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Laboratoire de l’Informatique du Parallélisme
École Normale Supérieure de LyonUnité Mixte de Recherche
CNRS-INRIA-ENS LYON-UCBL no 5668
The General Broadcast Scheduling
Problem with uniform length andoverlapping message sets
Master Thesis of
Sandeep Dey
Advisor: Nicolas Schabanel
4th July 2005
École Normale Supérieure deLyon
46 Allée d’Italie, 69364 Lyon Cedex 07, FranceTéléphone :
+33(0)4.72.72.80.37
Télécopieur : +33(0)4.72.72.80.80Adresse électronique :
[email protected]
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The General Broadcast Scheduling Problem with
uniform length and overlapping message sets
Master Thesis of
Sandeep Dey
Advisor: Nicolas Schabanel
4th July 2005
Abstract
The Broadcast Scheduing Problem consists of finding an infinite
schedulethat broadcasts news items so as to minimize the average
service timefor clients requesting subsets of the news items. In
the present day era,the demand for such type of a broadcasting
service is very high, sinceit allows for efficient dissemination of
data to a large number of passiveclients such as in satellites,
radios, cable TV networks etc.Previous work in this area
concentrated on scheduling news items whena client required only
one news item. Here we address a generalizationof the problem and
consider the situation when a client could requestfor any subset of
items. We conclude that the problem is NP-Hard andgive two new
approximation algorithms, a randomized algorithm withan
approximation factor of 2Hn and a 4 factor deterministic
algorithm.To complete the analysis, we come up with a new lower
bound.
Keywords: Data broadcast, Approximation algorithms,
Randomizedalgorithms, Lagrangian relaxation, Perfectly periodic
schedules
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1 Introduction
With the rapid growth of the Internet and its user base, and
with theavailability of high bandwidth links to almost all places,
networks have chan-ged the way data is delivered and distributed
between computers. Withthese technological improvements, the cost
of transeferring data has redu-ced greatly both monetarily and
bandwidth wise, due to which a lot ofnew applications like
distributing data on a stock exchange, traffic flow in-formations
and audio-video broadcasts have come up. These systems
havethousands of clients recieving data from a main central server
e.g ADSL TV,radios etc. As such huge amounts of data have to be
disseminated in realtime.
However these advances in communication along with the
increasing sizeof the network is testing the limits of a lot of
assumptions which were madeinitially to design distributed systems.
The principles and designs of thedistributed systems studeis here
need to be checked to keep them up to datewith the technological
advances.
An important change in network technologies has been the advent
ofsystems where servers should be capable of delivering large
amounts of in-formation to a huge number of users, especially in
popular events like theolympic games, etc. As a result new
innovative delivery technologies likesatellite communication and
cable networks have been devised to provideshared broadband
internet access.
Different from traditional networks, these new technologies have
one dis-tinguishing feature, they support broadcasting much more
naturally thanany systems before. In contrast to unicast, where an
object on interest toa lot of clients had to be transferred to all
of them individually, broadcastsends the object only once, thus
making very efficient use of the sharedbandwidth.
With the advent of wireless networks as one of the major
developingareas in computer networks today, broadcasting gets a new
meaning. Herethere are cases when the bandwidth from the source to
the client is muchmore than in the reverse way e.g. when handheld
wireless devices may beable to download traffic flow information
from a centralized wireless server.In these cases the client device
doesnt need an emitter if the download isthrough a broadcast.
Broadcast systems are push-based systems, where the client does
notrequest for data, but simply connects to the broadcast channel
shared by allthe clients e.g radios . Basically the server ”pushes”
the data to the clientsaccording to a schedule which does not
depend on the incoming requests.These schedules generally are
designed by user profiles which imply thepopularity of the
meessages relative to each other.
The data broadcasting protocols have a number of research and
com-mercial applications. Boston Community inormation system (BCIS,
1982)
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was one of the first application of these protocols to deliver
news and otherinformation to handheld radios. teletext and videotex
systems [1, 9] alsouse these protocols. The ATIS (advance traffic
information system) [17] pro-vides information to vehicules
specially equipped with computers, to recievetraffic flow data by
the same protocols and internet news delivery systemsmake use of
the same.
All the present problems deal with the the problem where either
demandof two clients is exactly the same or the demand of two
clients is totallydifferent. The idea we are driving to is a
customised news service, whereevery client has his choice to choose
any subset from a set of news items.Earlier on these kinds of
systems were not present, where information wasntsegmented, people
who wanted to find traffic flow on a particular streetwould get the
whole information in one broadcast.
We consider the issue of the broadcast newspaper. A broadcast
newspa-per is an internet based news service. The central server
broadcasts a set ofnews items. Different clients may want different
news items e.g one may wantto download the sports and the
entertainment news, while another one maylike to download the
sports and the national news. The problem which isposed , is how
often should one broadcast a news items. we will see that
thegeneral ideas of the past dont work. New broadcast scheduleing
algorithmsare needed to cope with the present problems. It is with
this idea, that weundertook a study of the theoretical base of
these practical problems andwere able to provide an efficient
algorithm.
1.1 State of Research
The research on data broadcast problems in a setting where all
messageshave the same length and the broadcast is done on a single
channel withdiscrete time, started in the early 1980’s [9, 1, 2, 4,
11]. Ammar and Wong [1,2] analysed the periodic schedules, gave an
algebraic expression for the Cost(defined as the average response
time to user), a lower bound and provedthe existence of a periodic
schedule which is optimum. Bar-Noy, Bhatia,Naor and Scheiber [4]
prove that problem with broadcast costs are NP-Hardand are able to
give a constant factor algorithm. Kenyon, Schabanel andYoung [11]
design a PTAS for the problem. In his Thesis [15],
Schabanelproposed several constant factor approximation algorithms
for non uniformlength, preemption and both together.
Our work draws very strongly from one another topic, perfectly
periodicschedules. Hammed and Vaidya [19, 18] propose the weighted
fair queuingto schedule broadcasts. Khanna and Zhou [12] show how
to use indexingwith periodic scheduling to minimoze busy waiting.
They also give an ap-proximation algorithm for designing periodic
schedules. Bar-Noy et al [5]introduce the tree schedule design and
the notion of perfect periodicity.
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1.2 Our Contribution
In this report, we address the issues of the broadcast
scheduling problemwhere clients are free to choose any subset of
messages. Schabanel [15, 16]worked on the special subcase of the
same problem where no two subsetsoverlapped. As mentioned earlier,
research has been done on broadcast sche-duling topics where each
message has its own demand probability and withpreemption (meaning
sets of messages are requested but no two sets overlap),but no
previous algorithms have been proposed in the present problem.
