A Polynomial-Time Approximation Scheme for the Minimum-Connected Dominating Set in Ad Hoc Wireless Networks

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1 A Polynomial-Tim Approximation Schm for th Minimum-Connctd Dominating St in Ad Hoc Wirlss Ntworks Xiuzhn Chng Dpartmnt of Computr Scinc, Gorg Washington Univrsity, Washington, DC Xiao Huang 3M Cntr, Building F-08, St. Paul, MN Dying Li Dpartmnt of Computr Scinc, City Univrsity of Hong Kong, Hong Kong, China Wili Wu Dpartmnt of Computr Scinc, Univrsity of Txas at Dallas, Richardson, Txas Ding-Zhu Du Dpartmnt of Computr Scinc and Enginring, Univrsity of Minnsota, Minnapolis, Minnsota A connctd dominating st in a graph is a subst of vrtics such that vry vrtx is ithr in th subst or adjacnt to a vrtx in th subst and th subgraph inducd by th subst is connctd. A minimum-connctd dominating st is such a vrtx subst with minimum cardinality. An application in ad hoc wirlss ntworks rquirs th study of th minimum-connctd dominating st in unit-disk graphs. In this papr, w dsign a (1 1/s)-approximation for th minimum-connctd dominating st in unit-disk graphs, running in tim n O ((s log s) 2 ) Wily Priodicals, Inc. Kywords: connctd dominating st; unit disk graph; polynomial-tim approximation schm; partition 1. INTRODUCTION Ad hoc wirlss ntworking has attractd mor and mor attntion rcntly [2, 6, 8, 12]. It will rvolutioniz information gathring and procssing in both urban nvironmnts and inhospitabl trrain. An ad hoc wirlss ntwork is an autonomous systm consisting of mobil hosts (or Rcivd Fbruary 2002; accptd July Corrspondnc to: D.-Z. Du; -mail: dzd@cs.umn.du 2003 Wily Priodicals, Inc. routrs) connctd by wirlss links. It can b quickly and widly dployd. Exampl applications of ad hoc wirlss ntworks includ mrgncy sarch-and-rscu oprations, dcision making in th battlfild, data acquisition oprations in inhospitabl trrain, tc. Two important faturs of an ad hoc wirlss ntwork ar its dynamic topology and rsourc limitation. In an ad hoc wirlss ntwork, vry host can mov in any dirction at any tim and any spd. Thr is no fixd infrastructur and cntral administration. A tmporary infrastructur can b formd in any way. Du to multipath fading, multipl accss, background nois, and intrfrnc from othr transmissions, an activ link btwn two hosts may bcom invalid abruptly. Thus, th communication link is unrliabl and rtransmission is quit oftn ncssary for rliabl srvics. Th rsourc constraints for an ad hoc wirlss ntwork includ battry capacity, bandwidth, CPU spd, tc. Ths two faturs mak routing dcisions vry challnging. Existing routing protocols rly on flooding for th dissmination of topology updat packts (proactiv routing protocols [5]) or rout rqust packts (ractiv routing protocols [9, 13]). Ntworkwid flooding (global flooding) may caus th following two problms: Broadcast storm problm [12]. Ntwork-wid flooding may rsult in xcssiv rdundancy, contntion, and col- NETWORKS, Vol. 42(4),

2 lision. This causs high protocol ovrhad and intrfrnc with othr ongoing communication traffic. Flooding is unrliabl [8]. In modratly spars graphs th xpctd numbr of nods in th ntwork that will rciv a broadcast mssag was shown to b as low as 80% [14]. To ovrcom or, at last, allviat ths problms, a virtual backbon-basd routing stratgy has bn introducd [2, 6, 16]). Th most important bnfit of virtual backbon-basd routing is th dramatic rduction of protocol ovrhad; thus, it gratly improvs th ntwork throughput. This is achivd by propagating control packts insid th virtual backbon, not th whol ntwork. Othr bnfits includ th support of broadcast/multicast traffic and th propagation of link quality information for QoS routing [15]. Basd on ths applications, w can summariz th ssntial rquirmnts for a virtual backbon