such views in- volve aggregate functions, which need duplicates to produce correct results. Unlike most work on the view maintenance problem that is based on ...
Incremental
Maintenance
Timothy
600 Mountain email:
Bell
Avenue, {Igrifin,
Murray
libkinl}
the
particularly volve
results.
problem
that
proach
may
of efficient contain
important
aggregate
correct
approach
when
functions, Unlike
use
present
a natural to bags
NJ 07974,
our
reasoning.
extension
to
We prove
that
materialized about
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equivalence
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computed.
changes
view
from
that
and more
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efficient,
algorithm
show
than that
we prove
with aggregate functions, on views that change.
allowing
their
that
correct
often
used
to
data
that
and
subsequently
base
relations
recomputed
can
compute be
systems views.
materialized queried
are changed, to ensure
base Views
(stored against. materialized
correctness
in If
one
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applies
views
of answers
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result
had
the
to be
a reading to
of
the
practical
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queries,
triggers
and
a rather
Bag
to queries
primitives
suggested
by
[11,
this
paper
adds
evaluation
the
nested deep
328
correct
relational
algebra,
relations results
bag basic
and
are present
on the
is
complexity
of this [20].
trying
basic algebra bag
the
24, 16]. algebras
turned bag
out
from
[11,
In 19].
essentially functions
to
continues
to hold
There
are a number
and
to
algebra.
algebra
aggregate
research
[17,
for
to
research
active
algebras
as the
basic the
also
set-theoretic In
database
basis
These
that
the
for
bags.
languages
the
accepted
we use the
to
of bag
It was also shown
since
employees
elimination
one particularly
formed 19].
and
two
duplicates
theoretical
design
[2]
be equivalent
Permission to copy without fee all or part of this material is granted provided that the copies are not made or distributed for direct commercial advantage, the ACM copyright notice and the title of the publication and its date appear, and notice is given that copying is by permission of the Association of Computing Machinery.To copy otherwise, or to republish, requires a fee and/or specific permission. SIGMOD ‘ 95, San Jose , CA USA @ 1995 ACM 0-89791-731 -6/95/0005.. $3.50
the
the projection
at least
obtained over
languages,
has been
then
(Employees).
operation.
results
between
if the
as duplicate
carry
do for
instance,
IIs.lary
from
They
computed,
eliminating
expensive
not
be
to
are
systems
important
For
when
Not
theoretical
and practical
model.
AVG
evaluation,
gap
underlying
be removed
salary.
do
database
is to
assumed
duplicates
practical
functions.
be wrong
query
has
is,
is particularly
aggregate
same
that
employees
cannot
the
topic
be
as the
of
would
up
bridge
the
must
as
maintenance
most
which
the
semantics
derived
on view set-valued,
salary
Duplicates
nicely
a database) some
views
has been
solution
constraint
are
of aggregate
average
evaluation
are
the
12, 5, 18, 26],
number
active
work
(multisets)
evaluation
of the
relations
to and
come
and
since, any
integrity
duplicates,
generally
management
to has
problem,
a large
of
be made relations
changes
relations
indicate,
However,
use bags
Introduction database
base
in
view
one often
view.
misleading
will
of the
Many In
base
[27, 4, 3, 6, 7, 14, 29,
slightly
relations
speeds 1
such
including
handle
faster
interacts
the
applicable
eliminated.
expect,
recomputing
approach
are
results
to be significantly
complete our
that
recomputing
one should
defines
implementation
Most
notion
a heuristic
must
the
finding
whole
monitors.
expressions.
than
that
to
maintenance
literature is
Instead
changes
to the
vtew
is
problems,
base
tuples
this
the
is based
a certain
only
rather
circumstances
space efficient also
view
is more
normal
from
algorithm
of
extensively name
the
expensive.
changes
that
on changes
problem
We
the
problem as the
recomputing
be very the
expression
studied
algebra
no unnecessary
is generally
propagation
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changes
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given
The
cor-
language.
of bag-valued
that
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scratch
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the change
it to
relational
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ensures
Although
computing
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propagates views.
it is correct
of minimality
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determine
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and it simplifies
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to view,
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The
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USA
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an algorithm
on reasoning
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such views
on the view
of advantages:
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proofs.
relations
saying
against
problem
need duplicates
work
to new language
operations
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queries
and based on equational
can be used by query
We
Hill
from
of materi-
This
on an algorithmic
has a number
rectness
maintenance
duplicates.
which
most
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extendible that
problem
that
Libkin
@research. att .com
against
views
I)uplicates
Laboratories
Abstract We study
with
Leonid
Griffin AT&T
alized
Views
of
expressive
when
power
of of
bag
languages
The
[11, 2, 19, 20, 21, 22, 31].
main
goal
for incremental algebra.
of this
maintenance
We advocate That
we
an
equation
this
primitive
applying made
view
literature.
Most
base
ad hoc the
when
is
The
been The
that
when
a similar
algorithmic
time
in
proved
as it
the harder
[12],
algebra
low-level
for
to
prove
a
of
not
clear
changes
in
difference
between
in section
7. An
how the
operation.
of
the
The
analysis
of [29],
a recursive
algorithm correctness
approach
That
This that
react
is similar
in
to be proved
that
maintenance
the
equational
out
Here
in
of
of an
problem
algorithmic
has a number
approach.
