Optimal Bandwidth/Delay Tradeo for
Feasible-Region-Based Scalable Multimedia Scheduling
Wei Zhao, Taruni Seth,y Michelle Kim,z and Marc Willebeek-LeMairz
Department of Computer Science yDepartment of Computer Science
University of Maryland
Columbia University
College Park, MD 20742
New York, NY 10027
[email protected]
[email protected]
Multimedia Networking Department
IBM T.J. Watson Research Center
Yorktown Heights, NY 10598
fyoon,[email protected]
z
Abstract
Feasible region is a simple and optimal framework
for scheduling the transmission of data with deadlines.
In this paper, we establish the fundamental relationship between the bandwidth requirement and the initial delay in feasible-region-based transmission scheduling. The relationship represents the optimal bandwidth/delay tradeo. In the process, we identify the
essential bandwidth, the exact bandwidth lower bound
regardless of initial delay. Ecient algorithms are
given to calculate the essential bandwidth as well as the
optimal bandwidth/delay tradeo. The results extend
the previous results on feasible-region-based scheduling,
most of which derived in the context of video trac
smoothing. When applied to the problem of scheduling
scalable multimedia, we show that the feasible region
framework enables the integrated scheduling of presentation and transmission and is a new and promising approach of dealing with scalable multimedia. We establish the presentation feasibility condition and present
some initial results on scheduling spatial scalability.
1 Introduction
Distributed multimedia applications have inherent
temporal constraints, typically expressed in terms of
the imposed deadlines, on the delivery of multimedia
information over the network. In the case of video,
for example, frames must arrive before their scheduled
Supported in part by the Army Research Laboratory under
Cooperative Agreement DAAL01-96-2-0002.
display time to ensure playback continuity. Feasibleregion-based scheduling is an elegant framework for
treating the problem of transmission scheduling of objects with delivery deadlines. Problem parameters including object sizes, deadlines and receiver buer size
collectively determine a feasible region within which
any valid transmission schedule curve must lie. The
problem thus is reduced to constructing a transmission curve within the feasible region with certain
application-specic properties.
A number of researchers have studied the problem of
feasible-region-based scheduling in the context of video
smoothing techniques [13, 8, 15, 7]. In a network with
strict end-to-end Quality of Service (QoS) guarantees
[9, 12, 4], a smoother transmission schedule results in
more ecient resource allocations. In [15] for example,
an algorithm was introduced to calculate an \optimal"
transmission schedule, where the optimality is dened
in terms of the \smoothness" based on the theory of
majorization.
We study the eect of initial delay on the transmission schedule bandwidth and establish the optimal
bandwidth/delay tradeo. The relationship is essential for applications to make optimal bandwidth/delay
tradeo decisions, for example, in environments with
transmission channel cost or bandwidth constraints. In
the process, we identify a threshold delay, or critical delay, beyond which the bandwidth of a valid transmission schedule can not be further reduced. We dene the
corresponding exact lower bound on bandwidth as the
essential bandwidth. Algorithms are presented to calculate the entire optimal bandwidth/delay tradeo in
( ) time, the same complexity as generating a single
O N
transmission schedule. For any point on the optimal
bandwidth/delay tradeo curve, the corresponding optimal transmission schedule can be directly obtained
without any extra computation. In essence, we are able
to generate the entire spectrum of optimal transmission
schedules with continuous bandwidth/delay tradeos,
in the same ( ) time.
We apply the feasible region framework to the
problem of bandwidth-constrained scalable multimedia
scheduling. Scalable multimediais a general framework
of multimedia presentations consisting of multimedia
objects with spatial and/or temporal scalabilities, and
is an eective way of handling receiver heterogeneity
and limited or changing bandwidth situations [6]. In
scalable multimedia, actual presentations are generated through presentation scheduling with the given
constraints and scalabilities. Based on our results of
optimal bandwidth/delay tradeo, we derive the exact feasibility condition for scalable multimedia presentation with a given bandwidth constraint. Through
the use of feasible regions, we integrated the problems
of presentation scheduling and transmission scheduling
into one integrated scheduling. We believe that feasibleregion-based scheduling is a promising and eective
approach for studying scalable multimedia scheduling
problems. As preliminary work in this direction, we
consider the case of spatial scalability. We show that
the problem of nding the \maximum" feasible presentation is NP-hard. However, we can nd a \maximal"
feasible presentation very eciently.