We propose two seperate algorithms for the beforementioned
problem.The first one is a randomized approximation algorithm which
has an approxi-mation factor of 2Hn, Hn being the harmonic
function. The next algorithmis a deterministic approximation
algorithm with an approximation factorof 4. Both the algorithms are
simple and offer an intutive viewpoint of theproblem at hand.
1.3 Organisation of the report
The outline of the rest of the report is as follows :– The next
section deals with the notations and the preliminaries invol-
ved with the problem. First we introduce the model on which we
definethe problem and then the proper notations employed while
giving theproofs and the algorithms. Then we go on to prove the
NP-Hardnessof the problem, which in turn implies that no polynomial
time algo-rithm will exist unless P=NP. Moreover we look into the
relations ofa periodic and a non-periodic schedule.
– The 3rd section composes of our original contribution. In this
sectionwe propose the randomized and the deterministic
approximation al-gorithm which we have developed during the course
of the internship.We also discuss the importance of the lower bound
obtained and itsderivation.
– The last section consists of the summary and the conclusions
and somefurther problems along with some suggestions for further
research onthose problems.
2 Notation and Preliminaries
In the present section, we introduce the model on which we base
theproblem. Once the model is established, we go on present the
broadcastingproblem on this model along with a couple of notations
which will facilitatethe presentation of the proofs later on.
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2.1 Model and Notation
Consider that there is a news station. This news station
broadcasts newsof all kinds but there is always a limit on the
number of news items that thestation will broadcast on a given day.
The clients recieve the news simplyby starting their radio sets.
Since clients may want to listen to the news anytime of the day,
hence the station keeps on broadcasting the news items allover the
day not necessarily in a periodic manner. The clients which
recievethe news may want news of a particular kind e.g.
international, national,sports, entertainment, buisness etc. Any
news item can belong to one ormany of these categories. A client
after switching on his set, waits for sometime until a news item in
the news category he is interested in starts broad-casting. He
waits until all the news items in this particular category havebeen
broadcasted and then switches the set off. The waiting time for
thatclient is the total amount of time that his radio set was
switched on. At anytime a lot of clients are accessing the news
broadcast, but for simplicity wemake two assumptions.
– News are organized in possibly overlapping categories.– A
client only waits for only one news category (Strictly speaking
this
is not a restrictive assumption but if a client wants news from
twocategories like sports and entertainment, then we can make
anothernews category which has both sports and entertainment news
in it)
– The number of clients switching on their radio sets at any
time isuniform the whole day.
– The probability of clients waiting for a particular category
of news isthe same at all times of the day.
– The broadcast duration of all the news items is the same and
newsitems are broadcasted one after another without any delay. We
alsoassume that the broadcast time for any news item is unit time.
If thenews item has a length larger than the unit, then we can
break the newsitem into unit size and hence return back to our
basic assumptions.
A schedule is simply a way of broadcasting the news items one
afteranother. The waiting time for a message at time t is the
length of timeafter which it is first seen after time t. The
average waiting time for a newscategory is average of the waiting
times of clients waiting for that newscategory which switch on the
sets over the whole day. The Cost of a scheduleis the average of
waiting times of all clients all over the day. Our goal is
tominimize this quantity.
Now since the description of the problem has been given. We now
des-cribe our model formally. We call the news items messages from
now on.
A set of n messages (news items) is given M = {M1,M2, . . .
,Mn}. Aset of subset(news categories) of M , ζ = {S1, S2, . . . ,
Sk} where Si ⊆ M , isgiven. S will be a variable denoting an
element of ζ. A set (pi)i=1,2,...,k isalso given such that with
each of the subsets Si is associated the probability
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WT(M )6
t=0
MMM M1 M3 M4 M2 M7 M6 1 M1 M3M4M267M4
WT(S) where S={M ,M ,M }62 1
Fig. 1 – New Periodic Schedule Constructed
pi that a random client will want access the messages of Si. We
use pS todenote the probability associated with the set S. So it is
evident that
k∑i=1
pi = 1 (1)
We define a schedule Sch as an infinite sequence Sch = s0s1 . .
. wheresj ∈ {1, 2, . . . , n} for all j. If sj = i, we say that the
message i is scheduled tobe broadcast at slot j. A schedule is
periodic if it is an infinite concatenationof a finite
sequence.
Consider the length of time after which message Mi appears for
the firsttime after time t. Call it WT (Mi, t) which is hence the
waiting time for themessage Mi at time t. The waiting time for all
the messages of S at time tis
WT (S, t) = maxMi∈S
WT (Mi, t) (2)
Hence the waiting time for a request arriving at time t is
Cost(Sch, t) =∑S∈ζ
pS × maxMi∈S
WT (Mi, t) (3)
So the average cost for a schedule comes out to be :
Cost(Sch) = limsupT→∞1T
T∑t=1
Cost(Sch, t) (4)
Cost(Sch) = limsupT→∞1T
T∑t=1
∑S∈ζ
pS × maxMi∈S
WT (Mi, t) (5)
We use E(WT (Mi, t)) as the expected waiting time in the case
when weare dealing with randomized algorithms.We use E(WT (Mi, t))
to denotethe expectation of WT (Mi, t). Notice that we have used
Cost as functionfor the schedule at time t and as an average. The
usage of Cost can beinterpreted from the arguments which we are
using so there is no ambiguityin analysis.
The problem is defined as :
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Definition 2.1. General Broadcasting Problem with uniform length
:Given a set of messages M = (Mi)i=1,2,...,n and a set of subsets
(Si)i=1,2,...,ksuch that Si ⊆ M , and a set of probabilities
(pi)i=1,2,...,k, our aim is to finda schedule Sch such that
Cost(Sch) ≤ Cost(Sch′) ∀ Sch′ (6)
where Sch′varies over the set of all schedules
We additionally define fi to be the frequencies associated with
a messagein a periodic schedule which is the number of times a
message is broadcasteddivided by the total number of slots for any
schedule. τi denote the inverseof fi.
τi =1fi
(7)
which implies τi is the ”period” of message Mi or the average of
the numberof slots between two successive broadcast of Mi.
2.2 NP Hardness of the problem
We show that the above mentioned problem is a NP-Hard problem
whichimplies that a polynomial time algorithm is unavailable
(unless P=NP), thisalso implies that our best hopes lie with
approximation and randomizedalgorithms.
Theorem 2.1. The General Broadcasting Problem with uniform
length isNP-Hard.
