Data gathering and personalized broadcasting in radio grids with interference

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grids with interference

Jean-Claude Bermond, Bi Li, Nicolas Nisse, Hervé Rivano, Min-Li Yu

To cite this version:

Jean-Claude Bermond, Bi Li, Nicolas Nisse, Hervé Rivano, Min-Li Yu. Data gathering and

personal-ized broadcasting in radio grids with interference. Theoretical Computer Science, Elsevier, 2015, 562,

pp.453-475. �10.1016/j.tcs.2014.10.029�. �hal-01084996�

Contents lists available atScienceDirect

Theoretical

Computer

Science

www.elsevier.com/locate/tcs

Data

gathering

and

personalized

broadcasting

in radio

grids

with

interference

✩

Jean-Claude Bermond

a

,

b

,

∗

,

Bi Li

b

,

a

,

c

,

Nicolas Nisse

b

,

a

,

Hervé Rivano

d

,

Min-Li Yu

e

a_{Univ.NiceSophiaAntipolis,CNRS,I3S,UMR 7271,SophiaAntipolis,France} b_INRIA,France

c_{InstituteofAppliedMathematics,ChineseAcademyofSciences,Beijing,China} d_{UrbanetInriateam,UniversitédeLyon,InsaLyonInriaCITILab,France} e_{UniversityoftheFraserValley,DptofMathsandStatistics,Abbotsford,BC,Canada}

a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

r

a

c

t

Articlehistory:

Received29December2012

Receivedinrevisedform8October2014 Accepted21October2014 Availableonlinexxxx CommunicatedbyG.Ausiello Keywords: Gathering Personalizedbroadcasting Grid Interference Radionetworks

Inthegathering problem,aparticularnodeinagraph,thebasestation,aimsatreceiving messagesfromsomenodesinthegraph.Ateachstep,anodecansendonemessagetoone ofitsneighbors(suchanactioniscalledacall).However,anodecannotsendandreceive amessageduringthesamestep.Moreover,thecommunicationissubject tointerference constraints, moreprecisely, two callsinterfere inastep, ifone sender isatdistance at mostdI fromthe otherreceiver. Givenagraphwith abase stationand aset ofnodes

having somemessages, the goalof the gathering problemis to computeaschedule of callsforthe basestation toreceiveallmessagesas fastaspossible,i.e.,minimizingthe numberofsteps(calledmakespan).Thegatheringproblemisequivalenttothepersonalized broadcasting problemwherethebasestationhastosendmessagestosomenodesinthe graph,withsametransmissionconstraints.

Inthispaper,wefocusonthegathering andpersonalizedbroadcastingproblemingrids. Moreover,weconsiderthenon-bufferingmodel:whenanodereceivesamessageatsome step, it must transmit it during the next step. In this setting, though the problem of determining the complexityof computingtheoptimal makespan inagrid isstill open, we present linear (inthe number ofmessages) algorithms that compute schedules for gathering with dI ∈ {0,1,2}. In particular, we present an algorithm that achieves the

optimalmakespanuptoanadditiveconstant2 whendI=0.Ifnomessagesare“close”to

theaxes(thebasestationbeingtheorigin),ouralgorithmsachievetheoptimalmakespan uptoanadditiveconstant1 whendI=0,4 whendI=2,and3 whenbothdI=1 andthe

basestationisinacorner.Notethat,theapproximationalgorithmsthatwepresentalso provideapproximationuptoaratio2 forthegatheringwithbuffering.Allourresultsare provedintermsofpersonalizedbroadcasting.

©2014ElsevierB.V.All rights reserved.

✩ _{ThisworkispartiallysupportedbyProjectSTREPFP7EULER,ANRVersoprojectEcoscells,ANRGRATELandRegionPACA.}

*

Correspondingauthorat:Univ.NiceSophiaAntipolis,CNRS,I3S,UMR 7271,SophiaAntipolis,France.

E-mailaddresses:[email protected](J-C. Bermond),[email protected](B. Li),[email protected](N. Nisse),[email protected](H. Rivano),

[email protected](M.-L. Yu).

http://dx.doi.org/10.1016/j.tcs.2014.10.029

1. Introduction

1.1. Problem,modelandassumptions

Inthispaper,westudyaproblemthatwas motivatedby designingefficientstrategiestoprovideinternetaccessusing wireless devices[8]. Typically,several houses in a village need accessto a gateway(for example a satellite antenna) to transmitandreceivedataovertheInternet.Toreducethecostofthetransceivers,multi-hopwirelessrelayroutingisused. We formulate this problemasgathering informationin a Base Station(denoted by BS)of a wireless multi-hop network when interferenceconstraintsare present. Thisproblemisalso knownasdatacollectionandis particularlyimportantin sensornetworksandaccessnetworks.

Transmissionmodel We adopt the network model considered in [2,4,9,11,16]. The network is represented by a node-weighted symmetric digraph G

= (

V

,

E

)

, where V is the set of nodesand E is the set of arcs. More specifically, each nodein V representsadevice (sensor,station,. . . )thatcantransmitandreceivedata.Thereisaspecialnode BS

∈

V called theBaseStation(BS),whichisthefinaldestinationofalldatapossessedbythevariousnodesofthenetwork.Eachnodemay haveanynumberofpieces ofinformation,ormessages,totransmit,includingnone. Thereis anarcfromu to v if u can transmitamessageto v.Wesupposethatthedigraphissymmetric;soifu cantransmitamessageto v,then v canalso transmit a messageto u.Therefore G represents thegraph ofpossiblecommunications.Some authors usean undirected graph(replacing thetwo arcs

(

u

,

v

)

and

(

v

,

u

)

byan edge

{

u

,

v

}

).However calls (transmissions)are directed:a call

(

s

,

r

)

is definedasthetransmission fromthenode s to node r,in which s isthesender and r isthereceiver and s and r are adjacentinG.Thedistinctionofsenderandreceiverisimportantforourinterferencemodel.

Hereweconsidergridsastheymodelwellbothaccessnetworksandalsorandomnetworks[14].Thenetworkisassumed to be synchronousandthetime isslottedintosteps. Duringeach step,atransmission(or acall)betweentwo nodescan transport atmostonemessage.Thatis,astepisaunitoftimeduring whichseveralcallscanbe doneaslongastheydo not interferewitheachother.Wesuppose thateach device isequippedwitha half duplexinterface:anode cannot both receive andtransmitduring astep.Thismodelsthenear-fareffectofantennas:whenoneistransmitting,it’sownpower preventsanyother signal tobe properlyreceived.Moreover, weassume that anode cantransmit orreceiveatmostone messageperstep.

