Proximity awareness and ad hoc network establishment in Bluetooth

Transcription

1 oimiy awanss and ad ho nwok salishmn in Bluooh hodoos Salonidis avin Bhawa Landos assiulas and ihad LaMai lial and Compu ninin Dpamn Univsiy o Mayland a Coll ak. A&-Las sah Floham ak NJ. IBM.J. Wason sah Cn Hawhon NY. Asa-- In n yas wilss ad ho nwoks hav n a owin aa o sah. Whil h has n onsidal sah on h opi o ouin in suh nwoks h opi o opoloy aion has no ivd du anion. his is aus almos all ad ho nwoks o da hav n uil on op o a sinl hannl oadas asd wilss mdia suh as 8. o I LANs. Fo suh nwoks h disan laionship wn h nods impliily (and uniquly) dmins h opoloy o h ad ho nwok. Bluooh is a pomisin nw wilss hnoloy whih nals poal dvis o om sho-an wilss ad ho nwoks and is asd on a quny hoppin physial lay. his a implis ha hoss a no al o ommunia unlss hy hav pviously disovd ah oh y synhonizin hi quny hoppin pans. hus vn i all nods a wihin di ommuniaion an o ah oh only hos nods whih a synhonizd wih h ansmi an ha h ansmission. o suppo any-o-any ommuniaion nods mus synhonizd so ha h pais o nods (whih an ommunia wih ah oh) oh om a onnd aph. Usin Bluooh as an ampl his pap is povids dp insihs ino h issu o link salishmn in quny hoppin wilss sysms. I hn inodus h Bluooh opoloy Cosuion oool (BC) an asynhonous disiud poool o onsuin sans whih sas wih nods ha hav no knowld o hi suoundins and minas wih h omaion o a onnd nwok saisyin all onniviy onsains posd y h Bluooh hnoloy. o h s o ou knowld h wok psnd in his pap is h is amp a uildin Bluooh sans usin disiud loi and is qui paial in h sns ha i an implmnd usin h ommuniaion pimiivs od y h Bluooh. spiiaions. Ind ms Fquny hoppin Bluooh opoloy onsuion san.. INODUCION An ad ho nwok is a wilss nwok omd y nods ha oopa wih ah oh o owad paks in h nwok. Almos all pimnal ad ho nwoks o da hav n uil on op o sinl hannl oadas asd 8. wilss LANs o I LANs. In suh nwoks all nods wihin di ommuniaion an o ah oh sha a ommon hannl usin a CSMA syl MAC poool. In addiion muli-hop ouin is usd as a mans o owadin paks yond h ommuniaion an o h sou s ansmi. Sin a sinl hannl is usd houhou h nwok h opoloy o h ad ho nwok is impliily (and uniquly) dmind y disan laionship amon h paiipain nods. his pap is aimd a addssin a nw polm whih aiss whn mulipl hannls a availal o ommuniaion in an ad ho nwok. h polm is ha o dminin whih suoup o nods should sha a ommon hannl and whih nods should a as lays and owad ai om on hannl o anoh. h hannl assinmn should don so ha all onsains posd y h undlyin physial lay a saisid whil nsuin ha h sulan aph omd y all nods is onnd. W addss an insan o h aov polm whih ous in Bluooh asd ad ho nwoks known as sans [8]. Bluooh is a pomisin nw hnoloy whih is aimd a suppoin wilss onniviy amon ll phons hadss DAs diial amas and lapop ompus. Iniially h hnoloy will usd as a plamn o als u in du ous o im soluions o poin-o-mulipoin and muli-hop nwokin ov Bluooh will volv. Bluooh is a quny hoppin sysm whih dins mulipl hannls o ommuniaion (ah hannl dind y a din quny hoppin squn). A oup o dvis shain a ommon hannl is alld a pion. ah pion has a mas uni whih sls a quny hoppin squn o h pion his wok is suppod y an IBM Univsiy anship Awad.

2 and onols h ass o h hannl. Oh paiipans o h oup known as slav unis a synhonizd o h hoppin squn o h pion mas. Wihin a pion h hannl is shad usin a slod im division dupl (DD) poool wh a mas uss a pollin syl poool o alloa imslos o slav nods. h imum num o slavs ha an simulanously aiv in a pion is svn. Mulipl pions an o-is in a ommon aa aus ah pion uss a din hoppin squn. ions an also inonnd via id nods o om a i ad ho nwok known as a san. Bid nods a apal o imshain wn mulipl pions ivin daa om on pion and owadin i o anoh. h is no siion on h ol a id nod an play in ah pion i paiipas in. A id an a mas in on pion and slav in anoh (md as M/S id) o a slav in all pions (md as S/S id). I is possil o oaniz a ivn s o Bluooh dvis in many din oniuaions. Fius and show wo ampl oniuaions in whih nods in a Bluooh nwok an aand. All nods a assumd o in adio poimiy o ah oh. Fi. shows an ampl in whih all nods a pa o a sinl pion. Fiu illusas anoh oniuaion in whih nod A is mas o pion nod is mas o pion nod B is an M/S id (mas o pion and a slav o pion ) nod D is a slav o pion and nod C is an S/S id (slav in pions and ). In onas o h aov wo oniuaions h nod inonnion opoloy in a sinl hannl sysm will a ompl aph (Fi. a) sin all nods will ha ah oh s ansmission. A A A ion ion B B B ion C D C D C D (a) () () Fiu : (a) Sinl hannl modl. ()() Din oniuaions aodin o h Bluooh mulipl hannl modl. Givn a ollion o Bluooh dvis an plii opoloy onsuion poool is ndd o omin pions assinin slavs o pions and inonnin hm via ids suh ha h sulin san is onnd. Suh a poool should asynhonous oally disiud and nods should sa wih no inomaion aou hi suoundins. h polm o onsuin disiud sl-oanizin nwoks has n addssd in h pas ([][][5][9]) u all h os so a w aimd a solvin h polm y assumin a sinl oadas hannl and a CSMA syl MAC poool. h polm is siniianly had o quny hoppin asd wilss sysms as will vidn in h la disussion. his pap is a is amp o addss h opoloy onsuion polm in h mulipl FH hannl sin imposd y h Bluooh hnoloy. In od o solv i w dsin ou poool in a oom-up ashion: Fis in sion w amin h wilss link povidd y Bluooh y psnin h asymmi snd-iv poin o poin link salishmn poool as dind in h Bluooh spiiaions. In sion w nhan his poool y poposin a symmi vaian o h link salishmn poool wh wo dvis alna indpndnly wn h snd and iv sa unil hy disov and onn o ah oh. Suh a poool is nssay o salishin a onnion wn a pai o idnial dvis o in siuaions whn any nal mans o slin iniial dvi sas a no availal. Sion inodus h Bluooh opoloy Consuion oool (BC) whih is an No h is no d amon slav nods sin slavs anno ha ah oh s ansmission.

