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USC Computer Science Technical Reports, no. 629 (1996)
(USC DC Other)
USC Computer Science Technical Reports, no. 629 (1996)
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Content
On the Complexit y of Resource Sc heduling for Co ordinated Displa y
of Structured Presen tations
Martha L EscobarMolano and Shahram Ghandeharizadeh
Computer Science Departmen t
Univ ersit y of Southern California
Abstract
With the structured approac h to represen ting video clips a presen tation consists of a collection of
bac kground ob jects and actors D represen tations constrained using spatial and temp oral constructs
along with rendering features eg shading audiences view p oin t While the spatial constrain ts dene
the p osition of displa y ed ob jects on the screen the temp oral constrain ts describ e when the ob jects are
rendered As compared with an alternativ e approac h termed streambased that conceptualizes a video
clip as a sequence of frames the structured approac h pro vides for b oth reusabilit y of ob jects in other
presen tations and eectiv e query pro cessing tec hniques for retriev al of relev an t data The displa y of
a structured presen tation is termed c o or dinate d when the displayof ob jects resp ects the presp ecied
temp oral and spatial constrain ts Otherwise the displa ymigh t suer from failures that translate in to
meaningless scenarios F or example a c hase scene b et w een a dinosaur and a jeep b ecomes meaningless
if the system fails to render the dinosaur when displa ying the scene
Assuming a m ultidisk hardw are platform congured with a xed amoun t of memory this study
sho ws the follo wing the computation of a resource sc hedule that supp orts a co ordinated displa y and
yields the minim um latency is an NPHard problem and giv en a system load the computation of a
sc hedule to c hange the placemen t of data across disk driv es in minim um time is an NPHard problem
This researchw as supp orted in part b y NSF gran ts IRI IRI NYI a w ard and CD A and a
unrestricted cashequipmen t gift from HewlettP ac k ard
In tro duction
One ma y representvideo using t w o alternativeapproac hes streambased and structured Gha With the
streambased approac h a video clip consists of a sequence of frames that are displa y ed at a presp ecied
rate eg frames per second to fo ol the h uman p erception to observ e motion With the structured
approac h EMG EM a video clip is represen ted as a collection of ob jects eg D represen tations
of a dinosaur with spatial and temp oral constrain ts eg p ositions of the dinosaur and the time of their
app earances along with their rendering features eg ligh tin tensit yviewpoin t Presen tly the structured
approac h is used to pro duce animated sequences F or example T o y Story Bat and Reb o ot Ber are animations generated using the structured approac h
Structured presen tations pro vide for b oth reusabilit y of information and dev elopmen t of eectiv e query
pro cessing tec hniques They enable a user to extract a c haracter eg a dinosaur a motion path eg
the tra jectory and timing of the dinosaurs motion from one presen tation and reuse it in another In
addition one can devise algorithms to supp ort pro cessing queries that reason ab out the temp oral and spatial
information of a structured presen tation T o illustrate consider the animation The Lion King Assuming it
w as represen ted using the structured approac h to retriev e the scene where Sim ba nds his father Mufasa
dead a user can p ose the follo wing query select scenes that con tain b oth Sim ba and Mufasa suc h that
Mufasa is static while Sim ba is mo ving The system lo cates the relev antdata b y analyzing the temp oral and
spatial constrain ts imp osed on the D represen tations of Mufasa and Sim ba
With structured presen tations the database con tains D represen tations of dieren t ob jects spatial
and temp oral constrain ts and rendering features Ac hallenging task for the system is to ensure a co ordinated
displa y where the system renders ob jects in a presen tation memory residen t at the p erio ds dened b y the
temp oral constrain ts If the temp oral constrain ts asso ciated with the presen tation are not satised then the
presen tation migh t suer from errors T o illustrate consider the sequence of p ostures p
p
n
that pro vide
the illusion that a dinosaur is w alking These p ostures migh t b e a collection of p ersisten t delta up dates on
the dinosaurs D represen tation c hanging its facial expressions morphing and mo ving its b o dy along a
curv e as a function of time TTa TTb Lev If the system fails to render the original D ob ject the
deltas will yield a partially missing c haracter with a c hanging face and mo ving b o dy parts
A resource sc hedule that supp orts a co ordinated displa y consists of ob ject retriev als ob ject migrations
and ob ject replications Once the displa y starts the sc hedule m ust render ob jects memory residen t at the
p erio ds dened b y the temp oral constrain ts T o satisfy these temp oral constrain ts the system can either
retriev e the ob jects immediately b efore their displa y prefetc h them at an earlier time and ha v e them memory
residentun til they are displa y ed or manipulate the placemen t of data to retriev e them from disks dieren t
from their original lo cation There are t w o alternativ es to manipulate the placementof data migration and
replication With migration the system remo v es the original cop y of the ob ject while with replication the
system k eeps b oth copies
The fo cus of this study is the computation of a resource sc hedule that satises the temp oral con
strain ts asso ciated to a presen tation and yields the minim um latency This computation motiv ated here b y
structured presen tations is fundamen tal for a database managemen t system that supp orts m ultimedia appli
cations suc h as structured presen tations co ordinated use of m ultiple streams in m ultimedia do cumen ts and
in teractiv e TV games This study sho ws that the complexit y of this computation is an NPHard problem
T erm Denition
m Num ber of time in terv als in the presen tation
t Duration of a time in terv al
Time In terv al i P erio d i i Instan t i The b eginning of time in terv al i
S i P ages in memory at instan t i
P i P ages con taining ob jects displa y ed during in terv al i
P
d
i
P ages in P i that reside in d
F i P ages retriev ed from disk on to memory during time in terv al i