Following[11,12,15,16,18]weassumethatnobufferingisdoneatintermediatenodesandeachnodeforwardsamessage assoonasitreceivesit.Oneoftherationalesbehindthisassumptionisthatitfreesintermediatenodesfromtheneedto maintaincostlystateinformationandmessagestorage.

Interferencemodel Weusea binaryasymmetricmodelofinterferencebasedonthedistanceinthecommunicationgraph. Letd

(

u

,

v

)

denotethedistance,that isthelength ofa shortestdirected path,fromu to v in G anddI be anonnegative

integer. Weassumethat whenanode u transmits,allnodes v suchthatd

(

u

,

v

)

≤

dI aresubjecttotheinterferencefrom

u’stransmission.We assumethat all nodesof G have thesame interferencerangedI.Twocalls

(

s

,

r

)

and

(

s

,

r

)

do not

interfereifandonlyifd

(

s

,

r

)

>

dI andd

(

s

,

r

)

>

dI.Otherwisecallsinterfere(orthereisacollision).Wefocusonthecases

wheredI

≤

2.NotethatinthispaperwesupposedI

≥

0.Itimpliesthatanodecannotreceiveandsendsimultaneously.

Thebinaryinterferencemodelisasimplifiedversionofthereality,wheretheSignal-to-Noise-and-InterferenceRatio(the ratioofthereceivedpowerfromthesourceofthetransmissiontothesumofthethermicnoise andthereceivedpowers of all other simultaneously transmitting nodes) has to be above a given threshold for a transmission to be successful. However, the valuesof thecompletion times that weobtain lead to lowerbounds onthe corresponding reallife values. Stateddifferently,ifthecompletiontimeisfixed,thenourresultsleadtoupperboundsonthemaximumpossiblenumber ofmessagesthatcanbetransmittedinthenetwork.

Gatheringandpersonalizedbroadcasting Ourgoalistodesignprotocolsthatefficiently,i.e.,quickly,gatherallmessagestothe basestationBSsubjecttotheseinterferenceconstraints.Moreformally,let G

= (

V

,

E

)

be aconnectedsymmetricdigraph, BS

∈

V and dI

≥

0 be an integer. Eachnode in V

\

BS isassigneda set (possibly empty)of messages that mustbe sent

to BS.A multi-hopschedulefora messageisa sequenceofcalls usedto transmitthismessageto BS. Asno bufferingis allowed, thisscheduleisdefinedbythestep ofthefirstcallandthepath thatthemessagefollowsin ordertoreachBS. The gatheringproblem consistsincomputinga multi-hopscheduleforeach messagetoarrivethe BS undertheconstraint that during anystepanytwo callsdonotinterferewithin theinterferencerange dI.The completiontime ormakespan of

thescheduleisthenumberofsteps usedforall messagestoreachBS.Weare interestedincomputingtheschedulewith minimummakespan.

Actually,wedescribethegatheringschedulebyillustratingtheschedulefortheequivalentpersonalizedbroadcasting prob-lem sincethisformulationallowsustouseasimplernotationandsimplifytheproofs.Inthisproblem,thebasestationBS hasinitiallyasetofpersonalized messagesandtheymustbesenttotheirdestinations,i.e.,eachmessagehasapersonalized destinationin V ,andpossiblyseveralmessagesmayhavethesamedestination.Theproblemistofindamulti-hop sched-ule for each message toreach its corresponding destination node under thesame constraints asthe gatheringproblem. The completiontime ormakespan ofthescheduleisthenumberofstepsusedforallmessages toreachtheirdestination andtheproblemaimsatcomputingaschedulewithminimummakespan.Foranypersonalizedbroadcastingschedule,itis always possibletobuildagatheringschedulewiththesame makespan.Hencethesetwoproblemsareequivalent.Indeed,

Table 1

Performancesofthealgorithmsdesignedinthispaper.Ouralgorithmsdealwiththegatheringandpersonalizedbroadcastingproblemsinagridwith arbitrarybasestation(unlessstatedotherwise).Inthistable,+c-approximationmeansthatouralgorithmachievesanoptimalmakespanuptoanadditive constantc.Similarly,×c-approximationmeansthatouralgorithmachievesanoptimalmakespanuptoanmultiplicativeconstantc.

Interference Additional hypothesis Performances

without buffering with buffering

dI=0 +2-approximation

no messages on axes +1-approximation

dI=1 BS in a corner and no messages “close” to the axes (seeDefinition 2) +3-approx. ×1.5-approx.

no messages at distance≤1 from an axis ×1.5-approximation

dI=2 no messages at distance≤1 from an axis +4-approx. ×2-approx. considera personalized broadcastingschedulewithmakespan

T

.Any call

(

s

,

r

)

occurringatstepk corresponds to acall

(

r

,

s

)

scheduledatstep

T +

1

−

k inthecorrespondinggatheringschedule.Furthermore,asthedigraphissymmetric,iftwo calls

(

s

,

r

)

and

(

s

,

r

)

donotinterfere,thend

(

s

,

r

)

>

dI andd

(

s

,

r

)

>

dI;sothereversecallsdonotinterfere.Hence,ifthere

isan(optimal)personalizedbroadcastingschedulefromBS,thenthereexistsan(optimal)solutionforgatheringatBS with thesamemakespan.Thereversealsoholds.Therefore,inthesequel,weconsiderthepersonalizedbroadcastingproblem. 1.2. Relatedwork

Gathering problems like the one that we studyin this paperhave received much recentattention. The papers most closelyrelatedtoourresultsare[3,4,11,12,15,16].Thedatagatheringproblemisfirstintroducedin[11]underamodelfor sensornetworkssimilar tothe one adoptedinthispaper.It deals withdI

=

0 and givesoptimalgatheringschedules for

trees.Optimalalgorithmsforstarnetworksaregivenin[16]andtheyprovideoptimalscheduleswhichminimizeboththe completion timeandtheaverage deliverytimeforallthemessages.Underthesamehypothesis,an optimalalgorithmfor generalnetworksispresentedin[12] inthecaseeach node hasexactlyonemessageto deliver.In[4](resp.[3]) optimal gatheringalgorithms fortreenetworksinthe samemodelconsideredinthe presentpaper,are givenwhendI

=

1 (resp.,

dI

≥

2). In [3]it is also shownthat the Gathering Problem is NP-complete ifthe process must be performedalong the

edges ofaroutingtree fordI

≥

2 (otherwisethecomplexity isnot determined).Furthermore,fordI

≥

1 asimple

(

1

+

_d2_I

)

factorapproximationalgorithmisgivenforgeneralnetworks.Inslightlydifferentsettings,inparticulartheassumptionof directionalantennas,theproblemhasbeenprovedNP-hardingeneralnetworks[17].Thecaseofopen-grid whereBS stands atacornerandnomessageshavedestinations inthefirstroworfirstcolumn, calledaxisinthe following,isconsidered in[15],wherea1

.

