ReviewLecture 30

Administrivia• Office hours 1:30 – 2:15 today• Final Exam May 16 8-11 a.m.• Location: 22 Warren• Topics since Midterm 2

– Transactions, concurrency control, locking, recovery

– Logical design, ER Modeling, Functional Dependencies, Normalization, Data Mining

– Guest lectures• Cumulative questions from semester

Review today

Review Tuesday

• Concurrent users introduce anomalies– Dirty reads (WR): T2 reads a value A that T1 wrote

but didn’t commit– Unrepeatable Reads (RW): T1 reads a value A that is

then written by T2– Lost Updates (WW): T2 overwrites a write by T1

• Serializable schedules: – A schedule that is equivalent to some serial execution of

the transactions.• Definition: Two operations conflict if:

– They are by different transactions, – they are on the same object, – and at least one of them is a write.

Concurrency

R(A) R(B)W(A) W(B)

R(A) R(B) W(B)

T1:

T2: W(A)

Conflict Serializability – Intuition• A schedule S is conflict serializable if:

– You are able to transform S into a serial schedule by swapping consecutive non-conflicting operations of different transactions.

• Example:

R(A) R(B)W(A) W(B)

R(A) W(A) R(B)W(B) W(A)

R(B)R(B)

R(A)

W(B)

W(A)

W(B)

R(A)

R(A) R(B)W(A) W(B)

R(A) W(A) R(B) W(B)

T1:

T2:

T1:T2:

Dependency Graph

• Dependency graph: – One node per Xact– Edge from Ti to Tj if:

• An operation Oi of Ti conflicts with an operation Oj of Tj and

• Oi appears earlier in the schedule than Oj.

• Theorem: Schedule is conflict serializable if and only if its dependency graph is acyclic.

Ti Tj

• A schedule that is not conflict serializable:

• The cycle in the graph reveals the problem. • The output of T2 depends on T1’s value of A,

and the output of T1 depends on T2’s value of B.

Another Example

T1 T2A

B

Dependency graph

T1: R(A), W(A), R(B), W(B)T2: T1: R(A), W(A), R(B), W(B)T2: R(A), W(A), R(B), W(B)T1: R(A), W(A), R(B), W(B)T2: R(A), W(A), R(B), W(B)T1: R(A), W(A), R(B), W(B)T2: R(A), W(A), R(B), W(B)T1: R(A), W(A), R(B), W(B)T2: R(A), W(A), R(B), W(B)

Review: Lock-Based Concurrency Control

Two-phase Locking (2PL) Protocol:– Each Xact must obtain:

• a S (shared) lock on object before reading, and • an X (exclusive) lock on object before writing.

– If an Xact holds an X lock on an object, no other Xact can get a lock (S or X) on that object.

– System can obtain these locks automatically– Two phases: acquiring locks, and releasing them

• No lock is ever acquired after one has been released• “Growing phase” followed by “shrinking phase”.

•Ensures acyclic dependency graphs•Allows only conflict serializable schedules

Review: Strict 2 Phase Locking

• Advantage: no other transaction reads anything you write until you commit.– e.g a transaction will only read committed data.

• Disadvantage: transactions end up waiting.

• Strict Two-phase Locking (Strict 2PL) Protocol:– Same as 2PL, except All locks held are released only

when the transaction completes

•Ensures acyclic dependency graphs•Allows only conflict serializable schedules•Allows only strict schedules

No values written by an Xact T can be read or overwritten until T commits or aborts.

Deadlocks• Deadlock: Cycle of transactions waiting for locks

to be released by each other.• Two ways of dealing with deadlocks:

– Deadlock prevention• Wait-die: new transactions aren’t allowed to wait• Wound-wait: old transactions don’t have to wait

• Deadlock detection– Create a waits-for graph:– There is an edge from Ti to Tj if Ti is waiting for Tj to

release a lock• Periodically check for cycles in the waits-for

graph

Deadlock Detection (Continued)

Example:

T1: S(A), S(D), S(B)T2: X(B) X(C)T3: S(D), S(C), X(A)T4: X(B)

T1 T2

T4 T3

S(A)S(D)

X(B)S(B)

S(D)S(C)

X(C)X(B)

X(A)

• Multi-granularity locking– Use database containment hierarchy to vary

granularity of locks • Full table insert: lock table vs read 1 row: lock

record

• Locking in indexes– don’t want to lock a B-tree root for a whole

transaction!– actually do non-2PL “latches” in B-trees

• CC w/out locking– “optimistic” concurrency control

Lock Management

Tuples

Tables

Pages

Databasecontains

Multiple Granularity Lock Protocol

• Each Xact starts from the root of the hierarchy.