F
d
i
P ages retriev ed from disk d on to memory during time in terv al i
U
d
i
P ages written to disk d during time in terv al i
K i P ages discarded from memory during time in terv al i
R
a
j
Read page a from disk j
W
a
j
W rite page a to disk j
D Disk driv es in the system
B Maxim um n um b er of pages read b y a driv e during a time in terv al
B
d
i
Disk bandwidth a v ailable at driv e d during in terv al i
C Num b er of memory frames in the system
n Num b er of clauses
k Num ber of v ariables
R Set of replications to b e sc heduled
N P erio d when R m ust b e sc heduled
A System load during N p Start up latency in time in terv als
T able List of terms used in this pap er and their denitions
In Section w e presen t the formal statemen t of the problem Section describ es related studies
Section sho ws that giv en a system load the computation of a sc hedule to c hange the placementof data
across disk driv es in minim um time is an NPHard problem Section pro v es that the computation of a
resource sc hedule that supp orts a co ordinated displaywith the minim um latency is an NPHard problem
In Section w e presen t conclusions and future researc h directions
Statemen t of the Problem
Our target platform for displa ying structured presen tations consists of D homogeneous disks and a xed
amoun t of memory W e assume that a disk page is the unit of transfer b et w een a disk driv e and memory The D disk driv es ma y retriev e D pages in to D dieren t memory frames sim ultaneously An ob ject migh t
b e either smaller or larger than a disk page When an ob ject is larger than a disk page it is represen ted as a
collection of pages W e discretize time in to xsized units termed time intervals The duration of eac h time
in terv al is denoted as t The b eginning of a time in terv al i is termed time instant i Figure When a user
requests a presen tation the system has adv anced kno wledge of the iden tit y of pages that should b e memory
residen t at sp ecic times to supp ort a co ordinated displa y This sc hedule is termed a display sche dule Denition A display sche dule is a sequence fP
P
m g of disk pages sets Where m is the duration
of the presen tation in time in terv als and P
i
is the set of pages displa y ed during in terv al i A resource sc hedule also dep ends on the placemen t of data across disk driv es An un balanced placemen t
of pages across the disk driv es increases memory requiremen ts It forces the system to use only the aggregate
Time
01 i i+1 m
Request Display
Arrives Starts
Display
Ends
Time
Interval i
S0 Si
Sm
Time
Intervals
-p
S-p
Time
Instant i
Figure Time in terv al and time instan t
bandwidth of the disks con taining referenced pages instead of the aggregate bandwidth of all disks This
under utilization of the disk bandwidth increase the n um b er of prefetc hed pages as compared with a system
that main tains a balanced placemen t of data The increase in n um b er of prefetc hed pages migh t increase
the memory requiremen t so that it is imp ossible to displa y the presen tation with the memory a v ailable to
the system An alternativ e to prefetc hing is to either replicate or migrate pages b efore they are referenced
so that the pages referenced sim ultaneously are ev enly distributed across the disk driv es
Denition A plac ement of data maps a page iden tier and a time in terv al in to one or more disk driv es
The state of memory ie pages o ccup ying memory frames at eac h instan t i denoted as S
i
can b e
dened in terms of pages sw app ed out of memory K
i
written to disk U
d
i
and those retriev ed from dieren t
disks F
d
i
Denition Giv en a system with D disks the state of memory at eac h instan t i is dened as
S
i
S
i K
i U
i U
D i F
i F
D i A resource sc heduler consumes a displaysc hedule a system conguration B C D and a placemen t
of data P to compute a sc hedule of page retriev als replications and migrations that satisfy the temp oral
constrain ts dictated b y the displaysc hedule
Denition Giv en a system with C memory frames D driv es with disk bandwidth that allo ws eac h disk
to retriev e B pages during a time in terv al an initial state of memory S
p
and an initial placemen tof data
P A r esour cesche dule consists of p m time in terv als m of these o v erlap with the displa y and p of them
either prefetc h pages in to memory or mo dify the placemen t of data across the disks in preparation for the
displa y In essence p denotes the incurred latency Asso ciated with eac h time in terv al i are
a collection of pages retriev ed from eachof the D disks during time in terv al i denoted as F
i
F
D i
a collection of pages written to eachofthe D disks during time in terv al idenotedas U
i
U
D i
a collection of pages sw app ed out of memory to accommo date these retriev als denoted as K
i
F urthermore the retriev ed written and sw app ed pages are sub ject to the follo wing restrictions
-4 -3 -2 -1 0 1 2 3
Display Starts Display Ends
R
1
e
Time
Pages
in
Memory
R
1
a
R
1
b
R
1
c
R
1
d
R
1
f
W
2
e
Request Arrives
R
2
e
a
e
aa a
b b b
c c c
dd
ee
f f
-4 -3 -2 -1 0 1 2 3
Display Starts Display Ends
R
1
a
Time
Pages
in
Memory
R
1
b
R
1
c
R
1
d
R
1
e
R
1
f
Request Arrives
a a aa a
b b b
cc c
dd
ee
f f
b
c
d
e
(b) (a)
Figure a Retriev al Sc hedule b Resource Sc hedule
i Once the displa y starts the set of pages in memory at eac h instan t i is a subset of those required b y
the displaysc hedule F or eac h i m P
i
S
i
and P
i
S
i ii The n um b er of pages retriev ed and written to a disk during a time in terv al do es not exceed B F or
eac h i p m and eac h d D jF
d
i
j jU
d
i
j B iii The n um ber of memory residen t pages at eac h time instan t is lo w er than the n um ber of a v ailable
memory frames F or eac h i p m jS
i
j C iv The retriev als resp ect the placemen t of data F or eachpage ain terv al iand disk d a F
d
i
implies
that d P
a i Where P
is the placemen t of data resulting from up dating P with migrations and
replications sc heduled b efore in terv al i If there is no manipulation of the placemen t of data ie for eac h time in terv al i and driv e d U
d
i
is empt y
then weha vea r etrieval sche dule This study demonstrate that the computation of a resource sc hedule that yields a minim um latency and
supp orts a co ordinated displa y of fP
P
m g in a system conguration B C D assuming an initial
placemen t of data P is NPHard
T o illustrate these concepts consider a displa ysc hedule that consists of three time in terv als P