5-approximationalgorithmispresented.

Other related resultscan be found in [1,2,6,7,10] (see [9] fora survey). Inthese articles databuffering isallowed at intermediatenodes,achievingasmallermakespan.In[2],a4-approximationalgorithmisgivenforanygraph.Inparticular thecaseofgridsisconsideredin[6],butwithexactlyonemessageper node.Anotherrelatedmodelcanbefoundin[13], wheresteady-state(continuous) flowdemands betweeneach pairofnodeshavetobe satisfied,inparticular,theauthors alsostudythegatheringinradiogridnetworks.

1.3. Ourresults

Inthispaper,we propose algorithms tosolvethe personalized broadcastingproblem(andsothe equivalent gathering problem)inagridwiththemodeldescribedabove(synchronous,nobuffering,onemessagetransmissionperstep,withan interferenceparameterdI).InitiallyallmessagesstandatthebasestationBS andeachmessagehasaparticulardestination

node(severalmessagesmaybesenttothesamenode).Ouralgorithmscomputeinlineartime(inthenumberofmessages) schedules with no calls interfering, witha makespan differing from the lower bound by a small additive constant. We first study the basic instance consisting ofan open grid where no messages have destination on an axis, witha BS in the cornerof thegrid andwithdI

=

0.This isexactly the samecaseas that consideredin [15].In Section 2we give a

simplelower bound LB.In Section 3we design forthisbasicinstance alinear time algorithmwitha makespanat most LB

+

2 steps, soobtaining a

+

2-approximationalgorithm fortheopen grid,whichgreatlyimproves themultiplicative 1.5 approximationalgorithmof[15].Suchanalgorithmhasalreadybeengivenintheextendedabstract[5];buttheonegiven hereissimplerandwecanrefineittoobtainforthebasicinstancea

+

1-approximationalgorithm.WeproveinSection4

thatthe

+

2-approximationalgorithmworksalsoforageneralgridwheremessagescanhavedestinationsontheaxisagain withBS inthecorneranddI

=

0.WeconsiderinSection5thecasesdI

=

1 and2.WegivelowerboundsLBc

(

1

)

(when BS

isinthecorner)andLB

(

2

)

andshowhowtousethe

+

1-approximationalgorithmgiveninSection3todesignalgorithms withamakespanatmostLBc

(

1

)

+

3 whendI

=

1 andBS isinthecorner,andatmostLB

(

2

)

+

4 whendI

=

2;howeverthe

coordinatesofthedestinationshaveinbothcasestobeatleast2.InSection6,weextendourresultstothecasewhereBS isina generalpositioninthe grid.Inaddition,wepoint outthat ouralgorithmsare 2-approximationsifthebufferingis allowed,whichimprovestheresultof[2]inthecaseofgridswithdI

≤

2.Finally,weconcludethepaperinSection7.The

2. Notationsandlowerbound

Inthefollowing,weconsideragridG

= (

V

,

E

)

withaparticularnode,thebasestationBS,alsocalledthesource.Anode v isrepresentedby itscoordinates

(

x

,

y

)

.ThesourceBS has coordinates

(

0

,

0

)

.Wedefinetheaxis ofthegridwithrespect to BS, astheset ofnodes

{(

x

,

y

)

:

x

=

0

}

or

{(

x

,

y

)

:

y

=

0

}

. The distancebetweentwo nodesu and v isthelength of a shortestdirectedpathinthegridandisdenotedbyd

(

u

,

v

)

.Inparticular,d

(

B S

,

v

)

= |

x

|

+ |

y

|

.

WeconsiderasetofM

>

0 messagesthatmustbesentfromthesourceBS tosome destinationnodes.NotethatBS is not adestinationnode.Letdest

(

m

)

∈

V denote thedestinationofthemessagem. Weused

(

m

)

>

0 todenotethedistance d

(

B S

,

dest

(

m

))

.We supposethat themessagesareordered bynon-increasingdistancefromBS totheir destinationnodes, andwedenotethisorderedset

M

= (

m1

,

m2

,

. . . ,

mM

)

whered

(

m1

)

≥

d

(

m2

)

≥ · · · ≥

d

(

mM

)

.Theinputofallouralgorithms

istheorderedsequence

M

ofmessages.Forsimplicitywesupposethatthegridisinfinite;howeveritsufficestoconsider a gridslightlygreater thanthe onecontaining all thedestinations ofmessages.Note thatourwork doesnotincludethe caseofthepaths,alreadyconsideredin[1,11,15].

We usethe nameofopengrid to meanthat no messageshavedestination onan axisthat iswhen allmessages have destinationnodesintheset

{(

x

,

y

)

:

x

=

0 and y

=

0

}

.

Notethat inourmodelthesourcecan sendatmostonemessageper step.Givenasetofmessagesthat mustbesent by the source, a broadcastingscheme consists in indicating foreach messagem the time at which the source sends the message m and thedirected path followed by thismessage.More precisely a broadcastingscheme is representedby an orderedsequence ofmessages

S = (

s1

,

. . . ,

sk

)

, wherefurthermoreforeach si wegive thedirected path Pi followedby si

andthetimeti atwhichthesourcesendsthemessagesi.Thesequenceisorderedinsuchawaythatthemessagesi+1 is

sentaftermessagesi,thatiswehaveti+1

>

ti.

As wesupposethereisnobuffering,amessagem sentatsteptm isreceivedatsteptm

=

lm

+

tm

−

1,wherelm isthe

lengthofthedirectedpathfollowedbythemessagem.Inparticulartm

≥

d

(

m

)

+

tm

−

1.Thecompletiontimeormakespan ofa

broadcastingschemeisthestepwhereallthemessageshavearrivedattheirdestinations.Itsvalueismaxm∈Mlm

+

tm

−

1.

Inthenextpropositionwegivealowerboundofthemakespan:

Proposition1.Giventhesetofmessages

M

= (

m1

,

m2

,

. . . ,

mM

)

orderedbynon-increasingdistancefromBS,themakespanofany

broadcastingschemeisgreaterthanorequaltoLB

=

maxi≤Md

(

mi

)

+

i

−

1.