The rest of this paper is organized as follows. In Section 2, we introduce the framework of feasible-regionbased transmission scheduling and the optimal smoothing algorithm from [15]. Section 3 presents the denitions and properties of essential bandwidth and critical delay. In Section 4, we derive the optimal bandwidth/delay tradeo. In Section 5, we consider the
problem of scalable multimedia scheduling, where the
feasibility condition and some preliminary results on
spatial scalability are presented. Finally, we conclude
the paper with a summary of our contributions and
future research.
O N
2 Feasible-Region-Based Transmission
Scheduling
Let the presentation consist of objects, each with
( ) amount of data and having a presentation deadline of ( ). In addition, there is a buer of size
at the client side to temporarily store received data.
The problem is visualized in Figure 1: the horizontal
axis represents time and the vertical axis represents
the amount of data received by the client. A transN
s i
t i
M
t
y
mission schedule is represented by a monotonically increasing function curve starting from = 0. The slope
of the curve at any point is therefore the transmission
rate at that instant. A lower bound curve ( ) represents the minimum cumulative amount of data that
should have been received
P by time . By the deadline
requirements, ( ) = t(i)t ( ). A corresponding upper bound curve ( ) represents the maximal amount
of data that can be received by time . ( ) lies
units above ( ), ( ) = ( )+ . Any schedule lying
below ( ) leads to buer starvation, while any schedule lying above ( ) leads to buer overow. Thus
( ) and ( ) dene the boundaries of a feasible region
(Figure 1(a)). A transmission schedule is valid if and
only if it lies within the feasible region.
y
L t
t
L t
s i
U t
t
L t
U t
L t
U t
M
M
L t
U t
L t
U t
y
y
U(t)
U(t)
L(t)
valid
L(t)
invalid
t
0
(a)
t
0
(b)
Figure 1. Feasible region and optimal smoothing algorithm
An optimal smoothing algorithm was given in [15].
The algorithm produces a piecewise linear transmission schedule that is as smooth as possible, with the
minimum peak rate and minimum rate variance. The
algorithm is straightforward: starting with = 0, the
algorithm looks for the longest linear segment within
the feasible region (Figure 1(b)). If the segment ends
on the lower bound curve, it generates a schedule segment ending at the latest upper bound point it touches,
and vice versa. The procedure is then repeated from
the new start point. We refer to an enhanced version
of the algorithm in [14] with running time ( ) as
algorithm A throughout the paper.
y
O N
3 Essential Bandwidth and Critical Delay
By allowing an initial delay, senders of multimedia
applications can use a work-ahead approach by starting transmission some time prior to the presentation
time, so that the transmission rate could be reduced.
We dene the concept of Essential Bandwidth as the
limit to which the peak transmission bandwidth can
be reduced by adding additional delays.
Denition 1 The Essential Bandwidth EB is the ex-
act lower bound on the peak-bandwidth of any transmission schedule, regardless of initial delay. Or, EB = d
inf fS2S
infd
0
( )
( )g
slope no less than the slope of the direct line linking
?U (s) . Thus we
( ( )) with ( ( )), which is L(t)t?s
?U (s) = . Note
have
( ) sup0stt(N ) L(t)t?s
that this holds for any schedule , we further have
EB = inf d0
( ( )) inf d0 = .
s; U s
t; L t
S
P eak S
(1)
P eak S
B
P eak S
d
B
B
y
S
where S (d) is the set of valid transmission schedules
with delay d, P eak(S ) is the peak-rate of schedule S .
L(t)
U(s)
Let S (d) be the optimal schedule obtained by algo-
rithm A with delay , we have
Lemma 1
( ( )) decreases monotonically with
d
P eak S
,
d
s
time
t
d
( ( 2)) ( ( 1)) 80 1 2 (2)
Proof : Construct a schedule 2 ( 2) from ( 1)
as follows, let the rst segment of be from time ? 2
to ? 1 with slope 0, then follow the schedule of ( 1 ).
Thus,
( ( 2)) ( )=
( ( 1 )). ut
P eak S
d
P eak S
d
;
d
S
d
S d
S
d
S
d
d
S
P eak S
d
P eak S
P eak S
d
d
Denition 2 The Critical Delay is the minimum
initial delay where EB is reached,
= inf f 0j
( ( )) = EBg
(3)
d
d
d
P eak S
d
The following theorem establishes essential bandwidth in closed-form.