fa b g P
fc d gand P
fe f g Assume that the system consists of four disks D eac h with the bandwidth
to retriev e one disk page during a time in terv al B Assuming that all the referenced pages reside on
disk one Figure a sho ws a retriev al sc hedule that supp orts a co ordinated displa y R
a
denotes that disk
page a is read from disk one In this gure a negativ e time instan t corresp onds to page retriev als p erformed
prior the displa y A page migh t either be retriev ed during the time in terv al prior to its displa y eg f or prefetc hed at an earlier time in terv al eg a Prefetc hing increases the memory requiremen ts of the
system F or example frames of memory are allo cated at instan t one a b c d e while the displa ysc hedule
dictates that only four should be allo cated a b c d The other page e is prefetc hed for later use and
increase the memory requiremen ts of the system
As illustrated b y this example an un balanced sc hedule of references to disks migh t result in formation
of b ottlenec k disks that requires the system to prefetc h pages while other disks remain idle In our example
while the bandwidth of four disks could accommo date the retriev al of four pages the system w as forced to
prefetc h pages b ecause they all reside on disk one The sc heduler ma y construct resource sc hedules that
utilize the idle disk bandwidth in order to minimize the n um ber of prefetc hed pages Figure b sho ws
one suchsc hedule With this sc hedule the system reads page e from disk one during time in terv al and
replicates or migrates it to disk t w o denoted as W
e
during time in terv al This allo ws the system to
free the memory frame o ccupied b y e at time instant and utilize disk n um ber t w o to retriev e e during
time in terv al one to satisfy the displaysc hedule With this sc hedule only memory frames are required at
instantone a b c d
The distinction b et w een replications and migrations in the resource sc hedule is the a v ailabilityof mi grated and replicated pages afterw ards When a page a is replicated from disk d
to d
the system can
retriev e a from either d
or d
after the replication is complete On the other hand if a is migrated instead
then the system m ust retriev e a from d
The purp ose of replications and migrations in a resource sc hedule
is to facilitate the retriev al of a page when it is referenced b y the displa y otherwise the system will do
w asteful w ork Therefore at least one of the page references in the displa ym ust retriev e the page from the
new lo cation The distinction b et w een replications and migrations v anishes when a page is referenced only
during one con tin uous p erio d in the displa y Because the new placemen t of the page is irrelev an t for the
displa y after suc h p erio d The displaysc hedules considered in the pro ofs presen ted in this study reference
a page only during one con tin uous p erio d Therefore the distinction b et w een replications and migrations is
not needed Henceforth w e refer to b oth of them as replications
The sc heduling of replications dep ends on the system load namely disk bandwidth and memory at
eac h time in terv al
Denition The system lo ad for a p erio d N sp ecies the system resources a v ailabilit y as follo ws F or
eac h time in terv al i N Disk bandwidth a v ailable at eachdriv e during in terv al i B
i
B
D i
P age frames a v ailable during in terv al i M
i
A replication consists of a disk page a source driv e and a set of alternativedriv es to place the replica
a sour ce f tar g et
target
n
g The execution of the replication can utilize in termediate driv es F or
example to execute the replication a f g the system migh t read a from disk at t
and write it to
disk then read it from disk and write it to disk at t
The adv an tage of using disk as an in termediate
stage is that it migh t reduce the memory requiremen ts In the absence of disk bandwidth for drivebet w een
t
and t
the system is forced to stage a in to memory during t
t
While for the case of using disk as in termediate stage the system do es not ha v e to stage a in memory b et w een the time in terv al when a is
written to disk and the time in terv al when it is read from disk The iden tit y of the disk page is irrelev an t
for the pro ofs w eth us omit it henceforth
Denition Giv en a collection of replications R f sour ce
f tar g et
target
n g sour ce
r
ftar g et
r
tar g et
r
n r
g g and a system load A for a p erio d N Asc hedule for replications R on A maps
t
1
i
t
2
i
t
3
i
read page write page
from source
i
to drive d
1
i
read page
from drive d
1
i
write page
to drive target
i
p
t
2k
i
i
..... .....
write page
to drive d
2
i
..... ..... .....
N
0
.....
Page in Memory
Figure Sc hedule of a replication
eac h replication sour ce
i
f tar g et
i
target
i
n i
g g in to a sequence ft
i
d
i
t
i
t
i
d
i
k i t
i
k i t
i
k i tar g et
i
p
t
i
k i
g Figure Suc h that
i The replications are sc heduled within the p erio d N and the reads and writes are sc heduled in the
rightorder F or eac h j k
i
and l k
i
t
i
j
N and j l implies t
i
j
t
i
l
ii The page is written to one of the target driv es p n
i
iii There is disk bandwidth a v ailable in sour ce
i
at in terv al t
i
and in tar g et
i
p
at in terv al t
i
k i
to read and
to write the page resp ectiv ely iv F or eac h j k
i
there is bandwidth a v ailable in disk d
i
j
to write the page at in terv al t
i
j
and to
read the page at in terv al t
i
j
v There is memory a v ailable to ha v e the page memory residen t during the follo wing p erio ds t
i
t
i
t
i
k i t
i
k i
vi The in termediate driv es are dieren t from the target driv es otherwise the system w ould be doing
w asteful w ork F or eac h j k
i
d
i
j
f tar g et
i
target
i
n i
g
This study demonstrates that the computation of a sc hedule that p erforms replications R in minim um
time based on a presp ecied system load A is NPHard
Related W ork
Sev eral researc hers ha v e studied the complexityof sc heduling problems GJ EIS GJS Their studies
assume a predened n um ber of jobs with sp ecic resource requiremen ts and duration Con v ersely in a
resource sc hedule that supp orts a co ordinated displa y the duration and resource requiremen ts of the jobs
render an ob ject memory residen t during a sp ecic time period are not predened T o ha vean ob ject
memory residen t the system migh t either retriev e the ob ject directly from the disk con taining it or manip
ulate the placemen t of data so that the ob ject can b e retriev ed from another disk The duration memory