Proof. Consider any personalized broadcasting scheme. For i

≤

M, let ti be the step where the last message in

(

m1

,

m2

,

. . . ,

mi

)

issent;thereforeti

≥

i.Thislastmessagedenotedm isreceivedatstepti

≥

d

(

m

)

+

ti

−

1

≥

d

(

mi

)

+

ti

−

1

≥

d

(

mi

)

+

i

−

1.SothemakespanisatleastLB

=

maxi≤Md

(

mi

)

+

i

−

1.

2

Note that this result is valid for any topology (not only grids) since it uses only the fact that the source sends at mostonemessageperstep.Iftherearenointerferenceconstraints,inparticularifa nodecansendandreceivemessages simultaneously,thentheboundisachievedbythegreedyalgorithmwhereatstepi thesourcesendsthemessagemiofthe

ordered sequence

M

throughashortestdirectedpathfromBS todest

(

mi

)

,i.e.themakespanLB isattainedby BS sending

allthemessagesthroughtheshortestpathstotheirdestinationsaccordingtothenon-increasingdistanceordering. AbetterlowerboundisprovidedinSection5forthecasewheredI

>

0.AsforthecasewheredI

=

0,thefollowingtwo

sectionsprovidelineartimealgorithmswithamakespanatmostLB

+

2 inthegridwiththebasestationinthecornerand amakespanatmostLB

+

1 whenfurthermorethereisnomessagewithadestinationnodeontheaxis(open-grid).Inthe casewheredI

=

0 andanopengridisused,ouralgorithmsaresimpleinthesensethattheyuseonlyverysimpleshortest

directedpathsandthatBS isneveridle.

Example1.Here,weexhibit examplesforwhichtheoptimalmakespanisstrictlylarger thanLB.Inparticular,inthecase ofgeneralgrids,LB

+

2 canbeoptimal.Ontheotherhand,resultsofthispapershowthattheoptimalmakespanisatmost LB

+

1 inthecaseofopen-gridsfordI

=

0 (Theorem 4)andatmostLB

+

2 ingeneralgridsfordI

=

0 (Theorem 7).Incase

dI

=

0 andinopen-grids,ouralgorithmsuseshortestpathsandtheBS sendsamessageateachstep.Wealsogiveexamples

forwhichoptimalmakespancannotbeachievedinthissetting.

Let us remark that there exist configurations for which no personalized broadcasting protocol can achieve better makespan than LB

+

1. Fig. 1(a) representssuch a configuration.Indeed, inFig. 1(a),messagemi has a destinationnode

vi fori

=

1

,

2

,

3 andLB

=

7.However,toachieve themakespanLB

=

7 fordI

=

0,BSmustsendthemessagem1 to v1 at

step1(because v1 isatdistance7 fromBS)andmustsendmessagem2 tov2 atstep2(becausethemessagestarts after

thefirststepandmustbesenttothedestinationnodeatdistance6)andthesemessagesshouldbesentalongshortest di-rectedpaths.Toavoidinterference,theonlypossibilityisthatBSsendsthefirstmessagetonode

(

0

,

1

)

,andthesecondone to thenode

(

1

,

0

)

.Intuitively,thisisbecauseotherwisetheshortestpathsfollowedbyfirsttwo messageswouldintersect insucha waythatinterferencecannotbe avoided.Aformal proofcanbeobtainedfromFact 2inSection3.2.Butthen,if wewanttoachievethemakespanof7,BShastosendthemessagem3 vianode

(

0

,

1

)

anditreachesv3 atstep7;butthe

directedpathsfollowedbym2 andm3 needtocrossandatthiscrossingpointm3 arrivesatastepwherem2 leavesandso

Fig. 1. Two particular configurations.

Inaddition,therearealsoexamplesinwhichBShastowaitforsomestepsaftersendingonemessageinordertoreach thelowerboundLB fordI

=

0.Fig. 1(b)representssuchanexample.Toachievethelowerbound7,BShastosendmessages

usingshortestdirectedpathsfirstlytov1via

(

3

,

0

)

andthenconsecutivelysendsmessagestov2via

(

0

,

4

)

andv3via

(

2

,

0

)

.

IfBSsendsmessagem4 atstep4,thenm4 interfereswithm3.Toavoidthisinterference,BScansendmessagem4 atstep5

andreachesv4 atstep7.

Thereare alsoexamplesinwhichnoscheduleusingonlyshortestdirectedpaths achievestheoptimalmakespan.1 For instance, considerthe grid with four messages to be sent to

(

0

,

4

),

(

0

,

3

),

(

0

,

2

)

and

(

0

,

1

)

(all on the first column) and let dI

=

0 (a moreelaborate examplewith anopen-grid is giveninExample 6(a)). Clearly, sending allmessages through

shortestdirectedpathsimpliesthatBS sends messageseverytwo steps.Therefore,itrequires7 steps.Ontheother hand, thefollowingschemehasmakespan6:sendthemessageto

(

0

,

4

)

throughtheuniqueshortestdirectedpathatstep1;send themessageto

(

0

,

3

)

atstep2 vianodes

(

1

,

0

),

(

1

,

1

),

(

1

,

2

)(

1

,

3

)

;sendthemessageto

(

0

,

2

)

throughtheshortestdirected pathatstep3 and,finally,sendthemessageto

(

0

,

1

)

atstep4 vianodes

(

1

,

0

),

(

1

,

1

)

.Notethatinthisexampletheoptimal makespanisLB

+

2.

3. Basicinstance:dI

=

0,open-grid,andBSinthecorner

Inthis section westudy simpleconfigurations calledbasicinstances. A basicinstanceis a configurationwheredI

=

0,

messages are sent in the open grid (no destinations on the axis) andBS isin the corner (anode with degree 2 inthe grid).Weseethatwecanfindpersonalizedbroadcastingalgorithmsusingabasicscheme,whereeachmessageissentvia asimpleshortestdirectedpath(withone horizontalandonevertical segment)andwherethesourcesendsa messageat eachstep(itneverwaits)andachievingamakespanofatmostLB

+

1.

