maxhmin d-cluster formation in wireless ad hac...

MaxHMin D-Cluster Formation in Wireless Ad Hac Networks

Alan D. Amis Ksvi Praknsh I Thai H.P. Vwng Dung T. Huynh Department of Computcr Scicncc

Universily of Texas at Dahs Richardson, Texas 75083-0688

[email protected], [email protected], ~vuong@nartelnetw~rks.com, [email protected]

1. INTRQDI1C'lTC)N

Ad hoc network (also referred to as packet radio networku) consist of nodes that move frecly and communicate with othcr nodes via wireless links. One way to sup~mrt cfficient cornmu- nication between nodes is to dcvciop R wirclcss backhne archi- tccturc Cl], [2], [4], 181. While all nodcs areidentical in their ca- pabilities, certainndes are clccted to forin thc backbonc. These nodes arc called cIustcrhc& and gateways. Clustcrhearta: arc nodm that me vested with thc rmponsibility of routing messages for all the nodes within their cluster. Gateway n& are nt&s a1 thc fringe of a cluster nnd typically communicate with gateway nodes of other clusters. The wireless backbone can be used eithcr to r o w packem, or to disseminate routing inkrmadon, or both.

Due to thc mobility of nodcv in an ad hoc network, the backbone must hc continuously reconshuctedin a timely fashion, as tlic nodes mow away Crom thcir associated cluslcrhe&. The clcction of clustcrheads ha? been a topic d many papers as de- s c r i W in i.11, [23.[8]. In di ofthcsc p;lpcrs the leader eleclim guarantcw that no n d c will he morc than one hop away from a leader. Furthermore, their time complexity is O(n), whcrc n is the niimkr of riodcs in Ihc; nelwork. Our work sliuld wjth the aim nf gcncralizing thc clustccring heuristics so that a node is cithcr a clusterhad or at most d hops away from a cluslcrhead.

We prow lliat constructing the minimkm d-hop dominaring

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set in an ad hoc network is NP-complete. Thcn, wc propose a new distrihuted leader clwtion hcuristic for ad hoc nclwork, guaranteeing that no node is more than d hops away from R icrldcr, whcrc d i s a value selccted for thc heuristic. Thus, this hcuristiu extends thc notion of cluster fom;ltioii. Existing I-hop clustrs are an inslance of the generic &hop clustcrs. The pro- poserl heuristic provides h d balancing among clustcheads to insurc a f i r distribution of load among c1ustcrIic;tdfi. Addition- ally, the hcuristic elcctl; cluaterhcads in such a inanncr as to fa- vor lheir re-ekeclion in future roundx, thereby reducing twit ion overheads when old clusterheads give way to new clusterhwds. Ifowever, it is also fair as a large number of nodes equally share the responsibility for acting as clnsterheads. Fuflhcmore, this hauistic liar time complexity of O(d) rounds which comparcs favorably to O(m) for carlicr heuristics [l], 141 for large mo- bile networks. T h i s rcduction in time complexity is obtained by increasing the concurrency in cornmiltication. Simulation cx- periment.; rapport thcsc claim?. Thus, it is nn iinprovcmcnt ovcr othcr known Iicuristics.

11. SYSTEM MODFI.

In an id hoc network all nddes arc alike and a11 arc mobilc. There arc no base stationfi to coordinate thc activities of sub- sets of nodes. Thercfore, all thc nodw have to coUcxtivcly rnake decisions. All mmmunicahn is over wirclcss links. A wire- leas link can bc cstablishcd between a pair of nadcs only if they are within wirelcsp m g c of each other. The Max-Min hcuristic only considers bidirectional links. It is mum& the MAC layer will "sk unidirectional links and pars bidirectional links to Max-Min. Beacons could be wed to determinc thc presence of neighboring nod&. Aftcr thc abscncc of some number of succcssivc beacons from a neighboring aode, il is concluded that the node is no longer R neighbor. WO nodcs that havc a wirelcss link will, hcnceforlh, be said to be 3. wircless hop away from each othcr. 'They are also said to be imincdiate neighbors. Communicstion bctween nodes is over a sir& shard channcl. The Multiple Acccss with Collision Avoidancc (MACA) protocol E143 may be used to allow asynchronous cornmunim~on while avoiding collisions and w&"itsions over R single wirc- IGSS channel, MACA utili-zw n fiequmt li, ScndK'Leur Yb Send ( R W L T S ) handshaking to avoid collision lxrween norlea.

A mdified MACA prritncol, MACA-BI my Invitation) [6]. supprcmw dl RTS and relies solely on CTS, iiwitations to trans-

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mit data. Simulation expriments show MACA-BI to bc s u p - rior to MACA and CSMA in multi-hop networks. Other potu- cols such a.. spatial TDMA [IO] may be U& til provide MAC layer communicaiicm. Spatial TnMA provides dctcrministic perhmance that is g o d if the number nf nodcs is kept rcla- Lively small. Hawevet. spatial TDMA requires that all nodcs be known and in a fixed localion to opcraic. Jn ad hoc nctwork thc nudes within each neighborhod are not known a priori. There- fore, spatial TI3MA is not a viablc solution initially. We suggerrt that MACA-Bl he used initially for this hcuristic to establifib clufiterheads and their associated neighborhoocb. Then tlic individual cluster m y transition lo spatial TDMA for inter-cluster and intra-clustm communication.

All nodes bruarlca.l thelr nu& identity periuddicdly to maintain neighborhaxi inlcEnty. h e lw mobiliiy, a nwMs neigh- horhml chnges with time. As h e mobilily of nod&? may nut bc predictable, changes in nctwork lopolugy over time are m- bitrary. lhwever, nodCS may nul be ~Wdrc a l changes in their neighhrhod. Therefom, cliistcrs and dustcrhcads must hc up- dated frequently to maintain accurate neiwork topology.

Dejirtitron I (d-neighborhood) - Thc d-neighborhood of a notb is the sef of all nodes that arc within d hops oC the node. This incMcs the node ilself. Thus, thc 0-ncighborliwd is only t hc node itself.