Démonstration. Schabanel in [15, 16], proved that the same
problem withthe preemptive condition, i.e no Si’s overlap i.e Si ∩
Sj = ∅, ∀i �= j, whereSi corresponds to the packet of messages Mi,
is a NP-hard problem. Henceit proved that the General Broadcast
Problem with uniform length is a NP-Hard problem.
2.3 Optimal periodic schedule are arbitarily close to the
op-timum
We prove that there exists a Periodic schedule whose cost is
less than �+ the optimum cost of the schedule for any �.
Theorem 2.2. Let Sch be any schedule for a general broadcasting
problem,then ∀� > 0 there exists a peiodic schedule PerSch�such
that
Cost(PerSch�) ≤ Cost(Sch) + � (8)
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Previous Schedule Sch
Seq
All messages from 1 to n
repeat the prior Sch.
Fig. 2 – New Periodic Schedule Constructed
Démonstration. The basic trick is to take a subsequence from
Sch whosecost is close enough to the optimum and after some
additions use it as acycle for a periodic schedule. The Avg Cost
refers to the average cost tilltime T .By definition
Avg Cost(Sch, t) =1T
T∑t=1
Cost(Sch, t) (9)
andCost(Sch) = limsupT→∞ Avg Cost(Sch, t) (10)
Hence for all � , there exists a T ′ , such that for all T >
T ′
Avg Cost(Sch, T ) ≤ Cost(Sch) + �2
(11)
so we choose a T� > T ′ such that
2�(n + Cost(Sch, 0)) × n − n ≤ T� (12)
so
(n + Cost(Sch, 0)) × nT� + n
≤ �2
(13)
We take the sequence Seq of messages of Sch from the beginning
to slotT� and at the end of the Seq, add a small sequence Seq′ of
messages oflength n consisting of all the messages M1,M2, . . .
,Mn,where the messagesare added in order of their first appearance
in Sch after time T� in scheduleSch. We make a new periodic
schedule PerSch� with period T� + n wherethe beforementioned
sequence to be repeated. This is a periodic schedule.The costs of
the slots of Seq remain the same or decrease since the
addedsequence of Seq′ provides all the messages earlier than in
Sch.
For t ≤ T�Cost(PerSch�, t) ≤ Cost(Sch, t) (14)
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For T� < t ≤ T� + n
Cost(PerSch�, t) ≤ Cost(Sch, 0) + n (15)
since for the slots there the waiting time is less than or equal
to the waitingtime of the first slot (periodic schedule).Since this
is a periodic schedule thecost is equal to the Avg Cost of the
first T� +n time slots. The contributionif the first T� slots is
Avg Cost(Sch, T�) and the later n slots is less than orequal to
Cost(Sch, 0) + n
Cost(PerSch�) =Avg Cost(Sch�, T�) × T� + (Cost(Sch, 0) + n) ×
n
T� + n(16)
Cost(PerSch�) ≤ Avg Cost(Sch, T�) × T�T� + n
+(Cost(Sch, 0) + n) × n
T� + n(17)
Cost(PerSch�) ≤ Cost(Sch) + �2 +�
2(18)
Cost(PerSch�) ≤ Cost(Sch) + � (19)
Corollary 2.1. OPTperiodic schedules = OPTnon periodic
schedules
Hence proved that there exists a periodic schedule whose cost is
arbitarilyclose to cost of the optimum schedule.
3 Our Contribution
Our contribution to this problem basically lies in giving
efficient al-goithms for the general case of the broadcast
schedulin problem, where theconsumer gets to choose what news items
he wants to choose, includingmultiple choices from a predefined set
of genres.
This problem is a NP Hard one, which implies a polynomial time
algo-rithm to find the optimum schedule can not be found unless P =
NP. ForNP Hard or NP Complete optimization problems, the best known
algorithmstend to be approximation or randomized algorithms.
Approximation algo-rithms refer to algorithms for optimization
problems which output resultsguaranteed to be less than a factor of
the optimum solution. This factor ter-med as the approximation
factor could be anything starting from a constantto a polynomial in
the size of the problem. Randomized algorithms on theotherhand are
algorithms which uses a string of random bits in its algorithm.
We have proposed two new algorithms for the beforementioned
problemout of which one is an approximation algorithm with a
constant approxi-mation factor of 4 and another randomized
approximation algorithm whichgives an approximation factor of 2 ×
Hn where
Hn =1n
+ . . . +12
+11
(20)
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Consider a minimization problem and an approximation algorithm.
Toshow that the output from the algorithm is less than a given
factor α of theoptimum solution, we need to define a lower bound
and show that
1. the lower bound is less than the cost of the optimum
solution.
2. the output solution has a cost less than α times lower
bound.
Even to judge the effectiveness of a randomized algorithm, a
lower boundis needed, the difference from the above being that,
instead of using theoutput of the algorithm to compare to the lower
bound, we use the averageof all outputs to compare to the lower
bound.
The rest of this chapter is organised as follows. The first
section showsthe importance of lower bound and the lower bound we
chose to analyseour algorithms. In the second section we present
our randomized algorithm,analyzing it with the help of our lower
bound. The third section brings usto the construction of a constant
factor approximation algorithm.
3.1 Bounding the optimum value
The use of a appropriate lower bound is central to the analysis
of anapproximation or a randomized algorithm of an optimization
problem. Weexplain the relation of lower bound and the
approximation factor in a littlemore detail.
Given an minimization problem P characterized by– D a set of
input instances– S(I) the set of all feasible solutions for an
instance I ∈ D– f a function which assigns value to all solutions f
: S(I) → R
We have to find a optimum solution OPT (I) which has been the
mini-mum value of f when compared to all other feasible
solutions.
If we device an approximation algorithm A for the aforementioned
pro-blem with a approximation factor α, then we should make sure
that,
f(A(I))f(OPT (I))
< α (21)
where A(I) is the solution obtained from the algorithm. Since we
do notknow what value the optimum has, so we try to bind it on the
lower side bya lower bound L.
Consider that we have two lower bounds L1 and L2 and assume L1L2
= βwhere β is not a constant, but an increasing function of n, the
size of input.If the approximation algorithm A is analysed with the
help of the lowerbound L2 then the approximation factor will come
out to be α, while if itsanalyzed by the help of the lower bound
L1, then the approximation factoris αβ which is definitely better
than the aforementioned bound e.g. a constant
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factor algorithm may be wrongly analysed as a O(n) factor
algorithm. So thelower bound is quite critical in the analysis of
the approximation algorithm.