3.1. Basicschemes

A messageis said to be sent horizontally to its destination v

= (

x

,

y

) (

x

>

0

,

y

>

0

)

, ifit goes first horizontallythen vertically,thatisifitfollowstheshortestdirectedpathfromBS tov passing through

(

x

,

0

)

.Correspondingly,themessage is sent vertically toits destination v

= (

x

,

y

)

, ifit goesfirst vertically then horizontally, that is ifit follows theshortest directedpathfromBS tov passingthrough

(

0

,

y

)

.Weusethenotation amessageissentindirectionD,whereD

=

H (for horizontally)(resp.D

=

V (forvertically))ifitissenthorizontally(resp.vertically).Also, D denotes

¯

thedirectiondifferent fromD thatis D

¯

=

V (resp. D

¯

=

H )ifD

=

H (resp. D

=

V ).

Definition1 (Basicscheme).Abasicscheme isa broadcastingschemewhereBS sends amessageateach stepalternating horizontal and vertical transmissions. Therefore it is represented by an ordered sequence

S = (

s1

,

s2

,

. . . ,

sM

)

of the M

messageswiththeproperties:messagesi issentatstepi andfurthermore,ifsi issentindirectionD,thensi+1 issentin

directionD.

¯

Notation: Notethat, by definitionofhorizontalandverticaltransmissions, thebasicscheme definedbelowuses shortest paths.Moreover,assoonaswefix

S

andthetransmissiondirectionD ofthefirstorlastmessage,thedirectedpathsused intheschemeareuniquelydetermined.Hence,theschemeischaracterizedbythesequence

S

andthedirectionD.Weuse whenneeded,thenotation

(S,

first

=

D

)

toindicateabasicschemewherethefirstmessageissentindirectionD,andthe notation

(S,

last

=

D

)

whenthelastmessageissentindirectionD.

Fig. 2. Cases of interference.

3.2. Interferenceofmessages

Ouraimisto designanadmissible basic schemeinwhichthemessages arebroadcastedwithoutanyinterference.The followingsimplefactshowsthatweonlyneedtotakecareofconsecutivetransmissions.Inthefollowing,wesaythattwo messagesareconsecutive ifthesourcesendsthemconsecutively(oneatstept andtheotheratstept

+

1).

Fact1.WhendI

=

0,inanybroadcastschemeusingonlyshortestpaths(in particularinabasicscheme),onlyconsecutive

messagesmayinterfere.

Proof. By definition,a basicscheme uses onlyshortest paths. Letthe messagem be sent atstep t and themessage m at step t

≥

t

+

2. Lett

+

h (h

≥

0) be a step at which the two messages have not reachedtheir destinations. As we use shortest directed paths the message m is sent on an arc

(

u

,

v

)

with d

(

v

,

BS

)

=

d

(

u

,

BS

)

+

1

=

t

+

h

−

t

+

1, while messagem issentonan arc

(

u

,

v

)

withd

(

v

,

BS

)

=

d

(

u

,

BS

)

+

1

=

h

+

1.Therefore,d

(

u

,

v

)

≥

t

−

t

−

1

≥

1

>

0

=

dI and

d

(

u

,

v

)

≥

t

−

t

+

1

≥

3

>

dI.

2

We now characterizethe situations whentwo consecutivemessages interferein abasic scheme.Forthat we usethe followingnotation:

Notation: InthecasedI

=

0,ifBSsendsindirectionD

∈ {

V

,

H

}

themessagem atstept and sendsthemessageminthe

otherdirectionD,

¯

atstept

=

t

+

1,wewrite

(

m

,

m

)

∈

DD if

¯

theydonotinterfereand

(

m

,

m

)

∈

/

DD if

¯

theyinterfere.

Fact2.Letm andmbetwo consecutivemessagesinabasicscheme.Then,

(

m

,

m

)

∈

/

DD if

¯

andonlyifthepaths followed bythemessagesinthebasicschemeintersectatavertexwhichisnotthedestinationofm.

Proof. Supposethedirectedpathsintersectinanode v thatisnotthedestinationofm.Themessagem sentatstept has not reachedits destinationandsoleavesthenode v atstept

+

d

(

v

,

BS

)

;butthemessagem sentatstept

+

1 arrivesat node v atstept

+

d

(

v

,

BS

)

andthereforethetwomessagesinterfere.

Conversely ifthetwo directedpaths used form andm donot crossthenthe messagesdonot interfere. Ifthepaths intersect onlyinthedestinationdest

(

m

)

ofm, thenm arrives indest

(

m

)

onestepafterm hasstoppedindest

(

m

)

andso thetwomessagesdonotinterfere.

2

Remark1.NotethatFact 2doesnotholdifwedonotimposebasicschemes(i.e.,thisisnottrueifanyshortestpathsare considered).Moreover,weemphasizethatthetwopathsmayintersect,butthecorrespondingmessagesdonotnecessarily interfere.

Insomeproofsthroughoutthepaper,weneedtousethecoordinatesofthemessages.Therefore,thefollowingequivalent statementofFact 2isofinterest.Letdest

(

m

)

= (

x

,

y

)

anddest

(

m

)

= (

x

,

y

)

.Then

• (

m

,

m

)

∈

/

HV ifandonlyif

{

x

≥

x andy

<

y

}

;

• (

m

,

m

)

∈

/

VH ifandonlyif

{

x

<

x and y

≥

y

}

.

Fig. 2showswhenthereisinterferenceandalsoillustratesFact 2forD

=

H (resp.V )incase(a)(resp. (b)). 3.3. Basiclemmata

Lemma1.If

(

m

,

m

)

∈

/

DD,

¯

then

(

m

,

m

)

∈ ¯

DD and

(

m

,

m

)

∈

DD.

¯

Proof. ByFact 2, if

(

m

,

m

)

∈

/

DD,

¯

then thetwo directed paths followed bym and m in thebasic scheme(in directions D and D respectively)

¯

intersect inanode differentfrom dest

(

m

)

.Then, thetwo directedpaths followedby m andm in thebasicscheme(in directions D and

¯

D respectively) donotintersect.Hence, by Fact 2,

(

m

,

m

)

∈ ¯

DD.Similarly, thetwo directedpaths followedbym andm inthebasicscheme(indirections D andD respectively)

¯

donotintersect.Hence,by

Fact 2,

(

m

,

m

)

∈

DD.

¯

2

Notethatthislemmaisenoughtoprovethemultiplicative ₂3 approximationobtainedin[15].Indeedthesourcecansend atleasttwomessageseverythreesteps,intheorderofm1

,

m2

,

. . . ,

mM.Moreprecisely,BS sendsanypairofmessagesm2i−1

andm2i consecutivelyby sending thefirst onehorizontally andthesecond one verticallyif

(

m2i−1

,

m2i

)

∈

HV,otherwise

sendingthefirstoneverticallyandthesecondonehorizontallyif

(

m2i−1

,

m2i

)

∈

/

HV (sincethisimpliesthat

(

m2i−1

,

m2i

)

∈

VH).Thenthesourcedoesnotsendanythingduringthethirdstep.Sowecansend2q messagesin3q steps.Suchascheme hasmakespanatmost 3₂LB.