requiremen t and disk bandwidth requiremen t of the manipulation of the placemen t of data are not pre
dened A replication or migration mighttak e sev eral steps to reac h its destination F or instance consider
the migration of an ob ject in disk d
to disk d
The system migh t either migrate the ob ject directly from
d
to d
or migrate the ob ject from disk d
to d
and then from d
to its nal destination d
F urthermore
the ob ject m ust b e memory residentbet w een consecutiv e reads and writes in a migration or replication eg
bet w een read from d
and write to d
These p erio ds the disk driv es used and the n um b er of steps in
the migration or replication are not predened
F or a singledisk arc hitecture there is an optimal resource sc hedule gr e e dy for co ordinated displa yof
structured presen tations EMGIng This optimal sc hedule minimizes b oth the memory requirementateac h
instan t and the latency and can b e computed in time O n lg n In a m ultidisk arc hitecture this optimal
sc hedule can b e applied as follo ws Giv en a displaysc hedule and an initial placemen t of data without replicas
ie there is only one instance of eac h page the pro jection fP
d
P
d
m g of the displaysc hedule on eac h
disk driv e d is as follo ws P
d
i
fa j a P
i
a resides in dg The union of the retriev al sc hedules computed
b y greedy on eac h pro jection fP
d
P
d
m g yields a nal retriev al sc hedule for the displa y Ho w ev er this
retriev al sc hedule do es not ha vean y impact on the placemen t of data across disks driv es And manipulation
of the placemen t of data migh t reduce the memory requiremen t and the latency Replication Sc heduling
This section demonstrates that the computation of a sc hedule that performs replications R in minim um
time based on a presp ecied system load A is NPHard W e rst pro v e that deciding whether there is
a sc hedule for replications R on a system load A dened o v er a period N is NPComplete W e rst
in tro duce a p olynomial algorithm SA TR epSc that transforms an y instance C
C
n
v
v
k
of SA T
in to an instance N A R of the replications sc hedule problem
An instance of SA T is dened as a collection fC
C
n
g of n clauses o v er a set fv
v
k
g of k
v ariables The problem is deciding whether there is a v ariable assignmen t that mak es all clauses true
Without loss of generalit y assume that there is not a clause in the SA T instance with disjuncts v
i
and v
i
for some v ariable v
i
if this is the case remo vesuc h clauses b ecause they are true for an y truth assignmen t
SA TRepSc
Input C
C
n
v
v
k
Output R A N N Let N be n k
A Let A be dened as in Figure Thic k time instan ts denote that the system has memory frames
a v ailable at that instan t
F or i f N gM
i
The system has at least one memory frame a v ailable at instant M
Thin time instan ts denote that the system has memory frame a v ailable at that instan t
F or i f N gM
i
The lab els on time in terv als denote disk driv es with a v ailable bandwidth during the in terv al If a lab el
has a sup erscript then the disk has bandwidth a v ailable for the retriev alwrite of at least one page
during the in terv al Otherwise the disk has bandwidth a v ailable for only one page retriev alwrite
during the in terv al The expressions v
i
C
j
and v
i
C
j
denote that v
i
and v
i
are disjuncts of C
j
resp ectiv ely T o illustrate consider time in terv al there is disk bandwidth a v ailable in driv es w
and
d
If v
C
then disk d
hasalsobandwidth a v ailable during in terv al The other disk driv es do
not ha v e bandwidth a v ailable during in terv al The system load for in terv al is as follo ws M
for eac h j f w
d
d
g B
j
B
w B
d
if v
is a literal in C
then B
d otherwise
B
d R Let R b e dened as follo ws F or eachv ariable v
i
s
i
f t
i
u
i
g is a replication in R F or eac h clause
C
j
d
j
f d
ji
j v
i
C
j
g f e
ji
j v
i
C
j
g is a replication in R T o illustrate the transformation consider the example in Figure The patterns in Figure c are used
to denote the alternativ es to sc hedule replications asso ciated to v ariables and clauses Note that the driv es
w
j
cannot participate in anysc hedule b ecause they are neither a source nor a target Moreo v er they cannot
be an in termediate driv e b ecause they ha v e bandwidth a v ailable only during one time in terv al Sc hedules
of replications comp ete with eac h other for disk bandwidth and memory F or example f d
g an
alternativeto sc hedule the replication asso ciated to C
comp ete with f d
d
t
g an
alternativetosc hedule the replication asso ciated to v
for bandwidth of d
at in terv al and for memory at
instant In tuitiv elya v ariable assignmentthat mak es v
true is equiv alen t to select the replication denoted
b y the line under v
with the pattern of v
Figure c as the sc hedule for the replication asso ciated to v
Then the replication denoted b y the line under v
with the pattern of C
Figure c can b e selected as
the sc hedule for the replication asso ciated to C
they w ould not b e comp eting for disk bandwidth of driv e
d
nor for memory at instant It is easy to see that SA TRepSc can b e p erformed in p olynomial time
SA TRepSc denes a resources a v ailabilit y A and a collection R of replications so that the p ossible
sc hedules on A for replications in R follo w a sp ecic pattern F or the case of replications asso ciated to
v ariables there are only t w o p ossible sc hedules on A per v ariable
Lemma Let R A N b e the output of SAT RepS c C
C
n
v
v
k
and r s
i
f t
i
u
i
g be
the replication in R asso ciated to v ariable v
i
There are only t w o alternativ es to sc hedule r on A .
.
.
read s
i
write
e
1i
read
e
1i
write
e
ni
read
e
ni
write u
i
(b)
4 n 1 + () i ⋅⋅
s
i
4 n 1 + () i 1 – () ⋅⋅
.
.
.
read
write
d
1i
read
d
1i
write
d
ni
read
d
ni
write t
i
4 n 1 + () i 1 – () ⋅⋅ 2 n 1 + () ⋅ +
(a)
4 n 1 + () i 1 – () ⋅⋅ 2 n 1 + () ⋅ +
e
1k
d
1k
d
11
d
2
,
d
11
d
n1
d
n1
d
1k
d
1k
d
2
,
d
nk
d
nk
s
1
d
1
,
t
1
s
k
d
1
,
t
k
Time
Intervals
0
N
v
1
v
k
v
k
4 n 1 + () ⋅
if v
1
C
1
∈
otherwise
2 n 1 + () ⋅
.
.
.
.
.
.
.
.
.
.
.
.
s
k
if v
k
C
1
∈
otherwise
.
.
.
.
.
.
.
.
.
.
.
.
.
w
1
+
w
2
+
w
2 n 1 + () ⋅
+
w
N
+
w
N 1 –
+
w
N
k
--- -
+
.
.
.
.
.
.
.
.
.
.
.
.
s
1
if v
1
C
2
∈
d
11
otherwise
e
11
d
2
,
e
11
e
n1
e
n1
s
1
d
1
,
u
1
v
1
if
v
1
¬ C
1
∈
otherwise
.
.
.
s
1
if v
1
¬ C
2
∈
e
11
otherwise
if v
k
C
2
∈
otherwise
e
1k
e
1k
d
2
,
e
nk
e
nk
s
k
d
1
,
u
k
s
k
if v
k
¬ C
1
∈
otherwise
.
.
.