111. PREVloLIS WORK AND DESIGN CIIUICES

There are two heuristic d&gn approaches for managemcnt uPad hoc networks. The first choice is to havc all nodm maintain knowledgenf the network and manage fhhemselvas 171, [12], [13]. This circumventfi he nced to sclect iearlers or develop cliisfers. Howcver, it impores a significant communication responsibility on individual nodes. Each n d e must dynamically maintain routes to tho ml of thc nod= in the network. With large nctworks the numbcr nftneasagcs ncedcd to maintain routing tables may cause congmtian in the network. Wltimatcly this trafic will generate fiugc delays in mcssa3e propagation from onc n d e to anQthcr. Thic approach will not be considered in the remainder of this papcr.

The sccond approach is to idcntify a subset of nddeu within thc network and vmt them with the cxlra responsibility of being a leadcr (dustcrhcad) of certain nodes in thcir proximity. The clurskrhds are responsible for managing communication hetween nodes in their own neighhrhnod a s well as routing information to ulher clustcrheatk in othcr neighborhoods. Typjcally, backbonm are conslructed to connect ncighborhoadu in the nctwork. Past Bolutions of this kind hnvc created 8 hierarchy where cvcry node in tho network wm no inore than I hop away from a clustcrhcntl [l:l, l.41, 1101. In largc network8 this approach may gcncrde a large nuinber o f clustcrhwds and cvcntually lead t o thr: samc prohlcm as statcd in the first design approach. There- fore, it is clesirdAe to have control over thc clustcrhmd densiiy in the nctwork.

hrthcrmore, mmc of the prcvious clustcrinp solutions have rciied on synchronous clocks for exchange of data belwcen nodes. In thc Linkerl Cluster Algorithm 111, E A , nodes com- municatc using TDMA frames. Each ~ramc has a dot for wch nodc in thc nctwnrk IO commirnicatc, avoiding ctilisions. For

every node to havc knowledge of all nodcs in it neigliborhwd it rqiiirtx 2n TDMA lime slots, whcre n i s the number of norlca in the network. A node 2 becomcs a clusicrhearl if ot lcast onc of the following conditions is satisfied: (i) 2 ha-, the highest identiiy among all nodes within 1 wirelcss hop of it, (ii) s does not have the highe8t identity in its 1-hop ncighhrhocd, hut lherc cxista at Iwt one neighboring node y such hat z is thc highwt idcntity nodc in U'S I-hop neighborhrmI. Later the E A heuristic waf, rcvised E51 to dccreasc the number of clusterheads pducctl in Ihc original LCA. In this rcviscd dition of LCA (LCAZ) ;1 node is mid to be uomrecl if i t is in the I-hop neighburhood of a no& thal has declared itsclf to be a dustcr- head. Siarting from L e lowest id nodc to thc highmt id n d e , 3

node declares itsdf to bc a clusterhad if among thc non-covcrcd nodes in its 1-hop neighborhood, i t has thc lowcst id.

Thc ECA hcuristic was dcvclopcd and inlcnded to be used with small networks of Icss llian 100 nrddcr;. In lhis case thc dehy hetween n t h transmissions is minimat and m;ly he (01- mted. However, as thc numher of nodes in thc wtwork grow larger, LCA will. impose prater delays beiwccn node transmissions in the TDMA communicahon whcmc and may be unac- ccptahlc. Additionally, it has bccn show [ 151 that IIS cnmmu- nications incrcase tlic mount of skew in a fiynchronqus timer also increases, thcreby degrading thc pcrformai~cc of he ovcrall system or introducing additional dday and overhead.

Other solutions base thc clcciion of cluslerheads on dcgrm uf conncctivity [I I], no[ n d c id. Each nodc broadcasts thc nodcs that it cm bcar, including itself. A node is elccted ns a cbistcr- hcad if it is the highest conncctcd nodc in a11 of the uncovered neighboring nodm. In the cam UT ;t tie, the lowest or highest id may bc used. As the network topology chmgcs this approach can rc8ult in a high turnover oE chifiterhads 183. This is in-

dwirablo due to the high overhad associated with clusterhcd change over. D3ata stnrcturcl; have to be maintained for each node itl the cluster. h new clustcrheads are elected thcse dah slnlctures must he pwsd from thc old clustcrhead to thc ncwly elected clustcihead. Rc-election of clusterheads corild minimize this nclwnrk traffic by circumventing thc need to send thcse data structures.

1v. CONTRlHIITIONS

The main objective Wac to develop a heuristic that would clecl multiple leadcm in large ad hoc networks or ihnumnds ofnodcs. Addil'lonally, we wished to generalize the cluster definition tcl a collection of nodm that are up io d hops awq from a clusicr- head, where d 2 1, i.e., a &hop dominaling set. First, we show that forming a minimum &hop dominating .set is NP-completc. Then wc propose a hairistic to sdve the prablcm. Somc of ihc design gods and contributiom of this heuristic arc: I. Nodes asynchronoudy run the heuristic: no necd for syn- chronized clocks, 2. Limit the number of messagm wnt betwecn nodcs 10 O(d), 3. Minimize thc number and s k of thc dish stnictures reipired to imylerncnl the hcnristic, 4. Minimizc the numbcr of dustclheads as n iiiiiction of d, 5. Formatiun of backbune rising gatcways, 6. Rc-elect clusterheads when posWc: stuhility.