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to
to the
of advantages
precise
semantics
quently, ier
to
using prove
algorithm. recursive variant
new
the equational correctness
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of our
to a later
approach primitives
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change see in allows
minimality Such phase
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in
copies
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ours.
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proof
rin
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menus
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order
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t is a
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we seek to construct
names T
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S),
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value
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maintenance
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executed).
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us to recompute
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solutions? should
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us to check
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state
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example,
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it allows
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resulting that
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solutions
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maintenance
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value
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evaluate 2.2
the
evaluated
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s, if s is defined
in S and
to
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pre(t,
of the
Z
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denotes
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so we consider
of applying all
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multisets Ri.
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s we have
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evaluate
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evaluating
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and
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to the
[19]. S,
the
relation
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of S. In particular,
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of interdefinability
consult symbols
base
becomes
state
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T))
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as S min
characterization
represent
into
can
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example,
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to check
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and
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language
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names
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we
is true
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E
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count(z,
+
inserted
in state
determine
=
Count(m, s ~ T)
A VRI)
as a function
s)
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=
z
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for
at
language,
as follows:
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from
the
but
let
of
count(x,
couni(~,
operations,
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operation
e(S, T))
ap(s))
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of
number
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when
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elimination
cartesian
for
count(~,
of count(z,
state,
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to be considered
+
Rn
menus minimum
the
{R,
selection
duplicate
the
S)
S.
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used,
projection
Sxs
define
count(~,
For
to any
(see
to adopt
to make
language
applicable
transactions
we prefer
in order
of a particular
readily
abstract
is easy
bag
time
transactions,
paper
of transactions,
independent
same
In this
form
s
To
for specifying
[1, 28, 30]).
results
grammar.
to languages
for example
on tuples
to we VS are
1. T7S L S =b $:
then
will
contrast
a solution
the
not
that
delete
(1)
that
may
will
meeting
strongly
relational
insist
transaction
do
condition
be called
to to
tuples
Figure
are in S.
a tuple
and
be
increase
both
[29],
the
Note
it
disjoint
weakly
conditions
mtnimai.
case
S
be called
does
from
not
AS
multiplicities
make a
of elements
in S. We
will
argue
especially
that
desirable
interact
with
aggregate
TOTAL((S =
not
acceptable.
way
functions.
or
in
strong)
which
the
a v
intended
application
a deletion
upward
in
equations
concerning
minimality
of the
Theorem
1
figure
repeated
can
propagate
upward.
Consider
the
an-
(for
an
how
the
read
to
Note
as
propa-
that
the
no assumptions
change
bags. 2 are
applications any
expression
are
a A
: when us
of Fzgure
By 2 we
tell
involves
equations
propaga-
emphasizes
an expression.
of these
in
or
equations
they
correctness
The
bag) simply
of these rules,
change
subexpressions
annotation
rewrite
changes
for
Some
(for
This
Example.
we have
equations
correct. of
number U =
the of
rules
changes
S M T.
Suppose
that + TOTAL(AS) minimal
minimal
example,
with bag).
is
changes
For example,
or strongly)
all strongly For
(weak
the
- TOTAL(VS))
a (weakly
Again,
to
A T7S) w AS)
(TOTAL(S)
assuming
minimality due
our
expressions.
insertion
gate
in
since
bag
left-to-right
(1)
that,
2 contains
in
notated
it.
while
(2)
sense
4 : We
meeting
manamal, and
=b
reinsert
A solution
delete
tion
~sminvs
2.
We only
pre(t,
solution.
solutions
The
are equally
U)
=*
changes
expressed
((S=
vS)
M AS)
69 ((T
A VT)
to S and T can be propagated
as changes
to
W AT). upward
and
U as follows:
pair
w AS)
((S -LvS)
w (( TLvT)
u AT)
P6
and AQ is a strongly
minimal
win
it for
by using
The
main
algorithm
and
solutions
efficient
than
3
However,
maintaining of
this
generating to
demonstrate
Q) =Q
solution.
goal
for
minimal
= pre(t,
view
paper
is
(at the
that
the
recomputing
to
present
time)
(at
((( SLVS)
~ AS)
;;
(((s ‘V$
~ (T ‘vT))
~;
((((s ‘VL’$)
‘b
((((s UT) L (vsmin
=E,
(U LVIU)WAIU
M AS) w AT
~(VTmin
T)) u AS) u AT
S)) L (vTmin
u AS) w AT
T))
where
problem
AS
vlU
general
more
=
u AT.
The
(VSmin
S) w (VTmin
last
is simply
step
T)
and
AIU
an application
=
of the
rules
run-time),
Preliminaries which This
ing
our
section
presents
change
propagation
agation
algorithm
sented are
in
[29],
used
to
on
equational
theory
algorithm.
the
based
For
the
for
‘(bubble
expression.
A change
relational
algebra
a collection
up”
change
example,
underlyprop-
was
sets
[29] uses the
to
the
top
applied bag
Repeated
pre-
of equations
are
one delete
that of an
a solution,
one.
