The Serializability of Concurrent Database Updates

transcript

The Serializability of Concurrent

Database UpdatesCHRISTOS H. PAPADIMITRIOU

(Journal of the ACM, Volume 26, 1979)

Presented by: Yuri Meshman

For any comments or updates contact: syurim@gmail.com

Transactions History (Preliminary definitions) SR is NPC (stencil of proof)

Polygraph SR Subsets in P

DSR The Class Q Two-Phase locking and P3 Topography of SR, S, Q, SSR, DSR, P3 and 2PL

Restrictions on Read Write Set (when SR is in P) Schedulers of Histories Discussion – extension to transactions with multiple steps –

not going through

Contents

Transactions History (Preliminary definitions) SR is NPC (stencil of proof)

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Contents

Transactions History

Transaction-Bank Example

Read (A, t)t = t - 100Write (A, t)

Read (B, t)t = t + 100Write (B, t)

Read (A, s)s = s - 100Write (A, s)

Read (C, s)s = s + 100Write (C, s)

Lucy Ricky A B C

300 600600

500 500500

300 500500

300 500600

300 + 600 + 600 = 1500

400 500500

Central Bank Registry

Transaction-Bank Example

Read (A, t)t = t - 100

Write (A, t) Read (B, t)t = t + 100Write (B, t)

Read (A, s)s = s - 100Write (A, s)

Read (C, s)s = s + 100Write (C, s)

Lucy Ricky A B C

400 600600

500 500500

400 500500400 500500

400 500600

400 + 600 + 600 = 1600

Central Bank Registry

Definition

, , ,h n V S

1 2 1 3 2 3, ,h R x R W x y R x W x y W y

positive integer, , , , ,

permutation of - a finite set of variables

- a funcion : 2

nR W R R W

1 3 2 1 2, , , ,...S R S R x S R S W S W x y

,i i iT R W

Transactions HistoryFrom Databases to

Programs 1 2 1 3 2 3, ,h R x R W x y R x W x y W y𝑡11← 𝑥

𝑡 31←𝑥F

y←t32

𝑡 32←f 31 (t 31 )

Transactions HistoryFrom Databases to

Programs 1 2 1 3 2 3, ,h R x R W x y R x W x y W y𝑡11← 𝑥

𝑡 31←𝑥F

y←t32

𝑡 32←f 31 (t 31 )

equivalent histories if and only if their program schemata are equivalent.

For:- Any set of domains for the variables - Any set of initial values- Any interpretation for the functions The values of the variables are identical after the execution in both histories.

Histories EquivalenceProposition 1. Two histories and are equivalentiff they have the same set of live transactions, and a live reads from in iff reads from in .

1 2 1 3 2 3, ,h R x R W x y R x W x y W y

1 1 1 2 1 3 2 3 2 2, , , ,n n n nh R W x y R x RW x y R x W x y W y R x y W

Histories EquivalenceProposition 1. Two histories and are equivalentiff they have the same set of live transactions, and a live reads from in iff reads from in .

Corollary. Equivalence of histories can be decided in time.

SerializabilityA history is serial if for all . immediatly precedes .

A history is serializable if there is a serial history such that

Notation:

Transactions History SR is NPC (stencil of proof)

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SR is NPC

Polygraph - IntroductionDefinition 1. A polygraph is a digraph together with a set of bipaths, pairs of arcs – not necessarily in A – of the form such that .

Alternatively, a family D of digraphs. A digraph is in D iff , and for bipath , contains one of .

v𝑢𝑤

SR is NPC

Polygraph- IntroductionDefinition 1. A polygraph is a digraph together with a set of bipaths, pairs of arcs – not necessarily in A – of the form such that .

Alternatively, a family D of digraphs. A digraph is in D iff , and for bipath , contains one of .

v𝑢𝑤

SR is NPC

Polygraph – Acyclicity Definition 2. A polygraph is acyclic if there is an a cyclic digraph is in D .

v𝑢𝑤

SR is NPC

Transactions PolygraphGiven .

If reads a variable from : Furthermore if is writing to :

𝑇 1 𝑇 2

𝑇 3𝑇 4

𝑇 5 𝑇 6

𝑇 𝑛+ 1

𝑇 𝑛+ 2

is live transactions of

𝑇 7

𝑊 4[𝑦 , 𝑧 ]𝑊 2 [𝑥 ]𝑅1 [ 𝑥 , 𝑦 ]𝑅3 [ 𝑦 ]𝑊 7 [ 𝑦 ]𝑊 5 [ 𝑧 , 𝑠 ]𝑅6 [ 𝑠 ]𝑊 6 [𝑧 ]…

SR is NPC

Transactions PolygraphLemma 1. Two histories are equivalent iff are identical

Lemma 2. is serializable iff is acyclic

𝑇 1 𝑇 2

𝑇 3𝑇 4

𝑇 5 𝑇 6

𝑇 𝑛+ 1

𝑇 𝑛+ 2

𝑇 7

𝑊 4[𝑦 , 𝑧 ]𝑊 2 [𝑥 ]𝑅1 [ 𝑥 , 𝑦 ]𝑅3 [ 𝑦 ]𝑊 7 [ 𝑦 ]𝑊 5 [ 𝑧 , 𝑠 ]𝑅6 [ 𝑠 ]𝑊 6 [𝑧 ]…

SR is NPC

The proof stencilTheorem 1. Testing if h is serializable is NP-complete, even if h has no dead transactions.