if v
k
¬ C
2
∈
otherwise
Figure System Load
v
1
d
11
d
2
,
d
11
d
21
d
21
e
11
e
11
e
21
e
21
d
12
d
2
,
d
12
d
22
d
22
e
12
e
12
e
22
e
22
d
13
d
13
d
23
d
23
e
13
d
2
,
e
13
e
23
e
23
s
1
d
1
,
t
1
s
2
t
2
s
3
d
1
,
t
3
Time
Intervals
0
N
s
1
u
1
s
2
d
1
,
u
2
s
3
u
3
v
2
v
1
v
2
v
3
v
3
N
3
--- -
2 N ⋅
3
---------- -
C
1
v
1
v
2
¬ v
3
∨∨ =
C
2
v
1
v
2
v
3
¬ ∨∨ =
C
1
v
1
v
2
v
3
C
2
d
2
d
21
d
22
e
23
,, {} →
d
1
d
11
e
12
d
13
,, {} →
s
1
t
1
u
1
, {} →
(a)
(b)
(d)
w
1
w
2
+
w
3
+
w
4
+
w
7
+
w
8
+
w
5
+
+
w
6
+
w
9
w
10
+
w
11
+
w
12
+
w
15
+
w
16
+
w
13
+
w
14
+
w
17
w
18
+
w
19
+
w
20
+
w
23
+
w
24
+
w
21
+
w
22
+
w
25
w
26
+
w
27
+
w
28
+
w
31
+
w
32
+
w
29
+
w
30
+
+
+
+
w
33
w
34
+
w
35
+
w
36
+
+
s
2
t
2
u
2
, {} →
s
3
t
3
u
3
, {} →
v
1
:
v
3
:
C
1
:
C
2
:
v
2
:
(c)
Figure Example of reduction of a SA T instance in to a replications sc heduling instance a SA T instance
b Corresp onding set of replications c P atterns to denote alternativesc hedules of replications asso ciated
to v ariables and clauses d Corresp onding system load
Pro of There are only t w o time in terv als with bandwidth a v ailable at driv e s
i
Consider the case when
the sc hedule starts with read the page from s
i
at in terv al n i Because there is not memory
a v ailable at instant n i then the next step m ust b e to write the page to driv e d
i
during
in terv al n i The next op eration to sc hedule m ust b e to read the page from d
i
during
in terv al n i b ecause there will not b e other in terv al with bandwidth a v ailable for driv e
d
i
afterw ards A similar argumen t can b e applied to conclude that the subsequen t steps in the sc hedule are
to write the page from d
i
during in terv al n i and then read it from d
i
during in terv al
n i and so forth The nal step in the sc hedule m ust b e to write the page to driv e t
i
at
in terv al n i n b ecause there will not b e memory a v ailable to hold the page at
instant n i n In sum one alternativ eto sc hedule the replication asso ciated to v
i
is
the sequence in a Similarlyw e can sho w that the other alternativ e is the sequence in b
F or the replications asso ciated to clauses there are only c p ossible sc hedules on A for a clause with c
disjuncts
Lemma Let R A N b e the output of SAT R epS c C
C
n
v
v
k
and r d
j
f d
ji
j v
i
C
j
gf e
ji
j v
i
C
j
g b e the replication asso ciated to clause C
j
Let l n i and s j There are only c c n um b er of disjuncts in C
j
p ossible sc hedules for the replication asso ciated to C
j
ffl s d
ji
l s gj v
i
C
j
g
ffl n s e
ji
l n s gj
v
i
C
j
g
Pro of The sc hedule for the replication asso ciated to C
j
m ust start with a read from d
j
F rom SA TRepSc
w e conclude that there are exactly c time in terv als in A with bandwidth a v ailable in disk d
j
Moreo v er the
time in terv als with bandwidth a v ailable for disk d
j
are fl s j v
i
C
j
g f l n s j v
i
C
j
g Let r b e the time in terv al when the read is sc heduled There are t w o cases r fl s for some iand
v
i
C
j
or r l n s for some i and v
i
C
j
Consider case F rom the construction
of A T ransformation SA TRepSc w e conclude that there will b e bandwidth a v ailable at driv e d
ji
during
in terv al r and there will not b e memory a v ailable at instan t r Moreo v er d
ji
is an alternativ e target
for the replication Therefore the sc hedule m ust nish with a write to disk d
ji
at in terv al r Similar
argumen t can b e applied to case In conclusion the p ossible sc hedules for the replication asso ciated to
C
j
are the c alternativ es describ ed ab o v e
Lemma Let R A N b e the output of SAT R epS c C
C
n
v
v
k
If there is a truth assign
mentfor v ariables fv
v
k
g that mak es all clauses C
C
n
true then there is a replication sc hedule R
on A during p erio d N
Pro of Let a b e a truth assignmentthat mak es all clauses C
C
n
true Consider the follo wing sc hedule
for the replications in R i F or eachv ariable v
i
if a v
i
is true then consider the sc hedule in Lemma b for the replication
asso ciated to v
i
Otherwise consider the sc hedule in Lemma a
ii F or eac h clause C
j
let v
i
be the v ariable suchthateither a v
i
istrueand v
i
C
j
or a v
i
is false and
v
i
C
j
Let l n i and s j If a v
i
istrueand v
i
C
j
then consider the
sc hedule fl s d
ji
l s g for the replication asso ciated to C
j
If a v
i
is false and v
i
C
j
then
consider the sc hedule fl n s e
ji
l n s g for the replication asso ciated to C
j
T o pro v e that the ab o v e is a replication sc hedule for R on A it suces to sho w that the sc hedules for
eac h replication do not o v erlap eac h other ie they do not comp ete for disk bandwidth nor memory The
sc hedules for replications asso ciated to v ariables span disjoin t p erio ds of time F or eac h i and j suchthat
i j the follo wing time in terv als are disjoin t
n i n i n n i n n i n j n j n n j n n j Similarlythe sc hedules for replications asso ciated to clauses span disjoin t p erio ds of time
Supp ose that the sc hedule for a v ariable v
i
o v erlaps the scheduleforaclause C
j
Therefore either v
i
or
v
i
mak es C
j
true If a v
i
is true then the sc hedule for v
i
spans the p erio d n i n n i and the sc hedule for C
j
spans the p erio d n i j n i j Ho w ev er these t w o p erio ds are disjoin t Hence it con tradicts the assumption that the sc hedules for v
i
and
C
j
o v erlap Similarly for the case where a v
i
is false w e can conclude that the sc hedules w ould not o v erlap
Therefore If there is a truth assignmen t for v ariables fv
v
k
g that mak es all clauses C
C
n
true then there is a replication sc hedule R on A during p erio d N W e no w pro v e the other direction if the replication sc hedule yielded b y SAT R epS c has a solution
then the input SA T instance has a solution
Lemma Let R A N be the output of SAT R epS c C
C
n
v
v
k
If there is a replication
sc hedule for R on A then there is a truth assignmen tfor fv
v
k
g that mak es all clauses fC
C
n
g
true
Pro of The sc hedules for the replications asso ciated to v ariables in S AT R epS c follo w either pattern of
Lemma Therefore a v alid truth assignmen t a is as follo ws a v
i
is true if the execution of the replication
asso ciated to v
i
follo ws the pattern in Lemma b and is false if it follo ws the pattern in Lemma a
Wenowsho w that a mak es all clauses fC
C
n
g true Supp ose that there exists a clause C
j
suc h
that all its disjuncts are false The sc hedule of the replication asso ciated to C
j
m ust be either Lemma
a f n i j d
ji
n i j g if v
i
C
j
or b
f n i n j e
ji
n i n j gif v
i
C
j
Supp ose that the sc hedule of C
j
is as describ ed in a Then the sc hedule of the replication asso ciated to v
i
m ust follo w the pattern in Lemma b Otherwise there w ould b e a conict for the disk bandwidth of
d
ji
bet w een the sc hedules for C
j
and v
i
Therefore a v
i
is true according to the denition of a describ ed
ab o v e Ho w ev er as stated in a v
i
C
j
then that C
j