3 asynhonous disiud onnion salishmn poool ha nds h poin o poin symmi poool o h as o many nods. his poool is asd on a lad lion poss wh ah nod uss a imou o indpndnly did aou h lad lion minaion. h imou dlay ao inodus a onss-dlay ado o h nwok omaion. By usin h dlay analysis o sion w show in sion 5 how o s hoos h poool paams in od o imiz h poailiy o omin a onnd san whil minimizin dlays. Finally sion 6 povids a uu wok disussion and onlusions.. LINK SABLISHMN IN BLUOOH : BACKGOUND h Bluooh Basand Spiiaion [] dins h Bluooh poin o poin onnion salis hmn as a wo-sp podu. Fis nihohood inomaion is olld houh h Inquiy odu. h ain podu is susqunly usd o salish h onnions wn nihoin dvis. Boh h Inquiy and ain podus a asymmi posss; hy involv wo yps o nods (whih w all snds and ivs) ah pomin din aions. Duin Inquiy snds disov and oll nihohood inomaion povidd y ivs. Duin ain snds onn o ivs disovd duin a pvious inquiy podu. Duin h inquiy o pain podu alhouh snds and ivs us h sam (inquiy o pain) quny hoppin squn i is likly ha hy will ou o phas sin ah uni sas a a din hop quny divd om is loal lok valu. his (unavoidal) phas din inodus a phas unainy amon h dvis paiipain in h podu. o ovom his phas unainy snds and ivs hop a din spds. A iv hops a a slow a ov h ommon quny pan lisnin on ah hop o snd mssas and h snd ansmis a a muh hih a lisnin in wn ansmissions o an answ in hop o disovin h quny a iv is unly lisnin o. Givn wo unis on opain as a snd and h oh as a iv h m Fquny Synhonizaion dlay (o FS dlay) s o h im unil h snd ansmis a h quny h iv is unly lisnin on. vn i h wo podus hav h sam synhonizaion mhanism a din is ha duin h pain podu h snd is o ypass h FS dlay y simain h phas o h iv. I pain is pomd dily a h Inquiy podu h snd has aquid h lok valu o h iv uni and an us i o dmin is phas and onn o i insananously. h unional din wn h Inquiy and ain odus lis in h us o a univsal FH squn in h is and a ommon poin o poin FH squn in h sond. Usin a univsal inquiy hoppin squn a snd nod ivly oadass an Inquiy Ass Cod (IAC) pak ha an had only y iv nods ha lisn o suh a pak. Duin h pain podu y usin h iv s pa hoppin squn a snd nod iniias onnion salishmn y ivly uniasin a Dvi Ass Cod (DAC) pak ha an had only y h ospondin iv dvi. hus h Inquiy odu involvs many unis wh a snd an disov mo han on ivs whil h pain podu involvs only wo unis wh a snd pas and onns o a spii iv... h Bluooh Asymmi poool o link omaion Aodin o h Bluooh Basand spiiaion h poool sas y h snd sain in h INQUIY sa and h iv in h INQUIY SCAN sa. As was dsid in h pvious sion h is an iniial FS dlay unil h snd his h quny h iv is lisnin o. Upon ivin h IAC pak h iv aks o o an amoun o im ha is uniomly disiud wn and N h num o qunis in h inquiy o pa hoppin s is qual o o sysms opain in uop and US and 6 o sysms opain in Japan Spain and Fan. h snd an ov h ni inquiy hoppin quny s in im ova N 65us whih is ms (ms) o h 6 () hop sysm.

4 ms. his happns in od o pvn h onnion polm ha would ais i h w wo ivs lisnin on h sam hop quny. I oh o hm spondd immdialy h spons mssa would ald and h snd would no iv i. W all h im whil h iv aks o h andom Bako dlay (o B dlay). Whn h iv uni waks up i sas lisnin aain a h hop i was lisnin o o akin o. A a sond FS dlay (sam as h is on) a sond IAC pak is ivd om h snd. hn h iv snds ak o h snd an FHS pak ha onains:. h iv s addss: his is usd y h snd o div h DAC o h iv and h pa hoppin squn i will us la in od o pa h iv.. h iv s lok valu: his is usd o sima h phas o h iv and hus limina h FS dlay duin h pain podu ha ollows. h imin diaam in Fiu summaizs h poin o poin onnion salishmn podu wn h wo unis. h dashd aows dno vns on ah uni s imlin and ah vn is numd in h od i happns duin h onnion salishmn podu. h imin diaam shows ha h iv ns h AG SCAN sa a sndin h inquiy spons FHS pak o h snd. Whn h snd ivs h FHS pak i ns h AG sa and uss h lok inomaion in h FHS pak o snd a DAC pak on h quny h iv is lisnin o in h AG SCAN sa. hn h iv sponds immdialy wih a DAC pak and h snd snds an FHS pak o h iv. h iv uss h FHS inomaion o dmin h hannl hoppin squn and h phas o h snd and oms h slav o h poin o poin onnion. I hn aknowlds h FHS pak wih anoh DAC pak. As soon as h snd ivs h aknowldmn i oms h mas o h onnion and may sa hanin daa wih h synhonizd iv-slav. () Sa in h INQUIY sa (6) n h AG sa (7) Connion salishd Snd Link Fomaion Dlay IAC IAC FHS DAC DAC FHS DAC iv FS dlay B dlay FS dlay ain dlay ms - ms - ms. 65 ms () Sa in INQUIY SCAN sa () Go o slp () Wak up (5) spond and n AG SCAN sa (7) Connion (7) Connion salishd salishd Fiu : h Bluooh asymmi link omaion poool. By osvin Fiu w an asily idniy h link omaion dlay omponns. h inquiy podu dlay onsiss o a is FS dlay h B dlay and a sond FS dlay ha is akin pla whn h iv wais o h sond IAC pak a i waks up. h pain podu dlay is nliil sin i immdialy ollows h inquiy podu. (As soon as h is DAC pak is ivd y h iv h s o h sps a happnin in onsuiv 65µs slos). hus w an appoima h link omaion dlay usin h ollowin quaion:

