i28
^o^\
OCT 201987
ALFRED
P.
WORKING PAPER
SLOAN SCHOOL OF MANAGEMENT
HIERARCHICAL TIMESTAMPING ALGORITHM
Meichun Hsu
Stuart E. Madnick
Aoril 1987
1876-87
MASSACHUSETTS
INSTITUTE OF TECHNOLOGY
50 MEMORIAL DRIVE
CAMBRIDGE, MASSACHUSETTS 02139
HIERARCHICAL TIMESTAMPING ALGORITHM
Meichun Hsu
Stuart E. Madnick
ADril 1987
1876-87
.I.T.
OCT
LIBRARIES
2
1987
RECEIVED
Hierarchical Timestamping Algorithm
By
Meichun Hsu
Harvard UniversityCambridge MA 02138
and
Stuart E. Madnick
Massachusetts Institute of Technolog^y
Cambridge, MA 02139
Acknowledgement: Work reported herein has been supported, in part, by the Naval Electronic
Systenis Command through contracts N0039-83-C-0463 and N0039-8o-c-0571.
ABSTRACT
The Hierarchical Timestamping Algorithm is proposed for handling database concurrency
By analyzing transaction conflicts and partitioning the database into hierarchical parti-
control.
which transactions will access discriminantly using different synchronization protocols, the
algorithm can offer significant performance gain. It also reduces the need for transactions to leave
traces (e.g.. locks, timestamps) when accessing a data element.
tions to
It is shown that the algorithm is correct in terms of transaction serializability.
This is done
by showing that the algorithm enforces a topological order among transactions. This research
advocates the potential benefit of application analysis in enhancing performance of concurrency
control algorithms where the level of concurrency is vital to system performance.
Introduction
1.
The two
basic approaches to database concurrency control are the two-phase locking tech-
nique [Eswaran76j and the timestamp ordering technique (BernsteinSO, Reed78].
endorse serializability as their criterion of correctness.
deadlock free and
in
drawback of
assigned to transactions without
among
interferences
is
While having the advantages of being
no need of unlock instructions when a transaction
ing algorithm has the
rigidly
much
transactions.
Both techniques
is
timestamp-
finished, the
obeying the order of timestamps which have been
consideration of the static or the dynamic nature of actual
In this paper, the Hierarchical
Timestamping (HTS) approach
described which makes unique use of the nature of the application conflicts.
By decomposing
the database into partitions to which the transactions will access discriminantly, the approach can
offer significant
its
prospect
when
performance gain
handling database concurrency control.
reducing the need for a transaction to leave a "trace"
in
(e.g.,
Equally important
steinSO, BernsteinSlj
a lock or a timestamp)
among
SDD-1
[Bern-
as a vehicle to discover, a priori, certain (static) conflict patterns
among
transaction classes that
may
terminology) to be used.
transactions has been proposed
in
the research of
enable a more flexible timestamp protocol
(e.g..
Protocol
may
more generalized timestamping algorithm that takes better advantage
be provided via conflict analysis,
presented
in
The
SDD-
.\long a difl^erent dimension, a novel
at the
of develop-
of information that
method has been
[Viemont82' which blends timestamp ordering and two-phase locking
chooses to switch to one or the other
currency.
in
1
However, with an orientation towards a distributed database system
where timestamps are not stapled with data elements, the SDD-1 approach stops short
ing a
is
accessing data elements.
Conflict analysb
I's
in
most opportune time so as to increase
multi-version timestamping algorithm has also been developed and
in
one and
level of con-
shown
to pro-
vide a higher level of concurrency than the conventional single-version timestamping algorithm
|Reed78, Reed79. Bernstein83
[Bernstein83.
,
Papadimitriou84J,
Theoretical aspects of multi-version databases are discussed
whilf
one-pre\ious-version
concurrency
control
methods
in
in
_= 2 =—
[BayerSO,
and
methods
multiple-previous-version
Chan82,
[StearnsSl,
in
Simulation studies of multi-version methods have been reported
Chan85].
and the
Garcia-Molina82]
in
[Harder86, Carey86]
results are generally favorable.
Using conflict analysis and multi-version database as vehicles, the algorithm proposed
HTS
paper, called the Hierarchical Timestamping Algorithm, or the
algorithm, allows a transac-
tion to use an "array" of timestamps, rather than a single timestamp, to synchronize
Which timestamp
its
accesses.
a transaction will use to synchronize an access to a particular data element
depends on which "data partition" the data element belongs
hierarchical database decomposition proposed in [Hsu83j.
Kedem83]
protocol [SilberschatzSO,
level of
in this
is
to.
The method
is
based on the
In comparison, the structural locking
a non-two-phase locking protocol which aims at increasing
concurrency by reducing the amount of lime the locks on the high-level nodes of a tree
must be held by each transaction.
certain sequence and
of the tree.
it
in this
requires that the transactions access the nodes of the tree in
involves lock and unlock protocols for each access to the high-level nodes
The hierarchy
chy discussed
It
referred to in their study
is
entirely different
from the kind of hierar-
paper.
Before going into the detailed mechanism of the algorithm, the basic idea behind the algo-
rithm can be illustrated by referring to an example schedule of read and write steps from three
transactions
t^.t^,
and
tr,'.
io{\V,b), t^(R.a), t^R.b). ^(U•,6), t^(\V.c)
where t^(R,a)
refers to a request
the basic timestamping algorithm,
abort.
b
is
stamped with
a
tr.
t^
will
to read data element a,
t^
is
and t^(W,b) to
smaller than that of
be denied permission to write data element
read timestamp which
Under our proposed method, however,
related to that containing data element
later),
<,
Suppose the timestamp of
write data element 6, and so on.
that time
from transaction
6
in
then the algorithm allows transaction
if
is
t^'s),
Using
(since at
and forced
the data partition containing data element
a certain
tr.
greater than that of
b
tr,.
way
to access b
a
to
is
(the exact nature to be explained
with a
pseudo' read timestamp
TS
earlier
than
aborted.
t^'s
timeslamp so that
Furthermore,
this
write request can proceed without being delayed or
/,'s
pseudo read timestamp
that no transactions will any longer write
leave
TS
with
6
TS may
be engineered to be early enough so
with even earUer timestamps, eUminating the need to
b
as a read timestamp. Therefore this read access can be accomplished without
having to leave any "trace" for concurrency control purposes. With these properties, the proposed
algorithm has the potential of increasing the level of concurrency while reducing the costly over-
head of leaving access traces.