3 3

7. Distribute rqmnsibility of managing clusters is q u d y dis- rributed among all nodm fairness.

Due io the large number of n o t h involved, it is desirable io let the n&s opcrate asynchnmmsly. The clock synchronim- tion overhead i s avoided, providing additional processing say- ings. Furthermore, the number of messages mt from each node is ltmited to a multiple of& the maximum number of hops away from the nearest clusterhead, rather than n, thc number of nodes in the network. This g u m t e e s a good controlled message complexiiy for the heuristic. Additionally, bemuse d is an input value to thc heuristic. there is control avcr the number nf clnsterheads ciwtd or the density of clustcrhmds in the network. The amount of rcsourca necded at each node is minimal, con- sisting nf four simplc rulcs and two data ntructures that mainlain node information over 2d rounds of communication. N d a arc candidates to be clwiterheads based on their n d c id rather than their degree of connectivity. As thc network topoIogy c h m p slightly the node's d e p c of comectivity is much more likely to changc than the nodc's id relalive to its neighboring nodcs. As will be dcscribcd below, if a n d c A is d c largest in the d- neighhrhrwd of another node B, thcn nude A will be elected a clusterhead, even though node A may not bc thc largest in ik CG neighborhod. This providcs a smooth and deliberate transition of clusterhcadii rather than an emtic exchange of leadership. This last dcsign goal i s intended to help minimka the amount of data that must be p a d from an outgoing clusterhead to a new one when them is a changc over.

v. NP COMPLETBNESS OP D-HOP DOMINATING SS1

An ad hoc nctwark c ~ n be modeled lt'l a graph G = (V, E). wherc two nocfcs we conneccted by an edge if they can "municatc with each oiher. If all nodes arc Imted in the planc and have the cramc transmission radius r. then C is called B unit disk graph. Clearly, unit disk graphs are the simplest model of ad hoc netwclrkx.

A sct S of nodcs in G = (V, E } is called a &hop dominating sei iC every n d e in V is at most d (d > 1) hops away from a vertex in S. Minimnm &hop dominating sei is the problem d determining tor a graph G and an inreger I: 2 0 whether G IIM a dominating set of size 5 k. In this section we show lhat the minimum &hop domnating S C ~ problem is NF-complele. In fact, we will prove hat minimum d-hop dominating set is NP- complete even for unit disk graphs.

Theorcm: Minimum d-hop dominating set is NP-complete for unit disk graphs.

Proof: Since it i s obvious that the minimum d-hop domioat- ing set problcin is in NP, i t remains to show that it is NP-hard. We will construct a rcduction from thc (1-hop) dominating set prohlcrn for planar graphs with maximum degree 3 which WFXP

shown ko be NP-cnmplcte in 191. To his end, we make usc of the lollowing result which shaws how planar graphs can be effi- cicntly embedded into the Ruclidian plane [16]:

A planar graph with maximum d e g w 4 can be emheddtd in rhe plane using O(lVl) areu in such a w u , ~ t h t its veriimv are a( integer coordinaies and iis edges are drawn so that ihey are mark up of line segmenrv oJform r=i or y=j, for integers i

anclj. Morcover, according to [3] such embeddings can be con-

slr~cted in linear timc. Thus, in constructing OUT reduction wc may afisumc that we

are given a graph G = (V, E ) that is ctnbodded in the plane according to 1161. We construct in pdynomial time a unit disk graph G' -. (V', E') with radius d such that G has a dominating set 5' of size 5 k if and only if G' hw a d-hop dominating set S' of size 6 b', where le' is determined from G and k.

Construction of the anit disk graph G': Defines = 1/(2d+l)unitastheradiusof~eunitdia~kraph

GI. For cach unh length in G wc add (ad+ 1) new intennediate vertice,~ in equal distance. Thus, far cach original edge (U, PI) in G of length l,,,, we add (2d -k 1) x Z,,, intermediate verrictls. Moreover, wc add (d - 1) adfit19 vertices ul , . . ~, E- quencially from original verlar p1 at each distance d w shown in Pigure 1. Obviously the resulting graph G' = (V, E') is a unit disk graph with radius 6, and G' can IE constructerf from G in polynomial time.

e uti-1 Vd-I

if and only if G' hns d-hop dominating set S' nf skzc

wherc IU," is the length of Ihc edge ( ~ , e ) in G. Proof of Claim: For the only if direction suppose that G has

a dominating set S orsize k. We construct the d-hop dominating get S' in 0' a6! foIlows. S' contains all vcrtices in S. Moreover, for e w h original cdgc (pa, U) we add certain intermediate vertices to S' according to the following rules: Rule I: if PL (or PI) is in S, we add a total of intermediate vertices such that consccutivc vertices arc (2d + 1) haps apart starting from U(@). Rule ZZ: if hoth and w are in S, we add a total of Iu," intcrmcdiate vertices such that consecutive verlices arc (2d + 1) haps apart starting from U.

Rule Ill: if both U and v arc nut in s, wc add a tolal of la , , intcrmcdiate vertices such that consccutivc vcrticw are (2d+ 1) hops apart swing from position d. An example of these rules is shown in Figure 2. Clearly, we have

We now prow hot thc sct S' is a &hop dominating SCL First observc that there is one intcrmdate vertex in S' for cvery (2d + 1) consecutivc jntemediatc vertices on my original edge,

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so the intcrmcdiatc verliccs arc ciihcr in S' or at mnst d h o p from a vcrtcx in S'. If an original vertcx p1 i R in S, then it is also in s' hy Rulc 1. Therefore, ilr: mtxilirrty vcrticcs wi, . . , , ~ d - l

arc d hops away from w, a vcrtcx in S'. Otherwisc, if U is not in 3, then thcrc is a neighboring vcrkx p1 which is in S due to thc facl that S is a dominating sct. Considcr [he original cdgc (u,v). According 10 Rulc I, the intcrmcdiate vcrtex at pcisition (2d-k 1) x I,,, is in S'. Thus, vertex v and ib auxiliary vcrtices 211,. . . , wd-1 me at most d hops away frnm lhc vcrtcx at position (2d + 1) x I,,,,, which is in S'.

We havc shown that any (original, auxiliary, or inicrmediatc) vcrtcx in G' i s either in S' or at most d hops away rrom a vertcx in S'. Hcncc 9' is a d-hop dominating sct in GI.

We now show thc if dircckion. To this cnd, s u p p c h a t S' is a &hop dominaling set of ti^^ k' in GI. An mtmded edge(@', d) is an cxlension of an original d g e (ul U) that includes all inkcr- mdiate vcriicm as wcll as the auxiliary vcrtices ul, . . . , ud-1

and V I , . . . , Vd-1. For thc sake of convenicncc, thc wt of vcrliccs in S' [hat arc on an cxknded d g c is denotcd by Sk,v. We construct the doininating scl S for ~ h c graph G as as follows. For cach cxtended d g c (U', w ' ) we remiive verticcs from S' according to thc following rulcx: Rule 1.- I f only pd (w) is in S' (Sec l'igurc I): Remtivc all verticcs in sL,+ cxccpt U (U).