5 FS B () wh FS and B a uniom andom vaials in [ ] quaion () h link omaion dlay an a mos h -hop sysm and 69.75ms o h 6-hop sysm. and [ ] ova spivly. Aodin o ov a ms ms ms o. A SYMMIC OOCOL FO LINK FOMAION h asymmi poool povidd y h Bluooh spiiaion yilds a vy sho onnion salishmn dlay povidd ha h snd and iv ols a p-assind. Whn wo o mo uss a yin o salish links wn hi Bluooh dvis in an ad ho ashion hy will no al o pliily assin snd and iv ols. hy will jus pss a uon and p o onn wih hi ps. hus h should a symmi mhanism ha oms onnions in an ad ho ashion wihou any plii snd o iv ol p-assinmn. A way o do his is y oin h wo nods o alna indpndnly wn h snd (INQUIY sa) and iv (INQUIY SCAN sa) ols and y o onn aodin o h asymmi poool duin an ovlap inval wh hy m in opposi sas. In Fiu Uni A has alady sad alnain and Uni B sas alnain a som aiay im. h md shdul is podud y min h sa swihin ims o h wo unis ino a sinl on whih an sn as an on-o poss. By usin sa alaion a onnion will salishd a a andom dlay whih in pinipl will la han h on o h asymmi poool. h ason is ha sain a ah on inval o h md poss h wo unis will onn a a andom inval FS B ivn ha hy oh main id a hi (omplmnay) sas o an amoun o im a han. Ohwis hy hav o wai o h n on inval. h im om im up o h poin wh h wo unis om o a omplmnay sa o a suiin amoun o im is ssnially h link omaion dlay wn h wo unis.... S I S I S I S Uni Uni I S I S I Md Shdul Fiu : A symmi link omaion poool: Nods alna wn snd and iv sa unil hy onn. h a som insin qusions aisin om h poposd alnain sas hniqu. Fis o all wha should h alnain shdul? Should h sas alna in a piodi o andom ashion? I an analyially povn ha h man onnion im is inini whn ah uni hans sas

6 dminisially (s Appndi A). Inuiivly i h sa sidn invals a id h invals o h md poss in Fiu will id as wll. hn h onnion im will dpnd on h id phas din o h wo dvis. I his phas is vy small hn h on invals in h md poss will vy small and h link omaion dlay vy la sin h unis will us aiaily many on invals unil hy inally onn. Alnaivly a andom shdul an imposd on h sa sidn ims. In Appndi B w povid an ad ho link omaion dlay modl and show ha whn ah uni alnas indpndnly wn INQUIY and INQUIY SCAN wih h sa sidn ims ollowin a ommon andom disiuion w an analyially alula h man and vaian o h link omaion dlay. h way o alula h onnion s up dlay is o dmin h d and pd o h md shdul poss ivn ha h wo nods alna indpndnly aodin o an idnial disiuion Z. In Appndi B w show ha h man and vaian o h link omaion dlay o h symmi poool a ivn y: [ ] ( [ > ] [ ])( p) [ ] p [ ] (.) [ ] ( Va[ > ] Va[ ])( p) Va Va[ ] Va p [ ] (.) wh p [ ]. (.) h alnain sas hniqu is a mhanism ha uaans an ad ho poin o poin onnion wn wo Bluooh dvis. Whn mo han wo dvis is and wish o om a san on h ly a poool should dvisd on op o his mhanism ha nsus ha h sulin nwok will ulill h quimns and suu o a Bluooh san. his poool should also iin in ms o nwok salishmn dlay. W will us h symmi link omaion dlay modl divd in Appndi B in od o ahiv his.. BC: A DISIBUD SCAN FOMAION OOCOL Ou moivaion o h san omaion polm aiss om a onn-snaio o an ad ho nwok salishmn. Suppos ha h a many uss in a oom ha wish o om an ad ho nwok usin hi Bluooh nald dvis. ah us psss a sa uon and wais o h dvi o show on h sn a nwok onnion salishd mssa a a sho piod o im. A his mssa appas h us will al o han inomaion wih any oh us in h oom. h dsipion o his appliaion aually onains h lmns o a sussul onnion salishmn poool: Nwok onnion salishmn should pomd in a oally disiud ashion. his mans ha ah dvi sas opain asynhonously on is own and i iniially dos no hav any knowld aou h idniis o num o nods in h oom. A omplion h poool mus uaan a onnd san. Connd mans ha h should a las on pah wn any wo nods in h nwok. h nwok s up dlay should minimizd suh ha i is olal y h nd us. In nal h a no siions adin h inal om o h san. h only quimns a ha: h should pions ha hav on mas and lss han svn slavs and ha pions a inonnd houh S/S o M/S id nods. vy nod mus al o ah vy oh nod in h sulin nwok i.. h nwok mus onnd.

7 In addiion o saisyin onniviy a dsial au o h poool would o al o shap h nwok opoloy aodin o san omaion iia imposd y spii appliaions. Fo ampl h sam nod may nd o hav din ols in din appliaions. Also i may possil o a nod o hav mo siiv d onsains han svn du o is own nau as a dvi; o ampl a palm pilo would no hav h possin pow o a mas o a svn slav pion. San omaion iia ould also in h om o ai dmands ha nd o saisid y h nods paiipain in h nwok onsuion poss. hs iia should akn ino aoun duin h opoloy onsuion poss i hy is. h polm o dinin san omaion iia is isl an opn sah issu ha is havily dpndn on h nvisiond appliaions. Alhouh w do no addss i in his pap ou appoah aks i ino aoun y ollin inomaion aou all nods paiipain in h poss a a sinl poin o aual onnion happns. BC is asd on a lad lion poss. Lad lion is nally an impoan ool o akin symmy in a disiud sysm. Sin h nods sa asynhonously and wihou any knowld o h oal num o paiipain nods in h nwok onsuion poss an ld oodinao will al o onol h nwok omaion and nsu ha h sulin opoloy will saisy h onniviy quimns o a Bluooh san. In h asn o any san omaion iia and in od o dsin a simpl and as poool w popos and jusiy h ollowin daul popis ha h sulin nwok will saisy:. A id nod may onn only wo pions. (Bid d onsain): A id nod owads daa om on pion o anoh y swihin wn hm in a im division mann. Givn ha ah poal dvi may hav limid possin apailiis a imum id d o wo livs a nod o in an ovloadd ossoad o muliply oiinad daa anss.. Givn h num o nods N h sulin san should onsis o h minimum num o pions possil. h impa o his is simila o h moivaion o solvin h polm in [5] o indin h minimum num o ous in an ad ho nwok. A minimum num o pions yilds an asi nwok o onol.. h sulin san should ully onnd. his mans ha vy mas will onnd o all oh mass houh id nods. Sans a pd o han and omd ov im. A ully onnd san in is iniial sa povids hih ousnss aains opoloy hans. Also no ouin is ndd in his oiinal sa sin vy mas an ah vy oh mas houh a id nod and vy slav an ah vyody ls houh is own mas.. wo pions sha only on id (ion ovlap onsain). his ondiion is usd in od o povid a mans o minain asily h onnion salishmn poool and alulain h minimum num o pions. I wo mass la wish o sha anoh id wn hm hy an do so y mans o a id noiaion poool. h poool onsiss o h phass: has I: Coodinao lion Duin his phas h is an asynhonous disiud lion o a oodinao nod ha will vnually know h oun idniis and loks o all h nods paiipain in h nwok onsuion poss. ah nod has a vaial alld VOS whih is s o as soon as h nod is powd up. A iniializaion h nod sas alnain wn h INQUIY and INQUIY SCAN sa. Any wo nods and y ha disov ah oh will om a poin o poin onnion n a on-o-on ononaion and ompa hi VOS vaials. h nod wih h la vaial is h winn o h ononaion. I h wo nods hav qual VOS vaials h winn is h nod wih h la Bluooh addss. Wihou loss o naliy suppos ha is h winn and y is h los. h los y snds all h dvi FHS paks o h nods i has won so a o h winn i as down h onnion and ns h AG SCAN sa. In his way i will no al o ha inquiy mssas any mo u only pa mssas om nods ha will pa i in h uu. his aion has h o liminain h los om h oodinao lion poss and ppain i o h n phass o h poool.