Coming back to the central problem, previous research done on
this topichas yielded a number of approximation algorithms. But
since none of theprevious work done was concerned with overlapping
message sets, hencepreviously considered lower bounds are not very
appropriate in any analysishere.
We use the notations developed in section 2.1. The estimated
waitingtime for any single message Mi is τi on average. Without
loss of generali-zation lets assume that S = {M1,M2, . . . ,Mq}.At
any point of time, theestimated waiting time for the set of message
packets S is greater than thewaiting time for any packet Mi for all
i in {1, 2, . . . , q}.
WT (S, t) ≥ WT (Mi, t) (22)WT (S, t) ≥ max
Mi∈S(WT (Mi, t)) (23)
WT (S, t) denotes estimated waiting time at time t for set of
messages S.The cost function of a schedule Sch at time t is the sum
of the cost of
individual message packets S
Cost(Sch, t) =∑S∈ζ
pS × WT (S, t) (24)
Hence,Cost(Sch, t) ≥∑S∈ζ
pS × maxMi∈S
(WT (Mi, t)) (25)
where ζ is the set of all message subsetsUsing theorem 2.2,
without loss of generality, we can restrict ourselves
to lower bound the cost of periodic schedules to get a lower
bound in theoptimum cost ? Let Sch be a schedule with period T .
Since the cost is theaverage over a period hence
Cost(Sch) =1T
T∑t=1
Cost(Sch, t) (26)
Cost(Sch) =1T
T∑t=1
∑S∈ζ
pS × WT (S, t) (27)
Cost(Sch) =1T
∑S∈ζ
pS ×T∑
t=1
WT (S, t) (28)
Cost(Sch) =∑S∈ζ
pS × 1T
T∑t=1
WT (S, t) (29)
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Let the message Mi be broadcasted N(Mi) times during a period
andlet thetime slots when Mi is broadcasted be Ti,j where i ∈ {1,
2, . . . , n} andj ∈ {1, 2, . . . , N(Mi)}.
Hence, for all Mi ∈ S
1T
T∑t=1
×WT (S, t) ≥ T2i,1 + (Ti,1 − Ti,2)2 + . . . + (Ti,N(Mi)−1 −
Ti,N(Mi))2
2 × T(30)
Classically the above term is minimum when the messages are
evenly distri-buted in the schedule with equal periods between them
i.e to say every
τi =T
N(Mi)(31)
. Since we already know the frequency fi = 1τi (that just
implies the totalnumber of occurences of message Mi in the
schedule),
1T
T∑t=1
WT (S, t) ≥ 12× 1
fi∀ i (32)
1T
T∑t=1
WT (S, t) ≥ 12× τi ∀ i (33)
1T
T∑t=1
WT (S, t) ≥ 12× max
Mi∈Sτi (34)
Cost(Sch) ≥ 12
∑S∈ζ
pS × maxMi∈S
(τi) (35)
(36)
hence we define the lower bound as follows
L =
minτ1,τ2,...,τn≥012
∑S∈ζ pS × maxMi∈S(τi)
such that 1τ1 +1τ2
+ · · · + 1τn ≤ 1As above we have already shown that L is
actually less than the Cost of theoptimum schedule.
We further show with the help of an example that relaxing
furthermorethe bounds by replacing the max by the average, so that
we can compute thelower bound easily by lagrangian relaxation
rather than ellipsoid algorithm,would have yielded a lower bound L′
arbitarily bad as compared to theoriginal L. As argued beforehand,
the analysis of the algorithms would nothave been proper, and the
approximation factor would not be tight.
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Let us define the relaxed lower bound as
L′ =
minτ1,τ2,...,τn≥012
∑S∈ζ pS ×
∑Mi∈S
τi|S|
such that 1τ1 +1τ2
+ · · · + 1τn ≤ 1(37)
Example : Let the message set M = {A1, . . . , An, B1, . . . ,
Bk}. The setof message setsζ = {{A1}, {A2}, . . . , {An}, {A1, . .
. , An, B1}, {A1, . . . , An, B2}, . . . ,{A1, . . . , An, Bk}} So
the set of messages consists of two different types,one which have
just one message of the type Ai i.e. Si = {Ai} and
havingprobability p = 12n , the other packet type has all the A
packets and oneBi packets i.e S
′i = {A1, . . . , An, Bi}. The second type of sets have each
probability p = 12k .Since all Ai are symmetrical, hence we
claim that in an optimum to the
lower bound L′, frequencies are symmetrically distributed i.e.
τ1 = τSi =τSj ∀ i, j and τ2 = τS′i = τS′j ∀ i, j. Suppose this is
not so, i.e there exists anoptimum schedule where for some i, j we
have τSi �= τSj . The cost functionis a convex function of the
frequencies τ . If we exchange the two valuesτ§i and τSj , we have
a new schedule with the same Cost, but since the costfunction is a
convex one, hence linear combinations of these two scheduleswill
have a cost less than or equal to the original Cost ,which presents
acontradiction, hence the previous assumption is wrong.
Consider the lower bound L′.
L′ =
minτ1,τ2≥012 × τ1 + 12 × τ2+n×τ1n+1
such that nτ1 +kτ2
= 1
Considering the above constraints, we find out the minima of the
func-tion by partial differentiation using Lagrangian relaxation by
introducing anadditional variable λ redefining the objective
function(function to be mini-mized) as
f(τ1, τ2, λ) =12× τ1 + 12 ×
τ2 + n × τ1n + 1
− λ(1 − nτ1
− kτ2
) (38)
Since the partial differentiation of f with respect to all the
variables, atthe minima will be zero for all variables, hence :
∂f∂τ1
= 0 ⇒ τ1 =√
2(n+1)n2n+1 × λ
∂f∂τ2
= 0 ⇒ τ2 =√
2(n + 1)k × λ
∂f
∂λ= 0 ⇒ n
τ1+
k
τ2= 1
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Using the above two results in the third result, we get
λ = (
√k +
√n(2n + 1)√
2(n + 1))2
Since we are interested in asymptotical values when n, k go to
infinity , weomit negligible terms to focus on the asymptotical
values of the parameters.
After solving the above equations, the lower bound comes out to
be :
L′ = (√
n +
√k√n
) × (√
k√n
+√
n) (39)
L′ = n +k
n+
√k (40)
Consider now the newly constructed lower bound L
L =
minτ1,τ2≥012 × τ1 + 12 × max(τ1, τ2)
such that nτ1 +kτ2
= 1
To find out the minimum of the function, we have to consider
three seperatecases :
– if at the minimum τ1 = τ2
τ1 = τ2 = n + kMin = n + k
– if at the minimum τ1 > τ2
L = τ1
Since τ1 > τ2 and nτ1 +kτ2
= 1so τ1 > n + k, hence Min > n + k.