The organization
tamping Algorithm
section four,
2.
we
is
of this paper
described
is
as follows. In the following section, the Hierarchical Times-
The proof
in detail.
of correctness
discuss the optimality aspect of the algorithm.
is
given in section three.
In
Section five concludes the paper.
The Hierarchical Timestamping Algorithm
The
tions.
We
HTS
Algorithm requires the decomposition of a database into a number of data
construct a data partition hierarchy which
tions subject to certain constraints.
basically a partial order of the data parti-
is
Hierarchical Timestamping Algorithm
The
parti-
is
a timestamp-
based concurrency control algorithm parameterized by a chosen data partition hierarchy and
corresponding transaction classification.
data partition, the
HTS
When
its
the database decomposition consists of a single
Algorithm degenerates to the conventional multi-version timestamping
algorithm.
We
will in this section first discuss the
for constructing a
method
database partition hierarchy.
of transaction analysis
Then
in
Section 2.2
and the mechanbm
we present
the algorithm
The purpose
of transaction
itself.
2.1.
Transaction Analysis
Let a database be decomposed into a number of data partitions.
analysis in our algorithm
is
to facilitate
which the hierarchical timestamp protocol
correct construction of a data partition hierarchy on
is
based.
The data
partition hierarchy, constructed
off-line, is basically
a partial order of the data partitions subject to additional graph-theoretic con-
and constraints related
straints
and any
set of transactions, a correct data partition hierarchy can be
(e.g.,
can outperform the basic multi-version timestamp algorithm) depends on an intelligent
it
choice of database deomposition and data partition hierarchy.
constructing a correct data partition hierarchy
2.1.1.
found and the hierarchical
However, performance and optimality of the protocol
timestamp protocol applicable.
whether
Note that given any data base partition
to transaction analysis.
is
mechanics of
In this section, the
discussed.
Transaction analysis against a database partition
Analysis of a set of update transactions T, against a database consisting of a set of data
partitions results in a data partition graph, where the nodes are the data partitions, and the arcs
are assigned in such a
D-
if
and only
elements
in
if
way
that there
an arc from a data partition
one can find a potential transaction
and accesses
Z),
is
(i.e.,
in
Z),
to another data partition
the database system that updates data
reads or writes) data elements
in
D
.
That
is,
D^—*D-, i^j,
indicates that there exist transactions in the system that would potentially link updates of data
elements
in
D, to the content of data elements
Definition.
Let
be a decomposition of
is
7",
D
a digraph denoted as
set of directed arcs
(f T", s.t.
The data
to
.
into data partitions Z)[,Z)o
DPG(P .T^]
jommg
A
D^.
data partition graph of
a(
Of^Z)
partition graph
Note
/
is
^^i?,
where w(t),
r{t)
(The access set a{t)
is
iff
all
ad hoc queries.
Let
w.r.t.
P
P
T,
and a
there exists a transaction
and a(t) are the write
set, the
read set
the union of r(t) and w{t).)
a tool for capturing the pattern of conflict
that, in our algorithm, there
is
among
transactions
no need for read-only transactions
participate in the transaction analysts, eliminating the difficulties of pinning
nature of
P
with nodes corresponding to the data partitions of
these nodes such that, for iV;, D^—'-D
set of transaction
to be run in the system.
P
be a set of update transactions to be performed on a database D.
w(t)r^D^^4> and
and the access
in
down, a
priori, the
Constructing a data partition hierarchy
2.1.2.
Once
a data partition graph
is
defined for a database, a data partition hierarchy, which
assigns a partial order to the set of data partitions, can be derived as follows.
partition hierarchy, denoted as
in
P
,
and arcs between
DPH[P ,T^)
(1)
P
Given a data decomposition
Definition.
is
DPH[P ,T^), b
partitions,
denoted as
a semi-tree (a semi-tree
and a data partition graph
a data
any acyclic graph where nodes are data partitions
—,
is
DPG(P ,T^),
such that
an acyclic digraph where there exists at most one
path between any pair of nodes,) and
If
(2)
there exists a directed path
between D- and D^
directed path between D, and D^in
Note that
tive arcs in
is
P
and T,
,
the existence of a
P
and a data partition graph
DPH
guaranteed.
is
G
order of the data partitions satisfies the definition of
DPH(P .T,)-
remainder of the paper, the notation
DPH
than a data partition D^, denoted as
DPH.
or
In general,
and
Z),
2.1.3.
D
we say
D->D',
if
DPG(P ,T^).
of the data partitions in
refers to a particular
chy chosen to base our hierarchical timesiamp protocol.
We
satisfy the
above
However,
This directly follows from the fact
that the transitive reduction of any acyclic digraph
In the
there exists a
necessarily a transitive reduction: there are no transi-
There may be multiple data partition hierarchies that
definition given a database decomposition
given any
DPG{P,T^), then
DPH{P.T,).
a data partition hierarchy
DPH{P .Tu).
in
P
that
a total
is
data partition hierar-
say that a data partition D-
there exists a directed path
—*D-
D^—*
that data partition D, and data partition D^ are related
if
higher
is
either
in
D-=D^
are connected by a directed path
Transaction Classification
Given a data partition hierarchy
the data partition(s) they write into
exists a transaction
t(T
DPH
,
it
is
possible to classify transactions in T, based on
Based on the definition of data partition hierarchy,
which writes into more than one data partition, then
all
if
there
the data parti-
tions that
writes into
it
must be connected by a directed path
Therefore, the rule for transaction classification
Assign each update transaction
We
denote
The
I
being assigned to
as follows:
to the highest data partition D-
as /fD,,
must be related
hierarchy.
and D-
called the
is
(read or write)
partition of
t.
to the
home data
partition of that transaction in the data
in
the next section.
is
related to the D, in
t
accesses
DPH.