8 h winn inass is VOS vaial y VOS(y) and oninus on h lad lion poss y sumin alnain wn INQUIY and INQUIY SCAN. I h a N nods paiipain in h san omaion h will N- on-o-on ononaions. h winn o h N- s ononaion will h oodinao nod and h s o h nods will in h AG SCAN sa waiin o pad y a nod ha has inomaion aou hm. has II: ol Dminaion h oodinao ha was ld duin phas I has h FHS paks (i.. idniisloks) o all h nods and hn knows h oal num o nods N ha paiipa in h nwok onnion salishmn. A h sa o phas II h oodinao hks i h num o nods ha i has disovd duin phas I is lss han ih. I his is h as i pas and onns o all o h nods in AG SCAN and on pion is omd wih h oodinao as h mas and all h oh nods as is slavs. In his spial as h poool minas a his poin. I h num o nods is a han svn hn mo han on pion mus omd and inonnd via id nods. Givn h loal viw o h nwok h oodinao an did on h ol ha ah nod will pom in h inal san. I h paiipain nods impos spii san omaion iia hy an ommuniad o h oodinao duin h lion poss in addiion o h FHS inomaion and an aid i in dminin h ols o h nods in h inal san. By usin h daul iia id a h sa o his sion h oodinao is alulas h num o pions. h minimum num o mass in od o h sulin san o ully onnd an alulad y h ollowin laion (s Appndi C): N N 6 () As w osv om h aov laion h daul shm woks o a num o nods lss han o qual o 6 du o h dsid popis - dsid a h innin o his sion. A la num o nods may lad o a daul shm ha dos no qui a ully onnd san. A alulain h oodinao sls isl and - nods o h dsinad mass and ( ) oh nods o h san ids. Consqunly h oodinao qually disius o h dsinad mass h mainin nods o hi pu slavs. A h ol assinmn o ah mas (inludin isl) h oodinao has a onniviy lis s (SLAVSLIS() BIDGLIS()) onsisin o h mas s assind slavs and ids. ah ny o hs liss onains FHS paks (idniisloks) so ha h dsinad mas an la pa is onniviy lis s insananously. hn h oodinao onns o h dsinad mass i sld y pain hm. (all ha a h nd o phas I all mainin nods w in h AG SCAN sa). hus a mpoay pion is omd insanly wih h oodinao as h mas and h dsinad mass as h slavs. h oodinao ansmis o ah dsinad mas is onniviy lis s insus h dsinad mass o sa phas III and onsqunly as down h mpoay pion and sas phas III as a mas nod isl. has III: h aual onnion salishmn Duin his phas ah mas pas and onns o h slavs and ids dind in is SLAVSLIS() and BIDGLIS() spivly. As soon as a nod is noiid y is mas ha i is a id i wais o pad y is sond mas. Whn his happns h id nod snds a CONNCD noiiaion o oh mass. No ha aodin o quaion () is always lss han svn. hus h mpoay pion an always omd.

9 Whn a mas ivs a CONNCD noiiaion om all is assind ids a ully onnd san o pions is uaand o omd and h poool minas. I is vidn ha h mos im onsumin pa o h poool is h lad lion phas. has II and has III involv only pain and onnin whih is happnin almos insananously du o h pvious disovy phas. h iky pa o h poool is aually h phas I minaion. Idally i should sop as soon as h oodinao is ound. Bu how dos a nod know ha i is h inal winn o h lion poss? All nods hav a sa alnaion imou piod AL_IMOU ha is s on a nod is powd up and s ah im i wins an on o on ononaion. Whn AL_IMOU pis h nod assums i is h ld oodinao and ha all oh nods a in h AG SCAN sa waiin o pad. (a) () () Alnain nod Coodinao/Mas Nod in AG SCAN / Slav Bid Nod (d) () A B B is a slav o A Fiu : h onnion salishmn poool o a s o N6 nods. (a) Sa o has I: All nods sa alnain yin o disov hi nihohood. () A h nd o phas I h oodinao has n ld. Sin N6 h oodinao ompus and sls h mass ids and slavs aodinly () has II: Coodinao oms a mpoay pion wih h dsinad mass and snds hm hi onniviy liss. (d) has III: ah mas pas h nods spiid wihin is onniviy lis. () Final san omaion. h qusion ha is aisd now is wha is a ood valu o AL_IMOU? A vy la valu will sul in a nod havin won h ompiion and oninuin alnain wihou knowin i is h only on l. his will sul in a vy slow phas I (and hn a vy slow onnion salishmn poool). On h oh hand a vy small imou valu may sul in a as wh mo han on nods assum hy a h oodinao and hn a poool ha will sul in a disonnd san. W addss h aov polm y makin h ollowin osvaion. Whn h a N nods alnain and yin o disov and onn o ah oh h im o h is onnion o happn is nally lss han h im i aks i h w only wo o hm yin o onn. Aodin o h link modl divd in Appndi B ivn a disiuion and h man sa sidn im h man onnion salishmn im an analyially alulad o h wo-nod as and his valu an usd o dmin h AL_IMOU im o ah nod.