Preliminary: 3SAT is NPC

Lemma 3. 3SAT is NPC when formulas are non circular (no repeating variables)

Theorem 1 proof: reduction from 3SAT to polygraph acyclicity

SR is NPC

Construction of =x or

SR is NPC

Construction of =x or

𝑎𝑠

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SR Subsets in P

DSR-conflict serializability Definition 3. Let , . We write if , for some , and:a) for some orb) , or

𝑅1 [ 𝑥 ]𝑅2 [ 𝑥 ]𝑾 𝟏 [𝒙 ]𝑾 𝟐 [𝒚 ] 𝑹𝟏 [𝒚 ]𝑹𝟐 [𝒙 ]𝑊 2 [ 𝑥 ]𝑊 1 [ 𝑦 ]𝑅2 [𝑥 ]𝑹𝟏 [𝒚 ]𝑾 𝟐 [𝒙 ]𝑊 1 [ 𝑦 ]𝑅2 [ 𝑥 ]𝑊 2 [ 𝑥 ]𝑅1 [ 𝑦 ]𝑊 1 [ 𝑦 ]

Let be the transitive reflexive closure of . Since is symmetric is an equivalence relation

SR Subsets in P

DSR-conflict serializability Proposition 2. If then

is a proper subset of

h is D-serializable(DSR) iff there exists a serial history h such that

SR Subsets in P

DSR-conflict serializability We can associate with a history h a digraph D(h) as follows: is an arc if:a) and , orb) and orc) and

Theorem 2. A history h is DSR iff D(h) is acyclic.

SR Subsets in P

DSR-conflict serializability

Corollary 2. A history is DSR iff we can find real numbers with the following propertiesa) and , then b) and thenc) and

Corollary 1. Checking whether a history is DSR can be done in

SR Subsets in P

The class QDefinition 4 . A history h is in Q if there exist non integer, distinct real numbers such that:a) b) and , then c) and

SR Subsets in P

The class QTheorem 3. If h is in Q, then h is DSR.

We can associate with a history h a digraph D’(h) a super graph as follows: is an arc if:a) b) and orc) and

h=𝑅1𝑅2𝑅3 [ 𝑥 ]𝑊 1 [ 𝑥 ]𝑊 2 [𝑦 ,𝑧 ]𝑊 3 [𝑦 ]

SR Subsets in P

Two Phase locking and P3Definition 5. h is two phase locked ( iff there exist distinct non-integer real numbers (the lockpoints) s.t:a) b) and , then c) and

SR Subsets in P

Two Phase locking and P3

The inclusion is proper:

SR Subsets in P

Two Phase locking and P3Definition 6. Let G(h) be an undirected graph corresponding to D(h). A cycle - s.t. is an edge - is bad if and

SR Subsets in P

Two Phase locking and P3Definition 6.(continued) is a guardian of if is part of a bad cycle in G(h). h obeys the P3 protocol () if when is a guardian of we do not have

SR Subsets in P

SSRDefinition 7. is Serializable in the Strict Sense h if there is a serial history such that and , implies .

𝑄⊂𝑆𝑆𝑅

SR Subsets in P

Topography

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Restrictions on Read Write Set (SR is in P)

Theorem 8. If for w for then is serializable iff is DSR.

Restrictions on Read Write Set (SR is in P)

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Schedulers of

Histories 2PL and P3 correspond to simple schedulers.

A trivial scheduler will generate only serial histories.Scheduler task becomes interesting and important when he must transform the arriving schedule to a non equivalent output which is closest to the input.

Definition 8. A schedule, for (histories), is such that

Schedulers of

Histories Assumptions: works online (receiving transactions) Optimism – doesn’t change order of transactions

seen so far, if can be extended to a serializable history.

efficient if computable in polynomial time.

Q: Which classes of histories have efficient schedulers. NoEfficient scheduler if the input class is

Schedulers of

Histories Should be computed efficiently by proposed scheduler.

Check for if the output starts with - to check SR efficiently.

NoEfficient scheduler if the input class is in P?

Schedulers of

Histories Corollary 1. Unless P=NP, SR has no efficient

scheduler.

Corollary 2. S, 2PL, P3, Q, DSR have efficient schedulers.

Thank you

The Serializability of Concurrent Database Updates

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