Theorem 1 Essential bandwidth EB satises,
( ) ? ( ) 0g
EB = maxf sup
?
s<tt N
?
= maxf i<jN
max ( ( ()))?? (( )( ) ) 0g (4)
L t
0
(
U s
t
)
s
L t j
1
B
B
t N ;L t N
L
B
B
B
L
S
P eak S
U t
B
P eak S
B
B
U
L
B
B
B
B
B
(b)
(a)
Figure 3. Schedule with peak slope B
U t i
Now we show the rst part of the equation. Let
?U (s) . We rst show EB .
= sup0stt(N ) L(t)t?s
Consider any two time instants 0 ( )
(Figure 2). Since ( ( )) is on the upper bound
curve, any schedule must cross time below or at
( ( )). Similarly, must cross time above or
at ( ( )). Therefore, any piecewise linear segment
connecting the two crossing points must have a peak
B
B
s
t
t N
s; U s
S
s; U s
d
U
;
t i
Proof : The second part the equation is easily veried.
Due to the right-continuity of ( ) at stepping point
( ), the term ( ( )? ) is equal to ( ( ? 1)).
U t i
from s to t
We then show EB by constructing a transmission schedule with peak rate from end point
( ( ) ( ( )) backwards. The schedule follows the
as long as its slope is at most . Whenever its
slope exceeds , we extend the schedule with a segment of slope (Figure 3(a)), as long as it remains in
the feasible region. We claim the end point of this segment must be on . The same procedure is repeated.
We now have a valid schedule with peak rate ,
thus EB = inf d0
( ( )) ( ) = . Figure 3(b) shows why the segments with slope never
penetrate rst. Otherwise, we have a segment linking and with a slope greater than , contradicting
's denition.
ut
where s and t are real numbers and i and j are integers.
t i
S
;
U t i
t j
Figure 2. Schedule
s
S
t
t; L t
Following mathematical conventions, \inf" denotes the highest lower bound and \sup" denotes the lowest upper bound.
The theorem gives a simple iterative ( 2 ) algorithm for calculating EB . Actually, EB can be computed easily in ( ) time using algorithm A , but with
a dierent starting point.
Theorem 2 EB can be computed in ( ) time, EB =
( 0 ), where 0 is the output schedule of algorithm
A starting at ( (1) ( (1)? )).
Proof : Extend the rst segment of 0 toward = 0.
In case the intersection is greater than zero, extend it
O N
O N
O N
P eak S
S
t
;U t
S
y
to time zero along the time axis (Figure 4(a) and 4(b)).
Construct schedule by concatenating the added segment(s) and 0 . is a valid transmission with an
initial delay 0 0 and
( )=
( 0 ). Thus
0
EB ( )=
( ).
S
S
S
d
P eak S
P eak S
P eak S
P eak S
the entire schedule into pieces delimited by the convex change points, we have a series of contiguous concave regions with only concave internal change points.
We will show that the bandwidth/delay relationship
is completely determined by the essential bandwidth
EB and the rst concave region of (0). Since both
EB and (0) can be calculated in ( ) time, the entire bandwidth/delay relationship can be determined
in ( ) time.
Lemma 2 All segments in (0) except the ones in
S
y
y
S
O N
O N
S
the rst concave region have slopes not exceeding EB .
start
start
U(1-)
S’
U(1-)
S’
S
0
-d’
t
t(1)
S
0
d’=0
t
t(1)
(b)
(a)
Figure 4. Backward extension of S 0
P
P eak S
d
d
t
S
P eak S
S
y
d
U t
;U t
t
;U t
S
d
S
P eak S
d
t
d
P eak S
concave region, it suces to show that the rst segment
in each concave region except the rst one has slope
at most EB. Suppose there is a rst segment of a
concave region with slope greater than EB. must
start from , by denition. If ends on , then we
have a segment with slope greater than EB connecting
to , contradicting Theorem 1. If ends on ,
then 's extension will cross (by algorithm A ), also
resulting in a slope contradicting Theorem 1.
ut
Let the time coordinates of the + 1 change
points on the schedule (0) in the rst concave region (including boundary) be (0 ( 1) ( 2 )
( K ))
and let the bandwidth (slope) of the corresponding segments on (0) be ( 0 1
K?1), where i =
L(t(c +1 ))?L(t(c )) is the slope of the th segment. All
t(c +1 )?t(c )
internal change points are on and 0
1
K?1 (Figure 6).