– if at the minimum τ1 < τ2
L =τ1 + τ2
2τ2 > n + k
Now we use the same Lagrangian relaxation as used earlier, so
here fis defined as
f(τ1, τ2, λ) =τ1 + τ2
2− λ(1 − n
τ1− k
τ2)
13
-
∂f∂τ1
= 0 ⇒ τ2 =√
2nλ
∂f∂τ2
= 0 ⇒ τ2 =√
2kλ
∂f∂λ = 0 ⇒ nτ1 + kτ2 = 1
Using the first two equations to derive the two above relations
betweenτ1 and λ and similarily between τ1 and λ and then using
these relationsin the final third equation results in
τ1 =√
n(√
n +√
k)τ2 =
√k(√
n +√
k)
Min =(√
k +√
n)2
2
Hence the lower bound comes out to be (√
k+√
n)2
2 if k > n and k + n other-wise.
So if n < k < n2 , LL′ = O(√
k/n). Consider the analysis of a constantfactor approximation
algorithm A with the above example in context. Withthe new lower
bound L,
Cost(A)L
= c (A Constant) (41)
Cost(A)L′
≥ c ×√
k/n (42)
So if analysed by the lower bound L′ the constant factor
algorithm, will beanalysed as a
√k/n-factor algorithm.
We have proved that a wrong lower bound will result in bad
analysisof approximation algorithms. We have proposed a lower bound
based onanother objective function.
L = minτ1,τ2,...,τn
12
∑S∈ζ
pS × maxMi∈S
(τMi)
where 1τ1 +1τ2
+ · · · + 1τn ≤ 1The only thing which remains is how to find the
optimum in polynomial
time. We use ellipsoid method mentioned in [13, 10, 14, 8] to
solve the abovementioned set of equations. The method works since
the present equationsare convex.
3.2 Randomized Algorithm
3.2.1 Introduction
Randomized algorithms are a common solution to optimization
problem.They work effectively in a manner that they are not very
complex in their
14
-
construction, but generally, their solutions are sufficiently
close to the opti-mum, although the analysis associated with
finding out the efficiency of thealgorithm are a little heavy.
Most earlier works done on this topic have come up with a
randomizedalgorithm for some specific subproblem of the general
broadcasting problem.For obtaining a randomized algorithm, the most
general approach has beenthe following.
– The lower bound on the cost of a schedule is first derived by
a sui-table objective function. We minimize the objective function
and bythis process we end up at a lower bound on the cost.
Intutively sincethe lower bound and the optimum are supposed to be
close, hence weuse the same frequencies that we obtain while
minimizing the objec-tive function. Let the objective function be
Obj(τ1, τ2, . . . , τn) and letCOST (Sch) be the cost of the
schedule which has the frequency ofMi as 1τi
Obj(τ1, τ2, . . . , τn) < COST (Sch) (43)inf
τ1,...,τnObj(τ1, τ2, . . . , τn) < inf
All SchedulesCost(Sch) (44)
infτ1,...,τn
Obj(τ1, τ2, . . . , τn) < OPT (45)
So finding a minimum to Obj(τ1, τ2, . . . , τn) gives us a lower
bound onoptimum cost for a schedule.
– Next, we assign the use the frequencies thus obtained and
device a ge-neral randomized algorithm where the probabililty of a
message beingbroadcast is equal to the frequency obtained by
previous methods.
3.2.2 The Algorithm
Algorithm 1 Randomized approximation algorithm for the broadcast
sche-duling problemInput: – n messages M1,M2, . . . ,Mn, k sets S1,
S2, . . . , Sk of messages and
demand probabilities (pi)i=1...k associated with each of the
message setsSi.
– A distribution of frequencies (fi)i=1...n, such that f1 + f2 +
. . .+ fm = 1while t > 0 do
Pick i ∈ {1, 2, . . . , n} with probability fiBroadcast Mi
end while
The frequencies fi mentioned in the above algorithm are obtained
fromthe objective function by the help of ellipsoid method, as
discussed in theprevious section.
15
-
3.2.3 The Analysis
Theorem 3.1. The randomized algorithm when using the frequency
distri-bution fi = 1
τ′i
yields schedules with an approximation factor of Hn on the
expected cost, where 2Hn is the harmonic function Hn = 1n +2n +
. . . + 1
Démonstration. The input given to us is– A set of n messages M
= {M1,M2, . . . ,Mn}– ζ = k sets S1, S2, . . . , Sk of messages, Si
⊆ M– Demand probabilities (pi)i=1...k associated with each of the
message
sets SiWe use the objective function
Obj(τ1, τ2, . . . , τn) =12
∑S∈ζ
pS × maxMi∈S
(τi)
LB = minObj(τ1, τ2, . . . , τn) whenn∑
i=1
1τi
≤ 1
and find the minimum for this function. If we get the minimum of
theobjective function at 1
τ′1
, 1τ′2
, . . . , 1τ ′n
and lower bound be LB. Working on
the intution that the objective function and the cost function
are close, werandomly broadcast each message Mi with probability 1τ
′i
.
Lemma 3.1. The expected waiting time or E(WT (S, t)) for any set
S isless than Hn × maxMi∈S(τi) for the schedule resulting from the
algorithm
E(WT (S, t)) < Hn × maxMi∈S
(τ ′i)
where WT (S, t) is the expected waiting time for the set S of
messages.
Démonstration. Consider a set S of J messages S = {M1,M2, . . .
,MJ} wi-thout loss of generality. Also we may assume that we start
at t and arewaiting for all the J messages. At any time interval
the message Mi isbroadcasted with a probability 1
τ′i
. The event of observing the J messages
is actually is a succession of J distinct events, one for each
time, a mes-sage appears from S for the first time after t. Let the
events be namedevent1, . . . , eventJ . Let the time for the eventi
be ti, and let the time inter-vals between two events eventi and
eventi+1 be termed as Intervali+1 so
Interval1 = t1 (46)Intervali = ti+1 − ti (47)
tJ − t = Interval1 + Interval2 + . . . + IntervalJ (48)E(tJ − t)
= E(Interval1) + E(Interval2) + . . . + E(IntervalJ) (49)
16
-
tJ is the last event or the point where all the messages have
been re-cieved. So tJ − t is the expected waiting time at t, but
since the messagesare broacasted with the same probabilities at all
times, hence the expectedwaiting time does not change.