• Special SIX lock used when reading many records, and updating a few. – SIX lock conflicts are all S and IX conflicts

(e.g. only compatible with IS locks).• To get S or IS lock on a node, must hold

IS or IX on parent node.• To get X or IX or SIX on a node, must

hold IX or SIX on parent node.• Must release locks in bottom-up order.

Multi-Granularity Example• Rules

– Each Xact starts from the root of the hierarchy.– To get S or IS lock, must hold IS or IX on parent.– To get X or IX or SIX, must hold IX or SIX on

parent.– Must release locks in bottom-up order.

Tuple 1

Sailor Table

Page 1

Database

Page 2

Tuple 2 Tuple 4Tuple 3

T1 wants to read & change tuple 2

T2 wants to read all of Page 1

•T1 gets IX lock on DBMS, Sailor, Page 1

•T1 gets X lock on Tuple 2 & changes it

•T2 gets IS lock on DBMS, Sailor

•T2 tries to get S lock on Page 1, but S conflicts with IX lock. T2 blocks.

What if T2 had started first?

IS IX SIX

IS

IXSIX

S X

S

X

T1:IX

T1:IX

T1:IX

T1:X

T2:IS

T2:IS

T2:wait

Multi-Granularity Example• Rules

– Each Xact starts from the root of the hierarchy.– To get S or IS lock, must hold IS or IX on parent.– To get X or IX or SIX, must hold IX or SIX on

parent.– Must release locks in bottom-up order.

Tuple 1

Sailor Table

Page 1

Database

Page 2

Tuple 2 Tuple 4Tuple 3

T1 wants to read & change tuple 2

T2 wants to read all of Page 1

•T2 gets IS lock on DBMS, Sailor

•T2 gets S lock on Page 1

•T1 gets IX lock on DBMS, Sailor

•T1 tries to get IX lock on Page 1, waits

IS IX SIX

IS

IXSIX

S X

S

X

T1:IX

T1:IX

T1:waits

T2:IS

T2:IS

T2:S

Locking in B+ Trees

24 30

7 9 14 15

13

20

1. Higher levels of the tree only direct searches for leaf pages.

2. For inserts:– a node must be X locked only

if a split can propagate up to it from the modified leaf.

– Example:insert 9vs insert 15

16

15

16

• We can exploit these observations to design efficient locking protocols that guarantee serializability even though they violate 2PL.

Simple Locking in B+ Trees• Search: Start at root and go down;

– S lock node.– Unlock its parent.

• Insert/Delete: Start at root and go down, – X lock node. – If node is safe, release all locks on ancestors.

• Safe node: Node such that changes will not propagate up beyond this node.– Inserts: Node is not full.– Deletes: Node is not half-empty.

Example

ROOT

A

B

C

D E

F

G H I

20

35

20*

38 44

22* 23* 24* 35* 36* 38* 41* 44*

T1: Search 38T2: Insert 45T3: Insert 25

23

T1:S

T1:S

T1:S

T1:S

T2:X

T2:X

T2:X

T2:X

• Search: – S lock node.– Unlock its

parent.• Insert/Delete:

– X lock node. – If node is

safe, release all locks on ancestors.

T3:X

T3:X

T3:X

T3:X T3:X

Optimistic CC (Kung-Robinson)

• Locking is a conservative approach in which conflicts are prevented. Disadvantages:

– Lock management overhead.– Deadlock detection/resolution.– Lock contention for heavily used objects.

• If conflicts are rare, we might be able to gain concurrency by not locking, and instead checking for conflicts before Xacts commit.

Kung-Robinson Model• Xacts have three phases:

– READ: Xacts read from the database, but make changes to private copies of objects.

– VALIDATE: Check for conflicts.

– WRITE: Make local copies of changes public.

Tj

R V W

14

Buffer Pool

23 27

14 23Tj private copies

Read

s fro

m

Writes to

Writes back

• Validation, and Write phase are done inside a critical section!– i.e., Nothing else goes on concurrently.

Validation Phase

• Tests conditions that are sufficient to ensure that no conflict occurred.– If conflict did occur, restart transaction.

