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**Temporal Databases S. Srinivasa Rao April 12, 2007**

[Part 1 based on Ch23 of C.J. Date (slides by Prof. Ghafoor, EE 562)] [Part 2 based on slides by Prof. Arge, I/O-algorithms]

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**Outline Part 1: Introduction to temporal databases**

Part 2: Temporal index: Persistent B-tree and its applications

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Introduction Temporal database: a database that contains historical data as well as current data. – Note: ‘historical’ is a misleading term – temporal databases may contain data regarding the future as well as the past. Extreme case: data is only inserted, never deleted from a temporal database (eg. vehicle position data in the ‘project’). So far, we have studied the other extreme - i.e. ‘snapshot’ databases. Distinguishing feature: the element of time.

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Introduction Temporal data: encoded representation of timestamped facts. – Each tuple must include at least one timestamp. Problem:What about queries that produce results that are not temporal? i.e. result of query is outside the domain of (temporal) database. eg. Get names of all people who have supplied something in the past. Redefine temporal database: database that includes, but is not limited to, temporal data.

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Motivation Queries on time-varying data are difficult to express in SQL. Temporal databases provide build-in support for recording and querying such information. It is possible to use SQL to evaluate these queries, but performance is poor.

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**Motivation Most applications manage temporal data.**

If a temporal database is used for such data: Schemas, including integrity constraints are simpler. Queries are simpler Application code is less complex easier to understand easier to produce easier to maintain

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Applications Most applications of database technology are temporal in nature: Financial apps.: portfolio management, accounting & banking, stock market analysis, audit analysis Record-keeping apps.: personnel, medical records, inventory management, legal records (commercial laws change frequently) Data Warehousing: historical trends for analysis Scheduling apps.: airline, car, hotel reservations and project management Scientific apps.: weather monitoring, chemical process monitoring

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**Intervals An interval [s,e] is a set of times from time s to time e.**

Does interval [s,e] represent an infinite set? Assumption: Timeline is a finite sequence of discrete, indivisible time quanta. Time Quanta: smallest unit of time system can represent. Timepoints/point: time unit considered indivisible for our purpose. An interval is treated as a single type, not as pair of separate values. Interval can be open/closed w.r.t. start point/end point. eg. [d04,d10],[d04,d11),(d03,d10],(d03,d11) all represent the sequence of days from day4 to day10 inclusive.

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**Operators on Intervals**

Temporal predicate operators: i1 = [s1,e1]; i2 = [s2,e2] – i1 BEFORE i2 (e1<s2) – i1 MEETS i2 (s2 = e1) – i1 EQUALS i2 (s1 = s2 AND e1 = e2) – i1 OVERLAPS i2 (s2 < s1 < e2 OR s1 < s2 < e1) i1 i2 i1 i2 i1 i2 i1 i2

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**Operators on Intervals**

– i1 DURING i2 (s2 < s1 AND e2 > e1 ) – i1 STARTS i2 (s1 = s2 AND e1 < e2) – i1 FINISHES i2 (e1 = e2 AND s1 > s2) Additional operators: – i1 MERGES i2: (i1 MEETS i2 OR i1 OVERLAPS i2) – i1 CONTAINS i2: (i2 DURING i1) i1 i2 i1 i2 i1 i2

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**Scalar and Relational Operators**

DURATION(i) - returns the number of time points in i eg. DURATION ([d03,d07]) returns 5 i1 UNION i2 returns [MIN(s1,s2),MAX(e1,e2) ] if (i1 MERGES i2) otherwise undefined i1 INTERSECT i2 – returns [MAX(s1,s2),MIN(e1,e2)] if (i1 OVERLAPS i2)

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**Aggregate Operators EXPAND(X):**

Where X is a set. The output is also a set. Used to generate time quantum intervals. – The expanded form of X is the set of all intervals of the form [p,p] where p is a time point in some interval in X. e.g.: X1 = { [d01,d01],[d03,d05],[d04,d06] } X2 = { [d01,dp1],[d03,d04],[d05,d05],[d05,d06] } X3 = { [d01,d01],[d03,d03],[d04,d04],[d05,d05],[d06,d06] } Then EXPAND(X1) = EXPAND(X2) = X3

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**Aggregate Operators COLLAPSE(X):**

