EFFICIENT CONSTANT K-DOMINATING SET APPROXIMATION

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1 EFFICIENT CONSTANT K-DOMINATING SET APPROXIMATION SEMESTER THESIS Jauary 7, 005 Author Patrce Müller Supervsor Thomas Moscbroda Professor Roger Wattehofer Abstract Calculatg a mmal domatg set s kow to be NP-complete It would be ce to have a approxmato whch s ot much bgger but could be calculated much faster I ths paper a algorthm s proposed whch costructs a k-domatg set It s mathematcally proved that the algorthm produces a costat approxmato to the MDS ad termates very quckly INTRODUCTION - - ALGORITHM - - DOMINATING SET - - K-DOMINATING SET RELATED WORK EXPECTATION HIGH PROBABILITY HIERARCHICAL ANALYSIS K-DOMINATING SET EXPECTATION HIGH PROBABILITY HIERARCHICAL ANALYSIS PHASE I ( ) 43 PHASE II ( ) OVERALL BOUND CONCLUSION FUTURE WORK ACKNOWLEDGMENTS REFERENCES Itroducto I wreless ad-hoc etworks collaborato amog odes s a extremely mportat task I small workg evromets wth oly a few odes the same trasmsso rage very smple ad prmtve algorthms lke floodg ca be used As the etworks get bgger ad more complex these basc approaches to solvg problems caot be used aymore There eeds to be establshed a overlay etwork layer o top of TCP/IP Wthout t scalablty ad relablty caot be guarateed Ths paper s focusg o domatg sets ut dsk graphs There s already a lot of work avalable dealg wth domatg set problems However a relable etwork mght eed to have more tha just oe domator per ode I ths paper t s show how a k-domatg set wth a costat approxmato ca be costructed usg oly local formato Algorthm The frst algorthm preseted below about costructg a domatg set s take from [] To exted the acheved result from domatg set to k-domatg set the algorthm was slghtly modfed Domatg Set At the start-up of a ode t eeds to assg tself a radom umber whch wll be ts uque detfer UID practce f the radomzato s usg a large eough rage of umbers Startg wth a small trasmsso rage ad the state of a domatg set every ode queres all odes rage about ther UID ad selects the ode wth the hghest UID as ts domator Selectg here meas that the ode forms the selected ode about ts selecto The ode recevg ths formato must accept to become a domatg ode f t s ot already the domatg set After such a domator beg foud ad formed the selectg ode ca remove ts domatg state I the specal case where the selectg ode s also the ode wth the hghest UID the t wll just rema the domatg state For establshg a smaller domatg set the above descrbed selecto process s appled recursvely to all odes stll the domatg state The trasmsso rage s doubled after every roud ad s of the form δ = log, for > 0 Itally every ode s the domatg set ad may fall out of t at ay roud Thereafter the algorthm stops o that ode Obvously certa odes ever drop out of the domatg set or there would be o domatg set at all After loglog rouds the algorthm must stop ad Wth costat approxmato a approxmato to the mmal domatg set s meat Ths approxmato s worse oly by a costat factor to the MDS s the total umber of odes the maxmum trasmsso rage of the ode executg the algorthm - -

2 the ode wll rema the domatg set f o ode wth th hgher UID was foud At the loglog roud the trasmsso rage s set to δ = After ths roud o other ode ca cover all the odes the processg ode was resposble for Eve the full trasmsso rage caot be eough aymore to cover all o-domatg odes K-Domatg Set The same dea as above for buldg up a domatg set s used to costruct a k-domatg set Every ode chooses a radom umber as ts UID ad starts wth a small trasmsso rage to fd the k hghest odes Those k odes are selected ad must become domators after gettg otfed If the ode executg the algorthm tself s amog the k hghest odes the t wll rema the domatg set Otherwse t ca remove ts domatg state If less tha k odes are avalable the curret trasmsso rage the all avalable odes wll be otfed to become a domator as ths wll be the best possble approxmato to a k- domatg set The same herarchcal techque as above for the domatg set algorthm s the appled o the remag domators The trasmsso rage wll be doubled every roud utl the ode s o loger a domator or reachg δ = as half of the maxmum trasmsso rage ad the the algorthm stops At each step the k hghest odes wll be forced to the domatg set ad the executg ode wll become a o-domatg ode f t s ot amog the k hghest odes 3 Related Work May deas for ths paper s work come from [] The followg three subcategores were aalyzed very carefully ad the modfed ad exteded to proof the requested propertes for the k-domatg case I all the aalyss squares are supposed to be the vsble rage of the wreless devces Ths s a method for smplfyg the aalyss It does ot exactly match the actual trasmsso rage whch would rather be a rregular crcle depedg o the earby evromet The squares could as well be replaced by crcles ad the outcome of all the results would ot be affected up to a costat factor 3 Expectato To get the expected umber of domators after oe step of the domatg set algorthm the close proxmty of oe vsble rage s aalyzed I Fgure 4 the model for the vsble rage two dmesos s show It s assumed that there are o more tha m odes the vsble rage L of S ad that the sde legth of S s oe half of the trasmsso rage Havg ths setup t was show [] that the expected umber of domators sde S s boud by O ( m) Ths s a very mportat property as t shows a drastc reducto domators by executg the domatg set algorthm to all the odes L 3 Hgh probablty It was also show that the reducto of odes sde S holds wth hgh probablty To be more precse the probablty that S cotas more tha m l m + domators s l m bouded by O( m ) ( m m) Ths meas that there are more tha O l domators S wth a very small probablty that goes towards zero quckly By kowg ths the proposed algorthm guaratees a reducto the umber of domators Furthermore the calculated expectato s more trustful by kowg the result wth hgh probablty 33 Herarchcal aalyss I the paper [] t s sad that the recursve algorthm wll evetually produce a costat approxmato O( d ) to the mmal domatg set wth d domators by usg at most steps To show ths property the expectato of ( m) O for oe step was terated o tself Ths calculato caot be regarded as correct as the expected reducto of the remag domators may chage after oe roud of executo The frst roud s ot ad caot be cosdered to be depedet from the followg rouds For aalyzg the herarchcal algorthm ad showg a costat approxmato of the mmal domatg set a dfferet approach eeds to be used 4 K-domatg set Costructg a domatg set ca be a very mportat task wreless ad-hoc etworks Geerally there s o structure amog wreless devces as they ca be located aywhere I certa areas the desty may be very hgh ad other parts t may be very sparse Buldg up a domatg set ca help to solve problems lke routg The stadard domatg set has the property that every ode has at least oe ode ts rage as ts domator If we cosder usg such a domatg set for stace for solvg a routg problem t s easy to see that the whole system s ot very fault tolerat If the domatg set problem ca be exteded to be more relable eve f odes are ustable the problems may be solved o top of ths k-domatg evromet by esurg a hgher relablty The algorthm for costructg such a k-domatg set s already gve above I ths secto t s to prove ts propertes The frst step s to show that the algorthm reduces the amout of domators Ths s doe by aalyzg the expected umber of domators after the executo of oe step of the algorthm I a secod aalyss t s show that ot oly the expectato holds but that a slghtly hgher result holds wth hgh probablty Ths proof s essetal to be covced that the algorthm almost caot be ufavorable I a thrd subchapter the whole algorthm usg ts herarchcal structure s aalyzed Ths leads to the fal result ad does show that the proposed algorthm fulflls the wated requremets 4 Expectato At a frst step to aalyze the algorthm for k-domatg sets the reducto domators s calculated To do ths oe roud of the algorthm s take a close look at I Fgure 4 the model used ca be see The expectato for the umber of domators sde S after oe roud s calculated All the areas are squares to smplfy the setup The sde legths are equal to / There s a cetral area S ad a eghborg area S' I total there are eght equal eghbor- - -