• Each Xact is assigned a timestamp at end of READ phase, just before validation begins. – Also keep track of xact phase begin & end

times

• Compute – ReadSet(Tj): Set of objects read by Xact Tj.– WriteSet(Tj): Set of objects modified by Tj.

Validation Test 1 for Tj: no overlap• For all i and j such that Ti < Tj, check that Ti

completes write phase before Tj begins read phase.

Ti

TjR V W

R V W

Implies a serial order for Ti and Tj; Ti came first.

Validation Test 2 for Tj: Overlapping read phase

• For all i and j such that Ti < Tj, check that:

– Ti completes before Tj begins its Write phase +– WriteSet(Ti) ReadSet(Tj) is empty.

Ti

TjR V W

R V W Ensures Tj does not read any object written by Ti. Implies a serial order; Tj might write same set of objects,

but writes are in a serial order; Ti’s writes come first.

Validation Test 3 for Tj: Overlapping write phase

• For all i and j such that Ti < Tj, check that:

– Ti completes Read phase before Tj does +– WriteSet(Ti) ReadSet(Tj) is empty +– WriteSet(Ti) WriteSet(Tj) is empty.

Ti

TjR V W

R V W Ensures Tj does not read or write any object written by Ti. Implies a serial order; Tj reads and writes are to different

objects than those written by Ti.

Optimistic CC Overhead• Must record read/write activity in ReadSet

and WriteSet per Xact.– Must create and destroy these sets as needed.

• Must check for conflicts during validation, and must make validated writes ``global’’.– Critical section can reduce concurrency.– Scheme for making writes global can reduce

clustering of objects.

• Optimistic CC restarts Xacts that fail validation.– Work done so far is wasted; requires clean-up.

Write-Ahead Logging (WAL)

• The Write-Ahead Logging Protocol: Must force the log record for an update before

the corresponding data page gets to disk. Must force all log records for a Xact before

commit. (or, a transaction is not committed until all of its log records including its “commit” record are on the stable log.)

• #1 (with UNDO info) helps guarantee Atomicity.

• #2 (with REDO info) helps guarantee Durability.

• This allows us to implement Steal/No-Force buffer management policy

Buffer Management summary

Force

No Force

No Steal Steal

No REDO

No UNDO UNDO

No REDO

UNDOREDO

No UNDOREDO

Force

No Force

No Steal Steal

Slowest

Fastest

PerformanceImplications

Logging/RecoveryImplications

WAL & the Log

• Each log record has a unique Log Sequence Number (LSN). – LSNs always increasing.

• Each data page contains a pageLSN.– The LSN of the most recent log record

for an update to that page.• System keeps track of flushedLSN.

– The max LSN flushed so far.• WAL: Before page i is written to DB

log must satisfy:

pageLSNi flushedLSN

LSNs pageLSNs

RAM

flushedLSN

pageLSN

Log recordsflushed to disk

“Log tail” in RAM

flushedLSN

DB

Log Records prevLSN is the LSN of the previous log record written by this Xact (so records of an Xact form a linked list backwards in time)

Possible log record types:• Update, Commit, Abort• Checkpoint (for log

maintenance)• Compensation Log

Records (CLRs) – for UNDO actions

• End (end of commit or abort)

LSNprevLSNXIDtype

lengthpageID

offsetbefore-imageafter-image

LogRecord fields:

updaterecordsonly

Other Log-Related State• Two in-memory tables:• Transaction Table

– One entry per currently active Xact.• entry removed when Xact commits or aborts

– Contains XID, status (running/committing/aborting), and lastLSN (most recent LSN written by Xact).

• Dirty Page Table:– One entry per dirty page currently in buffer pool.– Contains recLSN -- the LSN of the log record

which first caused the page to be dirty.

The Big Picture: What’s Stored Where

DB

Data pageseachwith apageLSN

Xact TablelastLSNstatus

Dirty Page TablerecLSN

flushedLSN

RAM

LSNprevLSNXIDtype

lengthpageID

offsetbefore-imageafter-image

LogRecords

LOG

Master record

Example

GDE

Page 1 LSN:2

ABC

Page 2 LSN:4

HIJ

Page 3 LSN:6

OPQ

Page 4 LSN:8

Buffer Frame 1 Buffer Frame 2 Buffer Frame 3

1. T1 update 2 (DEF)(assume written to disk)2. T2 update 3 (KLM)3. T2 update 1 (QRS)4. T1 update 2 (WXY)5. T2 commit6. T1 update 4 (RST)7. SYSTEM CRASH