The Hierarchical Timestamp Protocol
The
timestamp protocol (HTS)
hierarchical
by a chosen data partition hierarchy
DPH
is
a concurrency control algorithm parameterized
and the corresponding transaction
the conventional multi-version timestamp algorithm
t,
home data
Let tiD,. Then the data partition that contains a data element that
Property.
tamp
writes into.
it
This property completes the background for establishing the hierarchical
timestamp protocol to be described
2.2.
the data partition hierarchy.
following property establishes that the data partition that contains a data element that
a transaction accesses
partition
Z),
t
is
in
to a transaction
denoted as
accesses
may
I[t].
t
when
is
initiates.
We
will call this
algorithm assigns a timesinitiation
timestamp of
be different from
I(t).
home
In particular, the
timestamp to access data elements
partition will normally be
true for accesses to lower partitions.
somewhat smaller than
The exact mechanism used
is still
I(t),
in
its
data
while the
to calculate these
preserved.
Basic Definitions
Two
functions, called the
Decrement function, or the function DEC. and the Increment
function, or the function I.\'C. are devised to
data partitions different from
tamps
HTS
timestamp the
"access timestamps" will be such that the overall serializability
2.2.1.
the
Like
However, the timestamp the transaction actually uses to synchronize
partitions higher than Ts
reverse
it
(MVTS),
classification.
for
its
own home
compute the timestamps a transaction uses
partition.
Function
and
for
accessing higher data partitions,
!.\'C
DEC
is
to access
used for computing times-
lower partitions.
These definitions
assume that a data partition hierarchy
Two
values,
and C(t),
/(/)
DPH
are
given.
is
in
the system.
We
time,
denoted as
I(t).
associated with each transsaction
assume that the timestamp assigned to a transaction b
its
initiation
t
Therefore every data access request issued by a transaction must occur after
tion
also assigned a
is
actually finishes;
saction
is
In typical cases, C(t)
is
Each transac-
when
the time
t
no data access request would normally be issued at time after C{t) by tran-
i.e.,
However
t.
commit timestamp C(t)>I{t).
I{t).
there are exceptional occasions (as explained in procedure Clater)
forced to be a smaller value than the time
when
t
actually finishes.
It is
when C(t)
sufficient for
our pur-
pose to have I(t)<C(t).
m'
,
Definition.
A
Definition.
The function
i.e.,
transaction
/,
m'=/,(m), such that m'
active at time
is
t
m
if
I{t)<.m and C{t]'>m
defined for a data partition D-,
,
is
maps
.
a time
m
to another time
the initiation time of the oldest active transaction assigned to
m. Formally,
D, at time
'm
if
m
there exists no tiD^ active at time
/.(m)='
.Min{I(t)) otheru'ise
Z),
and
Z)
for all ttO^ active
Let the Decrement function
Definition
titions
,
,
where
w
D£C._,(m) =
if
D
DEC^
'>D,.
DEC^ ^
recursively
time m.
at
be a function defined for a pair of data par-
maps
a lime to another time as follows.
D=D-
Ij(m)
\l
D,^Dj
IS
m DPH
DEC^-[DECij^{m)) otherwise, where D--*D^-*....-*D^
That
is,
cessively
(i e.,
example,
if
We
the function
along the
DEC^
critical
^
maps
path of
a time
DPH)
the critical path between D, and D^
will
now
inverse of functions
m
and DEC^^.
to the initiation time,
in
DPH.
DEC-j(m),
of suc-
the oldest active transaction assigned to D^.
is
describe the functions C, and
7,
in Z),
is
D,—-Di^
ZA'C'^-,,
— Dj,
then
For
DEC, j{m)=I^{I^{m)).
that can be considered conceptually the
_= 8 =—
Let C,:m
Definition.
data partition and C,(m)
if
is
— m'
be a function which
maps
a time
m
to another m'
where D-
is
a
determined as follows.
m,
there exists no t(D^ active at time
Im
Max(C{t)) otherwise,
That
C{m)
is,
is
the latest
for all teD- active
commit time
of
all
time m.
at
transactions assigned to
Z),
that started
before or at time m.
While the
DEC
maps
function
a time in a lower partition to the initiation time of
INC
transaction assigned to a higher partition, the
the
commit time
of
Definition.
D yD
,
some transaction assigned
The Increment
denoted as INC^(m).
tm
INC-,{m) =
if
is
function
maps
function, defined for a pair of data partitions P, and
a function which
maps
Now we
£)y,
where
a time value to another such that
D.=D,
C-{m)
Hierarchical
a time in a higher partition to
to a lower partition:
if
D,— D^
ts
in
DPH
INCi^-{INCji^(m)) otherwise, where £)•-....
2.2.2.
some
— r'^^—Z)^
is
in
DPH
Timestamp Protocol
introduce the hierarchical timestamp protocol, where synchronization timestamps
are calculated according to functions
DEC
and
INC
above.
Hierarchical Timestamp Protocol For Update Transactions:
For every database access request from an update transaction tcD^
the following protocol
Protocol
If
(1)
E
D
is
data granule dtD-,
observed;
(for accessing a partition eqxial to
=D^, then (equivalent
If it is
for a
home
partition)
to the basic multi-version
timestamping)
a read request, then grant the latest version before I{t) of d, and leave I{t) as the
read timestamp of this version of d
if its
current read timestamp
is
smaller than I(t).
_= g =_
If it
(2)
is
a write request, then
the read timestamp of the latest version before I(t) of d
if
smaller than /(/), then create a
new
number
version of d with version
is
Otherwise abort
/(<)
t.
Protocol
If
H
(For accessing Higher partitions)
Dj>D-
then grant
access to the latest version before
t
DEC^
(lit)) of d.
Protocol L (For accessing Lower partitions)
If
D,>D-. then
If
(1)
it
is
a read request, then grant the latest version before INC^ j{I(t)) of d, and leave
INC^j(I(t)) as the read timestamp of this version of d
its
if
current read timestamp
is
smaller than IXC,j(I(t)).
If it is
(2)
d
is
a write request, then
than
smaller
/A'C,
if
(7(0).
INC-JI(t)). Otherwise abort
2.2.3.
