1) approximately.

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Comparison with a regular B+ tree. A regular B+ tree with no augmentation information will take less

little number of pages (for internal nodes) as Pi would be (M+k)/(k+p) (#children is one more than #keys). As fanout increases, the difference between a IB+ and B+ is very insignificant.

Height comparison:

hIB+-tree/hB+-tree= (ln Pi(B+-tree)+ ln ln2)/(ln Pi(IB+tree)+ ln ln2)

Which is very close to 1. For p=8, k=4, M=2048, Pi(IB+tree) is 128, and Pi(B+-

tree) becomes 171. In this case, the height ratio becomes 1.06 app.

Observation: The height of B+-tree and IB+-tree structure built on the same set of interval data will most likely be the same.

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AD-trees & AD*-trees We propose a one-dimensional AD*-tree structure for indexing transaction

time intervals with respect to their finish points.

AD* tree structure is simply an augmented AD-tree.

AD-tree is built on finish points of the data intervals and is augmented with minimum starting point information to obtain AD* structure.

Since AD* and AD-trees are similar, we just discuss the features of AD-tree.

Since the finish points of the insertions are ordered, we do the insertion at the right most node.

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AD-tree properties

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AD-tree with k=4 after an insertion and a deletion

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Deletion in AD-trees figure.

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AD-tree vs MB+-tree

Lemma1: The minimum number of keys in an AD-tree of height h with order k (maximum fanout) is (k-1)kh-2 + 2, where k,h>=2.

Lemma 2: The minimum number of keys in an MB+-tree of height h with order k (maximum fanout) is (floor(k/2)*kh-2) + 2, where k,h>=2;

Theorem: The worst case density of an AD-tree of height h with order k is more than that of an MB+-tree of the same height and the same order, (k>=3).

Experimental results have shown that the height of AD-tree increases slower than the MB+-tree when we assume they index the same set of keys and have the same parameters.

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R-trees 1D R-tree

Each internal node entry contains a minimum bounding interval (MBI) and a child pointer.

The deletion, insertion, and search algorithms of the general R-trees are not changed.

2D R-trees, each interval is mapped to a point in a 2D-space where the dimensions are the starting pt. and the finish pt.

In each internal nodes, each entry contains a child pointer and a MBR that encloses the 2D points indexed below in the corresponding subtree.

R-tree does not assume, no consider any ordering among the data intervals as in IB+ tree.

2D R-trees require more storage and perform worse for common intersection queries.

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Temporal relationships between intervals

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Querying the past state …… Queries on intervals employ different temporal operators. Depending on the querying requirement, different operators can be

applied. Operators: after, met by, right overlaps, left covered by, right covered

by, right covers, equals, covered by are well supported by IB+-tree. Operators: before, meets, left overlaps, left covers, right-covered by,

equals, covered by, left covered by are well supported by AD*-tree. All the operators either invoke an intersection or an inclusion search in

1-D R-tree structure. All the operators are uniformly supported by R-trees

For 2D R-trees, each of the operators corresponds to a 2D query region as shown in fig.

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Temporal relationships in 2D space.

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Experimental results. AD-trees

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Experimental results …. Fig a.

MB+-tree performs better than B+tree when its height is smaller than the height of B+tree. When they have the same height, B+tree performs better than MB+tree.

Fig b. B+ outperforms MB+tree when they have same height. AD-tree outperforms both the structures.

In terms of insertion performance, AD-tree and MB+tree are almost same, but B+tree performs worse than the other two.

When deletions are not strictly from the left hand MB+-tree performs better than B+-tree in all categories.

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Experimental results IB+ & R-trees

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Experimental results IB+ & R-trees Insertion and deletion in IB+trees are simpler and less costly

compared to R-trees. In Fig a.

For 2D R-trees, this operation is a window query so it is not very efficient. For IB+ trees, this is related to the fact that the starting points of all

qualifying intervals fall in the query interval. Thus it performs best. In fig b.

For 1D R-trees, all nodes whose MBIs include the finish point of the query interval have to be retrieved. This becomes costly as MBI size increases.

Similarly with 2d R-trees.

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Experimental results IB+ & R-trees

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Experimental results IB+ & R-trees In Fig a.

1D R-tree performed the best with 30-40% edge over IB+-tree and 30% over 2D R-trees.

Sequential search method and the range search method of IB+-tree performed very close each other with the sequential search method slightly better than range search method.

In Fig b. 1D R-tree performed the best. Exponential distribution affects the way the augmented information in an

IB+-tree is utilized. IB+-tree range search performed better than the sequential search method.

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Indexing valid time databases For valid time data, we need to use indexing structures that support

dynamic insertions and deletions. IB+-trees can also support indexing on valid time intervals, including

the intervals that span into the current time and into future. We make the following assumptions:

Valid time intervals should have an absolute (fixed) starting time point, and the finish points of the valid time intervals can either be an absolute time point or the current time variable now.

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Indexing valid-time intervals