3 g areas to S Therefore oly oe eghborg area eeds to be aalyzed relato to the cetral area The other seve eghborg areas do have the exact same propertes L S' S 3 Fgure 4: Vsble rage L of S Frst the umber of odes omated from sde S must be examed The sde legth of S s assumed to be half the trasmsso rage of the odes executg the algorthm Due to ths fact all the odes resdg sde S are mutually vsble Every ode kows about all the other odes presece ad also kows all ther UIDs The algorthm omates from every ode k odes to become ther domators As a result of the mutual vsblty o more tha k odes sde S ca become domators from odes wth S (4) E k Secod the umber of odes omated from outsde S must be examed Let x be the umber of odes S ad y sde S' x = S, y = S' Ay ode p S ca be omated by a q S' oly f p has ts UID wth the k-largest of q's vsblty The radom selecto of UIDs at the begg of the algorthm gves a uformly dstrbuto of UIDs o S ad S' Therefore the probablty for ode p beg wth the k-hghest odes of q's vsblty s k (4) P + y Havg x odes S the expected umber of odes omated S from S' s xk (43) E + y Those pots are omated to be domators from pots S' As there are y pots S' there s a absolute maxmum of omated pots sde S from pots q S' of (44) E yk Wth the two expectatos E ad E the overall expectato for omated pots from S' ca be calculated The varable m s the total umber of odes L before the algorthm starts (45) E xk = m, yk + y x = km, y as xy, 0 + y k x+ y+ ( ) < k m To get the total expectato the odes omated from sde ad those omated from outsde of S must be added There are equal squares aroud S A ew costat c s used to smplfy the equato (46) E = E+ E < k+ k m < c m From ths result the upper boud of expected domators after oe roud ca be see easly It s (47) E O( m) 4 Hgh probablty I ths subchapter a stroger boud tha the expectato s show By kowg oly the expectato the dstrbuto of all the possble cases remas ukow It would be ce f almost all the possbltes were close to the expectato I ths case the expectato seems to be a very reasoable boud But what f the dstrbuto s ot so favorable? There mght be a very hgh probablty for almost o reducto domators ad a hgh probablty of a very good reducto If the algorthm happes to produce the ufavorable reducto t does t help much to kow about the expectato sce a good reducto s amed at Showg a good reducto wth hgh probablty s much more covcg that the proposed algorthm works wth all crcumstaces ad at all tmes As a frst step to get a hgh probablty boud the probablty that the umber of odes omated S from S' ca reach a certa bechmark s s calculated Oce ths fucto s evaluated a bechmark ca be searched Ths bechmark wll be somewhere above the umber expectato ad must lead to a very low probablty The probablty that s (or more) pots S ca be omated from S' s eeded To get ths probablty the two areas S ad S' are aalyzed detal Lemma 4: Out of the s+k- hghest odes S S' at least s must be S for S' beg able to omate s or more domators S Proof: If there are less tha s of the s+k- hghest odes S tha the odes S' have o possblty to omate s domators S They are forced to select more odes S' as ther domators If the s odes S are chose from a smaller group tha the s+k- hghest odes ot all the possbltes for havg selected s domators S are covered Ths comes from the fact that each ode S' chooses the k hghest odes as domators Therefore t s possble that a ode chooses the frst k- domators S' ad the k th domator S For ths reaso the s+k- hghest group must be cosdered - 3 -

4 Fgure 4: Dstrbutg the s+k- hghest odes Kowg Lemma 4 the probablty ca be calculated by aalyzg the combatos The umber of possbltes for choosg the s pots S s x (4) s The umber of possbltes for choosg the k- pots S' s y (49) k The umber of possbltes for choosg the s+k- hghest pots amog the odes S S' (cotag x,y odes respectvely) s x + y (40) s+ k At ths pot the probablty ca be wrtte by usg #umber of 'good' cases #umber of total cases whch leads to the equato s+ k x y ( )( s+ k) = s (4) p = x+ y ( s+ k) I the above equato the top part adds up all the possbltes for havg more tha s odes S out of the s+k- hghest odes The lower part couts all the possbltes for placg the s+k- hghest odes amog the x+y odes of S S' By testg the probablty of equato (4) a fast degradato ca be observed Usg the same asymptotcal boud as [] for the proof wth hgh probablty the also the same low probablty s foud It was proofed by Thomas Moscbroda that wth a expoetally small probablty (4) k- s S S p m log m a maxmal boud of O max mlog m, k ( ) (43) ( ) exsts as m gets large By kowg ths result the proposed algorthm for the k-domatg set seems to be very assurg to decrease the umber of domators sde S sgfcatly wth oe step Whe the umber of odes s hgh the resultg umber of domators wll be < mlog m wth very hgh probablty The umber of domators ca always reach k whch s just a costat umber ad ca be assumed to be small Now a upper boud wth hgh probablty was show Ths boud s just a lttle hgher