K u k ii: IPboth U sid v are in S' (Sec Figure 2): Rcmovc a11 verticm in SL,v exccpi U ancl v. Rule If/: Nnnc of U and PI is in S': If lS~ ,o l 2 I , , , + 1, lhcn add vcrtcx U to S' and rcmove all vcrliccs in Sh8u. Othcrwisc rcmovc d l Sh,v.

Observc that thc niimhcr of intcrmedialc verticcs in S' froin cdch original cdge ( u , ~ ) is at l eu t iu,v bcoawc wc havc a toral of (2d -t 1) x I , $ , intermdiate vcrtices on Llic cdgc. Morcover, when applying the ilhrivc iiilcs, Ihc total numher of vcrlices rc- movedis at lcilst i,,, for cach cxtended cdgc (U', w'). Thercforc the sixc or the rcsulting SCI S is

To verify that thc sct S i s a dominating fiet in 1hc original graph G, wc just nccd la provclhat every original vcrtcx i s cithcr in S or adjacent with a verlcx in S. Tci ihis cnd, considcr m y original ve~lcx U which i s not in S wc havc Collowing cascs: C,"u.sc I : U is a tlcgrcc 1 vcrlcx witti il ncighhr PI ( S a Figure 3). T I IJ is in S', lhcn v is also in S by Rulc 1. Othcrwisc, on ihc exkndcdedgc (U', U'), thcrcwc (Zd-tl)xl,,,+l venicc: which arc at most d h o p from a vcrtex in s& Thcrcforc lbqh,vl 2 I,,v -t 1 and ZI is in s by Rulc HI. Caw 2: U is a ncighhr of dcgrec 1 vcrtcx v Samc rcwning as in C x c 1 .

Cuse 3: PL is a neighhor of at least two dcgrcc 2 vcrticcs x and

IC cithcr 5 (or U) is in S', lhcn 5 (or g) is also in S by Rulc I. Olhcrwisc, on thc cxtendcctcdgcs (d, U') and (U', p'), thcrc are (2d 4- 1) x I , , , i- (2d -t- 1) x lu,u vertices which arc at most d hops away h m a vcrlcx in Sk+ U S;,+. Duc to tlic auxiIiary V C r h 2 S 211,. . . , Ud-1 We have is;," U sL,uI 3 is,u f I,,,, -b 1. That I d s Lo either lSL,ul 2 I,,- -I- 1 or IS;,J 1 6u,u i+ 1. From Rulc 1 I I, we can scc that cihcr original vcrlcx x or y is in S. Thus S is a dominating wt in G. This completfir thc proof of Ihe theorem.

(Sce Figurc 4).

i u1

VI, HEum'i-Ic

A. Data Sfructures

l'hc Iicuristic runs Cor 2d rounds of information cxchangc. Each node mainlains two arrays, WINNER and SENDER, cach OC size 26 nodc ids: rmc id pcr round of information exchange.

Thc WINNBR is Be winning n d c id of a particnhr round and uscd to rletcrminc thc clustcrllcad for a ode, as descrihcd helow in thc Basic Idea.

Thc SENDER is thc n tde that scnl the winning n d c id for a pariiculm round and is uscd to detclmine [he shortest palh hack lo the duskrhcad, oncc the clustcrhcad is sclcctcd.

H. Basic Idea

n d c id. This is Collowcd by thc Floodmm phasc. Initially, cach n d c scls its WINNER lo be cqud to ib own

lkfnifion 2 (Flmdmax) - Each node 1ocalIy broadcasts its WINNER vdur: to all of ih 1-hop neighbors. After al l neighboring nodcs have been heard from, for a singl; round, the nodc chooses the largest value among its own WINNER d u c and the values received in the round as icS new WINNER. This process continues far d rounds.

Defiiztion 3 [H&nm) - This follows Ploodrnax and also lasts d rounds. It ia (he same as Hoodmax except a node chooses the malla at rather than the largest value as i ts new WINNER.

Definition 4 (Overtake) - As flooding occurs in the network, WINNER values are propagated to neighboring n&. At the end of each flooding m n d a node decides to maintain. i i s cur- rem WIN" value or changc 10 a value that was received in the previous flood round. Overtaking is the act of a new value, diffcrent from the node's own id, being selected based on the outcome of the information exchange.

A n d c pair is any node id that occurs at leut once a3 a WINNHR in both the lBt (Floodmax) and Znd (Floodmin) d rounds of flooding for an individual nude.

We simulate rounds of the flooding algorithm by having every node send and receive the equivalcnt of a synchronous round of masages. This is accomplished by requiring each node to send a round r message tagged with T as the round numb. After a node has receivcd round T messages from all its neighbors it may proceed with round T transirion and ultimately to round T -k 1.

The heuristic has four logical stages: first the propagation of larger node ids via floodmax, second the propagation of smaller node ids via floodmiri, third the &termination of clusterheads, and fourth the linking of clusters.

The first stage uses d rounds of IlaodmaK to propagate the largest node id in each node's d-neighborhood. At the conclu- a i m of the floadmax the surviving node ids are the elected clw- terheads in the network. Nodes rccord their winning node for each round. Floodmw is a greedy algorithm and may rarult in an unbalanced loading €or the clusterheads. In fact, there may he cases where clusterhcad B is disjoint from its clustcr as a result of bcing overtaken by clusterhead A. "herefore, a node must realize not only if i t is the largest in its d-neighborhodbut also if it is the largest in any other node's d-neighborhood. This i s similar io the strategy employed in [l]. The semnd stage uses d rounds of fi &n to propagate the maller node ids that have not been overtaken. T h i s allows the relatively smaller clusterheads the oppomnity to (i) allnw them to regain nodes within their d-neighborhood and, (ii) realize that they are the largest node in another nde's d-neighborhood. Again each node records tlic winning node for each round.