Notethat ingeneral,

(

m

,

m

)

∈

DD does

¯

notimply

(

m

,

m

)

∈ ¯

DD,namelywhen thedirected pathsintersect onlyinthe destinationofm whichisnotthedestinationofm.

Lemma2.If

(

m

,

m

)

∈

DD and

¯

(

m

,

m

)

∈ ¯

/

DD,then

(

m

,

m

)

∈

DD.

¯

Proof. ByFact 2,

(

m

,

m

)

∈

DD implies

¯

thatthepathsfollowedbym andm(indirectionsD andD respectively)

¯

inthebasic schememayintersectonlyindest

(

m

)

.Moreover,

(

m

,

m

)

∈ ¯

/

DD impliesthatthepathsfollowedbymandm (indirections

¯

D andD respectively)intersectinanodewhichisnot dest

(

m

)

.Simplecheckshowsthatthepathsfollowedbym andm (indirectionsD and D respectively)

¯

mayintersectonlyindest

(

m

)

.Therefore,byFact 2,

(

m

,

m

)

∈

DD.

¯

2

Lemma3.If

(

m

,

m

)

∈

/

DD and

¯

(

m

,

m

)

∈ ¯

/

DD,then

(

m

,

m

)

∈

DD.

¯

Proof. ByLemma 1

(

m

,

m

)

∈

/

DD implies

¯

(

m

,

m

)

∈

DD.

¯

Then wecanapply theprecedingLemma 2withm

,

m

,

m inthis ordertogettheresult.

2

3.4. Makespancanbeapproximateduptoadditiveconstant2

Recallthat

M

= (

m1

,

. . . ,

mM

)

isthesetofmessagesorderedbynon-increasingdistancefromBS.Throughoutthispaper,

S



Sdenotesthesequenceobtainedbytheconcatenationoftwosequences S and S.

In[5],weuseabasicschemetodesignanalgorithmforbroadcastingthemessagesinthebasicinstancewithamakespan atmost LB

+

2. We give herea different algorithm withsimilar properties,buteasier to proveand whichpresents two improvements:itcan beadapted tothecasewherethedestinations ofthemessagesmaybeontheaxes (i.e.forgeneral grid)(seeSection4)anditcanberefinedtogiveinthebasicinstanceamakespanatmostLB

+

1.Wedenotethealgorithm by TwoApprox

[

dI

=

0

,

last

=

D

](

M)

;foran input setofmessages

M

ordered bynon-increasingdistances fromBS,anda

direction D

∈ {

H

,

V

}

,itgivesasoutput anordered sequence

S

ofthe messagessuch thatthe basicscheme

(S,

last

=

D

)

hasmakespanatmostLB

+

2.Recallthat D isthedirectionofthelastsentmessagein

S

inDefinition 1.

ThealgorithmTwoApprox

[

dI

=

0

,

last

=

D

](

M)

isgiveninFig. 3.Itusesabasicscheme,wherethenon-increasingorder

iskept,ifthereisnointerference;otherwisewechangetheorderalittlebit.Todothat,weapply dynamicprogramming. Weexaminethemessagesintheirorderandatagivenstepweaddtothecurrentorderedsequencethetwonext uncon-sideredmessages.Weshowthatwecanavoidinterference,onlybyreorderingthesetwomessagesandthelastoneinthe currentsequence.

Remark2. Notice that, there are instances (see examples below) for which Algorithm TwoApprox computes an optimal makespanonlyforonedirection.Hence,itmaysometimesbeinterestingtoapplythealgorithmforeachdirectionandtake thebetterofthetwoobtainedschedules.

Becauseofthebehaviorofabasicscheme,thedirectionofthefinalmessageandofthefirstone aresimplylinkedvia theparityofthenumberofmessages.Hence,wecanalsoderiveanalgorithmTwoApprox

[

dI

=

0

,

first

=

D

](

M)

thathasthe

firstdirectionD ofthemessageasaninput.

Example2.Here,we giveexamplesthat illustratetheexecutionofAlgorithmTwoApprox.Moreover, wedescribeinstances forwhichitisnotoptimal.

ConsidertheexampleofFig. 4(a).Thedestinationsofthemessagesmi(1

≤

i

≤

6)are v1

= (

7

,

3

)

,v2

= (

7

,

1

)

,v3

= (

3

,

3

)

,

v4

= (

2

,

4

)

,v5

= (

1

,

5

)

andv6

= (

2

,

2

)

.HereLB

=

10.LetusapplyAlgorithm TwoApprox

[

dI

=

0

,

last

=

V

](

M)

.Firstweapply

Input:M= (m1,. . . ,mM),thesetofmessagesorderedbynon-increasingdistancesfromBS andthedirectionD∈ {H,V}ofthelastmessage. Output:S= (s1,. . . ,sM)anorderedsequenceoftheM messagessatisfying (i)and(ii)(SeeinTheorem 1)

begin 1 Case M=1: returnS= (m1) 2 Case M=2: 3 if(m1,m2)∈ ¯DD returnS= (m1,m2) 4 else returnS= (m2,m1) 5 Case M>2:

6 letOp=TwoApprox[dI=0,last=D](m1,. . . ,mM−2)

(p isthelastmessageintheobtainedsequence) 7 Case 1: if(p,mM−1)∈DD and¯ (mM−1,mM)∈ ¯DD returnO (p,mM−1,mM)

8 Case 2: if(p,mM−1)∈DD and¯ (mM−1,mM)∈ ¯/DD returnO (p,mM,mM−1)

9 Case 3: if(p,mM−1)∈/DD and¯ (p,mM)∈ ¯DD returnO (mM−1,p,mM)

10 Case 4: if(p,mM−1)∈/DD and¯ (p,mM)∈ ¯/DD returnO (p,mM,mM−1)

end

Fig. 3. Algorithm TwoApprox[dI=0,last=D](M).

Fig. 4. Examples for Algorithms TwoApprox[dI=0,last=D](M)and OneApprox[dI=0,last=V](M).