From each concave change point at ( i), =
12
? 1, we extend the schedule segment with
slope i toward = 0, let the intersection be ? i .
Let K = 0 and 0 = 0. Now each change point at
( i ), = 1 2
, determines a bandwidth interval ( i i?1] and a delay interval [ i?1 i). It shall
become clear shortly that a one-to-one projection between them represents the optimal bandwidth/delay
tradeo. Let the bandwidth interval containing EB be
( k k?1], k EB k?1. From the the corresponding change point at ( k ) we start a segment with
slope EB towards and landing on = 0 at ? 0. It will
be shown that 0 is exactly the critical delay dened
earlier ( 0 = ).
We construct a set of transmission schedules
f 0 ( )j0 0g. Each 0 ( ) is a schedule with
initial delay constructed as follows: the rst segment of 0 ( ) goes from (? 0) directly to a change
point ( ( i ) ( ( i))), then follows the schedule (0)
until it ends. The change point ( ( i ) ( ( i ))) is the
one that determines the delay interval containing ,
2 [ i?1 i). By construction, the rst segment of
0 ( ) has a slope at least EB.
P
We now show that longer delays do not result in a
lower bandwidth than
( 0 ). By the monotonicity
lemma, we only need to show
( ( )) ( 0)
0
for any . The rst segment of ( ) must
cross the horizontal line = ( (1)? ) at or to the
right of ( (1) ( (1)? )) (Figure 5). Now construct
a 00 from ( (1) ( (1)?)) by following the horizontal line and then following ( ) after the intersection.
Since 0 is the optimal schedule from ( (1) ( (1)?)),
( ( )) =
( 00 ) ( 0 ). The algorithm
runs in the same linear time as algorithm A .
ut
P eak S
Proof : With segments of decreasing slopes in each
;U t
P eak S
y
U
U
P
L
L
P
P
U
L
K
S
;t c
S
; : : :; t c
B ; B ; : : :; B
i
i
i
;t c
B
i
i
L
B
> B
t c
;
B
start
t c
S’
S’’
y
0
-d -d’
t
t(1)
Figure 5. Construct
S
d
d
i
;
; : : :; K
B ;B
S*(d)
00 by merging into S (d)
d
B ;B
B
<
;d
B
t c
y
4 Optimal Bandwidth/Delay Tradeo
Let the optimal schedule obtained by algorithm
(0). Using the notions in
[15], each point at which the slope of the schedule
changes is either a concave change point where the
rate decreases, or a convex change point where the
rate increases. All concave change points lie on the
lower bound curve whereas all convex change points
lie on the upper bound curve (Figure 6). By dividing
A with delay = 0 be
d
S
i
; : : :; K
B
U(1-)
> ::: >
B
d
d
d
S
d
d
d
d
S
d
d
S
t c
d
d;
;L t c
S
t c
;L t c
d
d
S
d
d
;d
B
y
S*(0)
B0
B1
F
B2
Concave
EB
B3
-d3
-d*(-d’)
B B2
-d
-d2
B1
B0
0 t(c1) t(c2)
Concave
EB
Concave
d
d
d
d
( 0 ( )) equals the slope of its rst segment.
Proof : It suces to show that algorithm A with initial
delay generates 0 ( ). We rst show that the rst
segment of 0 ( ) lies in the feasible region. To see
that is above , we note that (0) lies above , and
that lies above (0) by construction. To see that
is below , we use Theorem 1 and a similar argument
as in its proof (Figure 3(b)): if violates , we have
a slope from to greater than EB.
Starting from (? 0) (Figure 6), algorithm A looks
for the longest segment within the feasible region. Due
to the construction, any segment with slope lower than
touches earlier, any segment with slope higher
than touches earlier. Hence is indeed the rst
segment selected by algorithm A . Algorithm A then
works from the end-point of , also a change point on
(0), generating the same schedule afterwards.
( 0 ( )) is dominated by the rst segment, with
slope at least EB. Slopes in the rst concave region are
less than its rst segment and slopes in other concave
regions are less than EB (Lemma 2).
ut
P eak S
d
d
S
P
S
P
P
d
S
L
S
L
U
P
U
U
L
d;
P
L
P
U
P
P
S
P eak S
d
Lemma 4 The critical delay can be derived as follows,
EB
= 0L t c
(5)
?