10 5. IMNS 5.. mulain Bluooh W hav implmnd BC on op o an isin pooyp implmnaion ha mulas h Bluooh nvionmn on a Linu plaom. h ason o usin an mulao insad o h Bluooh dvis hmslvs is aus un Bluooh unis do no suppo h pion swihin union and hn anno opa as ids. In addiion an mulao povids a hih d o liiliy in sin h sysm o vaious paams and an aod sin h poool o a la num o nods. ah Bluooh hos is implmnd as a poss ha mainly onsiss o wo inain moduls. h Bluooh Basand (BB) modul mulas in sowa h Inquiy ain and pion swihin podus as dind in h Bluooh Basand spiiaion []. h BC modul inas wih h BB modul houh h HCI onol spiiaion unions as dind in []. h us o HCI unions allows us o la pla h Bluooh sowa modul wih a hadwa modul whn h idin apailiis om availal in hadwa. h wilss mdium is simulad y a N -hop hannl poss whih is usd o h han o IAC and FHS paks duin h inquiy and pain podus. h N -hop hannl poss also dmins h quny hoppin ollisions ha a happnin wn h dvis and mulas h FS dlays. No ha his hannl poss is no simila o a CSMA hannl sin h snds o ivs anno pom ai snsin o any kind o inllin ako. W also assum ha all h dvis a wihin an o ah oh. his is a loial assumpion o nwokin many sho-an wilss dvis in a sinl oom. his a is mappd in ou ahiu y havin all Bluooh hos posss iniially onnd o h N -hop wilss hannl poss and uin h opoloy onsuion poool. 5.. Dminin AL_IMOU Usin h h iodi_inquiy_mod HCI ommand [] i is possil o poam Bluooh unis o alna wn INQUIY and INQUIY SCAN sas wih uniomly disiud sa sidn ims. In his as h d o h md poss (s Fiu ) whn ah uni has sa sidn ims uniomly disiud in [] is: F ( ) [ ] () y usin () and () in (.)(.)(.) w an alula analyial pssions o h man and vaian o h link omaion im o h symmi poool as a union o h man sa sidn im µ [ Z ] o ah uni (ivn y quaions (B.) and (B.7) in Appndi B spivly): µ µ µ [ < ] µ µ [ ] µ µ [ ] ( ) µ ( 8) µ < ( ) ( ) µ 8 µ µ µ wh [ < ] is ivn y h laion (B.) Appndi B: (5) [ ] µ ( 6µ ) < aan (6) 8 µ 6µ πµ 6

11 Va [ ] [ ] Va < 6 8 [ ] Va < 6 8 ( 6 ) ( ) ( 6 ) ( ) wh Va [ < ] is ivn y (B.6) Appndi B: Va [ < ] ln ( ) π ln ln 5 aan ( ) aan ( ) π 6 Givn (.) and (.) w hoos AL_IMOU aodin o h mpiial laion: AL _ IMOU [ ] Va[ ] (5) Fiu 5 shows h man onnion salishmn im and h sandad dviaion in h onnion salishmn im in h poin o poin as whn h sa sidn ims a uniomly disiud. W osv ha o vy alnain man sa sidn im h sulin sandad dviaion is almos qual o h man onnion im. his mans ha h link omaion im disiuion is no nd aound h man. his jusiis h inlusion o h Va ( ) m in ou mpiial omula. avsd (ms) al_man (ms) man sandad dviaion

12 Fiu 5: Mans and sandad dviaion dlays o h poin o poin onnion salishmn im wh nods alna wih sa sidn ims aodin o a uniom disiuion. h m was mo sul and was dmind only a pomin pimns and osvin h poool havio on many uns. I sms lik h ollowin as was happnin vy qunly: A h N- nd ononaion h winn A would sa alnain y sin AL_IMOU whil h was on nod B in SL mod (and all h s in AG SCAN). h wo nods A and B would sa yin o om h N- s onnion only a nod B wok up! h addiional m is h upp limi o h ako inval and hus liminas h onn aou his as. In ou pimns w hoos a man sa sidn im o 6ms whih aodin o quaion (5) and Fiu 5 yilds h smalls AL_IMOU valu o 57.ms 5.. oool oman h poman mis assoiad wih h poool a h nwok onnion s up dlay and h poailiy o poool onss whih dpnds on h valu o AL_IMOU. h hih his valu is h hih h poailiy o poool onss u also h lon i will ak h nwok o onn. h nwok onnion s up dlay masud in h pimns is always h im o l h lad (phas I duaion) sin phas II and III inlud only insananous pain and onnions idal (ms) N No os p p Fiu 6: Ava idal onnion salishmn im o vaious appliaion snaios. Unis alna aodin o uniomly disiud sa sidn ims wih man ms. h no os uv in Fiu 6 shows h man nwok onnion salishmn dlay idal whn all nods sa alnain a h sam im. By idal w man h im wh h oodinao is aually ld. h nod isl will assum i is h oodinao whn is im pis a im AL_IMOU. hus h aual nwok onnion im aual will : aual AL _ IMOU (6) idal h uv shows a dlay ha is inasin slowly wih h num o nods ha paiipa in h nwok omaion. h ason is ha h a many on-o-on ononaions ouin in paalll unil h oodinao is ld. his is aually a dsial ass o a nwok salishmn poool. W wouldn o ampl lik h dlay inasin linaly wih h num o nods. W osv ha h dlay ans om s o s o a s o nods ha span om N o N.

13 h no os uv yilds vy small dlays paly aus all nods sa paiipain in h nwok omaion a h sam im insan. In a mo alisi snaio wh human uss push uons in od o onn o h nwok h nods will no nssaily sa alnain a h sam im. W modl h uon pushin as a oisson poss in a Ws appliaion window. A h is us ah us i will aiv wihin an iid (unad) ponnially disiud im L i i K N in h s appliaion window as shown in Fiu 7. h aphs p and p in Fiu 6 show h idal nwok omaion dlay whn ah us is pd o aiv a h is us wihin s and s in h ava aodin o h unad ponnial disiuion. As h man valu inass h sysm oms mo asynhonous and lss paalll on-o-on ononaions ou a ah im insan. his has an o inasin h dlay o onnion salishmn. Nvhlss h poool s immuniy o h inas o N is psvd. his is illusad y a onsan dlay os wn h uvs o h sam num o N. s aival nd aival d aival... Nh aival... L L L N- Fiu 7: h push uon aival poss. h imou may viwd as a pnaly ha has o paid in od o hav a disiud aloihm. A la AL_IMOU valu will saisy h onss ondiion wih hih poailiy (hih imou iiny ) u will aumula a la a ovhad in h aual nwok onnion im aual. Fiu 8 illusas his ad-o y dmonsain h imou iiny as a union o din andida valus o AL_IMOU 5. Fo all appliaion snaios h imou iiny iniially inass apidly as a union o h imou and hn ahs a sady sa. I is la ha h valu o AL_IMOU wh h uvs sa sailizin is a 5ms whih is vy los o h valu 57.ms hosn y ou mpiial omula (5). h ominaion o Fius 6 and 8 povid paial uidlins o h dsin usin h opoloy onsuion poool. W 9 imou iiny (%) No os p p imou (ms) 5 hs pnas a h ava o h imou iiny in h ass o N5 nods.