tha the value expectato Due to ths fact the algorthm performs well all practcal cases 43 Herarchcal Aalyss I ths secto the whole k-domatg set algorthm s aalyzed I the prevous sectos oly oe step of the algorthm was executed The whole process cotues though by doublg the trasmsso rage after each roud ad proceedg wth the remag domators I [] oly the expectato of domators was recursvely calculated to show a costat approxmato at the ed Ths calculato s msleadg as cosecutve rouds are ot fully depedet For the mathematcal aalyss ths secto the hgh probablty result s used All the possbltes combed wth ther umber of remag domators are added up Ths wll gve the fal expectato of domators uder a hgh probablty codto after the algorthm termates The reducto domators from dow to a costat approxmato s dvded to phases I Fgure 43 t ca be see that the frst phase reduces the domators to a loglog approxmato ad the secod phase brgs the algorthm to ts stop at a costat approxmato Fgure 43 Herarchcal algorthm dvded to phases for mathematcal aalyss 43 Phase I ( ) Startg from a total umber of domators the whole evromet the umber gets reduced at each step wth a hgh probablty Stll there remas a slght possblty that the umber does t get reduced very much At the begg there are 0 = domators ad after the th roud there rema domators For roud + oly the 'good' ( + ) cases up to roud are of terest As soo as a 'bad' ( ) case happes the executo stops for the aalyss ad couts the umber of domators before the 'bad' case Followg the frst 3 rouds are lsted Roud ( ) p = log o 0 o phase I phase II Remember that at the begg of the k-domatg set algorthm all odes are marked to be the domatg set ad rema domators utl they are ot omated aymore at some roud At ths stage the terest of the aalyss s o all odes a square wth a sde legth of half the maxmal trasmsso rage ad ot oly o same specfc area S aymore Oly ths aalyss wll lead to the desred costat overall approxmato - 4 -

5 ( + ) p = log o 0 o log o Roud ( ) see Roud ( + ) p = log o log 0 o log o ( ++ ) p = log o log 0 log Roud 3 ( ) see Roud ( + ) see Roud ( ++ ) p = log o log log 0 3 ( +++ ) p = log o log log 0 3 log From the above frst 3 rouds t s ow easy to deduce the expected umber of domators for the 'bad' cases the frst phase To do ths the sum of the probabltes multpled wth the umber of domators of all the cases wth a 'bad' fshg roud must be calculated To smplfy the calculato the probabltes are augmeted a lttle bt Istead of usg the true ad hgh probabltes of havg the 'good' cases before the abortg 'bad' case, those probabltes are set to Usg ths smplfcato the expected umber of domators from the frst phase ca be calculated by > loglog > loglog (44) E[ Phase] = = log log = 0 = 0 Now two propertes must be show for the fucto (44) to receve a costat expectato Frst t must be show that the term S s suffcetly small ad secod t must be show that there are ot too may addtos Lemma 4: > S < Proof: For a that s large eough t s true that log > As s ot a costat > ca be assumed to meet the prevous requremet Out of ths follows mmedately that S = log < < O steps applyg Lemma 43: There are oly ( ) log recursvely to tself utl the result s smaller or equal to Proof: Usg Maple t was show that the umber of recursos s growg proportoally slower tha for S = 0,0,0 The umber of recursos wth 000 odes s ad log log(000) 93 Ths meas there are about 6 tmes more recursos as wth Due to the structure of the fuctos aalyzed they have o dscotutes ad wll ot chage ther behavor the large Therefore the umber of recursos s boud as Lemma 43 says Now the expectato for the frst phase ca be calculated usg fucto (44) leadg to the fal expectato of the frst phase c c (45) E[ Phase] S < O() = 0 As show for Lemma 43 3 c 6 s true f 0 Therefore the above result proofs that the umber of domators from phase s boud by O () 43 Phase II ( ) For the secod phase as see Fgure 43 the umber of domators must be reduced from to As the aalyss here s cotued from the prevous result the umber of domators s At ths stage whe the umber of domators s gettg fewer the the probablty for 'bad' cases s gettg bgger It s ot possble just to sum up all the cases as the frst phase or the expected umber of domators from the secod phase wll be too hgh To proceed the secod phase s dvded to X rouds As t would be ce to have the algorthm acheve the costat approxmato wth rouds the maxmum umber of rouds s set to (46) X Each roud ow has a falure rate of (47) log m, 3 m m = m [ ] = < ( m ) P falure + Out of ths equato follows mmedately the success rate at each roud whch s equals to Psuccess, m 3 (4) [ ] ( ) m = m log m + From here o the secod phase s dvded to cases The frst oe s that there are X 4 successful rouds o the total of X rouds The secod case s the oe wth > X 4 successful rouds For solvg the frst case the Cheroff boud s very helpful Cheroff boud: (49) P E ( δ ) µ e δ µ Applyg the Cheroff boud to the frst case the probablty for havg X 4 successful rouds ca be calculated wth For the umercal aalyss the logarthm to the base 0 was used The falure rate s equal to the probablty of havg a 'bad' case that partcular roud - 5 -

6 X X (40) X X 4 P E P E e e 4 = = Usg the above probablty the expected umber of domators from the frst case ca be calculated To smplfy the calculato the worst case s take where o reducto has take place sce reachg domators Therefore the expectato s X E case e loglog (4) [ ] As X was boud by the assumpto (46) the above equato ca by smplfed as follows (4) [ ] loglog Ecase e loglog ( log ) () = = log O Now t s also show that the frst case of the secod phase bouds the umber of domators by O () It remas to show that the secod case does t leave too may domators ether Cotug wth the domators reduced to the algorthm executes ad due to the precodto there are more tha X 4 rouds These 'good' rouds all lower the umber of domators to m = + m log m It was show that the recursve applcato of the above mplct reducto fucto ca be boud wth a expoetally decreasg explct fucto 34 (43) mlog m m, m 5504 Ths makes t possble to use the explct fucto for estmatg the upper boud of expected remag domators for the secod case The probablty s just set to as ths wll lead to a suffcetly small boud X 4 34 Ecase loglog (44) [ ] ( ) ( )( ) = ( ) ( 34 ) ( ) ( log ) ( ) ( log ) () 4 log log log (I) = log log, k log (II) O (III) I the above calculato for the secod case several propertes were used These propertes are lsted here (I) X 4loglog (II) 4 log (III) X X O() Wth (44) t s show that the secod case of phase reduces the umber of remag domators below the boud of O () k 433 Overall boud Due to the splt up of ths herarchcal aalyss the dvdual parts eed to be combed to get the overall boud for the whole algorthm The total process was dvded to 3 dfferet parts I the frst phase the possble domators due to falures were estmated I the secod phase two cases had to be cosdered The frst case s ufavorable as t has few successful rouds The umber of possble domators stll stays small as the possblty for ths ufavorable case s relatvely small The last case s the 'good' case whe there are may favorable rouds I ths case t s almost obvous that the umber of domators gets reduced as desred Summg up all the expectatos the overall boud s foud by (45) E O() + O + O = O () () () Cocluso I ths paper t s show how a k-domatg set ca be costructed usg oly local formato The proposed herarchcal algorthm performs ths task ad t could be show that t acheves a overall boud of O () domators each ut dsk square of sde legth / Suppose a mmal domatg set cotas d domators the whole evromet The proposed herarchcal algorthm ca cover the whole evromet wth 4d ut squares of sde legth / ad all of them havg O () domators It follows mmedately that ths paper's algorthm costructs a k- domatg set usg oly local formato wth a O( d ) approxmato to the optmal domatg The tme requre for executo of the algorthm ca also be boud asymptotcally Both phases of the herarchcal aalyss termate O( ) Therefore the total algorthm also termates O( ) 6 Future work The mathematcal aalyss of ths paper focuses maly o the asymptotcal boud Certa fuctoal trasformatos ca oly be realzed whe the umber of odes s very bg It may be ecessary for future work to aalyze the algorthm detal o a 'reasoable' umber of odes depedg o the requred applcato Ackowledgmets For the supportve ad dspesable suggestos I would lke to thak my supervsor Thomas Moscbroda May crucal steps the mathematcal aalyss would ot have bee possble wthout hs help A very grateful thak you also goes to Faba Kuh who helped to fd the rght track several dscussos Refereces [] Je Gao, Leodas J Gubas, Joh Hershberger, L Zhag, A Zhu Dscrete Moble Ceters, pages -, 003 [] Fe Da, Je Wu A Exteded Localzed Algorthm for Coected Domatg Set Formato Ad Hoc Wreless Networks, pages -4,