is true This con tradicts the assumption that all
disjuncts in C
j
are false Similarlyw e can reac h a con tradiction when the sc hedule for C
j
is as describ ed in
b
Therefore a mak es all clauses fC
C
n
g true
Because the transformation SAT RepS c is a p olynomial time algorithm and Lemmas and w e
conclude the follo wing
Theorem Deciding whether there is a sc hedule for a set R of replications on a system load A o v er a
period N is NPComplete
Corollary Given a system lo ad A the c omputation of a sche dule to p erform a set R of r eplic ations to
change the plac ement of data acr oss disk drives in minimum time is an NPHar dpr oblem
Resource Sc heduling
This section sho ws that deciding whether there is a resource sc hedule for a giv en displaysc hedule that yields
the latency of one time in terv al is NPComplete W e rst in tro duce a p olynomial algorithm SA TR esSc
that transforms an y instance C
C
n
v
v
k
of SA T in to an instance fP
P
m g PBCD of
the resource sc hedule problem
SA TResSc
Input C
C
n
v
v
k
Output fP
P
m g PBCD Replication Instance
Let R A N be SA TRepSc C
C
n
v
v
k
System Conguration
Set the n um b er of disks in the system to the n um b er of dieren t lab els used in SA TRepSc
Let D m k n n k Let M axC ar d maxf cardinalit y of tar g et
i
j i n and sour ce
i
tar g et
i
is the replication
asso ciated to C
i
g
Let the disk bandwidth of eac h drivebe suc h that it can retrieveup to M axC ar d pages during an
in terv al
Let B M axC ar d
Let q B D
Set the memory capacit y of the system as follo ws
Let C q
DisplaySc hedule
Let fP
P
m g m N n k b e the displa ysc hedule in column P
i
of T able Placemen t of Data
Set the placemen t of pages on the disk driv es so that an y resource sc hedule m ust include a replica
tion sc hedule for R the system load after applying retriev al sc hedule g r eedy fP
P
m gB D Column
S
i
in T able w ould b e iden tical to A during the p erio d N Ev ery page resides in one disk ie there is no replicas
Set the placemen t of disk pages in the displaysc hedule as follo ws
Placemen t of pages retriev ed during in terv al a
a
q
w ould be placed on disk driv es dieren t from w
B pages on eac h driv e Because
D B q B the only driv ewith a v ailable bandwidth during in terv al is w
Placemen t of pages retriev ed during ev en in terv als in N F or the assignmentof the q pages retriev ed during ev en time in terv als b efore instan t Nw e
ha vet w o cases
There are three disks x y w
i
with a v ailable bandwidth during the in terv al in A assign the
rst D B pages to driv es dieren tfrom x y w
i
B pages to eachdriv e assign the next
B to driv e x the next B to driv e y and the last page to w
i
Then driv es x and y
w ould ha v e disk bandwidth a v ailable for one page retriev alwrite eac h And driv e w
i
w ould
ha v e disk bandwidth a v ailable for B retriev alswrites
There are t w o disks x w
i
with a v ailable bandwidth during the in terv al in A assign the
rst D B pages to driv es dieren t from x w
i
B pages to eac h driv e assign the last
B pages to driv e x Then driv e x w ould ha v e disk bandwidth a v ailable for one page
retriev alwrite and driv e w
i
for B retriev alswrites
Placemen t of pages retriev ed during o dd in terv als in N The assignmen t of the q pages retriev ed during odd time in terv als before instan t N and after
instan t is as follo ws Let x w
i
b e the disk driv es with a v ailable bandwidth during the in terv al in
A Assign the rst D B pages to driv es dieren t from x w
i
B pages to eac h driv e assign
the next B pages to driv e x and thelastpage to w
i
Then driv e x w ould ha vea v ailable
disk bandwidth for one page retriev alwrite and driv e w
i
for B retriev alswrites
Placemen t of pages retriev ed during N N k
The assignmen t of the q pages retriev ed during eachin terv al i is as follo ws Let t
i N u
i N be targets of the replication asso ciated to v
i N Assign the rst D B pages to driv es
dieren t from ft
i N
u
i N w
i
g B pages to eachdriv e assign the next page to s
i N
the next B to t
i N the next B to u
i N
and the last one to w
i
Then driv es
t
i N and u
i N w ould ha v e disk bandwidth a v ailable for one page retriev alwrite eac h driv e
s
i
w ould ha v e exceeded the disk bandwidth requirementb y one page and driv e w
j
w ould ha v e
bandwidth a v ailable for B retriev alswrites
Placemen t of pages retriev ed during N k m The assignmentof the q pages retriev ed during eachin terv al ti is as follo ws Let x
x
i
b e the
target disk driv es of replication asso ciated to C
ti N k Assign the rst D i B pages
to driv es dieren t from x
x
i
w
ti
B pages to eachdriv e the next page to d
ti N k the
next B to x
the next B to x
and so forth Finally assign the last i pages to w
ti Then driv es x
x
i
w ould ha v e disk bandwidth a v ailable for one page retriev alwrite eac h
driv e d
ti N k w ould ha v e exceeded the disk bandwidth requiremen t b y one page and driv e
w
ti
w ould ha v e bandwidth a v ailable for B i retriev alswrites
Observ ation F r om the tr ansformation SAT R esS cwe c an observe the fol lowing
L et R A N SAT RepS c C
C
n
v
v
k
The tr ansformation SAT R esS c pr o duc es a display
i F
i
S
i
P
i
a
a
q
a
q
a
q a
a
q
a
a
q
a
q
a
q a
a
q a
q
a
q a
q
a
q a
q
a
q a
q
a
q a
q a
q a
q
a
q a
q a
q a
q a
q a
q a
q N b
q b
q
b
b
q
b
q
b
q
N b
q b
q
b
q b
q
b
q b
q
N b
q b
q
b
q b
q
m T able Displa y and Retriev al Sc hedules fP
P
m g and f
S
S
m
g
sche dule fP
P
m g a system c ongur ation B C D and an initial plac ement of data P such that
Ther e is a r esour c e sche dule for fP
P
m g c onsisting of a r etrieval sche dule Ret f
S
S
m
g and a r eplic ation sche dule Rep such that i the system lo ad during p erio d N r esulting fr om
applying Ret is identic al to A and ii Rep is a sche dule for R on the system lo ad r esulting fr om
applying Ret Ret do es not pr efetch p ages Ther efor e for any r esour c e sche dule for e ach instant i i m S
i
S
i
T able F or any r esour c e sche dule that supp orts fP
P
m g ther e is not memory available at instants
N m for pr efetching nor r eplic ation Be c ause for e ach i N i m j
S
i
j C Weno w sho w that giv en an instance C
C
n
v
v
k
of SA T there is a onetime in terv al resource
sc hedule for SAT R esS c C
C
n
v
v
k
if and only if SAT R epS c C
C
n
v
v
k
has a so
lution
Lemma Let R A N be the output of SAT RepS c C
C
n
v
v
k
The transformation
SAT ResS c pro duces a displa y sc hedule fP
P
m g a system conguration B C D and an initial
placemen t of data P suc h that an y resource sc hedule for fP
P
m g that yields a onetime in terv al
latency m ust sc hedule replications R during time in terv al N
Pro of The system m ust replicate the pages that cannot retriev e during in terv als Nm Observ ation
b efore instan t N Hence for eac h i k the system m ust replicate a page from driv e s
i
to either
t
i
u
i
or w
N i
b efore in terv al N And for eac h i n the system m ust replicate a page from driv e d
