ThealgorithmisclearlyasetofFloodMaxalgorithmsforleaderelection.Infacttheboundaryoftheorthotopeineachdirectionaisgivenbythecoordinatesofthepointsonsuchboundarywhicharecharacterizedbythepropertyofhavingthemaximumandminimumvalueoftheathcoordinaterespectively.
Inordertoprovethattheexactnumberofcommunica-tionroundsneededisTFloodMEO,simplyobservethatitisexactlytheminimumtimeforalltheleaderstopropagatetheirinformationthroughallthenetwork.Hencethisistheminimumtimeforeverypossibleconsensusalgorithmtoconverge.Butthisisexactlythetimetakenby2dFloodMaxalgorithmsrunningsimultaneouslyandthereforethetimetakenbyFloodMEO.
In this paper we introduce the notion of optimization under control and communication constraint in a robotic network. Starting from a general setup, we focus our attention on the problem of achieving rendezvous in minimum time for a network of first order
theMEB(MEO)ofagenti(forsomei∈{1,...,n})hasnotchangedandthealgorithmhasnotconvergedyet.ThentherewillexistaT>diamGsuchthattheMEB(MEO)ofagentiwillchangetoanewvalue.Butthismeansthatthenewvalue,storedTroundsbeforebysomeotheragentj,tookanumberofcommunicationroundsgreaterthandiamGtoarrivefromjtoiandthiscontradictsthede nitionofdiameterofG.
T∈Nsuchthatfort=
A.TimecomplexityofCCMEBandCCMEO
InthepreviouslemmawehaveproventhatthecontrolandcommunicationlawsCCMEBandCCMEOachieveconsensus.Nowweaskhowfasttheselawsaredependingonthecontrolboundrctrandthenumberofagents.
Theorem5.2:Forrcmm∈R+,d∈N,considerthenetworkSwithcommunicationedgemapeitherEdiskorEcube.Thefollowingstatementshold:
(i)foru[i]∈B(0,rctr),i∈{1,...,n},thecontroland
communicationlawCCMEBasymptoticallyconvergestotheminimumtimerendezvouscentralized+solutionMTR(Ecmpl,B(0,rctr))asrctr→0(forall xedn).(ii)foru[i]∈C(0,rctr),i∈{1,...,n},thecon-trolandcommunicationlawCCMEOconvergestotheminimumtimerendezvouscentralized+solutionMTR(Ecmpl,C(0,rctr))forrctr→0(forall xedn).Moreover,itisaconstantfactorapproximationofMTR(Ecmpl,C(0,rctr)),i.e.,TC(Trndzvs,CCMEO)∈Θ(n
rTctr
+FloodMEO.
(3)
The rststatementisprovenbyobservingthatTFloodMEOdoesnotdependonrctr,therefore,asrctr→0+,TMEOconvergestotheoptimalvalueofthecentralizedcase.
Inordertoprovethesecondstatement,observethatdiam(p[1](0),...,p[n](0))≤(n 1)rcmmandTFloodMEO∈Θ(n).Theresultfollowsbysubstitutingtheseboundsin(3).
In this paper we introduce the notion of optimization under control and communication constraint in a robotic network. Starting from a general setup, we focus our attention on the problem of achieving rendezvous in minimum time for a network of first order
B.DistributedminimumtimerendezvousinonedimensionInonedimension(alltheagentsspreadonaline),wecan ndaconditiononrctrensuringthatthemove-toward-MBCalgorithmisthesolutionofMTR(Edisk,B(0,rctr)).
Theorem5.5:Ford=1,letimaxandimintheagentsinthenetworkSwiththemaximumandminimumpositions.Ifrctr<1
rctr
Proof:Considertheinputsequenceofthecentralizedsolutionfortheagentimin(andequivalentlyforimax).Itisu[imin](t)=rctrforallt<T 1andu[imin](T 1)=MBC(p[1](0),...,p[n](0)) p[imin](T 1).SincetherendezvoustimeisboundedbythetimethatimaxandimintaketoreachMBC(p[1](0),...,p[n](0)),weneedtoprovethat,aslongastheconsensusontheminimalenclosingballisnotreached,thenu[imin](t)= u[imax](t)=rctr.Duetothesymmetryoftheproblemwewillgivetheproofonlyforimin.Itcanbeeasilyshownthatforallt≥1such
[imin]
thatpmax(t)=p[imax](0)(consensusisnotreached),thefollowingholds:
[imin]imin]
p[max(t+1)>pmax(t 1)+rcmm.
.
Itfollows:
MBC[imin](t+1)=
1
2
imin][imin]
(p[(0))max(t 1)+rcmm+p
=MBC[imin](t 1)+
1rcmm.
2
Thisleadsto
rctr+
1
4
rcmm.
Theothertwoassumptionsensuretheconditionfort=0.