However,
solution
equation
to take
AS)-
the insertion
to the
insertion
Our
first
is more
step
is
to
rules
for
bag
the
for
familiar
accustomed
expressions.
into
Let
S and propagate
– T into
complicated
obey are
AS
AS
propagation not
T’=(S-T)u(/LS-T)
a
collection
expressions.
&l laws
to using
The
expressions of with
For bag expression,
since
boolean
do
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relational example
the
(Qmin
is not
J- T=(SLT)W(AS
immediately
following
of figure tells
into
a strongly
W
=b
A AIQ
deletions
2 guaranminimal
us that
any
minimal
(Q ~ VIQ) and
into bag.
a strongly
theorem
that VIQ)
all
one insert
rules
necessarily
a)
W =b (Q LvzQ) v~Q~Q=b4
c)
T72Q min
AZQ
one.
w AIQ. =
AIQ
V2U
to
=b
4.
from
a strongly
above,
V1 U and
minimal
Al
solution
to be
now
and
L(TLS)),
obvious.
A2U (AS
332
L
UAZQ
ApQ
to the example
(U min((~SminS)
(S WAS)
of the
b)
be transformed
taking
into
Then
Returning
that
becomes
which
=
such
they
collect
not
2 Suppose
v2Q
situation
algebra
set-valued the above
of
to
insertions
can be transformed
S – T.
define
order
all
but
(Q minVIQ).
it upward
in
and
application
tees
Theorem (Su
we
~T)
w AT
an
strongly
computationally
view
not
by Q.
maintenance
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does
given
compile
view
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;;
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u AT)
to be u AT)
L (U min((VSmin
S) w (VTminT)))
U by
P2.
ap(S -L ~S) =b LTP(S)J-(TP(VS) CTP(SM AS) =b &p(S) U ap(M)
P3,
~A(I$
L VI$)
=b
HA(S)
-
P4.
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~ /!h$)
=b
HA(S)
u ~A(~s)
PI.
P5,
(s~vs)~~=b(s~~)-(vsmins)
P6,
(SkJAS)MZ’ (s~vs)=7’ SL(’T-LVT)
P7. P8.
(S WAS)
P9. Plo.
Although
these
be greatly
expressions
simplified
=b(Su T)w AS =,(s-T)=vs =* (S= T) W((VTminT)=(T~S))
LZ’=b(S=T)
SL(TUAT)
Ph.
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(S u AS)minZ’
P13.
(S A VS)
P14.
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=,
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u (ASmin(T
maxT
AS)maxT
A (VSmin(S
=*
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w (AS
C(S ~ VL5’) =b ~(s) c(S w AS)
P17.
(S L~S)x
P18.
(S UAS)XT=6(S
~ (~(~smins)
=b E(S) w (c(AS) T=b(Sx
they
~ (S ~ VS))
T)
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equations
base
can
for
bag
expressions
relations,
tions.
to
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~ S)
T)&(VSx XT)
propagation
complex,
~S))
(SmaxT)
P15.
2: Change
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=*
P16.
are rather
w(AS~(T=S))
=~(S4Z’)LAT
P12.
Figure
~A(vsmins)
rather
this
transformations V3U
~f
A3U
(vS
~f
L AT)
(AS
W (vT
~ vT)
A AS)
V (AT
(rule
P8 in figure
and
k be numbers
the
(vT,
assumption
AT)
This
example
was used the
Second,
by
application
simplified
subexpressions Note
there
in
solution
is derived
by
a general
2.
Third, that
oniy
is considerable
propagation
rules
replace
S~(T~VT)
the
are for
with
freedom
change
propagation
correctness,
in the 2.
design
For
example,
views.
tional
we have
point
of view.
equation
in figure
operation
e and
n =
2,
designed
if one
V’
on changed
or
V u A.
Note
2 follows its
value
of its
form.
Intuitively,
vRis
ARis,
that
ing over them
that
the
the V =
pattern.
For
any
. . . . I&),
n =
1 or
abbreviations
changes,
they
then
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as either and are
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corresponding
value
We
V ~ v
are
“controlled”
be computed
S-v(t,
always by
elements
from
333
this the
maintaining
more
precise
in
Algorithm
algorithm
for
to a given
view
given
two
t and
expressions
Q)) 3.
S) for
a
~Q
a &t
and
AQ
U AQ.
mutually
such
(Q L V(t, in figure
computing maintenance
a transaction
compute
Q)
add(t,
derived
three
by
repeated
minimal
be made
Q) =5 (Q L vQ)
A(t,
for
Rls, make
recursive
that
for
any
functions
transaction
t
w A(t, Q). These functions For readability, we use the S M A(t,
S) and
of these
recursive
del(t,
S) for
s).
in
figure
by iterat-
is,
define
and
our
solution
pre(t,
first Q)
will
will
6.
presents
That
A
suitable
in
are generally
relations
and
Propagation
Q, we will
that
v
S
elements
ARis
of base
let n, m T and
~ (m ~ k) times fetch
and
for
intuition
m)
we only
size
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