At time ti, i messages have been already broadcasted after t.
For theevent to happen , any of the other J − i messages have to be
broadcas-ted after now. Since all the messages have broadcast
probabilities atleastminMi∈S(
1τ′i
), hence the probability of getting any of the J − i messages
isatleast (J − i) × minMi∈S( 1τ ′i ). Lets call it q.
If at every time interval an event can happen with a probablity
q, thenthe expected time at which the event first occurs is equal
to 1q .
Pr(event happens first at time t) = (1 − q)t−1q
E(first time event happens) =∞∑t=1
(1 − q)t−1q
E(first time event happens) =1q
applying this in the present case, we get.
Pr(eventi+1) = (J − i) × minMi∈S
(1τ
′i
)
E(Intervali+1) =1
(J − i) × minMi∈S( 1τ ′i )
E(Intervali+1) =1
J − i × maxMi∈S τ′i
E(tJ − t) = E(Interval1) + E(Interval2) + · · · +
E(IntervalJ)
E(tJ − t) = maxMi∈S
(τ′i ) ×
J−1∑i=1
1i
E(tJ − t) = maxMi∈S
(τ′i ) × HJ
E(WT (S, t)) = maxMi∈S
(τ′i ) × HJ
17
-
so the expected cost from the schedule comes out to be
Cost(Sch, t) =∑S∈ζ
pS × WT (S, t)
E(Cost) =∑S∈ζ
pS × E(WT (S, t))
E(Cost) =∑S∈ζ
pS × maxMi∈S
(τ′i ) × HJ
Lower Bound =12×
∑S∈ζ
pS × maxMi∈S
(τ′i )
Approximation Factor =E(Cost)
Lower Bound≤ 2Hn
So its proved that the approximation factor of the randomized
algorithmbeing analysed comes out to be 2Hn where n is the number
of messages
Further we prove that this is a tight bound with the help of
someexamples. Consider the example,There are n messages M1,M2, . .
. ,Mn, there is one set S = {M1,M2, . . . ,Mn}and PS = 1.
Obviously due to the symmetry of the question, we know that that
allthe frequencies should be equal to minimize our lower bound .
Further wecan even guess the most optimum schedule, a round robin
schedule of all themessages like M1 : M2 : . . . : Mn. The Cost
comes out to be n. If we takethe random schedule , the probability
for any message to be broadcasted isequal to 1n . hence the
expected waiting time equals 1 +
nn−1 + . . . +
n2 + n.
So the approximation factor comes out to be Hn = 1 + 12 + . . .
+1n .
The above example can be extended into an example where we have
ksuch subsets each having a n messages with probability 1k each.
Schaba-nel [16] worked on Preemptive cases like this and proved
that the optimumschedule is when all the messages are broadcasted
one after another and thenall the subsets are broadcasted one after
another. The cost of the optimumschedule comes out to be a
Optimum Cost =n × (k2 + 1)
2k(50)
Randomized Cost = nk × H(k) (51)Approximation Factor =
2H(k)k2
k2 + 1(52)
18
-
Hence, if k = n, then as k → ∞, the Approximation factor tends
to2H(n). This demonstrates that the bound is tight. So the
randomized algo-rithm is actually a 2Hn-factor algorithm.
3.3 Deterministic Approximation Algorithm
We propose a new 4-approximation algorithm for the general
schedulingproblem. While the older algorithm were essentially
derandomizations of therandomized algorithms, our approach consists
of using the perfectly periodicschedules, to construct a periodic
schedule while at the same guaranteeingan approximation factor of
4.
3.3.1 Introduction to Perfectly periodic schedule
Consider a system with n messages which have to be broadcasted
anda single bandwidth resource on which they have to be
broadcasted. So theyhave to share the bandwidth by time
multiplexing. A schedule for resourceallocation is called perfectly
periodic, if the resource gets allocated to anymessage i every once
after βi time slots.
The main question arises as to the usefulness of the perfectly
periodicschedules. Since these schedules are mathematically very
simple, hence theyresult in some particularly pleasing
consequences.
– They are very simple and easy to analyse.– The process of
inference of the schedule from the client’s viewpoint is
very simple , although this point is unrelated to the problem
presentlyunder consideration.
Even deciding if a given set of periods admit a Perfectly
periodic scheduleis NP-Hard if
∑i τi ≤ 1 [5]. So its impossible to hope for a polynomial
time
algorithm to find a perfectly periodic schedules. However, tree
scheduleshave been found to be very effective in calculations of
perfectly periodicschedules. A tree schedule is a schedule
represented by a tree, where theleaves correspond to clients, and
the period of each client is computed basedon the depth of the leaf
and the product of degrees if its ancestors.
The construction of a tree schedule by using the Shamir et al
algorithm.[6] can be used very effectively in the present problem
to construct a perfectlyperiodic schedule.
3.3.2 Construction of a Perfectly periodic Schedule
Schedule S is defined as an infinite sequence S = s0s1 . . .
where sj ∈{1, 2, . . . , n} denoting the n messages. A schedule is
perfectly periodic, if theslots allocated to each message are
equally spaced i.e for each message Mithere exists integers βi ≥ 1
and 0 ≤ oi < βi, such that i is scheduled in thejth slot if and
only if j = oi mod βi. βi is the period of the message i andoi its
offset.
19
-
Algorithm 2 Dispatching algorithm for a Schedule Tree
[6]Dispatch Function
Input: A schedule tree TA message Identifier
Output: a messageCODEv ← root(T )while v is not n leaf do
v ← Token(v)end while
Token FunctionInput: a non-leaf node uOutput: a node in T
CODElet e0, e1, . . . , ed−1 be the d outgoing edges of uif ei
has the token and ei = (u, v)move token to e(i+1) mod dreturn v
A schedule tree is defined as follows : a ordered tree can be
interpretedas a perfectly periodic schedule as follows. The message
of the scheduleform a bijection with the leaves of the tree. An
ordered tree is a rootedtree where the edges coming out from each
non-leaf u mode are numbered0, 1, . . . , deg(u)−1. The period or
root r is β(r) is 1. The period of a non-rootnode u is computed
recursively as
β(u) = β(par(u)) × deg(par(u))
where deg(v) is the degree of node v and par(v) denotes the
parent of nodev. To calculate the offset of a node, we define a
funtion h(u) which denotesthe position of a node among its siblings
and is equal to the number on theedge from par(u) to u. The offset
then can be easily described as
o(u) = o(par(v)) + h(v) × β(par(v))
The above described schedule will come out to be a perfectly
periodic sche-dule with the periods coming out as β(u).