Implementing
To implement
the
HTS
HTS
C,(rn),
and
it
9,
create
a
new
version
of
d with version number
t.
protocol,
we must
We
>0, generic
specify
will first
ProcI^, used for
(2)
constant time quantum
then
{!{t)) of
Protocols
C-{m) are computed operationally
computing
the read timestamp of the latest version before INC-
to a
how C(t)
assigned and how^ A('") ^^'^
is
introduce two procedures: (1) ProcC^, used for
computing I,{m).
data partition
remembers that the procedure has been invoked
Z),,
The procedure ProcC-(m) adds
to the
argument value.
m
for time value
a
In addition,
so as to constrain future
assignment of C(t) for teD,. The procedure Procl^(m) computes the initiation timestamp of the
oldest active transaction in
the
and
HTS
D
protocols are
be
D,—Dj—
Proclj[... [Proclaim]...),
Z),
at time
computed
m With PtocC and Prod
as follows
Let the path
in
DPH
—D,- The timestamp DEC^ ^(m)
and that
usi-d
in
Protocol L
is
defined, the timestamps used in
used
between two data partitions
in
computed
These procedures are consistent with thf definitions of the functions
Protocol
as
H
is
computed
£),
as
ProcCi^{...(ProcC-(Tn)...).
DEC
and
INC
given
in
the
;
_=
=—
10
previous subsection.
These procedures are described as
follows:
ProcC^(m): Procedure.
If
m+q-
less
is
Mark m;
/*
than current time, then return ("Abort Requestor");
Create a pseudo transaction
^
Return
<*(£>, s.t. I[t!
)=m and
computed
is
for liD^
there exists an invocation of ProcC\{m')
minimum
of m'+q^ over
In other words,
if
mit timestamp of tiD^
all
such m'
simply Ts
is
noted that
Z),
the time
If
t
is
finish time.
at least
quantum
7,
g, 's
where
I
finishes at time
<m —
m
then C(t)
q^
as follows:
asssigned the
is
Otherwise C(t)=m.
.
ProcC\ has not been invoked
(which also implies that
sactions in
C{l!)=mJrq-.
I{t)<m'
s t.
in £>,
When
them was invoked with an argument value which
is
*/
(m-\-q-).
The commit timestamp C(t)
If
m
remembers that ProcC- has been invoked with argument
is
during the
such invocations
at least
long), then t's
exist,
commit timestamp
t
and
time units before
q-
are selected to be large
time of
life
then the comat least
time of
It
enough such that most of the tranq^
after they start, the chance of having to abort such transactions due to aborted
all
time
<'s finishing
pushed backwards.
is
that require accesses to lower data partitions would do so within
cedures would be relatively small, and virtually
one of
units of time
ProcC- pro-
C{t) would be the same as the actual
finish
t.
The procedure Proel^(m)
is
defined as follows:
Proclaim):
If
there exists unfinished transaction
finished at the time Proclaim)
suspend requestor
Compute
till
t
is
t
at time rn
invoked,) and
it
s.t.
is
its finish
known
time
that C{t)
);
unknown
is
m
(i.e.,
samller than
finishes;
the initiation time of the oldest active transaction at time
Return {m'
is
as m'
t
is
not
m. then
.
— = 11 =—
It is
Ci{m') in
noted that Prod- would suspend requestor only
PTOcI-{m)
q^
s.t.
for appropriately large
9,'s
there exists a previous invocation of
transactions started before m' have not finished by the time
Note that Procl^(m)
invoked.
is
<m —
some m'
for
Z),
if
always invoked at a time later than m.
is
Therefore,
the chance of an invocation of Froc/,(m) needing to block
is
very
small.
To
to
ensure that the procedure /,(m)
compute an access timestamp,
Since /.("i)
is
all
implementable, we argue that when /.(m)
is
information needed to correctly compute /,(m)
always invoked at a time later than
information needed to correctly compute I,(m)
At time m,
we only need
to consider
mf
inserted at time
ing
regular transactions in
all
>m
is
it
only
making
Z),
pseudo transactions.
INC- {x) then there must
unfinished as
is
if
/('')>"'/
— 9,
m
it
,
with
to the behavior of
ProcC^
is
Therefore
>m—
If
9,.
Z),
m. For
all
Ts actual
are also
finish
known by time
2.2.4.
s.t.
t"
is
is
inserted as a result of comput-
I(f')=I(t') and at time
at time
mf
.
mf
it is still
Therefore the effect of
which b known by time m.
on the path between D^ and
If
D^
t'
with I(t')<m must be
such f
is
then due
known by time m.
regular and pseudo transactions that can influence the value of I,{m) are
m
all
ProcC\ the argument that ProcC- receives must be greater than the time when
transactions
smaller than
such f
segment
where D,
invoked. Therefore a pseudo transaction
all
argue that at time m,
Pseudo transactions f with I{f)<m would be
a request to lower data
y(x)
available.
I{t)<m would have been known. Therefore
exist a transaction t" in
computing ISC^.
sufficient to
is
invoked
available:
the pseudo f on I,{m) would be dominated by that of
inserted as a result of
is
is
time.
known
to be unfinished at time
known by time m due
Therefore
all
m
whose commit timestamps would be
to the rule for assigning C{t) to values other than
information on transactions needed to compute I,(m)
m
Remarks
To summarize,
known by
several observations are
made
of this set of protocols:
is
—
— = 12 =
If
(1)
DPH
the database partition hierarchy
col
E
and the hierarchical timestamp algorithm
will apply,
MVTS
consists of a single data partition, then only Proto-
H
E
The key
benefit of the
Protocol
H
or Protocol L, since
is
used
HTS
H
is
partition, Protocol
"cheaper" than either
does not require timestamping the data element accessed.
it
algorithm comes from choosing a database partition such that
much more than Protocol
L.
Since L\'C- j(I{t))'>I(t). Protocol L
sactions.
is
tends to increase the chance for transactions
larger
there
is
is
9,,
less probability for
designers
in
the better off are transactions in D,
them
to abort;
to lower data partitions. Offering tradeoffs
may
parameters as
reflect their
the most "expensive"
perceived priorities
the three, as
it
lower data partitions to abort. In fact, the
when they
and the worse
is
among
access lower data partitions, as
the transactions belonging
off are
intrinsic to hierarchical
among
all
timestamping. System
transactions by their choice of such
q-.