LSN Prev XactID Type pageID Before After

11 null T1 update 2 ABC DEF 12 null T2 update 3 HIJ KLM

ABC

Page 2 LSN:4

DEF

Page 2 LSN:11

13 12 T2 update 1 GDE QRS 14 11 T1 update 2 DEF WXY 15 13 T2 commit and end

16 14 T1 update 4 OPQ RST SYSTEM CRASH

KLM

Page 3 LSN:12

WXY

Page 2 LSN:14

To disk

GDE

Page 1 LSN:2

HIJ

Page 3 LSN:6

KLM

Page 3 LSN:12

QRS

Page 1 LSN:13

OPQ

Page 4 LSN:8

RST

Page 4 LSN:16

BEGIN_CHKPT

END_CHKPT

DEF

Page 2 LSN:11

Crash Recovery: Big Picture

Start from a checkpoint (found via master record).

Three phases. Need to do:– Analysis - Figure out which

Xacts committed since checkpoint, which failed.

– REDO all actions.(repeat history)

– UNDO effects of failed Xacts.

Oldest log rec. of Xact active at crash

Smallest recLSN in dirty page table after Analysis

Last chkpt

CRASH

A R U

End result – goal of recovery

GDE

Page 1 LSN:2

ABC

Page 2 LSN:4

HIJ

Page 3 LSN:6

OPQ

Page 4 LSN:8

1. T1 update 2 (DEF)2. T2 update 3 (KLM)3. T2 update 1 (QRS)4. T1 update 2 (WXY)5. T2 commit6. T1 update 4 (RST)7. SYSTEM CRASH

KLM

Page 3 LSN:12

QRS

Page 1 LSN:2

• T1 aborts• Roll back

updates if they made it to disk.

• T2 commits• Re-apply

updates if needed

Recovery: The Analysis Phase• Re-establish knowledge of state at checkpoint.

– via transaction table and dirty page table stored in the checkpoint

• Scan log forward from checkpoint.– End record: Remove Xact from Xact table.– All Other records: Add Xact to Xact table, set lastLSN=LSN,

change Xact status on commit.– also, for Update records: If page P not in Dirty Page Table,

Add P to DPT, set its recLSN=LSN.

• At end of Analysis…– transaction table says which xacts were active at time of

crash.– DPT says which dirty pages might not have made it to disk

Analysis

GDE

Page 1 LSN:2

DEF

Page 2 LSN:11

HIJ

Page 3 LSN:6

OPQ

Page 4 LSN:8

Buffer Frame 1 Buffer Frame 2 Buffer Frame 3

Xact ID

Last LSN

State

T1 11 U

ABC

Page 2 LSN:4

DEF

Page 2 LSN:11

KLM

Page 3 LSN:12

WXY

Page 2 LSN:14

GDE

Page 1 LSN:2

HIJ

Page 3 LSN:6

KLM

Page 3 LSN:12

QRS

Page 1 LSN:13

OPQ

Page 4 LSN:8

RST

Page 4 LSN:16

LSN Prev XactID Type pageID Before After

11 null T1 update 2 ABC DEF 12 null T2 update 3 HIJ KLM 13 12 T2 update 1 GDE QRS 14 11 T1 update 2 DEF WXY 15 13 T2 commit and end

BEGIN_CHKPT

END_CHKPTXact Table

T2 12 U T2 13 U

T1 14 U T2 15 C

1. Create entries the Xact table with xacts active at time of crash.

Analysis

GDE

Page 1 LSN:2

DEF

Page 2 LSN:11

HIJ

Page 3 LSN:6

OPQ

Page 4 LSN:8

Buffer Frame 1 Buffer Frame 2 Buffer Frame 3

Xact ID

Last LSN

State

ABC

Page 2 LSN:4

DEF

Page 2 LSN:11

KLM

Page 3 LSN:12

WXY

Page 2 LSN:14

GDE

Page 1 LSN:2

HIJ

Page 3 LSN:6

KLM

Page 3 LSN:12

QRS

Page 1 LSN:13

OPQ

Page 4 LSN:8

RST

Page 4 LSN:16

Page ID rec LSN

LSN Prev XactID Type pageID Before After

11 null T1 update 2 ABC DEF 12 null T2 update 3 HIJ KLM 13 12 T2 update 1 GDE QRS 14 11 T1 update 2 DEF WXY 15 13 T2 commit and end

BEGIN_CHKPT

END_CHKPTXact Table

Dirty Page Table

T1 14 U

2 11

3 12

1 13

2. Create entries in the Dirty Page table with pages that might not have made it to disk.

Phase 2: The REDO Phase

• We Repeat History to reconstruct state at crash:– Reapply all updates (even of aborted Xacts!), redo CLRs.