Now comes the question of dispatching the tree schedule, which
meansgiven the schedule tree how to construct a schedule from it.
We describe thealgorithm developed by Shamir et al [6]. The idea of
the algorithm is tofind the message to schedule by traversing the
tree with the help of tokensplaced on the tree edges. All non-leaf
nodes have a token placed on one oftheir outgoing edges. The
algorithm descends to a leaf node by following the
20
-
tokens down the tree. In addition, each time, the algorithm
crosses an edge(u, v) the token on this edge is moved to the next
sibling of v i.e. the nextchild of u after v.
The algorithm is easy to comprehend but a little difficult to
visualize.We can prove very easily that any leaf node u has a
period of β(u) =β(par(u)) × deg(par(u)). This is because u is
scheduled if the tokens are soarranged that the algorithm reaches
par(u) in β(par(u)) period and par(u)leads to u only once in
deg(par(u)) times, hence the result.
Our approach to the present problem is as follows. Consider we
havegot the frequency distribution for all the messages, but we
have to fit theminto a schedule. We try to construct a perfectly
periodic schedule for themessages. The first step in this direction
would be the construction of aschedule tree from the frequency
distribution. We propose an algorithm todo that in the next
section. After that using the algorithm presented beforewe
construct the schedule and analyse it. The schedule comes out to be
a4-approximation.
3.3.3 Construction of a schedule tree from a frequency
distribu-tion
We use the same notation, namely set of m messages {M1,M2, . . .
,Mn}make up the set of messages. Consider that the frequencies for
all the mes-sages has been given i.e (fi)i=1,2,...,n.
We round down the frequencies fi to a power of 2 and get a new
set offrequencies f
′i such that
f′i =
12j+1
if12j
> fi ≥ 12j+1 (53)
We can construct a perfectly periodic schedule with these f′i
with the
help of a dummy message M0 with its frequency being f′0 = 1
−
∑ni=1 f
′i .
The algorithm by Shamir et al constructs the schedule tree. We
definetwo kinds of nodes : leaf nodes and non-leaf node. Mi or
messages are at-tached as a leaf nodes, all others are supposed to
be non-leaf nodes. Weconstruct the Tree in stages, adding another
layer of nodes at depth i , atstage i. Some non-leaf nodes from the
previous stage serve as the parentsfor the leaf nodes Mi we attach
in this stage. For all other non-leaf nodesfrom stage i−1 , we
attach two nodes at depth i as its children. These nodein turn will
either serve as parents of leaf nodes or non-leaf nodes at depthi +
1. We stop when we have attached all the message nodes.
Theorem 3.2 (Shamir et al [6]). The algorithm constructs a
scheduletree T from any set of frequencies (fi)i=1,2,...,n
satisfying the condition∑n
i=1 fi ≤ 1
21
-
Algorithm 3 Construction of a Schedule Tree from
(fi)i=1,2,...,nInput: (fi)i=1,2,...,nOutput: a Schedule tree T
Procedure 1 : Change of FrequencyCODEfor 1 ≥ i ≤ n do
if 12j
> fi ≥ 12j+1 thenf
′i =
12j+1
end ifend for
Procedure 2 : Construction of Tree TCODEinitiate root rfor 1 ≤ j
≤ maxi −1 × log f ′i do
add all the nodes Mi with f′i =
12j
to the tree at the depth j as leavesfrom the non-leaf nodes from
the previous stage.Condition : no node should not have a outgoing
degree more than 2.Add remaining non-leaf nodes to existing
non-leaf nodes at
end forturn all remaining non-leaf nodes to leaf nodes with the
label M0
Démonstration. The Procedure 1 gives us a set of (f′i ) .
fi ≥ f ′i (54)fi < 2 × f ′i (55)
n∑i=1
fi ≤ 1 (56)n∑
i=1
f′i ≤ 1 (57)
We use the following lemma
Lemma 3.2 (Shamir et al [6]). All the messages Mi are inserted
to thetree T and at any stage j there are enough non-leaf nodes
left from theprevious stage to accept the leaf nodes Mi with f
′i =
12j
in this stage aschildren.
By the above lemma we have seen that the all the messages are
insertedinto the tree T which implies that the T can transformed
into a perfectlyperiodic schedule with the help of a dummy
variable. The dummy variableis used since the sum of all the
altered frequencies is not equal to one. So
22
-
we add a new dummy message such that the total sum of frequency
nowbecomes one.
3.3.4 Construction of the schedule and final analysis
The next step would be to construct a perfectly periodic
schedule PPSwith the help of Algorithm 2.
Theorem 3.3. The schedule PPS if constructed with the help of
the fre-quency set ( 1
τ′i
)i=1,2,...,n mentioned in the lower bound section, is a 4
approxi-
mation solution .
Démonstration. Before beginining the actual proof. We recall
the cost of theoptimum .
Lower Bound =12×
∑S∈ζ
pS × maxMi∈S
τ′i ≤ OPT (58)
We now find the cost of our schedule PPS in terms of
(fi)i=1,2,...,n.The waiting time for a message is 1
f′i
at the maximum. Now consider the
set S. Since this is a perfectly periodic schedule, hence the
messages repeatthemselves at regular intervals so at any point in
time, the waiting time willatmost be the maximum waiting time for
all the individual messages,
WT (S, t) ≤ maxMi∈S
WT (Mi, t) (59)
WT (Mi, t) ≤ 1f
′i
(60)
WT (S, t) ≤ maxMi∈S
1f
′i
(61)
Cost(PPS) ≤∑S∈ζ
pS × maxMi∈S
1f
′i
(62)
Cost(PPS) ≤∑S∈ζ
pS × maxMi∈S
2 × 1fi
(63)
Cost(PPS) ≤ 2 ×∑S∈ζ
pS × maxMi∈S
1fi
(64)
Now since we are using the frequency set f′i =
1
τ′i
, Hence the cost becomes
Cost(PPS) ≤ 2 ×∑S∈ζ
pS × maxMi∈S
τ′i (65)
OPT ≥ 12×
∑S∈ζ
pS × maxMi∈S
τ′i (66)
α =Cost(PPS)
OPT≤ 4 (67)
23
-
where α is the approximation factor.So finally we have been able
to prove, that the approximation algorithm
has an approximation factor of 4. But there is still a small
issue, our schedulehas been constructed with the help of a dummy
message which may as wellcover at he maximum half of the whole
time. We have incorporated it inour schedule. But if we were to
schedule some other message in place of thedummy message, the
waiting time for that message will decrease as will thetotal
waiting time and hence the cost.