At the conclusion of the floodmin. each node evaluates the round's WINNERS to best determine their clusterhead. In or- der to accommodate cas- where a node's id is overtaken by another node id, the s d c s t node id appearing in both of the f l d i n g skgw is chosen as thc clusterhcd. The smaller clusterhead i s chosen to provide load baiancing. However, in the WOIS~ case where c lwerhe~ A and clusterhead B are one hop away from one anfitha; clusterhcad €I will record its own node id as a WINNER only in the final round of flooding. Thcrc- fork if a ncdc m i v e a i ls own node id in the f l d m i n stage it knows that other nodcfi have elecled it their clusterhead so it

DeJinition 5 (Node Pairs)

dalarec; itself a clusterhead. Additionally, thore m y be sccnm- ios wherc a ncde is overtaken in tbc fI &ax abge by a set of nodes and then o~crtakcn by a completely diffmnt set of nodes in tho Aoodmin stage, nmc of which is its awn nda id. In this casc the node has no other aption but to select a ctustcrhead that is within d hops. The only known clusterhead that is within d hops is thc mER of the final round of floodmax.

finally, the gateway nodcs (nodes at the periphery of a clustcr) begin a convergecast message to link all nodes of the cluster to the clusterhead and, link the clusterhead to other clustem. Each gateway node will include its id and all othcr gateway nodes of other neighbohng cluskm in thc message. This will cstablish thc backbone ofthe network. During the convergecast it may be determined that a clusterhead resides on the path between a no& and its selected clusterhead, as shown in Figure 5 with nodes 3, 16, 28, and 48 electing clusterhead 200. In this case the cluslerhead closest to the. no& adopts it as a child. Wg- ure 5 shows the clusters formed when the heuristic terminate.

The propod heuristic providcs an optimal solution when the largest nodo ids are spaced d distance apart. However, even when the largest node ids are located in close proximity the heuristic provides a good solution at IOW cost in time and messages.

C. Clusterhead Selection Criteria

The mechanics of the heuristic arc quite simple. At some common epoch each n& initiates 2d rounds of f l d i n g . Each node maintains a logged entry of the results of each flooding round. The rounds am segmented into the Id d rounds and thc Pd d roundq. The Ink d rounds are a noadmax to propagate the hgcst node ids. After completion oFthe Is$ d rounds of f lod- ing the Znd d rounds of flooding begin, using the values that exist at each n d e after the lgt d rounds, The Znd d rounds of flooding WG a floodmin to allow the fimder node ids to reclaim some OF their territory. After mmpktion of the 2nd d roundrr each node looks at its logged entries for the 2d rounds of Bood- ing. T ~ G following rules explain the logical steps ofthc heuristic that each node runs on h e logged entries.

Rule 1: First, each nodc checkR to see if it has received its own original node id in the 2nd d rounds of f lding. If it has thcn it can &fare helf a clusterhead and skip the rest of this p k of the heuristic. Otherwise proceed to Rule 2.

Rule2: Elach node looks for node pairs. Once a node has identified all node pairs, it selects the minimum node pair to be the clusterhead. If a node pair does not exist for a node then proceed to Rule 3.

Rule 3; Elect the maximum node id in the Is$ d rounds of flooding as the clusterhead for this node.

D. Gateway Selection and Convergecast

After a node has; dcbxmincd its clusterhead bascd on Rules 1, 2, or 3, it communicates that i t is a member of the cluster to the clusterhead, In ordcr to minimize messages this information Is communicarcd from the Mnps or the cluatcr, gateway nod=, inward 10 thc clustcrhead. Furlhcrmore, a nade has no way to know if it is a gateway node. Therefore, after clusterhead fie-

lection mh node b"sts its elccted clusterhead to ail of its

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\38 La -2

Ncd= 1 IO 1 2 7 35 8 W 22 ZI 63 37 31 1'9 85 16 100 73 28 41 61 11 48 3 I S 36

neighbra. Only after hearing from all neighbors can a ndde determine if it is a gateway node. If all neighbors of a node have the same cluskrhead selection then this node is not a gateway node. However, if t h e are neighhringhodes with clusterhcad selections that are different, then these nodes are gateway nodes.

Once a node hss iden~fied itself as a gateway node it then begins a convergecast to ihc clustcrhead node scnding its node id all neighboring gatcway nodcs and their associated clusterheads. The SENDER data structure is used to determine who next to send the ccmvergecaql message. The proccsa continues with each node aiding itrr own node id such that when he dusterhead has heard from each of ita immediate neighbors it has a database of every node in its cluster. It i s not the intcnt of this heuristic to minimize the number of gateways.' %her this heuristic inaximizes the number of gateways resulting in H backbone with multiple paths between neighboring clwtcr- heads. This providRv halt koh", and eases congestion in the backbonenetwork.

Rule 4: Thcre are certain scenarioer, as shown in Figure 5, where this hcuristic will generate a clusterhcad that is on the path Mween a node and its electcd clusterhead. In this case, during rhe convcrgmast [he first clusterhead to receive the convergecast wiU adopt the made as one of its children. The clusterhead will immediately send B message to the nodc identifying itself as the new clusterhead.

ti. Correctness of Heuri.Ttic

The correclrms of the heuristic is solely dcpendant on nodes electing clusterheds that actually become clusterheads. The following m~mptions are used to first show that every node that survives tho f l o " stage of the heuristic becomes a dux- terhead. Then we will show that every node that is elected as a

'Rcsuicting thc number of gateways minimks the number of pths baween chxtrrheads [t2], IS].

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clusterhead doR~ in fact hecome a clusterhcad. Assumption 1: During the f f d m i n and floodmi algo-

rithms no node's id will propagate €arther than d-hops from the originating node itself (definition of flooding),

Awumption 2: AU nodm that survive the floodmm elect themshes clusterheads.

P m f of Assumption 2: The f l"ax will propagate thc individual node idt; outward creating a dominating set of node i&. This dominating set of nodc ids will consist of two classes. Class 1 nodes will be thosenode ids that are the largest in their o& neighborhood. Qd nodes will be those that are thc largest in at least one of their &hop ncighbors' &neighborhood. A Class2 node can not be a C l w l node.

Consider a Class1 nade id, say node A.. Node A will overtake each node that ia & h o p away from it during the flocdmax. Thefore, all nodm that are within the &hop covmgc area of node A will pcwcss nodc A's id value in thc WINNER dah structure.