i
to
either driv e in the target set of the replication asso ciated to C
i
or to w
N k i
b efore in terv al N The system m ust sc hedule replications R b efore instan t N Ret retriev es eac h referenced page during
N only once and do es not prefetc h pages Observ ation Therefore the disk bandwidth requiremen t
from a driv e during N is at least the disk bandwidth required b y Ret during the same p erio d Then the
source target and in termediate driv es in the sc hedule of a replication m ust ha v e disk bandwidth a v ailable
in A during N Otherwise the bandwidth requiremen ts of a disk drivew ould exceed the disk bandwidth
a v ailabilit y during N Th us disk driv es w
i
for i N m cannot b e a target driv e of a replication
Replications R m ust b e sc heduled after instan t otherwise the latency w ould b e higher than one time
in terv al Starting the sc hedule of a replication at in terv al w ould increase the latency b ecause the
retriev al of all pages in P
w ould require the disk bandwidth of all disks except w
and w
is not a source
driv e for an y replication In sum the system m ust sc hedule replications R during N Lemma Let R A N be the output of SAT R epS c C
C
n
v
v
k
Let fP
P
m g P B C D b e the output of SAT ResS c C
C
n
v
v
k
If there is a replication sc hedule RS for R
on A during N then there is a resource sc hedule that yields a onetime in terv al latency and supp orts a
co ordinated displayof fP
P
m g on a system conguration B C D and an initial placemen tofdata
P Pro of Construct a resource sc hedule as follo ws
Step Include retriev al sc hedule Ret in Observ ation Step Change the retriev als in Ret of replicated pages in R to b e retriev ed from their target driv es
in RS Step Include the sc hedule of replications RS This resource sc hedule supp orts a co ordinated displa y of fP
P
m g that yields a onetime in terv al
latency Topro v e the other direction w eshowthat sc heduling a replication r R as part of a resource sc hedule
for SAT R esS c C
C
n
v
v
k
requires at least the memory required b y the sc heduling of r on A
during N Where R A N is the output of SAT R epS c C
C
n
v
v
k
Giv en a sc hedule of a replication the time in terv als when the reads and writes are sc heduled determines
the memory requiremen ts The memory requiremen ts of t w o replications that coincide in the time in terv al
when a read and the next write is sc heduled are iden tical
Denition Giv en a replication sc hedule
ft
i
d
i
t
i
t
i
d
i
k i t
i
k i t
i
k i tar g et
i
p
t
i
k i
g
the memory r e quir ements of the replication sc hedule during p N is dened as the sequence
z p t
i
times
z t
i
t
i
times
z t
i
t
i
times
z t
i
k
i
t
i
k
i
times
z N t
i
k
i
times
that represen ts the n um b er of memory frames required bythe sc hedule at eachinstan t i i p N
Let R A N b e the output of SAT RepS c C
C
n
v
v
k
F or an y resource sc hedule for SAT ResS c C
C
n
v
v
k
there are t w o alternativ es to sc hedule a replication r in R sc hedule r based
on the system load A or mo dify the retriev al sc hedule Ret See observ ation to accommo date the
replication r F or the second alternativ e the system migh t sc hedule additional replications F or example
to sc hedule a replication of page a from driv e s to driv e t The system migh t utilize the disk bandwidth
used to retriev e a page b from s in Ret to read the page a from s at in terv al ti Then page b can either
b e prefetc hed at an earlier time in terv al or b e replicated from s toadisk u with a v ailable bandwidth at ti
so that b can b e retriev ed from u at ti If there is not memory to prefetc h b then the system is forced to
replicate b Therefore the sc hedule of the replication from s to t includes the sc hedule of a new replication
from s to u The additional replications also increase the memory requiremen ts Therefore their memory
requiremen ts should also b e considered to obtain the memory requiremen ts of the sc hedule
Denition Let fP
P
m g PBCD be the output of SAT ResS c C
C
n
v
v
k
Let
R A N be the output of SAT R epS c C
C
n
v
v
k
If the sc heduling of a replication r
in R
mo dies the retriev al sc hedule Ret in sucha w a y that additional replications r
r
n r
m ust b e sc heduled
Then the extension of r
is the set of replications fr
r
n r
g F or example supp ose that the system sc hedules replications r
and r
to accommo date replication r
Supp ose that the memory requiremen ts of the replications sc hedules are as follo ws The timing for replication
r
is for r
is and for r
is Then the
extension of r
is fr
r
r
g and its memory requiremen ts is
The memory requiremen ts of replications sc hedules dene a partial order on the sc hedules
Denition A replication sc hedule sr
is gr e ater smal ler than a replication sc hedule sr
if and only if
for eac h instan t i p N the memory requiremen t of sr
at i is greater smaller than or equal to the
memory requirementof sr
at i
Lemma Let fP
P
m g PBCD be the output of SAT R esS c C
C
n
v
v
k
Let
R A N b e the output of SAT R epS c C
C
n
v
v
k
Let sr be the sc hedule of the extension of r
in a onetimein terv al resource sc hedule for fP
P
m g where r is a replication asso ciated to clause C
j
Then there exists some replication sc hedule sr
on A for r suc h that sr
is smaller than sr Pro of It suces to consider the case when the system c hanges Ret to accommo date the sc heduling of r sr m ust end with a write page on driv e d
ji
or driv e e
ji
This write page m ust b e sc heduled at an in terv al l
suc h that there is memory a v ailable at instan t l Therefore the write page m ustbesc heduled during an o dd
time in terv al ie etc Supp ose that the write page is sc heduled during an in terv al l that do es not
ha v e bandwidth a v ailable for anydriv e in the target set Then there are t w o alternativ es to prefetc h
a retriev al from a target driv e that w as sc heduled during l in Ret or to replicate a page a retriev ed
from a target driv e during l in Ret to a driv e with a v ailable bandwidth in lsothat a can b e retriev ed from
another driv e during l The rst alternativ e is not p ossible b ecause the write a page op eration requires an
additional memory frame at instan t l to hold the page The memory is th us exhausted at instan t l then
there is not memory a v ailable to hold the prefetc hed page The second alternativ e is not p ossible either
b ecause the disk driv es with a v ailable bandwidth during o dd time in terv als eg u
i
t
i
e
ji
d
ji
do not ha v e
disk bandwidth a v ailable at an earlier time in terv al Therefore the replication of a w ould increase the disk
bandwidth requirementof suc h driv es to more than what is a v ailable during the p erio d l In sum the
write page op eration m ust be sc heduled during an odd time in terv al l that has bandwidth a v ailable for a
driv e in the target set Because there is not memory a v ailable at instan t l the page m ust b e read during