(
)
i
i EB i?
EB
Proof : The EB
case is veried easily. For the
d
> B0
( ( i ))
t c
B
<
B
1
> B0
the second case, note that the right side of the equation is actually the expression for 0 dened earlier,
thus we only need to show = 0. By Lemma 3
( ( 0 )) = EB, since EB is its rst segment slope.
0,
For any 0 ( ( )) EB, since its rst
segment has slope greater than EB.
ut
d
d
P eak S
d
d
d < d
P eak S
d
>
Theorem 3 The following one-to-one correspondence
F (Figure 7) exists between optimal transmission bandwidth 2 [EB ] and minimal initial delay 2
B
; B0
d
[0 ]. Furthermore, 0 ( ) is the optimal schedule for
delay and corresponding bandwidth = F ( ).
;d
d
S
d
d
B
From delay to bandwidth:
B
d
L
d*
Figure 7. Optimal bandwidth/delay tradeoff
function F
Lemma 3 0 ( ) is identical to the optimal transmission schedule ( ), for any 0 0; furthermore,
S
d2
t
t(c4)
Figure 6. Bandwidth/delay tradeoff calculation
S
d1
0
t(c3)
-d1
= F( ) =
d
(
EB
d
L(t(ci))
t(ci )+d
d
d
d
i?1 d < di
(6)
From bandwidth to delay:
0
= F ?1( ) = 0L(t(c )) ? ( )
i?1 (7)
i
i
B
Proof : From Lemma 3, the peak rate of the optimal
schedule 0 ( ) with delay is the slope of its rst segc )) , ( ) is the corresponding change
ment,which is Lt((ct()+
i
d
point of . The relationship follows.
ut
We point out that our results extend the previous
results of [15, 8], where an optimal schedule is calculated given an initial delay. We have an algorithm that
computes the entire optimal bandwidth/delay tradeo relationship, in the same ( ) time. The bandwidth/delay relationship is essential for applications in
making optimal tradeo decisions between bandwidth
reservation and presentation delay. Given a bandwidth
constraint, the minimal delay required to meet the
bandwidth constraint can be derived by the mapping
function. Vice versa, given a delay constraint, the corresponding optimal bandwidth can be computed. In
both cases, the optimal transmission schedule is directly obtainable. In essence, we have an algorithm
that computes the entire spectrum of optimal schedules with continuous bandwidth and delay tradeos in
( ) time.
d
B > B
B
S
i
d
t c
B
< B
B
d
i
i
t c
d
O N
O N
5 Bandwidth-Constrained Scheduling
of Scalable Multimedia
Recently, numerous research activities have been
reported integrating multimedia objects with varying timing requirements in a multimedia presentation
[1, 2, 11]. Their main results were centered around the
formal specication and modeling of multimedia composition and the development of synchronization tools
such as the automatic scheduling of presentation times.
A multimedia presentation may contain text, audio,
video, graphics, where the objects are related and can
be scaled temporally as well as spatially. We refer to
this type of multimedia presentation scalable multimedia. Scalable multimedia in a distributed networking
environment presents a new set of challenges. Most importantly, the lack and unpredictability of bandwidth
demand a more general framework where all these constraints can be dealt with.
Scheduling scalable multimedia consists of presentation scheduling and transmission scheduling. The former refers to the actual presentation content generation and the latter schedules the transmission of data.
In bandwidth-constrained environment, the bandwidth
limit imposes a constraint on the transmission schedule, which in turn restricts the possible presentation.
There have been approaches to treat the two problems
separately such as [1, 10]. There are also approaches to
concatenate the two stages into a feedback loop such as
[3], where presentations are generated until the resulted
transmission schedule satises the resource constraints.
We propose the approach of integrated presentationtransmission scheduling. Based on our results on optimal bandwidth/delay tradeo, we can translate the
imposed bandwidth constraint directly into the constraints on the presentation through the use of feasible region. Consequently, feasible presentations can be
generated by solving these constraints as a whole.
5.1
Given a presentation, we say it is feasible under
bandwidth constraint B if its essential bandwidth EB
does not exceed B. With EB not exceeding B, the optimal bandwidth/delay tradeo can be used to nd a
minimal delay and an optimal transmission schedule.
Let the optimal schedule with no delay be (0), three
possibilities exist:
S
1. EB ( (0)) B. The presentation is feasible and (0) is the optimal transmission schedule
with no delay.