At the mclusion of the R&in a Class1 node will elect itself a clusterhead, based on Aswmption I and Rule 1. Consider a Clad! node id, say node B. Although node B is overtaken by larger node ids, its node id continues to propagate out and con- sume all amallernode ids within &hop of node B. Thercfore, at the comptctiun of thc floodma node B's id and larger nade idc (Class1 or Class2) wiU coverthed-hopcovera~eareaof n&B. Therefore, thc Class2 id is the smallest s~rviving id in the &hop neighborhood of the originating Claw2 nadc.

Based on Assumption 1 we can conclude that the floodtnin p m c w wi11 successfuuIIy propagate rhe Class2 node id back to the originating Class2 node. A CIasQ nodc will eIect itself a clusterhead, based on Rule I .

Iherefbre, any node lhaf YYWZYG,T the jloodmaxsrage will elecr itsew a clusterhed.

L e m a I : If node A elects node B as its ciusterhcad, then node B becomes a clusterhead.

37 IEEE INFOCOM 2000

€”E The proof of the LRmma will consider all possible ways that a node may elect its clusterhead, and then prove that ilus node does in fact becumc n ciusterhcad.

Case 1: N d c A elecrS itself a a clusterhead based on Rule 1. If notle A Icceivcs its own id in the floodmin s~age, i t h o w s that othcr nodes have elcctcd it a clusterhead based on Assumption 2. Therefore, it clccts itself a clusterhed.

Case 2: Node A clccts node 3 its clusterhead based on Rule 2. Nodc A receives an entry for 3 in the floodrnin portion of the hckuistic. Therefore, based on Assumption 2 we concludc that B does hccome a clusterhead. We choosc the smdlcr node id pair to promotc: faimcss and distribute the load among elected clustcrheads.

C m 3: Node A elects node B it clusterhead based on Rule 3. Node A receives no node p& and must selw t the only node known to be a clusterhead. The mly node that is guaranteed to bc 11 dusterhead is the last WINNERof the f l o a c h ” This node survives thc f l d m x and again based on Assumption 2 i t will bccome a clustwhcad.

VII. ItLUSTRATIVF! EXAMPLES

Figure 5 shows an eKampIe of the nclwork topology generated hy he heuristic with 25 nodes. Herc we see four clusterheads elecled in closc proximity with one another, namcly nodes 65, ’73,8fi, and la[). This figure shows how cluster division has the effecl of drawing a line between clustorheads and sprilting h e nuda? among themselves. Additionally, Hg~rc 5 demonstrates the need for Rub 4. RS nodes 3, lfi, 28, and 48 have elected nodc 100 tts Lhcir ciustcrhcad hut must pms hough other ch te r s an thcir convcrgcltst to nodc 100. On applicatiun of Rulc 4, clustcrhead 85 instructs nodes 3, 10, and 48 to join its duski. While clusterhead 73 inxtrucls node 28 to joins its cluster.

Figure 6 shows the resulting network toplogy after slightly perlurbing thc network in Pigurt: 5 . Here we sec that hree of tho previous four clusterheads arc rc-elcctcd. The fourth clustcrhcad, node ti5 from Figure 5, is overtaken hy clusterhead 85.

Fig. 6. %cluster fonnnlion apler tapaloay cllaryc

A. Pdhohgicnl Cuse

‘lhcre is a Ixiown cuiifiguration whim the p r o p t d lrcurlc- tic fails tu providc a good solution. This conkguration is whcri no& ich are monotnniclly increming ur decreasing in a smight line. In this cilsc. the d-k 1 srnallesl node ids belong to the same

cluster ;1s shown in Figure 7. All other n& become cfustcr- hearla of thcmsdves only. Again, while this is riot optiinrI it still guarantecs that no node is more than d hops away fram a clusterhead, Furthermore, this configuration is highly unlikely in a real world application. However, this ig a topic of futurc work to be performed with thiq heuristic.

B. Rme, Message and S m g e Complexiiy

Each node propagates node ids for 2d nands to ckct clusterhcads. A convergecast is then initiated to inform [he clustcrhead of its children. Sincc no node is mare than d hops from its cIwterhCad the convcrgccast will be O(d) munds of messages. Therefore, the time complexity of thc heuristic is 0(2$ + d) rounds = O(d) munds.

The time complexity and the number of transmissions re- quircd to achievc a locd broadcast (to all neighbors) for a single round is dependent on the auccess of the data link layer pro tocoi. While the MACA-€31 h ~ 6 bcen shown to be xuperior to MACA and CSMA [6J i t firill suffers from Ihe hidden terminal problem and may r e q u b re-transmissions to complete a round. A data link protocol similar to MACA-BI that resolvc.4 completely the hidden terminal problctn irr an wea of additional rcscarch and not the intcnt of this paper.

Bnch node has to maintain 2d nadc ids in its WlNNER data structure, and the same number of node ids in its SENDER data atruclure. Thus, the sloragc complexity is O(ii!). This coin- pares Favorably with heuristics like [l], [Z] where identities of all neighboring nodes is maintained and thc storage complexity is O(n).

VIII. SIMU1,ATION EXPERIMENT9 AND RRSIILTS

We conducted simulation experiments to evatuate the performance of the proposed heuristic and comparc ( Imc finding againsl three heuristics, the original Linked Ciustcr Algorithm &CA) 111, the rcvised Linked Cluster Ngorithm (LCA2) [51. and the Bighest-Connectivity (Degrec) I1 11, [81 heuristic. Wc mumod a variety of wstem mnning with 100,200,400, and 600 nodes ta simulatc ad hoc nctworks wiU vfllying levels of nade dcnsity. ‘ h o nodes are said to have a wircless link betwmn [hem if they are within communication range of~ach ohcr. Vie performance W A ~ sitnulated with the communicatinn range of thc noder; set to 20, 25 and 30 lcnglh units. Additionally, the span of a cluster, i.e., thc maximum number of wircless hops h t w w a tiode and its clustciliead (4 wm SOL CO 2 and then 3 for each of thc simulation conibinalionu abovc. Tlic cntire simulation was conductcd in a 2130 x 200 unit region. Initially, each node was assigncd a unique n d c id and x, y cmrdinatcs within

the region. The nodes wcrc then allowed lo move at random in any c l i d o n at a s p d of not greater than 112 the wireless range of a nodc per second. Thc simulation ran for 2000 sccontls, and the network WW sampled every 2 monds. At each sample time the proposed Max-Min heuristic was run to determine clwiter- hcads and heir m i a k d chatear;. For every simulation run a number of statistics were measured for the entirc 2000 seconds oC simulation. Some nT the more noteworthy simulation statistics "stired wcre: Number of Clusterheads, Clusterhead Duration, Cluster Sizes, and Cluster Member Durafion. Thtse statistics provided a basis for cvaluating the performancc of the pro& heuristic.