in terv al l In conclusion an alternativ e sr
to sc hedule the replication asso ciated with C
j
in Lemma is smaller than sr Lemma Let fP
P
m g PBCD be the output of SAT R esS c C
C
n
v
v
k
Let
R A N b e the output of SAT R epS c C
C
n
v
v
k
Let sr be the sc hedule of the extension of r
in a onetimein terv al resource sc hedule for fP
P
m g where r is a replication asso ciated to v ariable
v
i
Then there exists some replication sc hedule sr
on A for r suc h that sr
is smaller than sr Pro of It suces to consider the case when the system c hanges Ret to accommo date the sc heduling of r sr m ust end with a write page a to either driv e t
i
or driv e u
i
As for the case of write page on a target driv e
in pro of of Lemma the write page on either driv e t
i
or u
i
mustbesc heduled during an o dd time in terv al
l that has bandwidth a v ailable for either driv e t
i
or u
i
Without loss of generalit y supp ose that it writes
the page on driv e t
i
Because there is not memory a v ailable at instan t l the page m ust b e read during
in terv al l Ho w ev er there is not a v ailable disk bandwidth for driv e s
i
during in terv al l Then the
system m ust either replicate a from s
i
to a disk with a v ailable bandwidth during l so that a can b e
retriev ed from the other disk or replicate a page b retriev ed from s
i
during l in Ret to a driv e with
a v ailable bandwidth during l so that a is retriev ed from s
i
and b from the new lo cation during l Then
the system m ust replicate a page a or b from s
i
to d
ni
Tosc hedule this replication the write a page on
driv e d
ni
m ust b e sc heduled at in terv al l b ecause it is the only o dd time in terv al b efore l with a v ailable
disk bandwidth for d
ni
Then the system has to sc hedule a read from s
i
at in terv al l b ecause there is
not memory a v ailable at instan t l If there is only one clause in the SA T instance then an alternativ e
sr
to sc hedule the replication asso ciated with v
i
in Lemma is smaller than sr If there is more than one
clauses in the SA T instance then there is not a v ailable disk bandwidth for s
i
at in terv al l Therefore
as b efore the system has to replicate a page from s
i
to either d
n
if there is a v ailable bandwidth in d
n
or
d
n i
Because there is not a v ailable bandwidth for driv e d
n
during an o dd time in terv al the system has
to sc hedule the replication from s
i
to d
n i
Similar reasoning can b e applied iterativ ely to conclude that
an alternativ e sr
to sc hedule the replication asso ciated with v
i
in Lemma is smaller than sr Weno w conclude the pro of of the other direction of the equiv alence of instances
Lemma Let R A N be the output of SAT R epS c C
C
n
v
v
k
Let fP
P
m g P B C D b e the output of SAT ResS c C
C
n
v
v
k
If there is a resource sc hedule that yields a
onetime in terv al latency and supp orts a co ordinated displayof fP
P
m g on a system conguration
B C D and an initial placemen tof data P then there is a replication sc hedule for R on A during N Pro of Supp ose that there is a resource sc hedule Sc for fP
P
m g that yields a latency of one in terv al
and there do es not exist a replication sc hedule for R on A during N Consider the follo wing sc hedule for
replications R on A Sa F or eac h replication r R consider a sc hedule sr
in Lemmas and suc h
that sr
sr where sr is the sc hedule of r s extension in Sc
Because there is not a replication sc hedule for R on A there m ust b e t w o replications r
and r
suchthat
their corresp onding sc hedules in Sa conict The only p ossibilit y of conict b et w een the sc hedules for r
and
r
in Sa is if r
is asso ciated to a v ariable v
i
and r
to a clause C
j
Because the other com binations do not
ha veo v erlapping p erio ds Without loss of generalit y supp ose that the sc hedule of r
in Sa span the p erio d
x x n where x n i and the sc hedule of r
the p erio d x j x j
Both sc hedules require a memory frame at instan t x j Then the sc hedules of the extensions
of r
and r
in Sc w ould also require t w o memory frames at instan t x j Ho w ev er there is
only one memory frame a v ailable at this instan t Therefore Sc is not a resource sc hedule for fP
P
m g
that yields a latency of one in terv al whic hcon tradicts our assumption ab out Sc The computation of SAT ResS c is p olynomial time Therefore b ecause of Lemmas and w e
conclude
Theorem Deciding whether there is a resource sc hedule that yields the latency of one time in terv al
for a giv en displaysc hedule is NPComplete
Corollary Computing the r esour cesche dule that yields the minimum latency for a given display sche dule
is NPhar d
The computation of a resource sc hedule is constrained b y the memory capacit y of the system An
increase of the memory capacit y migh t lead to a decrease in latency One question that arises is what the
minim um memory requiremen t is to render a resource sc hedule with a giv en latency Ho w ev er deciding
whether there is a resource sc hedule that yields a latency of one time in terv al on a system with memory
capacit y C is NPComplete Therefore computing the minim um memory capacit y is NPhard
Corollary Computing the minimum memory r e quir ements to r ender a r esour c e sche dule that yields a
given latency is NPhar d
Conclusions and F uture Researc h
A co ordinated displa y of a structured presen tation m ust satisfy the temp oral and spatial constrain ts asso ci
ated with eachobject Once the displa y starts ob jects m ust b e rendered at presp ecied times dened b y the
temp oral constrain ts W e studied the complexit y of a resource sc heduler that supp orts a co ordinated displa y
of structured presen tations for a m ultidisk arc hitecture W e sho w ed the follo wing the computation
of a resource sc hedule that supp orts a co ordinated displa y and yields the minim um latency is an NPHard
problem and giv en a system load the computation of a sc hedule to c hange the placemen t of data across
disk driv es in minim um time is an NPHard problem
One question that arises is whether the resource sc heduling for sp ecial cases of m ultidisk arc hitectures
suc h as a system with disks is still NPHard
References
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Description
Martha L. Escobar-Molano and Shahram Ghandeharizadeh. "On the complexity of resource scheduling for coordinated display of structured presentations." Computer Science Technical Reports (Los Angeles, California, USA: University of Southern California. Department of Computer Science) no. 629 (1996).
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Escobar-Molano, Martha L.
(author),
Ghandeharizadeh, Shahram
(author)
Core Title
USC Computer Science Technical Reports, no. 629 (1996)
Alternative Title
On the complexity of resource scheduling for coordinated display of structured presentations (
title
)
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