P eak S
S
2. EB B
( (0)). The presentation is still
feasible, but it has to be delayed by = F ?1 (B),
F is the bandwidth/delay tradeo function.
< P eak S
d
3. B EB ( (0)). The presentation is not
feasible, must scale down the presentation.
P eak S
L t j
s k
U t i
t j
t i
i < j
t j
N
M
t i
M
o
is the kth object ordered by its presentation time t(k),
and s(k) is the size of object ok .
Proof : The corollary follows from the denitions of
( ) and ( ) and Theorem 1.
ut
U t
L t
By Corollary 1, we map the bandwidth constraint
into a set of presentation constraints (8). Consequently, the problems of presentation and transmission
scheduling can be integrated into one scheduling problem and handled in a unied fashion.
5.2
Scheduling Spatial Scalability
Multimedia information typically can have dierent
quality, resolution, sampling rate, etc. that result in
dierent data sizes. Such spatial scalability is an effective way of adapting a presentation to certain bandwidth or other resource constraints.
Lemma 5 Essential bandwidth of a presentation never
decreases as a result of an increase in object's size.
Proof : Let the object being? increased be k . We ex(t(i)
amine the term L(t(tj())j )?U
?t(i)
o
before and after the increment. For
, the term does not change.
For , since and are cumulative, the relative positions of the two points thus the slope does
not change either. For
, the slope can only
increase. The lemma follows from Theorem 1.
ut
By Lemma 5, we can test the feasibility of the \minimum" presentation with each object having its minimum size to see if there exists a feasible presentation at
all. Within the feasibility constraints, it is desirable to
have a high quality presentation, which usually implies
that the sizes of the objects be as large as possible. We
call a feasible presentation \maximum" if its total size
is the largest over all feasible ones. Alternately, we call
a feasible presentation \maximal" if the increment of
any object's size results in infeasibility. Unfortunately,
nding a \maximum" presentation turns out to be NPhard, proof given in the appendix.
)
i < j < k
k
i < j
U
L
i < k
Feasibility Condition
<
Corollary 1 A presentation is feasible, if and only if,
P
( ( )) ? ( ( )? ) = ikj ( ) ? B (8)
( )? ( )
( )? ( )
for any 1 ; where is the buer size, k
j
Theorem 4 The problem of nding the maximum feasible presentation is NP-hard.
Nevertheless, a \maximal" feasible presentation can
be found eciently. The algorithm starts with the
\minimum" presentation mentioned earlier. At each
step, we enlarge the current presentation by increasing
a certain object's size, while maintaining its feasibility.
We examine the eect of increasing
the size of object
L
(t(j ))?U (t(i)?)
of
condition
(8). For
n on the term
t(j )?t(i)
or , the term does not change.
Only for
will the term increase in value.
To maintain feasibility, it should not exceed B for all
after the increment.
o
i < j < n
n
i < j
i < n
i < n
j
j
y
The algorithm iterates over objects from 1 to N .
During each iteration, it computes the current ceiling
line and the minimumdistance to , and then increases
the object's size according to (10).
o
o
L
Theorem 5 The above presentation scheduling algorithm nds a maximal feasible presentation.
Proof : The feasibility part is implied by construction.
The maximality part is shown by tracing the execution path. Let be the presentation generated. We
increase the size of any object k to form a new presentation 0. We show 0 is not feasible. Let the intermediate presentation just after the increment of object
k during the algorithm's execution be 00. is identical to 00 before ( ), but no less than 00 after ( ).
The only reason that the algorithm did not choose a
larger k in 00 was because it would raise from ( )
to ( ) over the ceiling line at ( ). Because and 00
have the same conguration before ( ), they share the
same ceiling line. Thus 0, resulted from an increase
in size of k to , will also have a lower bound curve
violating the ceiling line, making itself infeasible. ut
The running time is easily bounded. There are at
most iterations, one for each object. In each iteration, the brute-force calculations of ceiling line and
distances take ( ) operations. Choosing a size from
a set ( ) takes log
We
P j ( )j using a binary search.
have an ( 2 + log j ( )j) algorithm.
We
can
reP
duce the running time to ( + log j ( )j) by updating the ceiling line and the distance values instead
of calculating them from scratch in each iteration.