The collapsed form of X is the set Y of intervals of the same type such that (a) X & Y have the same unfolded form. (b) no two distinct members i1 and i2 of Y are such that (i1 MERGES i2) is true. e.g.: X1 = { [d01,d01],[d03,d05],[d04,d06] } X2 = { [d01,d01],[d03,d04],[d05,d05],[d05,d06] } X3 = { [d01,d01],[d03,d06] } Then COLLAPSE (X1) = COLLAPSE (X2) = X3

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**Relation Operators Involving Intervals**

PACK r on A: groups the relation r by all its attributes apart from A This is equivalent to WITH ( r GROUP {A} AS X ) AS R1 ( EXTEND R1 ADD COLLAPSE (X) AS Y ) {ALL BUT X } AS R2 : R2 UNGROUP Y UNPACK r on A: Replace COLLAPSE with EXPAND in PACK.

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**Example Given two temporal relations:**

S: Supplier S# was under contract during the interval During SP: Supplier S# was able to supply part P# during the interval During SP S# P# During S1 P1 [d04,d10] P7 [d05,d10] P3 [d09,d10] P5 [d06,d10] S2 [d02,d04] P9 [d03,d03] [d08,d10] S3 S4 P2 [d06,d09] [d04,d08] S S# During S1 [d04,d10] S2 [d02,d04] [d07,d10] S3 [d03,d10] S4 S5 [d02,d10]

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Example 1 Active supplier intervals: Get S#-DURING pairs for suppliers who have been able to supply at least one part during at least one interval of time, where DURING designates such an interval. PACK SP {S#,DURING} ON DURING SP S# P# During S1 P1 [d04,d10] P7 [d05,d10] P3 [d09,d10] P5 [d06,d10] S2 [d02,d04] P9 [d03,d03] [d08,d10] S3 S4 P2 [d06,d09] [d04,d08] RESULT S# During S1 [d04,d10] S2 [d02,d04] [d08,d10] S3 S4

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Example 2 Inactive (passive) supplier intervals: Get S#-DURING pairs for suppliers who have been unable to supply any parts at all during at least one interval of time, where DURING designates such an interval. PACK ( ( UNPACK S {S#,DURING} ON DURING ) MINUS ( UNPACK SP {S#,DURING} ON DURING ) ) ON DURING Shorthand: U_MINUS RESULT S# During S2 [d07,d07] S3 [d03,d07] S5 [d02,d10]

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**More Relational Operators**

USING ( AList ) ◄ r1 op r2 ► is a shorthand for: PACK ( ( UNPACK r1 on (AList) ) op ( UNPACK r1 on (AList) ) ) ON (AList) Where op is either UNION, INTERSECT, MINUS or JOIN Various comparison operators on relations are defined similarly. USING ( AList ) ◄ r1 rel-op r2 ► is equivalent to ( ( UNPACK r1 on (AList) ) rel-op ( UNPACK r1 on (AList) ) )

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Part 2 Persistent B-trees and applications

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Persistent B-tree In some applications we are interested in being able to access previous versions of data structure Databases Geometric data structures Partial persistence: Update the current version (getting a new version) Query all versions We would like to have partial persistent B-tree with O(N/B) space – N is number of updates performed update query in any version

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**Persistent B-tree East way to make B-tree partial persistent**

Copy structure at each operation Maintain “version-access” structure (B-tree) Good query in any version, but O(N/B) I/O update O(N2/B) space update i+3 i i+2 i+1 i i+1 i+2

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Persistent B-tree Idea: Elements augmented with “existence interval” and stored in one structure Persistent B-tree with parameter b: Directed graph Nodes contain elements augmented with existence interval At any time t, nodes with elements alive at time t form B-tree with leaf and branching parameter b (i.e., each node/leaf has at least b/4 and at most b children/keys in them) B-tree with leaf and branching parameter b on indegree 0 nodes If b=B: Query at any time t in I/Os

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**Persistent B-tree: Updates**

Updates performed as in B-tree To obtain linear space we maintain new-node invariant: New node contains between and alive elements and no dead elements

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**Persistent B-tree Insert**

Search for relevant leaf u and insert new element If u contains B+1 elements: Block overflow Version split: Mark u dead and create new node u’ with x alive element If : Strong overflow If : Strong underflow If then recursively update parent(u): Delete (persistently) reference to u and insert reference to u’