IltJnition 6 Mmher of Clusierhcads) - The mean number of cluslerhmch in a nelwt)rk for a samplcA We do not want IMI few clusbxheads, xi thcy will. bc overloaded with too many clustcr "he. Nor is it good to have a large numhcr of clusler- hesds, cadi rnanaging a very small cluster.

OeJnirzon 7 (Clwtciterhead Duration) - The mean lime for which once a node is elected a clusterhead, it stays BS B clusterhcad. This stalirrtic is a " a c e of stabilily, t he longcr the duration the morc stablc the system.

Defwirion 8 (Cluster Sizs) - Thc mean siu: of a cluster. This value is inverscly proportional to the Number of Clusterheah. Wc do not waiit clustcm tw large ihat they will overload their clusterhcads, or so mall that thc clusterheah arc idie a good part of the lime.

D,$nitiorr 9 (Cluster Mcmbcr Duration) - Thc mean con- tiguous time a node stays it membcr of n cluster before moving to another clustq2 clustcrheads illl: mndered cluster memhcm. also. This stahtic i s a " m e of R t M i t y like the Clusterhead Duration, but from the point of view of nodes that arc not clus- tcrhends.

X A , LCA2, and Degrcc bawd heidtics generate I-hop clusters. Therefore, to properly compare these heuristics with the propsd Max-Min heuristic i t was nccexsaty to perform B

d-clmurc on the connectivity topology bcfore riinning each oC these hcuristicr;. The d-closure yields a modified graph in which nodcs A and 6: fire I-hop ncighhors if they were at most &hop% away in thcactual topology graph. Here, d ixcither 2 or 3. When the LCA, LCA2, and Degree based heuristics arc run on this modified graph, they form cluHters where each nude is at most d wirclmv hops away fmm its clusterhcad. The LCA heuristic clects clustcrheads h a t may bc adjaccnt to one another whilc the LCA2 and Degree b x u l heuristics do not allow clusterheads to be adjwent tu one another. Thcrefore, thc tiekction of thwe three heilrisliwi should pmvide good coverage For hnchinarkhg the perl7"cc of the proposed Max+Min heuristic.

Ohscrving the simulation results of Rgwc 8 shows that Max- Min, LCM, and agree based heuristics never produce more than 33 clusterheads, when %hop clusters are formed and thc wireless rangeis cqud to 20 length units. Furihermore, as rhore nodes are d d c d thc numbcr of clusterhcads produced by these hcuristics rcinains almnsi unchanged. The LCA heuristic pro- duces a tnaxinium d 130 clusterhcitds. Observing the LCApIot shows that the. slope, approximately 0.17 for high denaily networks, will gncrate a clusterhead for evcvcry 5.8 ncwly a d d 4

a A clusrer is represenml by the identity of its climtCrticad.

nodes. This is an unnecesfdly l q c number of clusterhcads. Similar trends are exhibited for other combinations of hop count and wireless mge.

pigure 9 shows W - M i n with the highmt clustcrhead duration followed by LC.42, E A and finally the Dcgm hmed heurislic. Max-Min s h w s ;in i n c m c in dusterhead duration ;t9 rhc networkbcoomes mwc dense, whilc for LCA, KA2, and Degree the d w h n decreases as the system si7z incrcasm. This is nbt surprising for Degree il~i it is based on (kgm ofconiicctiv- ity, not node id. As thenctwork topology changes this approach can result in high tumover nf clustcrheds IS]. Similarly, in LCA and LCA2 a singlc link makc or break may move a Iower id n d c within or out of d-hops of a nude g, forcing i t LO transition be- lwecn clusterhead and normal node states. Such transition5 rimy dtro have R ripple effcct throughout the network. This adversely impacts the stability of clustcrs.

Cluster Weads in e Hqr Cmni, Ilango Gysbm l 4 O ' ' '

...a . ka2 --It

degree - - R -

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Figure 10 shows thc Degree bawd and LCA2 hcurisiics produce the largest cluster sizes followed by the Max-Min, and finally thc LCA heuristics. The Dqycc, LCAZ, and Max-Min heuristics produce clusrers whose s i ~ s increase by 3.1,3.1 and 2.3 nodes per 100 nodes respcclively. Whilc thc LCA heuristic cIustcr sixes are vcry flat and only increase slightly as the network density incrmscs. Combining the number of clusterheark and number of cluster si7es results we can see that thc LCA hcuristic is producing a largc number of small clusters ;ts: the system sixc gets larger. This indicates that the LCA hcuristic very oftcn suffers frum R pathological case whcrc a node hc- comes a clusterhead under somewhat false pretences. '@is can happcn when a nude becomes a clusterhead because i t is Ibc largcst node in one or its neighhor's neighborhoods.

Cbdor~ires in2 Hopcount 20 RangaSystem 251 I , I r I , I I I

io0 150 2 M 250 WJO 950 400 450 5m NumPsr 01 "I- in Ihe Svstsm

550 sm

Pig. 10. Impact oPnetwork dcwlty on clustcr slzc.

Figurc 1 I showsECA2 and Max-Min with thc highest cluster member durations Callowed by LCA and finally Degree. Here wc SCC that the LCA2 heuristic shows a slight jncrcase in clustcr mcmhcr duration tls the network becomes more dense, while thc ECA heuristic shows a slight decrease. Max-Min has become fairly flat at 3.7 ticconds for dense networks, while thc Degree hcuristic show a steady dccline to about 2 seconds. the samplhg ratc of the simulation.