(line 6) ism1 and as

(

m1

,

m3

)

∈

/

VH and

(

m1

,

m4

)

∈

HV,we get(line 9,case 3)

S = (

m2

,

m3

,

m1

,

m4

)

. Wenow apply the

algorithm withm5

,

m6.The value of p (line 6) ism4 andas

(

m4

,

m5

)

∈

/

VH and

(

m4

,

m6

)

∈

/

HV,we get (line 10,case4)

S = (

m2

,

m3

,

m1

,

m4

,

m6

,

m5

)

.ThemakespanofthealgorithmisLB

+

2

=

12

=

d

(

m1

)

+

2 achievedfors3

=

m1.

But,ifweapplytothisexampleAlgorithm TwoApprox

[

dI

=

0

,

last

=

H

](

M)

,wegetamakespanof10.Indeed

(

m1

,

m2

)

∈

VH and we get (line 3)

S = (

m1

,

m2

)

. Then as p

=

m2,

(

m2

,

m3

)

∈

HV and

(

m3

,

m4

)

∈

/

VH, we get (line 8, case 2)

S =

(

m1

,

m2

,

m4

,

m3

)

.Finally, with p

=

m3,

(

m3

,

m5

)

∈

HV and

(

m5

,

m6

)

∈

VH we get (line 7, case1) the final sequence

S =

(

m1

,

m2

,

m4

,

m3

,

m5

,

m6

)

withmakespan10

=

LB.

Consider the example of Fig. 4(b). The destinations of the messages m_i (1

≤

i

≤

6) are v_i, which are placed in symmetric positions with respect to the diagonal as vi in Fig. 4(a). So v₁

= (

3

,

7

)

, v₂

= (

1

,

7

),

. . . ,

v₆

= (

2

,

2

)

. So

we can apply the algorithm by exchanging the x and y, V and H . By Algorithm TwoApprox

[

dI

=

0

,

last

=

V

](

M)

,

we get

S = (

m₁

,

m₂

,

m₄

,

m₃

,

m₅

,

m₆

)

with makespan 10; by Algorithm TwoApprox

[

dI

=

0

,

last

=

H

](

M)

, we get

S =

(

m₂

,

m₃

,

m₁

,

m₄

,

m₆

,

m₅

)

withmakespan 12.

However there are sequences

M

such that both Algorithms TwoApprox

[

dI

=

0

,

last

=

V

](

M)

and TwoApprox

[

dI

=

0

,

last

=

H

](

M)

give amakespanLB

+

2.Consider theexampleofFig. 4(c)with

M

= (

m1

,

. . . ,

m6

,

m1

,

. . . ,

m6

)

.The

destina-tions ofm1

,

. . . ,

m6 areobtainedfromthedestination nodesinFig. 4(a)by translating themalonga vector

(

3

,

3

)

,i.e.we

move vi

= (

xi

,

yi

)

to

(

xi

+

3

,

yi

+

3

)

.SoLB

=

16 andAlgorithmTwoApprox

[

dI

=

0

,

last

=

V

](

m1

,

. . . ,

m6

)

givesthesequence

S

V

= (

m2

,

m3

,

m1

,

m4

,

m6

,

m5

)

withmakespan18 andAlgorithmTwoApprox

[

dI

=

0

,

last

=

H

](

m1

,

. . . ,

m6

)

givesthesequence

S

H

= (

m1

,

m2

,

m4

,

m3

,

m5

,

m6

)

withmakespan16.Notethatthedestinationsofm1

,

. . . ,

m6areinthesameconfigurationas

thoseofFig. 4(b).Now,ifwerunAlgorithm TwoApprox

[

dI

=

0

,

last

=

V

](

M)

onthesequence

M

= (

m1

,

. . . ,

m6

,

m1

,

. . . ,

m6

)

,

we get as

(

m5

,

m1

)

∈

VH and

(

m1

,

m2

)

∈

HV, the sequence

S

V



S

V

= (

m2

,

m3

,

m1

,

m4

,

m6

,

m5

,

m1

,

m2

,

m4

,

m3

,

m5

,

m6

)

with makespan 18 achieved for s3

=

m1. If we run Algorithm TwoApprox

[

dI

=

0

,

last

=

H

](

M)

on the sequence

M

=

(

m1

,

. . . ,

m12

)

,wegetas

(

m6

,

m1

)

∈

HV and

(

m1

,

m2

)

∈

/

VH thesequence

S

H



S

H

= (

m1

,

m2

,

m4

,

m3

,

m5

,

m6

,

m2

,

m3

,

m1

,

m4

,

m₆

,

m₅

)

withmakespan18 achievedfors9

=

m₁.

However we can find asequence witha makespan 16 achieving thelower bound witha basic schemenamely

S

∗

=

(

m1

,

m5

,

m2

,

m4

,

m3

,

m₁

,

m6

,

m₂

,

m₅

,

m₃

,

m₄

,

m₆

)

withthefirstmessagesent horizontally.

Theorem1.Givenabasicinstanceandthesetofmessagesorderedbynon-increasingdistancesfromBS,

M

= (

m1

,

m2

,

. . . ,

mM

)

andadirectionD

∈ {

H

,

V

}

,AlgorithmTwoApprox

[

dI

=

0

,

last

=

D

](

M)

computesinlinear-timeanordering

S = (

s1

,

. . . ,

sM

)

ofthe

messagessatisfyingthefollowingproperties:

(i) thebasicscheme

(S,

last

=

D

)

broadcaststhemessageswithoutinterference;

(ii) s1

∈ {

m1

,

m2

}

,s2

∈ {

m1

,

m2

,

m3

}

andsi

∈ {

mi−2

,

mi−1

,

mi

,

mi+1

,

mi+2

}

forany3

≤

i

≤

M

−

2 andsM−1

∈ {

mM−3

,

mM−2

,

mM−1

,

mM

}

,sM

∈ {

mM−1

,

mM

}

.

Proof. TheproofisbyinductiononM.IfM

=

1,wesendm1 indirectionD (line1).Sothetheoremistrue.IfM

=

2,either

(

m1

,

m2

)

∈ ¯

DD and

S = (

m1

,

m2

)

satisfiesallpropertiesor

(

m1

,

m2

)

∈ ¯

/

DD andbyLemma 1

(

m2

,

m1

)

∈ ¯

DD and

S = (

m2

,

m1

)

satisfiesallproperties.