as follows: (i) Th numbr of hosts in th backbon is minimizd; (ii) all hosts in th backbon ar connctd; and (iii) ach of th hosts not in th backbon has at last on nighbor in th backbon. This is clarly th ida of a minimum-connctd dominating st. A connctd dominating st on a graph is a subst of vrtics such that (a) vry vrtx is ithr in th subst or adjacnt to a vrtx in th subst and (b) th subgraph inducd by th subst is connctd. Th problm of a minimum-connctd dominating st (MCDS) is to comput a connctd dominating st of minimum cardinality. On th othr hand, w assum that an ad hoc wirlss ntwork contains only homognous mobil hosts. Each host is supplid with an qual-powr omnidirctional antnna. Similar assumptions ar takn by most rsarchrs in th fild of mobil ad hoc wirlss ntworking. Thus, th footprint of an ad hoc wirlss ntwork is a unit-disk graph. Indd, in a unit-disk graph, th vrtx st consists of a finit numbr of points in th Euclidan plan and an dg xists btwn two vrtics (points) if and only if th distanc btwn thm is at most on. According to th abov analysis, w formulat th problm of constructing a virtual backbon as th problm of an MCDS in unit-disk graphs. Th MCDS in gnral graphs was studid in [7], which proposd a rduction from th st-covr problm. This implis that, for any fixd 0 1, no polynomial tim algorithm with prformanc ratio (1 ) H( ) xists unlss NP DTIME[n O(log log n) ] [10], whr is th maximum dgr and H is th harmonic function. Th MCDS in unit-disk graphs is still NP-hard [4]. Th bstknown prformanc ratio of prvious polynomial-tim approximations is a constant 7 [1, 3, 11]. In this papr, w will propos a Polynomial Tim Approximation Schm (PTAS) for th MCDS in unit-disk graphs. An algorithm A is a PTAS for a minimization problm with optimal cost OPT if th following is tru: Givn an instanc I of th problm and a small positiv rror paramtr, (i) th algorithm outputs a solution with cost at most (1 )OPT, and (ii) whn is fixd, th running tim is boundd by a polynomial in th siz of th instanc I. If thr xists a PTAS for an optimization problm, th problm s instanc can b approximatd to any rquird dgr. 2. PRELIMINARIES A dominating st in a graph is a subst of vrtics such that vry vrtx is ithr in th subst or adjacnt to at last on vrtx in th subst. If, in addition, th subgraph inducd by a dominating st is connctd, thn th dominating st is calld a connctd dominating st. Th following is a wll-known fact about dominating sts and connctd dominating sts: Lmma 2.1. For any dominating st D in a connctd graph, w can find at most 2( D 1) vrtics to connct D. Morovr, if D* 1 and D* 2 ar, rspctivly, a minimum dominating st and an MCDS, thn D* 2 3 D* 1 2. W ar intrstd in th minimum-connctd dominating st in unit-disk graphs. Th unit-disk graph has th following proprty: Lmma 2.2. Suppos that a unit-disk graph G lis in an m m squar such that vry vrtx is away from th boundary with distanc at last 1/2. Thn, G has at most 4m 2 / connctd componnts. Proof. Lt x dnot th numbr of connctd componnts of such a unit-disk graph. From ach connctd componnt, w choos a vrtx and idntify it with th cntr of a unit-disk. (A unit-disk has diamtr on.) Such unit-disks ar disjoint and all li in th cll. Thrfor, w hav Hnc, x 4m 2 /. x 1/2 2 m 2. It has bn known that th MCDS has som polynomialtim approximation with a constant prformanc ratio [1, 3, 11]. Hr, w quot a rsult from [3]: Lmma 2.3. Thr xists a polynomial-tim approximation for th MCDS in unit-disk graphs, with prformanc ratio ight. In dsign of a PTAS for th MCDS, w somtims considr th following gnralization of th concpt of dominating st and connctd dominating st: Considr a graph G (V, E). Suppos that H is a subgraph of G. A subst D of vrtics in G is said to b a connctd dominating st in G for H if vry vrtx in H is ithr in D or adjacnt to a vrtx in D and, in addition, th subgraph of G inducd by D is connctd. NETWORKS