P
B
o
P
e(n+3)
e(n+2)
e(n+1)
o
e(n)
U(t)
L(t)
0
t n
L
t N
t n
Lemma 6 A presentation is feasible, if and only if, the
ceiling line at any object time t(n) bounds from above
the lower bound curve from t(n) to t(N ).
Proof : Condition equivalent to Corollary 1.
ut
Due to the cumulative nature of the bounding functions, an increment of object n raises from ( )
to ( ) by the same amount. To maintain feasibility, the raised from ( ) to ( ) must remain under
the ceiling line at ( ). Thus the maximum amount
that n can increase is bounded by the minimum vertical distance between the ceiling line at ( ) and from
( ) to ( ) (Figure 8). Let the ceiling line at ( ) be
= B + 0 , 0 is its intersection with the vertical axis.
The vertical distance from ( ( ) ( ( ))) on is,
( ) = B ( ) + 0 ? ( ( )) 0 for any (9)
We choose the maximal size that object n can increase
to without exceeding any ( ), ,
N
0 ( n )=max 2 ( n ), ( n ) + min
(
)
(10)
k=n
o
L
t n
t N
L
t n
t N
t n
o
t n
L
t N
y
t n
y
t k ;L t k
e k
t k
y
L t k
L
;
k
n
o
s
o
s
D o
e k
k
s
s o
n
e k
where 0 ( n) and ( n ) are the new and current sizes of
n , respectively, and ( n ) is the set of available sizes
of n .
s
o
o
s o
D o
o
L
t k
t k
P
P
P
o
t n
P
t k
t k
On the feasible region (Figure 8) determined by a
feasible presentation containing object n , we draw a
series of parallel lines with slope B from stepping points
on before time ( ). Observe that all points on the
from ( ) to ( ) must not exceed any of these parallel
lines to ensure feasibility. We call the lowest of these
parallel lines the ceiling line at ( ).
U
P
P
t N
t
t(n)
Figure 8. Effect of increasing the size of on
t
t k
o
y0
y
P
P
ceiling line
t n
P
o
P
N
O N
D o
O N
D o
D o
O N
D o
6 Conclusion
We identied the optimal bandwidth/delay tradeo
in feasible-region-based transmission scheduling problems. We dened the concepts of essential bandwidth
and critical delay that are properties of the feasible
region. The relationship between bandwidth and delay is essential for applications to make optimal bandwidth/delay tradeo decisions. We gave algorithms to
determine the entire bandwidth/delay tradeo curve
as well as the entire spectrum of optimal schedules in
( ) time. Applying feasible-region-based scheduling
to scalable multimedia, we proposed the idea of integrated presentation-transmission scheduling and gave
the exact presentation feasibility condition. Some preliminary results on scalable multimedia scheduling in
the presence of multimedia spatial scalability were presented. We believe feasible-region-based scheduling
represents a new and promising approach to the many
open issues in scalable multimedia, which are the main
focus of our ongoing and future research.
O N
References
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A NP-hard Proof for the Maximum
Feasible Presentation Problem
We show the problem is NP-hard by reduction
from the well-known NP-hard problem Partition [5].
The partition problem is the following, given nonnegative integers in set , is there a partition of such
that the sum of the two subsets are equal. From the
partition problem with = f i g = 1 2
, we
dene an instance of the maximum feasible presentation problem with the following parameters. There are
objects, each with two possible sizes ( i ) = f0 ig.
Set the time of presentation
P ( ) = and the buer size
= 2, where = Ni=1 i is the sum of the integers in . The bandwidth constraint is chosen to be
B = 0 5 ( ? 1). We claim that there is a partition of
if and only if the maximum feasible presentation has
a total size of 2.
It is clear that if there is a presentation of size 2,
then the set of i 's being chosen and the rest form
a partition. On the other hand, if there is a partition into 1 and 2 , we show that the maximum
feasible presentation has a total size of exactly 2.
First we claim that the size of a feasible presentation can not be more than 2. If it is, we have a
slope from (1 (1?)) to ( ( )) with slope at least
( 2 + 1 ? ) ( ? 1) = 1 ( ? 1) B. We then
show there exists a feasible presentation with a total
size of 2 constructed as follows. For each object i ,
we choose size i if it is in 1 , 0 otherwise. Thus we
have a presentation of size 2. Observe that the highest point on the lower bound curve has a value of
2 and the lowest point on the upper bound curve
also has value of = 2. Hence any line from
to has a slope of at most 0 B, conrming the
presentation's feasibility.
2
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