Proof of Correctness
3.1.
Proof Structure
To show
We
own home
More importantly, Protocol
needs to cover only read accesses.
Protocol
its
Protocol L uses timestamps which are different from the timestamps of the accessing tran-
(3)
3.
reduced to the conventional
algorithm.
Since no transaction will write a data partition higher than
(2)
is
first
that the above algorithm
id,
A
schedule
If
is
one must show that
a sequence of steps, each of
action, version of a data granule
sion of a data granule
version.
correct,
formulate the problem of testing serializability, and then apply
Definition.
saction
is
the action
bv the transaction.
is
is
If
denoted as d
,
which
is
in
>. The action can be read
serializability
it
to the
HTS
is
algorithm.
the form of a tuple <tran-
(r)
or write (w).
is
The
ver-
where d indicates the data granule and v indicates the
write, then the version of the data granule included in the step
the action
enforced.
read, then
is
created
the transaction reads the version of the data
_=
granule indicated
in
An example
the tuple.
=—
13
of a schedule
is
<<,,w,(/'>, <t2,T,d'>, <t2,v/,d^ >,
<t^,T,d'>.
<<,
Definition, .\ssume that a version order, denoted as
element d
that
j
is
the predecessor of a version k of d
<< <<
A
in
(1)
<(,,w,(f^
(2)
<t^j.d'
is
>
>
TG(S)
of a data
j
i
of d such
< t^.r.d' >
and
and <<2,w.(f
following
in
is
a directed graph, denoted as
S and the arcs, representing direct dependencies
iff
a predecessor oi d
The
version
exist according to the following rule:
between transactions,
is
A
k and there exists no version
transaction dependency graph of a schedule S
TG(S), where the nodes are the transactions
r„— /;
<<
j
given.
k.
i
Definition.
if
is
if
are in S for
some
d'
,
or
>
are in S for
some
d'
<
k
and there
exists
no d
j
theorem b adapted from the
d
,
where
d'
such that
the predecessor of d
\s
j
1-Serializability
<
o
<
.
{d'
k.)
Theorem proven
in
[Bern-
stein83]:
Theorem Given
a version order
Given the above
correctness of the
(1)
HTS
< <,
a schedule S
that direct dependencies
partitions that are related
(3)
Show
(ie,,
cvcle.
that
if
if
TG(S)
is
acyclic.
prove
algorithm:
Define a relation topologically follows, denoted as
Show
seriahzable
rule for testing for serializability, the following steps are devised to
are assigned to related data partitions in
(2)
is
in
if
between any pair of transactions that
DPH.
may occur
only between transactions that are assigned to
DPH.
a schedule S enforces the relation
tr,—*t^(TG{S) only
=>,
=>
on
all
direct transaction dependencies
';=>',•) then the transaction dependency graph TG(S) has no
—
— = 14 =
Show
(4)
HTS
that the
algorithm produces only schedules that enforce the relation
=>
on
all
direct transaction dependencies.
The proof
HTS
may
protocols
much
structure
of the proof
INC, not
is
important
in
understanding how the implementation detaik of the
be modified without having to reconstruct proofs from scratch.
is
In particular,
accomplished by referencing solely the definitions of the functions
their implementations.
of the proof in this paper.
The only exception
Only a small portion of
is
its
m
the last step
(i.e.,
step
(4)),
DEC
and
a key step
proof directly cites the implementation of
the protocol. This portion will be clearly identified.
3.2.
The
topologically-follows Relation and
We now
t^,
say that
A
relation topologically follows, denoted as
tr,
where
t^
topologically follows
=
t^^cD-,
tntD^.
/?,
In
D
and
(or
Z>,.
(2)
if
D,>D^
then !(t^)>DEC^,(I(tn)).
(3)
if
i?,.<Z)^.
then /(<„)<Z)£'C,^(/(^,))
related in
is
defined for a pair of transac-
We
iff
then /(?,)>/(/o),
if
Z)^.
=>,
are related in the chosen data partition hierarchy.
<j=>io)
(1)
Clearly,
Properties
define topologically follows, which assumes a given data partition hierarchy.
Definition.
tions
its
=>
is
defined only between transactions that belong to data partitions that are
DPH, because
Lemma. Given
a
otherwise the
DEC
function
DPH
t^eO,
and tncD^.
and
let
is
undefined
If
<2~*'p ^h^" ^,
^"'^
^j
^'"^
related in
DPH.
Proof.
Let d(Di^ be the contended data element that causes '2"^']
saction writes d. without loss of generality,
definition of data partition graph
Since
tr,
must
at
least
read
£),—( = )D^—( = )Z),£Z)56'.
d.
ba.'^.ed
assume
it
to be
*,
that writes d.
on transaction analysis, either
therefore
This means
let's
tiiat
either
if
D^—* Df^cDSG
.
or
Since at least one tran-
Then by
D^—*D^iDSG
Dj=Df..
the
or Dj^^D-.
This means that
D^v=^l\. then there exists a directed
path between
.
.
_=
D- and D,
DSG
in
.
By
path between D^ and D-
=—
15
the derivation of the data partition hierarchy, there must be a directed
in
DP//, which means that D^ and
theorem concerning the
In [Hsu86] the following
Z>,
are related in
DPH.
Q.E.D.
=>
partial ordering effect of the relation
was proven:
Theorem
1.
Let TG(S) be a transaction dependency graph of a set of update transactions
DPH,
run on a database. Let S enforce the synchronization rule that, given a
if
t2=>ty. Then TG(S) has no
In fact, a
that
if
cycles.
statement stronger than the above was proven
S enforces the rule that <o—^i«TG(5) only
~ -^ T"^ AL ^
if
(2~^TS
in
What was
[Hsu86].
aL^i ^^^"
TG(S) has no
TS maps each
proven was
cycle,
TS and
defined to be topologtcally follows with respect to two functions
the requirements that
(j^^'i^^^l-S^) only
transaction to a unique time value, and
AL-
.4L,
where
with
,
satisfies the
following properties:
(1)
ComposabiUty: AL. (AL,^(m
>D,>
(2)
))=.4L,j(r7i
)
,
all
all
Z),
m' where
=>
defined here
is
has been assigned the initiation timestamping
3.3.