14 Fiu 8: imou iiny o h h onn appliaion snaios. Fo ampl i h a nods nvisiond paiipain in h poool and w hoos an AL_IMOU qual o 5ms Fiu 6 shows ha h ava dlay pind y ah us will ouhly ms5ms5.5s and Fiu 8 shows ha a onnd san will omd wih a poailiy o 96.% in h as o h no os appliaion snaio. 6. CONCLUSIONS AND DISCUSSION In ad ho nwoks usin quny hoppin hnoloy nods an oupd ino mulipl ommuniaion hannls. his physial lay sin povids a nw way o viwin hih lay unions lik opoloy onsuion aloihms. Moivad y his nvionmn and usin h Bluooh hnoloy as ou sah vhil w is sudy h Bluooh sandad asymmi snd-iv poin o poin link salishmn shm and hn popos a symmi mhanism o salishin a onnion wihou any ol p-assinmn. Basd on h ad ho link omaion mhanism w psn BC a disiud opoloy onsuion poool wh nods sa asynhonously wihou any pio nihohood inomaion and sul in a nwok saisyin h onniviy onsains imposd y h Bluooh hnoloy. h poool is nd on a lad lion poss wh a oodinao is ld in a disiud ashion and onsqunly assins ols o h s o h nods in h sysm. BC was sd und a onn-snaio wh uss aiv in a oom and y o om a san y pssin a uon on hi Bluooh nald dvis. A ni au o h poool is ha h nwok omaion dlay is su-lina wih h num o paiipain nods (implyin ha h uss don nd o wai popoionaly lon whn mo uss a psn). Alhouh h dlay is small ah nod mus hav an sima o how lon i mus paiipa in h poool o assumin poool minaion. A onsvaiv sima o h imou will inodu unnssay dlays in nwok omaion whil an assiv sima may lav h nwok disonnd. Ou analysis o h dlay saisis o h symmi link omaion poool povids a ih sima o h appopia imou valu makin h poool as whil nsuin hih poailiy o san onndnss. houhou h dsin o BC ou aim has n o uild a poool whih an implmnd on op o Bluooh hadwa. Alhouh ou implmnion uns in a Bluooh mulad nvionmn whn h in-pion ommuniaion au is mad availal in h n las o h Bluooh hadwa w an s ou poool in an aual sin. W would lik o mphasiz ha h wok psnd h is h is appoah owads aklin h opoloy onsuion polm and povidin a ully unional poool in h Bluooh quny hoppin nvionmn. h is sill muh wok ha mains o don. Fo ampl h poool nds o ndd o h as whn no all nods a wihin ommuniaion an o ah oh. In his as a omplion o h lion poss h oodinao will lan aou all paiipain nods u no all o hm will aually wihin is an. Founaly h san an sill asily omd y kpin h lion phas I whil plain phass II and III y h ollowin simpl podu: A h oodinao is ld i pas and onns only o h nods i onond and won sin hs a h nods uaand o wihin is wilss an. On i has onnd o is on hop nihos as a mas i insus hm o sa pain assum h ol o mass and pa h sam sps usivly unil all nods a ovd. h sulin san is uaand o onnd and will hav a suu ood a h lad. Givn a s o nods wih zo knowld o ah oh ha nd o om quikly an iniial onnnd ad ho nwok BC ouss on minimizin h onnion dlay whil povidin onndnss wih hih poailiy. his is a dsid popy in appliaion snaios wh ad ho nwoks oninuously onn (ih) pom a oodinad union o a sho amoun o im (liv) and disonn sin h onnion sup dlays should a small aion o hs "ih-liv-di" yls. Kpin his nwok opaion modl in mind alnaiv mhods o opoloy onsuion nd o sudid and ompad in ms o dlay wih h on psnd h.

15 Finally in addiion o zo-knowld nwok iniializaion h omaion o an isin nwok in h a o dynami hans an viwd as a spaa u qually impoan issu. A nwok onnion a spaa opoloy mainnan and opimizaion poool nds o un in od o ak a o moiliy and/o nods nin and lavin h nwok and mak su ha h san is omd aodinly. Suh a poool alhouh ou o h sop o h un pap should h suj o uu sah os. 7. FNCS [] J. Haasn Bluooh Basand Spiiaion vsion. [] K. Flmin Bluooh Hos Conoll Ina Funional Spiiaion vsion. [] D. Bak and A. phmids h ahiual oanizaion o a pak adio nwok via a disiud aloihm I ansaions on Communiaions COM-9 (98) pp [].G. oazzi.. Saahik Sl-Oanizin Communiaion Nwoks I Communiaions Maazin Vol No Jan 986. [5] A. K. akh Slin ous in Ad Ho Wilss Nwoks SB/I Innaional lommuniaions Symposium (Auus 99). [6].G. Galla.A. Huml and.m. Spia A Disiud Aloihm o Minimum Wih Spannin s ACM ans. on oam. Lan. and Sysms Vol. 5 pp Januay 98. [7] M. Haol-Bal. Lihon D. Lwin sou Disovy in Disiud Nwoks ODC 999: p [8] J. Haasn Bluooh-h univsal adio ina o ad ho wilss onniviy isson viw no 998. [9]. amanahan. Hain opoloy Conol o Mulihop Wilss Nwoks usin ansmi pow adjusmn. [] K. ivdi oailiy & Saisis wih liailiy quuin and ompu sin appliaions ni Hall Nov. 99. ANDI A: SYMMIC OOCOL CONNCION SABLISHMN DLAY WHN H UNIS ALNA DMINISICALLY. Suppos ha ah nod alnas wn Snd (I) and iv (S) sa mainin in ah sa o a id piod. in o Fiu A. Uni A has alady sad alnain a som im in h pas and uni B sas alnain a som andom im a uni sad alnain. Sin is andom h onnion salishmn dlay whih is h dlay sain a up o h poin wh h wo unis onn will also andom. W will show ha h pd valu o is inini. Uni A N ( ) Uni B... S θ θ I I S S I I S Md Shdul θ θ θ Fiu A.: Unis A and B y o onn y alnain dminisially wn INQUIY and SCAN sas wih piod.