Finally, Figure 12 shows that Max-Min produccs the highcst percentage of re-clccted clusterhcads (consistent with Fiigurc 9). As a result Figure 13 shows that Max-Min elect? only a fraction or the total nutnhcr of ndcs as lcadcrs during the cnlire sim- ulalion nrn af 2000 scconds. This fiupprts the idca that Mrtn- Min will try lo rc-clcct exisling leaders. Thc LCA and Dcgrec ba7crl licuristics clecred cvery node, or one short of every d e , as Icadcr at leas1 oncc during each simulation run of' Zoo0 seconds. So, thcir plots arc supcrimpr)sction cach nthcr and cannot hc distinguished. Whilc LCAZ d w x not clcct every ncxle a clustcrhcad in cach simulationrun, i t s1ill clects amuch highcrnum- hcr of clustcrheads ihan Max-Min. It is not dcsirable 10 changc Icadcwhip ttm frequcntiy as [his causes ulc exchange of leadership databases to the RCW clusterhcads. This may ulimatcly

1 3.4 -

m x .--a.. . ka - -m- -

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iao 150 zoo 250 30[) 350 4w 4 9 500 m 6m

I'ig. I I . Impsct of iielwurk demlty on cluster ~ncrnber duratlon.

Nunbsr of Nod- in the System

cause congestion in the nclwork.

. I 1 1 1 1 1 1

150 20 250 900 3% 4W 450 5cO 6M Bw Number of N d g ~ in the Srjtam

i i g . 12. Impnct ofndwork daislty am rc-dcchd cliiGfCrJ1~d9.

The Mm-Min heuristic pwduces fewer dusterheuds, much larger dusters, and longer clusterhenddurntiofi on the average, !/tan the I X A heuristic. While the Degres based heurisfic does huve dight1-y larger chsier s ixq lhan the M a - M i n , it suffer.7 greatly in other cntegciries such us clusterhead durniirm, nrrd cluster member duration. ?ke I L A 2 heurisiic p w d ~ ~ e s clustcrheads lhaf are compnrtrble in nwntcr to thai of Max-Min. Iluwever, Max-Min hns clusterhcd durations that am appmxi- mntely 100% hrgcr [han that of LCA2 fv'urdensr: networkx Fur- thermore, the Max-Min clusterhud dumrion continutx io increase with increased network den.qiQ, while the LCA.2 heuristic du.plerheod durdion dtwreuses wdh increased network densiv. Based on these initial simaaltinrr results ,he M a - M i n heuristic provides h e les t d I uruund cluslerhsad leader eleclion charm-

Big 13. Impad dndwark denslty on number of n d m elactod as dustm Ireads dudog ihc enure shulptlon.

teristics. hibhrrc work is needed to determine the appropriate time

to trigger the Max-Min heuristic. If periodic triggers are too clusely spaced Lhen the system may run the heuristic ewn when tlicrc has bccn no noticcablc topology change to warrant run- ning thc heuristic. If the pciiorlic riggers mc too far apart then thc topolobT rnay changc withouk mnnitlg the heuristic, caw- ing n d c c In he stranded without a clusterhead. The triggerin& conrlitian should be complckly asynchronous and locdimd to a nodc’s cluster and its ncighhboring cIusters lo restrict execution of the heuristic to only affected ndes. Furthermore. the trig- gering schcmc g h d d account Cor topnlagy changcs during the progrcss of the heuristic. The Max-Min heiiristic has n tendency to re-elcct existing ciusterhcads. T h i s is desimble h r stability, hriwevcr it must he tempered with B load-bahnung scheme. Load-balancing allows rc-election of existing clusterheads until they ttwc exceeded thcir clusterhead duration budget. Th’hcrc clusterheads sliould then give way to allow other ndes to become clustcrheads.

Ix. PUSSrl3LR APPLICATIONS OP THE HEURISTIC

Ad hnc nctwoxks arc suilahle For taclical missions, cmcrgency response opcvitions, electronic c l a s s r ~ m networks, etc. A pas- sible application for this hellrisk is to IISC it in conjunction with Spatial TDW. Spatial TfDMA provides a very cfficieiit com- inunimtim protaco1 for clusters with row nodes. Howcver, thc nodex must be known and in a fixed Incalion. Hence, Spatial TDMA is not w i l y used in ad hoc networkti. The proposed harigtic may bc u . 4 io Jckrmim ihe clusters and the cluster- ha&, in he neiwork. At his pojnt sll of ihe nodcs within B

cluskr Rce known and assiime to be: fixd. This information may k used by Spatial TDMA to consttLlcl a 1DMA frame €or he individual clusters. Spatid TDMh will continuc as thc cqmmu- nication protocol until there is sufficient loplogy change that the p m p d hcuristic is run again ta form new chitcis.

The propmi hcuristic can bc u d for hierarcliicd muting

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purposm wherein clusterheads can maintain ranting informa tion. It can aLS0 be used for location management purpa . where the clusterhcRda receivc location updates and qucrics from other nodcs in thc system.

X. CONCLUSION A new heuristic for electing multiple leaders in mi ad hot; net-

worb has bccn presented, calied Max-Min Leader Rlcction in Ad Hoc Networks. Max-Min runs asynchronously eliminating the need and overhcad of highly aynchronized clocb. Thc maximum dishncu a n d c is from ita clusterhead hap becn general- iecd to be d hops, dowing cantroi and fle~ibility in the detcr- minatiotl of the clusterhead dcnsity. Futihcmarc, the number of mesagm ir; R muItiple of id rounds, providing B very good run time at the nerwork tcvcl. Sitnple dah structurcs have been uscd lo minimize the local rcsourca at each node. Rc-clcction of clusterheads is promoted to minimisrx transferal of databases and to provide stability. The solution is scalablc as i t generates ZI small number of clusterheads compared to gome other heuristics. Ah, a low variance in cluster sizes lcads to better load bdancing among thc clusterhen&. Finally, Lhis heuristic utilizes clusterhcitds and multiple gateway nodm to form a redundant backbone architecture to provide communication betwcen clas- (em.

maxhmin d-cluster formation in wireless ad hac...

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