• Scan forward from log rec containing smallest recLSN in DPT. Q: why start here?

• For each update log record or CLR with a given LSN, REDO the action unless: – Affected page is not in the Dirty Page Table, or– Affected page is in D.P.T., but has recLSN > LSN, or– pageLSN (in DB) LSN. (this last case requires I/O)

• To REDO an action:– Reapply logged action.– Set pageLSN to LSN. No additional logging, no forcing!

Redo

GDE

Page 1 LSN:2

DEF

Page 2 LSN:11

HIJ

Page 3 LSN:6

OPQ

Page 4 LSN:8

Buffer Frame 1 Buffer Frame 2 Buffer Frame 3

Xact ID

Last LSN

State

ABC

Page 2 LSN:4

DEF

Page 2 LSN:11

KLM

Page 3 LSN:12

WXY

Page 2 LSN:14

GDE

Page 1 LSN:2

HIJ

Page 3 LSN:6

KLM

Page 3 LSN:12

QRS

Page 1 LSN:13

OPQ

Page 4 LSN:8

RST

Page 4 LSN:16

Page ID rec LSN

LSN Prev XactID Type pageID Before After

11 null T1 update 2 ABC DEF 12 null T2 update 3 HIJ KLM 13 12 T2 update 1 GDE QRS 14 11 T1 update 2 DEF WXY 15 13 T2 commit and end

BEGIN_CHKPT

END_CHKPTXact Table

Dirty Page Table

T1 14 U

2 11

3 12

1 13

Step 1. Find lowest rec LSN in Dirty Page Table.

Redo

GDE

Page 1 LSN:2

DEF

Page 2 LSN:11

HIJ

Page 3 LSN:6

OPQ

Page 4 LSN:8

Buffer Frame 1 Buffer Frame 2 Buffer Frame 3

Xact ID

Last LSN

State

ABC

Page 2 LSN:4

DEF

Page 2 LSN:11

KLM

Page 3 LSN:12

WXY

Page 2 LSN:14

GDE

Page 1 LSN:2

HIJ

Page 3 LSN:6

KLM

Page 3 LSN:12

QRS

Page 1 LSN:13

OPQ

Page 4 LSN:8

RST

Page 4 LSN:16

Page ID rec LSN

LSN Prev XactID Type pageID Before After

11 null T1 update 2 ABC DEF 12 null T2 update 3 HIJ KLM 13 12 T2 update 1 GDE QRS 14 11 T1 update 2 DEF WXY 15 13 T2 commit and end

BEGIN_CHKPT

END_CHKPTXact Table

Dirty Page Table

T1 14 U

2 11

3 12

1 13

Step 2. Scan forward and redo all redoable log records.

Redo

GDE

Page 1 LSN:2

DEF

Page 2 LSN:11

HIJ

Page 3 LSN:6

OPQ

Page 4 LSN:8

Buffer Frame 1 Buffer Frame 2 Buffer Frame 3

Xact ID

Last LSN

State

ABC

Page 2 LSN:4

DEF

Page 2 LSN:11

KLM

Page 3 LSN:12

WXY

Page 2 LSN:14

GDE

Page 1 LSN:2

HIJ

Page 3 LSN:6

KLM

Page 3 LSN:12

QRS

Page 1 LSN:13

OPQ

Page 4 LSN:8

RST

Page 4 LSN:16

Page ID rec LSN

LSN Prev XactID Type pageID Before After

11 null T1 update 2 ABC DEF 12 null T2 update 3 HIJ KLM 13 12 T2 update 1 GDE QRS 14 11 T1 update 2 DEF WXY 15 13 T2 commit and end