16 Duin h onnion poss wo unis will in opposi sas o a id inval θ and in h sam sa o a id inval -θ. h phas dinθ dpnds on and sin is aiay θ is a uniom andom vaial in [ ]: [ Θ θ] FΘ ( θ) θ θ (A.). L us now i (and hn θ ) and assum wihou loss o naliy ha uni B inds uni A in an opposi sa a im. in o Fiu A. h wo dvis will hav a han o onn only duin h id on invals o lnh θ wh hy a in opposi sas. Duin an on inval h unis will y o om a onnion aodin o h asymmi Bluooh poin o poin onnion salishmn poool. h dlay o h asymmi poool is FSB wh FS is a uniom.v. in [ ov a] ova ms and B is a uniom.v. in [ ] ms (s sion ). Sin h ak-o dlay is muh a han h FS dlay h dlay o h asymmi poool an appoimad y B and hn: [ ] F ( ) and [ ] (A.). Duin an on inval i is lss hanθ hn h unis will onn ohwis hy will hav o wai unil h n on inval. hus h onnion salishmn poss an sn as a oin oss wih a suss poailiy in: p [ θ] ( ) θ θ (A.). F Givn θ h num N o on invals ha will ndd o a onnion happns is a omi andom vaial havin d: [ N k ] ( p) k θ (A.). hn h onnion salishmn im in his as will qual o N unsussul piods ( on o invals) plus h las sussul inval whos dlay is qual o : θ N o θ in id. akin h paions w hav: [ θ ] [ N θ] [ ] (A.5). Usin h a ha [ N ] salishmn im will : θ [ ] θ p p θ and usin laion (A.) and (A.) h ava onnion θ (A.6)

17 h ava onnion salishmn dlay an now ound y akin h paion ov all possil valus o θ and usin quaions (A.6) and (A.): θ θ θ [ ] [ [ ] [ ] ( ) θ θ Θ θ dθ dθ θ θ θ [ ] dθ (A.7). h inal θ dθ θ salishmn dlay [ ] in h HS o (A.7) nds o is inini. and hn h ava onnion ANDI B: SYMMIC OOCOL CONNCION SABLISHMN DLAY WHN H UNIS ALNA ANDOMLY. B.: GNAL MODL Assum ha ah nod alnas wn Snd (I) and iv (S) sa andomly. Mo spiially Z o indpndn o ah nod h inquiy (I) and inquiy san (S) invals onsis a andom poss { n } and idnially disiud (iid) andom vaials ah havin a disiuion ( z) F z and ini man valu [Z]. in o Fiu B. Uni has alady sad alnain a som im in h pas and Uni sas alnain a som aiay im a uni sad alnain. N ( )... S I S I S I S N ( ) I S I S I N( ) Fiu B.: Unis A and B y o onn y alnain wn INQUIY and SCAN sas aodin o a andom disiuion wih man [Z]. Fo Bluooh uni i w dno y N i ( ) h num o sa swihs om up o im. hus N i ( ) an sn as a nwal poss wih an undlyin disiuion wn swihs o F z ( z) unis alna indpndnly h ospondin nwal posss N ( ) and N ( ) a indpndn. Now onsid h poss N ( ) ha suls om min h wo indpndn nwal posss N ( ) and N ( ). ( ). Sin h wo N is a poss (no nssaily a nwal on) ha onsiss o all h sa swihs o oh

18 nods om im up o im. W dno y { n } iid invals wn h md sa swihs o ( ) h undlyin andom poss onsisin o h N. In nal h wo unis hav a han o disovin and onnin o ah oh only duin h im invals wh hy a in omplimnay sas. As is shown in Fiu B. his happns vy oh N. inval o h poss ( ) h a wo quipoal ass whih w ak ino aoun: A im h unis a in opposi sas. In his as hy will hav h han o disov ah oh only on odd invals i. On invals i h wo unis a in h sam sa i i and anno disov ah oh. A im h unis a in h sam sa. his mans ha in h is im inval o h md poss h will no han o h wo unis o m and hn his inval is a pu dlay ao in h onnion im. A his dlay h wo unis will sa havin h han o disov ah oh lik h pvious as. Assum ha a im h unis sa in opposi sas as dpid in Fiu B.. h d F ( ) o h md poss n ivn h iid N ( ) and ( ) N is ivn y ivdi in []: [ ] F ( ) F ( ) z F z ( ) [ Z ] [ F ( z) ] z dz (B.) hn h ospondin pd will qual o: and ( ) F ( ) ( ) ( ) [ Z ] [ Fz ( ) ] [ Z ] d z z [ Fz ( z) ] dz d (B.) [ ] ( ) d (B.) L us now s Yi i i i. (B.) { Y n } is an iid andom poss o omposi invals Y i ah onsisin o an on is inval an o sond inval i onn in a omposi inval whih is ivn y: i and. Duin h onnion salishmn poss h wo unis sa yin o Y i. hn hy will hav h han o onn a a andom dlay FS B (B.5) wh FS and B a uniom andom vaials in [ ] ms [ ] ms a ova ov and spivly. h unis will onn only i is lss han h on inval o omposi inval Y. I no hy will hav o wai unil h sa o h n omposi i i

19 opp inval Y i. hus h onnion im o h as wh h unis sa a opposi sas is ivn y: opp Y Y YN YN L (B.6) wh N is a andom vaial. h sum o Y vaials in h HS dnos h im o unsussul disovy amps and h las m is h poion o h las on inval whih osponds o h sussul onnion amp. ah on inval wihin a omposi Y inval may sn as a oin oss. A suss in h oin oss mans ha h wo unis hav suiin im o om a onnion aodin o h asymmi Bluooh onnion salishmn poool. Hn N is a omi andom vaial dnoin h num o ailus o h suss-onnion ous. hus h poailiy o suss is p [ ] and h man o N is simply [ N ] will : opp h man disovy im [ ] p. p i opp opp opp [ ] [ [ N ] [ ] [ ] [ N n] [ N n] n [ ] Yi [ N n] [ ] n [ Yi ] [ N n] n i n [ ] [ Yi ] [ N] [ ] [ i i ] [ N] [ ] ( [ > ] [ ]) [ N ] [ ] opp [ ] h quaniy [ > ] n ( [ > ] [ ])( p) is alulad as ollows: p (B.7) Fis w ind h d F ( ) ( ) ( ) F <. (B.8) F < hn h ospondin pd is ivn y: ( ) ( ) ( ) <. (B.9) F < ov a hn [ < ] [ [ < ] ( < ) ( ) d d. (B.) In h as wh h wo unis sa a h sam sa h is inval is an o on and h unis mus wai unil h n inval (whih will on ). hus h onnion salishmn dlay in his as will : sam opp Y Y L YN YN (B.)