.
It is
HTS
m
and
for
all Z),
,
Z?^
and D- where D-
>
and D^ where D^
Z),
and
for
all
times
m >m'
In essence, the relation
DEC^
times
D,.
Non-decreasing: AL,j{in)>AL^^{m') for
m
for
easy to verify that / and
DEC\
an instance of
/
=>ts
of transactions,
al ^'here the function
and
.4L,y
satisfy the requirements for
TS
has been assigned
TS and AL
Enforces topologically-follows
Given the above theorems stating that the relation
=>
can be used as a vehicle for order-
ing transactions for concurrency control purposes, the following
theorem completes our proof of
correctness.
Theorem
2.
Let S be a schedule thai
Then t^-^t^iTG{S) only
if
t^=>ty
is
permitted by the hierarchical timestamp protocol.
.
_=
Let '2*^; ^"'^ 'i^A-
Ptooj.
'2"^']
'*
=—
16
translated into the following three cases:
(1)
<o
reads a data element itD^^ written (created) by l^
(2)
fj
writes
-
version of) a data element diD^^ whose predecessor
was read by
version of) a data element diD^^ whose predecessor
was written
hwrites (creates a
^2
(3)
new
creates a
(i.e.,
(created) by
What
new
and the version created by
i^
has to be shown
lemmas which bind
that
is
the functions
three cases
in all
DEC
t^ is
and
AVC
read by another transaction.
i^^X-x
1.
DEC,.^(INC^,(m])<m.
Lemma
2.
DEC-
We
Lemma
m
first
active at time
m
in
D,
active at time
m
be
D,
,
t^,
let
least
as
old
as
/,(C',(m))</(<^)<m
between
D
definition of
DPH
in
DPH,
c>0.
for all
and
D
DEC
t„.
.
.
all i,r7i. If
and I^{C^{m))=I^{m)=m
.
If
the transaction with the largest
then C^{m)=C(t^).
saction active (since at least
at
>D,
prove that /,(C',(m))<m for
C^{m)=m
in
for all Z)^
in
prove these two lemmas.
then
time
,
D^>D,
=>:
1.
We
Proof.
^[m)+i)>m
,(INC,
will first
for all
two
together are used to transform the timing rela-
tionship imposed by the protocol to that defined by the relation
Ltmma
In all cases, the following
holds.
t^
where
is
t„
active),
was
there exists no transaction active at
there exists at least one transaction
commit timestamp among
Therefore, at time C^{m), there
and the oldest active transaction
active
at
time
and
D —*D,—-D^—>'
—/)„—/),.
L\C DEC,JL\r),{m))=
.
=
.
Q.E.D.
1,771.
Let the
must be
Therefore
critical
path
to the
I,{IJ-\IJI„{C,,{C,,{...{C,^{C^[m)))...))))..]).
-^;(A„(-(A2(^'.2(-(^',n(<^;("' )))•)•
have DEC,.(L\'C^^(m ))<I^[C^(m ))<m
at that time
Then bv expanding according
Let C,2(...(C',„(C;(m)))...)=m,2, then we have lJCJm,„))<m,.^.
^;(AJ-(A2('".2))
all
that are
at least one tran-
m, which means I(t^)<m.
Therefore we conclude that I,(C^(m))<.m for
be
is
all
Therefore
DEC,^(LWC-,(m))<
Continuing with the same reasoning, we
.
— = 17 =—
Lemma
2.
We
Proof.
(a) If
first
no transaction
prove that /,(C,(»n ))+«)>»" for
oldest active transaction,
(b) If
some transaction
the proof of
at time
m
Lemma
is still
concludes
if
is
C;{m)=m. Therefore
m
active at time
in D-,
then
let
That
Z),-D.,-D..-Z),3^
all
«,m.
IJC„(C,.(m,,))+e)>C,^{m,,).
That
is.
is,
"1,2=^,2(^1,3).
By
Now we
DEC,.(L\C^,(m]+i)=
above
the
I„(C,^(C,^{m,,))+i)>C,2{m,^)+i'
to both sides, /,o(/,i(Qi('-'.2('".3))+0)^ A2(*''.2('".3)+^') >'".3ing one derives
between
path
the
Z),
.
Theorem
2.
and
and
in
one
(b)
be
D^
and
let
we have
conclusion,
Therefore, applying
/.^
Continuing with the same reason-
I,(IJ.-(I„{C„(...(C,,{C.{m))...)+i)..))
are ready to complete the proof for
(a)
C,,{...{C,,(C.(m)))...)=m„
Let
Thai
)=rM,2.
Let
was
as
at time C(t^)+i, no transaction active
,
for
C-{m)+i the
way
be defined the same
t^
/,(C(/^)+f)>m or /,(C,.(m)+<)>m. Combining
is,
An-^r
)))...
at time
Therefore /,(C,(m)+£)>m
any, cannot have started on or before m.
I-(C-(m))>m
C,-2(Ct3..,(C,„(C(m
then
in Z),
we have C^(m)=C{t^). Therefore
1,
active;
that
m
active at time
Ls
There are two cases.
i,m.
all
>m.
For the three cases
Q.E.D..
identified pre-
viously:
(1)
In this case. D^>D).. Since
D
and
Z),
must be related, we have three cases and
each
for
case observing the hierarchical timestamp protocol leads to the conclusion that <2=>'i-
D->Di>D^.
In
obeying
this case,
HTS
leads to the assertion: ISC^j,(I{t^))>INCi,^[I{t^))
Therefore ISCji^(I{tn)]>INC^j^{I{l^))+i.
(0.1).
Applying function DEC^.^ to both sides we
obtain DEC^.[ISC^j,{I(t^)))>DEC\,(INC,^(I[t,))+i) (0.2).
right
hand
side of (0.2)
is
greater than I{t{).
From Lemma
Applying function DEC, j
2
above the
to both sides
have DEC,/DEC,,{L\'C^,{I{t^m>DEC,^(DEC,,(INC,,(I(t,))+i) >DEC,JI{ti)).
ever, the left-hand side of the
is
less
than or equal to
D >D >D,.