BEGIN_CHKPT

END_CHKPTXact Table

Dirty Page Table

T1 14 U

2 11

3 12

1 13

1. Reapply LSN 11 T1 update 2 (DEF)

DEF

Page 2 LSN:11

KLM

Page 3 LSN:12

2. Reapply LSN 12 T2 update 3 (KLM)

Redo

GDE

Page 1 LSN:2

DEF

Page 2 LSN:11

HIJ

Page 3 LSN:6

OPQ

Page 4 LSN:8

Buffer Frame 1 Buffer Frame 2 Buffer Frame 3

Xact ID

Last LSN

State

ABC

Page 2 LSN:4

DEF

Page 2 LSN:11

KLM

Page 3 LSN:12

WXY

Page 2 LSN:14

GDE

Page 1 LSN:2

HIJ

Page 3 LSN:6

KLM

Page 3 LSN:12

QRS

Page 1 LSN:13

OPQ

Page 4 LSN:8

RST

Page 4 LSN:16

Page ID rec LSN

LSN Prev XactID Type pageID Before After

11 null T1 update 2 ABC DEF 12 null T2 update 3 HIJ KLM 13 12 T2 update 1 GDE QRS 14 11 T1 update 2 DEF WXY 15 13 T2 commit and end

BEGIN_CHKPT

END_CHKPTXact Table

Dirty Page Table

T1 14 U

2 11

3 12

1 13

3. Reapply LSN 13 T2 update 1 (QRS)

DEF

Page 2 LSN:11

KLM

Page 3 LSN:12

4. Reapply LSN 14 T1 update 2 (WXY)

GDE

Page 1 LSN:2

QRS

Page 1 LSN:13

WXY

Page 2 LSN:14

Redo

GDE

Page 1 LSN:2

DEF

Page 2 LSN:11

HIJ

Page 3 LSN:6

OPQ

Page 4 LSN:8

Buffer Frame 1 Buffer Frame 2 Buffer Frame 3

Xact ID

Last LSN

State

ABC

Page 2 LSN:4

DEF

Page 2 LSN:11

KLM

Page 3 LSN:12

WXY

Page 2 LSN:14

GDE

Page 1 LSN:2

HIJ

Page 3 LSN:6

KLM

Page 3 LSN:12

OPQ

Page 4 LSN:8

RST

Page 4 LSN:16

Page ID rec LSN

LSN Prev XactID Type pageID Before After

11 null T1 update 2 ABC DEF 12 null T2 update 3 HIJ KLM 13 12 T2 update 1 GDE QRS 14 11 T1 update 2 DEF WXY 15 13 T2 commit and end

BEGIN_CHKPT

END_CHKPTXact Table

Dirty Page Table

T1 14 U

2 11

3 12

1 13

5. Reapply T2 commit (and we’ll write dirty pages to disk.)

DEF

Page 2 LSN:11

KLM

Page 3 LSN:12

WXY

Page 2 LSN:14

QRS

Page 1 LSN:13

QRS

Page 1 LSN:13

Phase 3: The UNDO PhaseWe undo actions of all active but not

committed xacts at the time of the crash.– May even need to undo some of what we did

in REDO phase!

ToUndo={lastLSNs of all Xacts in the Trans Table} a.k.a. “losers”

Repeat:– Choose (and remove) largest LSN among ToUndo.– If this LSN is a CLR and undonextLSN==NULL

• Write an End record for this Xact.– If this LSN is a CLR, and undonextLSN != NULL

• Add undonextLSN to ToUndo – Else this LSN is an update. Undo the update, write a CLR,

add prevLSN to ToUndo.Until ToUndo is empty.

CLRs will help us remember where we are in case of system crash during recovery.

Undo

GDE

Page 1 LSN:2

DEF

Page 2 LSN:11

HIJ

Page 3 LSN:6

OPQ

Page 4 LSN:8

Buffer Frame 1 Buffer Frame 2 Buffer Frame 3

Xact ID

Last LSN

State

ABC

Page 2 LSN:4

DEF

Page 2 LSN:11

KLM

Page 3 LSN:12

WXY

Page 2 LSN:14

HIJ

Page 3 LSN:6

KLM

Page 3 LSN:12

OPQ

Page 4 LSN:8

RST

Page 4 LSN:16

Last LSN

LSN Prev XactID Type pageID Before After

11 null T1 update 2 ABC DEF 12 null T2 update 3 HIJ KLM 13 12 T2 update 1 GDE QRS 14 11 T1 update 2 DEF WXY 15 13 T2 commit and end

BEGIN_CHKPT

END_CHKPTXact Table

ToUndo

T1 14 U

14 11

1. Add last LSN for all transactions in Xact table

QRS

Page 1 LSN:13

2. Recursively Process each last LSN in To Undo table.

DEF

Page 2 LSN:11

16 undoNextLSN=null T1 CLR 2 DEF ABC

ABC

Page 2 LSN:16

ABC

Page 2 LSN:4