What remains is the analysis of the running time of the
algorithm. Thefirst part of the algorithm which estimates the τ
′i uses ellipsoid algorithms
and is polynomial in time complexity, the second part which
dispatchingthe tree does not have a complexity more than O(n2)
since at any level,we need to maintain only n non leaf nodes. And
the maximum depth isgoing to be O(log(length of input)). And the
final step of making perfectlyperiodic schedule is O(n× length of
the schedule). So the total computationalcomplexity is
polynomial.
4 Conclusions
The broadcast scheduling problem gives a nice theoretical base
for analy-sing broadcasting problems. It also models other problems
like maintenancescheduling problem and the multi-item replenishment
problem [3].
We worked on the general broadcasting problem. We found out that
it isNP Hard. We designed a lower bound for the optimum cost, a
randomized2Hn-approximation algorithm and finally a constant factor
approximationalgorithm based on perfectly periodic schedules to
give us a algorithm withapproximation factor 4.
A couple of open questions and issues came out as a result of
this re-search :
– We did not consider the case when the lengths of individual
messagesare different from each other. Although this will be hard
to solve bythe methods developed in this report, but nonetheless,
the methodsmay work, the reason being that similar methods have
been used whiledealing with the variable length topic in other
settings.
– Derandomization of the randomized algorithm may be able to
shedsome more light on the exact relationships between τi and pi. A
greedyalgorithm devised by the help of the above mentioned method
will mostprobably have a better approximation ratio than we have
been able toacheive in this report.
– It will be interesting to see how extending the number of
channels tomake it a multi channel network, affects the efficiency
of the algorithmssuggested in this report
– last but not the least, recently several results have been
obtained for
24
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the online setting of the basic broadcast problem where the pi
arenot pre-determined but are considered dynamically [7]. These
resultsinclude impossibility results as well as competitive online
algorithms. Itwould be interesting to determine whether the
customized newspaperproblem is still tractable in this setting.
Références
[1] M. H. Ammar and J. W. Wong. The design of teletext broadcast
cycles.In Performance Evaluation, volume 5(4), pages 235–242, 1985.
Infor-mation available at
//www.cc.gatech.edu/fac/Mostafa.Ammar/.
[2] M. H. Ammar and J. W. Wong. On the optimality of
cyclictransmission in teletext systems. In IEEE Trans. on Comm.,
vo-lume COM-35(11), pages 1159–1170, 1987. Information available
at//www.cc.gatech.edu/fac/Mostafa.Ammar/.
[3] S. Anily, C. A. Glass, and R. Hassin. The scheduling of
maintenanceservice. Paper available at
//www.math.tau.ac.il/∼hassin/, Juil. 1995.
[4] A. Bar-Noy, R. Bhatia, J. Naor, and B. Schieber. Minimizing
serviceand operation costs of periodic scheduling. In Proc. of the
9th AnnualACM-SIAM Symp. on Discrete Algorithms (SODA’98), pages
11–20,1998. Paper available at //www.eng.tau.ac.il/∼amotz/.
[5] Amotz Bar-Noy, Aviv Nisgav, and Boaz Patt-Shamir. Nearly
optimalperfectly-periodic schedules. In PODC ’01 : Proceedings of
the twentiethannual ACM symposium on Principles of distributed
computing, pages107–116, New York, NY, USA, 2001. ACM Press.
[6] Zvika Brakerski, Vladimir Dreizin, and Boaz Patt-Shamir.
Dispatchingin perfectly-periodic schedules. J. Algorithms, 49(2)
:219–239, 2003.
[7] Jeff Edmonds and Kirk Pruhs. Broadcast scheduling : when
fairness isfine. In SODA ’02 : Proceedings of the thirteenth annual
ACM-SIAMsymposium on Discrete algorithms, pages 421–430,
Philadelphia, PA,USA, 2002. Society for Industrial and Applied
Mathematics.
[8] Rugenstein EK. active set strategies and an ellipsoid
algorithm forgeneral non linear programming problems. PhD thesis,
rensselear poly-technique inst, 2002.
[9] J. Gecsei. The architecture of Videotex Systems. Prentice
Hall, Engle-wood Cliffs, N. J., 1983.
[10] Ecker JG and Kupferschmind M. Introductions to operations
research.1988.
[11] Claire Kenyon, Nicolas Schabanel, and Neal Young.
Polynomial-timeapproximation scheme for data broadcast. In STOC ’00
: Proceedings ofthe thirty-second annual ACM symposium on
Principles of distributedcomputing, pages 659–666, New York, NY,
USA, 2000. ACM Press.
25
-
[12] Sanjeev Khanna and Shiyu Zhou. On indexed data broadcast.
In STOC’98 : Proceedings of the thirtieth annual ACM symposium on
Theory ofcomputing, pages 463–472, New York, NY, USA, 1998. ACM
Press.
[13] Shor NZ. Cut-off method with space extension in convex
programmingproblems. Cybernetics, 12(94) :6, 1977.
[14] Shah S. An ellipsoid algorithm for equallity constrained
non linear pro-gram. PhD thesis, rensselear polytechnique inst,
1998.
[15] Nicolas Schabanel. Algorithmes d’approximation pour les
télécommuni-cations sans fil : Ordonnancement pour la
dissémination de données etAllocation statique de fréquences.
PhD thesis, ENS Lyon, January 2000.Available at
//perso.ens-lyon.fr/nicolas.schabanel/schabanel phd thesis.
[16] Nicolas Schabanel. The data broadcast problem with
preemption. InLNCS Proc. of the 17th Symp. on Theoritical Aspects
of ComputerScience (STACS’2000), volume 1770, pages 181–192, Feb.
2000.
[17] S. Shekhar and D. Liu. Genesis : An approach to data
dissemination inAdvanced Traveler Information Systems (ATIS). IEEE
Data Enginee-ring Bulletin, Special issue on Data Dissemination,
19(3), Sept. 1996.Paper available at //www.cs.umn.edu/∼shekhar.
[18] N. H. Vaidya and S. Hameed. Log time algorithms for
scheduling singleand multiple channel data broadcast. In Proc. of
the 3rd ACM/IEEEConf. on Mobile Computing and Networking (MOBICOM),
Sept. 1997.Paper available at
//www.cs.tamu.edu/faculty/vaidya/Vaidya.html/.
[19] Nitin Vaidya and Sohail Hameed. Data broadcast scheduling :
On-lineand off-line algorithms. Technical report, College Station,
TX, USA,1996.
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