If M

>

2, let

O 

p

=

TwoApprox

[

dI

=

0

,

last

=

D

](

m1

,

. . . ,

mM−2

)

be the sequence computed by the algorithm for

(

m1

,

m2

,

. . . ,

mM−2

)

.By theinduction hypothesis, we mayassume that

O 

p satisfiesproperties (i) and(ii). In

particu-lar p issentindirection D and p

∈ {

mM−3

,

mM−2

}

.Now we provethat thesequence S

= {

s1

,

. . . ,

sM

}

satisfies properties

(i)and(ii).Property(ii)issatisfiedinallcases:forsi

,

1

≤

i

≤

M

−

3,asitisverifiedbyinductionin

O

;forsM−2,aseither

sM−2

=

p

∈ {

mM−3

,

mM−2

}

orsM−2

=

mM−1;forsM−1,aseithersM−1

=

p

∈ {

mM−3

,

mM−2

}

orsM−1

=

mM−1 orsM−1

=

mM

andfinallyforsM,assM

∈ {

mM−1

,

mM

}

.Forproperty(i)weconsiderthefourcasesofthealgorithm(lines7–10).Obviously,

thelastmessageissentindirectionD inallcases.Inthefollowingweprovethattherearenointerferenceinanycase.For cases1,2and4,

O 

p isbyinductionaschemethatresultsinnointerference.

Incase1,byhypothesis,

(

p

,

mM−1

)

∈

DD and

¯

(

mM−1

,

mM

)

∈ ¯

DD.

In case 2, since

(

p

,

mM−1

)

∈

DD and

¯

(

mM−1

,

mM

)

∈ ¯

/

DD, Lemma 2 with p

,

mM−1

,

mM in this order implies that

(

p

,

mM

)

∈

DD.

¯

Furthermore,byLemma 1,

(

mM−1

,

mM

)

∈ ¯

/

DD implies

(

mM

,

mM−1

)

∈ ¯

DD.

Forcase4,by Lemma 1

(

p

,

mM

)

∈ ¯

/

DD implies

(

p

,

mM

)

∈

DD.

¯

FurthermoreLemma 3,appliedwith p

,

mM

,

mM−1 inthis

orderanddirectionD,

¯

implies

(

mM

,

mM−1

)

∈ ¯

DD.

Forcase 3,

(

p

,

mM

)

∈ ¯

DD;furthermoreby Lemma 1,

(

p

,

mM−1

)

∈

/

DD implies

¯

(

mM−1

,

p

)

∈

DD.

¯

It remains toverifythat

if q is the last message of

O

,

(

q

,

mM−1

)

∈ ¯

DD. As

O 

p is an admissible scheme we have

(

q

,

p

)

∈ ¯

DD and since also

(

p

,

mM−1

)

∈

/

DD,

¯

byLemma 2appliedwithq

,

p

,

mM−1inthisorderanddirectionD,

¯

weget

(

q

,

mM−1

)

∈ ¯

DD.

2

Ascorollarywegetbyproperty(ii)anddefinitionofLB thatthebasicscheme

(S,

last

=

D

)

achievesamakespanatmost LB

+

2.WeemphasizethisresultasaTheoremandnotethatinviewofExample 2itisthebestpossibleforthealgorithm.

Theorem2.Inthebasicinstance,thebasicscheme

(

S,

last

=

D

)

obtainedbyAlgorithm TwoApprox

[

dI

=

0

,

last

=

D

](

M)

achievesa

makespanatmostLB

+

2.

Proof. Itissufficienttoconsiderthearrivaltimeofeachmessage.BecauseAlgorithmTwoApprox

[

dI

=

0

,

last

=

D

](

M)

uses

abasicscheme,eachmessagefollowsashortestpath.ByProperty(ii)ofTheorem 1,themessages1arrivesatitsdestination

atstepd

(

s1

)

≤

d

(

m1

)

≤

LB andthemessages2 arrivesatstepd

(

s2

)

+

1

≤

d

(

m1

)

+

1

≤

LB

+

1;forany2

<

i

≤

M,themessage

si arrivesatitsdestinationatstepd

(

si

)

+

i

−

1

≤

d

(

mi−2

)

+

i

−

1

=

d

(

mi−2

)

+ (

i

−

2

)

−

1

+

2

≤

LB

+

2.

2

3.5. Makespancanbeapproximateduptoadditiveconstant1

Inthissubsection,weshowhowtoimproveAlgorithmTwoApprox

[

dI

=

0

,

last

=

D

](

M)

inthebasicinstance(opengrid

withBS in thecorner) to achieve makespanatmost LB

+

1.Forthat we distinguishtwo casesaccording tothe value of lasttermsM whichcanbeeithermM ormM−1.Inthelatercase, sM

=

mM−1wealsomaintainanotherorderedadmissible

sequence

S

 of the M

−

1 messages

(

m1

,

. . . ,

mM−1

)

which can be extended in the induction step when

S

cannot be

extended.Bothsequences

S

and

S

shouldsatisfysometechnicalproperties(seeTheorem 3).

We denotethe algorithm asOneApprox

[

dI

=

0

,

last

=

D

](

M)

. Foran ordered input sequence

M

ofmessages andthe

direction D

∈ {

H

,

V

}

,itgivesasoutput anordered sequence

S

ofthe messagessuch thatthe basicscheme

(S,

last

=

D

)

hasmakespanatmostLB

+

1.AlgorithmOneApprox

[

dI

=

0

,

last

=

D

](

M)

isdepictedinFig. 5.AsweexplaininRemark 2,

wecanalsoobtainalgorithmswiththefirstmessagesentindirectionD.

Example3.Here, wegive examplesthatillustratetheexecutionofAlgorithmOneApprox.Moreover,we describeinstances forwhichitisnotoptimal.

ConsideragaintheExampleofFig. 4(a)(seeExample 2).LetusapplyAlgorithm OneApprox

[

dI

=

0

,

last

=

V

](

M)

.Firstwe

applythealgorithmform1

,

m2;

(

m1

,

m2

)

∈

/

HV,weareatline4 and

S = (

m2

,

m1

)

and

S



= (

m1

)

.Thenweconsiderm3

,

m4;

thevalue of p (line6) ism1;as

(

m1

,

m3

)

∈

/

VH and

(

m2

,

m4

)

∈

HV,we areincase 3.2 line11(p

=

mM−3). Sowe get,as

O



_{= (}

_m1

₎

_{, S}

_{= (}

_m1

_,

_m3

_,

_m2

_,

_m4

₎

_{.Wenowapplythealgorithmwithm5}

_,

_{m6;thevalueofp (line6)ism4;as}

₍

_m4

_,

_m5

₎

_∈

_/

_VH

and

(

m4

,

m6

)

∈

/

HV,we areincase4.1line13.Sowe get S

= (

m1

,

m3

,

m2

,

m4

,

m6

,

m5

)

.The makespanofthealgorithmis