20 h wo ass a quipoal so h andom vaial dsiin h onnion salishmn im in any as is: sam opp opp (B.) Usin (B.7) and akin paions ov (B.) w inally h pssion o h man onnion salishmn dlay: [ ] ( [ > ] [ ])( p) [ ] p [ ] (B.) Also sain om (B.) and (B.7) and applyin h vaian opao on oh sids w h pssion o h vaian o h onnion salishmn dlay: [ ] ( Va[ > ] Va[ ])( p) Va Va[ ] Va p [ ] (B.) W now pod in divin h analyial pssions o h man and vaian o h onnion salishmn dlay o spii dis iuions. W will us h appoimaion (insad o h a B FS B ) sin h ako dlay is on od o maniud a han FS. Wihou usin Va. h appoimaion muh mo ompliad pssions an oaind o [ ] and [ ] Aodin o h appoimaion: F ( ) ( ) and [ ] Va[ ] (B.5) Assum ha N ( ) and ( ) B.. Z IS ONNIALLY DISIBUD WIH AAM. N a oisson posss wih paam : [ N ( ) k ] i ( ) k! k i (B.6) W know ha y min wo oisson posss wih paam yilds a oisson poss wih is ponnially disiud wih paam : paam. hus h andom poss { } n F ( ) ( ) and [ ] Va( ). (B.7) Usin h appoimaion (B.5):

21 [ ] [ ] [ ] ( ) d d p [ ] p (B.8) Usin (B.9) and appoimaion (B.5) in (B.) w : [ ] ( ) ( ) < d d d d F [ ] ( ) ( ) lo di < (B.9) Usin (B.5) (B.7) (B.8) (B.9) in (B.) w inally hav o h ava onnion salishmn dlay: [ ] ( ) ( ) lo di (B.) In od o alula h vaian w mus is ind h quaniy [ ] Va < : [ ] ( ) ( ) < dd d d F [ ] d < (B.) Usin h a ha [ ] [ ] [ ] ( ) Va < < < and usin (B.9) and (B.) w : [ ] ( ) ( ) lo di d Va < (B.) Usin (B.5) (B.7) and (B.) in (B.) w inally hav h pssion o h vaian o h onnion salishmn dlay: [ ] ( ) ( ) lo 8 di d Va (B.)

22 By sin [ ] µ Z and plain ino (B.) and (B.) w inally hav h pssions o h man and vaian o h onnion salishmn dlay as a union o h man sa sidn im o ah uni: [ ] ( ) ( ) µ µ lo di (B.) [ ] ( ) ( ) µ µ µ µ lo 8 di d Va (B.5) B.. Z IS UNIFOMLY DISIBUD IN [B]. Assum ha h unis sa sidn im Z is uniomly disiud in [ ] as ollows: ( ) z z F z and ( ) z z o [ ] z and ( ) Z µ (B.6) y susiuin (B.6) o (B.) w ind h d o : ( ) [ ] F (B.7) and ( ) [ ] F > (B.8) h pd will : ( ) ( ) F d d 6 (B.9) and hn [ ] ( ) 6 d d (B.) Also [ ] ( ) 6 d d (B.)

23 And sin [ ] [ ] [ ] ( ) Va w : [ ] 8 6 Va (B.) h n quaniy w nd o div is [ ] p. In od o do his w hav o disinuish wn wo ass: Cas : [ ] [ ] ( ) ( ) [ ] d d (B.) Cas : [ ] [ ] ( ) d d [ ] (B.) o alula [ ] < Usin (B.9) (B.7) (B.9) and appoimaion (B.5) in (B.) w : [ ] ( ) ( ) < d d d d F 6 [ ] ( ) 6 aan 8 π < (B.) Usin (B.) (B.) (B.) (B.) in (B.) w h pssion o h ava onnion salishmn dlay: (B.) In od o ind h pssion aou h vaian w is alula: [ ] ( ) ( ) ( ) ( ) ( ) ( ) aan aan 8 8 π π

24 [ < ] ln ( ) π ln ln 5 aan ( ) (B.5) hn om (B.) and (B.5): Va [ < ] ( ) ln π aan ( ) ln ln 5 aan ( ) π 6 (B.6) h vaian is hn ivn y (B.): Va [ ] [ ] ( Va[ > ] Va[ ])( p) Va Va[ ] p [ ] Va 6 Va 6 wh Va [ < ] Va 8 [ < ] is ivn y (B.6). By sin [ Z ] [ < ] 8 ( 6 ) ( ) ( 6 ) ( ) (B.7) µ and plain ino (B.) and (B.6)&(B.7) w an h pssions o h man and vaian o h onnion salishmn dlay as a union o h man sa sidn im µo ah uni. ANDI C: OOF OF QUAION () Any san onsiss o mas nods slav nods ha lon only in on pion (md as pu slavs ) and slav nods ha lon o mulipl pions (md as ids). Givn a num o Bluooh nods Nand h ollowin ondiions:

25 . h sulin san is ully onnd (vy mas is onnd o all oh mass via a id).. wo mass sha only on id nod.. A id nod may onn only wo pions.. h imum num o slavs p pion is svn. w will pov ha h minimum num o pions is ivn y h laion: N N 6 oo: Suppos w i h num o pions (mass) in h san. ah pion i has n i slavs onsisin o s i pu slavs and i ids. hus: n i s i (C.) i i wh is h num o pions in h san. Du o h Bluooh spiiaion d onsains h imum num o slavs p pion is svn : ( 7 i ). (C.) n i Aodin o ondiions and ah mas should onnd o all oh mass (ondiion ) houh only on id nod (ondiion ). hus ah mas will hav i - ids and s n ( ) pu slavs. i i Also h oal num o mass in h san is and h oal num o ids should ( ) (ondiion ). ho h ollowin laion holds: i ( ) si N s i 7 ( ) i Wh h sum ms o h LHS a h oal num o assind mass pu slavs and id slavs in h san spivly. quaion (C.) ls h allowal valus o and N asd on h san omaion ondiions -. W s ha o a id h is an assoiad an o valus o N ha an ovd dpndin on h possil ss o valus s i. Fo ampl pion an aommoda om N up o N8 nods. mass an ov om N9 o N5 nods (wh h wo mass a onnd y a ommon id and ah mas has si pu slavs). N6 nods anno suppod y only mass aus h ondiions - will violad and quaion () will no hold. h imal s ( ) i s i (C.) 7 yilds h imum N ha an suppod y a spii. Fo h valus o h imal s quaion (C.) oms:

26 i [ 7 ( ) ] 7 N ( ) N (C.) Solvin (C.) o N w h imum num o nods ha an suppod y a spii minimum num o pions wihou violain quaion (C.): N ( 7 ) ( ) (C.5) Aodin o ondiions - w wish ah mas o onnd o all oh mass houh aly on id nod. Hn h imum num o pions is 8 whih is h as o vy mas havin svn id slavs o all oh (svn) mass. hus h imum num N is ivn y (C.5) o N 6. Usin (C.5) o up o o 8 w na h odd s N o ospondin nums N : N { } (C.6) Solvin (C.5) o and kpin h - oo soluion w : N ( N ) N N (C.7) Sin (C.7) is h invs union o (C.5) o any valu in h s N (C.7) yilds an in. Also w an asily s ha is a sily inasin (dis) union o N. ho any wo onsuiv nums N and N in N will ospond o wo valus and spivly wih. Sin is sily inasin union o N any valus o N no in N in h odd s { N K } S N. hus h valus o S alon wih minimum num o pions. ha a usd in (C.7) will yild a al num wn and N a aually h valus o N ha an suppod y a Hn y usin any valu o N in (C.7) and oundin h sulin al num o h n in will always yild h dsid valu o h minimum num o pions ha an suppo N: N N 6