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**Persistent B-tree Insert**

Strong overflow ( ) Split u into u’ and u’’ with elements each ( ) Recursively update parent(u): Delete reference to u and insert reference to v’ and v’’ Strong underflow ( ) Merge x elements with y live elements obtained by version split on sibling ( ) If then (strong overflow) perform split into nodes with (x+y)/2 elements each ( ) Recursively update parent(u): Delete two insert one/two references

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**Persistent B-tree Delete**

Search for relevant leaf u and mark element dead If u contains alive elements: Block underflow Version split: Mark u dead and create new node u’ with x alive element Strong underflow ( ): Merge (version split) and possibly split (strong overflow) Recursively update parent(u): Delete two references insert one or two references

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**Persistent B-tree Insert Delete done Block overflow Block underflow**

Version split Strong overflow Strong underflow Merge Split -1,+1 -1,+2 -2,+2 -2,+1 0,0

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**Persistent B-tree Analysis**

Update: Search and “rebalance” on one root-leaf path Space: O(N/B) At least updates in leaf in existence interval When leaf u dies At most two other nodes are created At most one block over/underflow one level up (in parent(u)) During N updates we create: leaves nodes i levels up blocks

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**Summary/Conclusion: Persistent B-tree**

Update current version Query all versions Efficient implementation obtained using existence intervals Standard technique During N operations O(N/B) space update query Persistant B-tree stuff hard to follow in general

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**Interval Management Problem:**

Maintain N intervals with unique endpoints dynamically such that stabbing query with point x can be answered efficiently As in (one-dimensional) B-tree case we are interested in space update query x

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**Interval Management: Static Solution**

Sweep from left to right maintaining persistent B-tree Insert interval when left endpoint is reached Delete interval when right endpoint is reached Query x answered by reporting all intervals in B-tree at “time” x space query construction using buffer technique Dynamic with insert bound using logarithmic method x

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**Internal Memory Logarithmic Method Idea**

Given (semi-dynamic) structure D on set V O(log N) query, O(log N) delete, O(N log N) construction Logarithmic method: Partition V into subsets V0, V1, … Vlog N, |Vi| = 2i or |Vi| = 0 Build Di on Vi Delete: O(log N) Query: Query each Di O(log2 N) Insert: Find first empty Di and construct Di out of elements in V0,V1, … Vi-1 O(2i log 2i) construction O(log N) per moved element Element moved O(log N) times amortized

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**External Logarithmic Method Idea**

Decrease number of subsets Vi to logB N to get query Problem: Since there are not enough elements in V0,V1, … Vi-1 to build Vi Solution: We allow Vi to contain any number of elements Bi Insert: Find first Di such that and construct new Di from elements in V0,V1, … Vi We move elements If Di constructed in O((|Vi|/B)logB |Vi|) = O(Bi-1logB N) I/Os every moved element charged O(logB N) I/Os Element moved O(logB N) times amortized

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**External Logarithmic Method Idea**

Given (semi-dynamic) linear space external data structure with I/O query I/O construction (– I/O delete) Linear space dynamic data structure with I/O insert amortized Dynamic interval management x

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**Planar Point Location Static problem:**

Store planar subdivision with N segments on disk such that region containing query point q can be found I/O-efficiently We concentrate on vertical ray shooting query Segments can store regions it bounds Segments do not have to form subdivision Dynamic problem: Insert/delete segments (we will not discuss this) q

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Static Solution Vertical line imposes above-below order on intersected segments Sweep from left to right maintaining persistent B-tree on above-below order Left endpoint: Insert segment Right endpoint: Delete segment Query q answered by successor query on B-tree at time qx space query q

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**Static Solution Note: Not all segments comparable!**

q Note: Not all segments comparable! Have to be careful about what we compare Problem: Routing elements in internal nodes of leaf oriented B-trees Luckily we can modify persistent B-tree to use regular (live) elements as routing elements However, buffer technique construction cannot be used Only I/O construction algorithm Cannot be made dynamic using logarithmic method

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**References External Memory Geometric Data Structures**

Lecture notes by Lars Arge. Section 1-4 I/O-efficient Point Location using Persistent B-trees Lars Arge, Andrew Danner and Sha-Mayn Teh

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