In
above expression
I(t,J.
this ca^e (O.l)
(a)
=
DEC/^ -{INC^
i,{I(t^))).
How-
Lemma
which by
Therefore I{t2)>DEC--(I(t^)), which means t.= >t^.
holds also.
Apply DECj,,
DEC),{Aj(L\C)i^{I{tr,))))>DEC\J!XC,JI{t\))+().
to both side of (0.1)
Right hand side
is
>/(/,).
we
1
(b)
we have
Left hand
)
_=
side
(2)
<DECj,{I{t^)).
is
=—
18
Di>D^>D^.
In this case INC\^{I(t^))<DEC-i,(I{t,)).
Apply DEC,,^
to both sides
D
In this case,
yOi^.
By
same reasoning
the
in
D->D^>Di^ and D->Dj>Di.. Now we consider
at which
performs writing
f„
P£'C-^(/(<,))<4(m)<m
Combine
ProcCj{I(tr,))).
to both sides
DEC;,
Otherwise
^°^
with
this
we
write
the
m
get I{tr.)>DEC^^(I(t^)),
have
/(<,)<m and
been
rejected
D-=Di,.
(a.2)
L
Protocol
the
(in
Apply
i,(I{tr,))>DEC-i,(I(t^]).
to=>ty
i.e.,
i.e.,
,
by
case,
would
INC.
have
request to read d
(a) If (j's
this
request
we
(0.3)
In
for the cases of
Dj'>Df.>D-. Let the time
had started at or before
/[
Dj>D^.
(a.l)
(0.3).
the cases where
Consider two cases,
.
m. which means
arrives at or before time
'^'-^ *(^('2))^'"
m
be
rf
(c)
'2=>'r
i.e.,
we can show t^=>t^
(1)
t^=>t^.
INC-^{I{t^))+i<DEC.^(I[t^)).
i.e.,
we have l{t^)<DECj^{I[t^)).
means
which
DEC^,l{t„))>I(t^),
Therefore
In this case.
J-
INCfore
Since
i^{I(tr,))=I(tf,).
/|fc(
w )</('o)),
makes a request
when
except
^o satisfies
m
pushed backwards to be before
to be
DEC-
procedure for computing
after
is
tr,
m
.
.
at time
m,
t^ is
active at time
conditions under which
|^(I{t^)) for
m'
<m
^^'s
would have to block
<o
already existed, and by Protocol
<2 it
(3)
In
this case,
are already
is
it
suffices to
created by
shown
Proof of Theorem
in (1)
2
is
in
HTS
prove that
t^.
if
^j
not to have
Since
in
tr,=
^2=>^
>t^ when
this case
in
to
mdependent
(2)(a).
is
ProcIi^(m'
t^ finishes,
m
.
By same reasoning
which
There-
in (a.l)
m, then the version created by
chooses to read a version before that created by
and therefore ^n=>^,.
largely
except for arguments
protocol
H
d arrives after
must be INC^j^{I{tr.))>DEC,i^(I{t^)). Therefore
predecessor
Prod
(b) If <,'s request to read
till
request to read d arrives before
fore I^{m)<I(t,). Therefore L\Z)i^(I[tr,))>I;^(m)>DEC,^{I{t^)).
(and there-
commit timestamp
However when such conditions hold the
contradictor}' to the given that
we have <o=>^,.
its
m
tn
(b)
creates a
Z),>£>^and
new
D^>D^.
the
version of d whose
two relevant cases
Q.E.D.
of the implementation procedures for
The dependency
leave read limef^laiu^r'^
arises
If
HTS
ProcC and
due to our desire to allow the
H
modified such that Protocol
H
is
_=
19
=_
also required read timestamps, then the proof can be constructed entirely
functions
DEC
and
INC and
from the definitions of
their properties without resorting to implementations.
Discussion of Performance
4.
Performance of a concurrency control algorithm can be assessed along two dimensions. One
is
the computational overhead, such as setting locks and timestamping data elements.
is
the level of concurrency, or "optimality" of the algorithm.
The other
Along both dimensions, the
algorithm potentially performs better than the NTV'TS algorithm,
if
HTS
a data partition hierarchy can
be found such that the frequency of accesses from transactions assigned to lower partitions to
data elements contained
other words,
is
used
if
higher data panition? dominates that of the reverse direction.
in
the data partition hierarchy
much more
is
chosen such that Protocol
often than Protocol L (the "expensive" protocol,)
H
In
(the "cheap" protocol)
then a net saving
may
be
achieved to warrant the use of this algorithm.
The HTS algorithm
requires a transaction analysis and the overhead of maintaining selective
information on transaction initiation and commit limes to enable computation of the
B
and enforcement of the
However, the fact that Protocol
function.
tamping the data elements makes
it
is
HTS
function
does not require times-
very attractive for situations where transactions assigned to a
lower partition need to access a certain quantity of data
the
H
A
algorithm can potentially produce a net savmg
in
in
a higher data partitions.
computational overhead
if
Therefore
Protocol
H
used "often enough."
Now we
analyze the optimality aspect of the algorithm.
currency control algorithm
is
In [Kung79j, optimality of a con-
defined as the degree to which the algorithm allows for "serializable
input schedules" to proceed without being mterrupted
Intuitively,
an input schedule, which
interleaved sequence of read and write requests issued by a set of transactions,
serializable (NrV'SR)
if
there exists a
way
is
is
an
mulli-version-
of assigning versions to these read and write requests
such that the resulting transaction dip('n(i''n(> graph
i-
acyclic
For reasons of implementability,
.
_=
=—
20
no known multi-version concurrency control algorithm
However, the degree
proceed without interruption.
H
is
HTS
comparing the
used
much more
measure of
HTS
extreme case when Protocol L
Lemma
acyclic, then
S.
If
in
is
this
number
if
of restarts
Protocol
(i.e.,
MVTS
tran-
algorithm.
statement, the following theorem confirms that,
not used at
the data partition graph
among
all,
the
HTS
algorithm dominates the NfV'TS
DPH
DPG{P ,T^)
of
some database decomposition P
DPH{P ,T^).
the set of correct data partition hierarchies
such that Protocol L
exists
can be argued that
it
schedules
terms of optimality.
in
assigning to
algorithm,
MVSR
concurrency.
algorithm would be smaller than that under the
While further studies are needed to quantify
algorithm
N-fV^SR input schedules to
level of
frequently than Protocol L, then the expected
saction aborts) under the
in the
MVTS
algorithm to the
all
which an algorithm allows
to
to proceed without interruption can be used as a useful
In
will allow
is
not needed to process update transactions
the partial order of data partitions exhibited in
DPG{P ,T^)
in
T,
.
there exists
is
DPH
(This can be shown by
DPG{P ,T^),
and since no path
from a higher data partition to a lower data partition, there exists no
accesses from transactions assigned to higher data partitions to the lower data partitions, and
L
therefore Protocol
Theorem
is
Let
3.
partition hierarchy
never needed.)
F
DPG{P ,T,]
be a database decomposition and
DPFI be such
that Protocol
L
is
never used
in
Let the data
be acyclic.
processing transactions
in
T,
Let 'S(.\f\'TS) denote the set of serializable input schedules that are allowed for execution by
NPk'TS without interruption, and
hierarchy
DPH
S(HTS)
where both Protocol
H
that by the
HTS
algorithm under the data partition
and Protocol E are used.
Then S{\{\'TS)
is
a subset of
S(HTS).
Proof
and
(1)
We
must show that
(2) there exists at least
(1)
one schedule allowed by
Let a schedule ScS{M\'TS).
Then
any schedule allowed by NfXTS must be alloHt-d by HTS,
Want
there must exist a step {l^,w,d)
to
in
show
6'
HTS
that
that
that
is
not allowed by
S(S(HTS).
HTS
will abort.
Suppose
We
will
\r\TS.
this
is
not true.
show that
this
is
_=
Lei t^(D^.
impossible.
Since Protocol L
HTS
/(<,).
Therefore
HTS
that will cause the
is
case,
no read timestamp
(2)
equal to that
is
it
suffices to
{t^,r,d)
all
is
left
show that there
exists
no (tj.r,d)<(t-,w,d)
where t^eD^, either
0^=0^
or
I(ti)
D^>D^.
by HTS. In the former case, the read timestamp
D,>D^ m DPH. and
of S{.\fVTS).
The above
SiS(HTS) and S
be achieved, but a design
H
if
left
DPG(P ,T^)
is
is
HTS
is
afD,,
<,eZ?,,
member
not a
in
is
acyclic, then the
terms of
level of
HTS
P
can be found such
algorithm
will definitely per-
concurrency.
When
such design cannot
found that enables construction of a data partition hierarchy where
much
HTS
higher than that of Protocol L, the
expected to perform better than the
NATS
algorithm can
still
rithm to a simple Iwo-data-partition database.
In the analysis,
it
Ls
HTS
The
city of the case enables the use of simple analysis to obtain formuli for the rate of blocking
in the
algo-
assumed that the higher data
heavily contended for by transactions than the lower data partition.
two-phase locking (2PL) due to conflict
be
algorithm.
[Hsu83b,, a simple analytical model was established for an application of the
b more
by
the set of update transactions
such a way that a non-trivial database decomposition
in
form better than the M\'TS algorithm
usage of Protocol
d.
Q.E.D.
that the data partition graph
partition
Clearly
with
5
/(<,).
result implies that, given a database application,
r, can be designed
In
I{t^]<I[t^).
in
In the latter
Consider the following schedule: S={t ^,r ,a),{tn.r .a),{t„,w ,a),{t„w ,,h), where
t^iDj. biDj.
before
is
never used, the timestamp used to synchronize (t^,w.d) by
by M\''TS, which must be smaller than
left
.r,d)
{t
algorithm to leave a read timestamp greater than
true because for
This
is
=—
Let (t^,r ,d)<{t,,w.d) denote that fact that step
(tf,w,d) in S.
must be
21
simpli-
under
higher data partition,
algorithm and the rate of
makes assumptions that
are in general in favor of
A
abort under the
the
algorithm
2PL approach and
the effect of the
the
HTS
much more
HTS
The
analysis
discriminate^ against the
algorithm
in
HTS
approach.
The purpose
is
to demonstrate
relieving contention in the higher data partition,
heavily contended data partition
in
our case.
The
result,
reported
presumably
in
[Hsu83b],
_=
shows
that, in general, the rate of blocking
where A/Fj
is
the
number
22
=—
under 2PL
proportional to
is
+ar,*MP^*\fPr,.
a ^*,\fP^
system at any time and type-1 transac-
of type-1 transactions in the
tions are those assigned to the higher data partition, A/P2 that of type-2 transactions
transactions are those assigned to the lower data partition, and
in
D, accessed by a typical transaction assigned
abort under
HTS
is
proportional only to a, A/P,'.
parameters, one can conclude that
5.
to D,
HTS
,
at
where
is
the
i'=l,2.
number
and type-2
of data elements
In contrast, the rate of
While the absolute values depend on other
would be preferred when Oj'A/Pj
is
large.
Conclusion
The Hierarchical Timestamp Algorithm can be considered
as a generalized
timestamp
rithm parameterized by a data partition hierarchy constructed via transaction analysis.
It
algo-
relies
on an understanding of the structural disciplines of an application system and represents an
attempt to take advantage of these
disciplines.
By being
tures
and being
tural
approach to concurrency control holds promise
flexible in
manipulating the basic control tools
databases where the level of concurrency
The
It
sensitive to the existence of such struc-
is
vital to
for
(e.g.,
timestamps, locks) this struc-
improving applications or activities
in
system performance.
thrust of this paper thus goes beyond proposing a
new concurrency
control algorithm.
demonstrates the potential benefit of exploiting the knowledge and the structure of the applica-
tion
systems to implement more
efficient
also points out an area of research
and more tailored concurrency control mechanism.
which examines the problem of how
to
It
design transactions so
that concurrency control can be more efficient without compromising the key requirements on the
data.
It is
believed that transaction design performed with concurrency control problems in mind
could produce a structure of applications that
be implemented.
is
significantly less costly for concurrency control to
_=
6.
23
=—
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\2kl
051
^
I-ault
Tolerance Computer Systems,
June
Date Due
Lib-26-67
MIT LIBRARIES
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