3 FIG. 2. Cntral ara and boundary ara. FIG. 1. Squars Q and Q. 3. MAIN RESULTS In this sction, w will construct a PTAS for th MCDS in unit-disk graphs. Th gnral pictur of this construction is as follows: First, w divid a spac, containing all vrtics of th input unit-disk graphs, into a grid of small clls. For ach small cll, tak th points h distanc away from th boundary (th cntral ara of th cll). Thn, w optimally comput a minimum union of connctd dominating sts in ach cll for connctd componnts of th cntral ara of th cll. Th ky lmma is to prov that th union of all such minimum unions is no mor than th MCDS for th whol graph. Thn, for vrtics not in cntral aras, just us th part of an 8-approximation lying in boundary aras (within distanc h 1 away from th boundary, with som ovrlap with th cntral aras) to dominat thm. This part, togthr with th abov union, forms a connctd dominating st for th whol input unit-disk graph. Finally, using th shifting argumnt (i.., shift th grid around to gt partitions at diffrnt coordinats) to mak sur thr ar at last half good partitions having good approximations ovrall. Nxt, w work out dtails along th abov lins: For th input connctd unit-disk graph G (V, E), w initially find a minimal squar Q containing all vrtics in V. Without loss of gnrality, assum that Q {(x, y) 0 x q, 0 y q}. Lt m b a larg intgr that w will dtrmin latr. Lt p q/m 1. Considr th squar Q {(x, y) m x mp, m y mp}. Partition Q into a ( p 1) ( p 1) grid so that ach cll is an m m squar xcluding th top and th right boundaris and, hnc, no two clls ar ovrlapping ach othr. This partition of Q is dnotd by P(0, 0) (Fig. 1). In gnral, th partition P(a, b) is obtaind from P(0, 0) by shifting th bottom-lft cornr of Q from ( m, m) to( m a, m b). For ach cll as an m m squar, w dnot by C (d) th st of points in away from th boundary by distanc at last d, for xampl, C (0) is th cll itslf. Fix a positiv intgr h whos valu will b dtrmind latr. W will call C (h) th cntral ara of and C (0) C (h 1) th boundary ara of (Fig. 2). For simplicity of notation, w dnot B (d) C (0) C (d). Not that, for ach cll, its boundary ara and cntral ara ar ovrlapping with th width on. For ach partition P(a, a), dnot by C a (d) (B a (d)) th union of C (d)(b (d)) for ovr all clls in P(a, a). C a (h) and B a (h 1) ar calld th cntral ara and th boundary ara of P(a, a). For a graph G, dnot by G (d)(g (d)) th subgraph of G inducd by all vrtics lying in C (d)(b (d)) and by G a (d)(g a(d)) th subgraph of G inducd by all vrtics lying in C a (d)(b a (d)). Lt G (V, E) b an input connctd unit-disk graph. Considr a subgraph G (h). This subgraph may consist of svral connctd componnts. Lt K b a dominating st in G (0) for G (h) with minimum cardinality such that, for ach connctd componnt H of G (h), K contains a connctd componnt dominating H. In othr words, K is a minimum union of connctd dominating sts in G (0) for connctd componnts of G (h). Now, w dnot by K a th union of K for ovr all clls in partition P(a, a). By Lmma 2.3, w can comput, in polynomial tim, a connctd dominating st F for an input connctd graph G within a factor of 8 from optimal. St A a K a F a(h 1). [Not that w considr F as a graph without dgs. According to th abov dfinition, F a(h 1) F B a (h 1).] Lmma 3.1. For 0 a m 1, A a is a connctd dominating st for input graph G. Morovr, A a can b computd in tim n O(m2). Proof. A a is clarly a dominating st for input graph G. W nxt show its connctivity. Not that for any connctd componnt H of th subgraph G (h) for som cll in partition P(a, a), if a connctd componnt E of F a(h 1) has a vrtx in H, thn E must connct to th connctd dominating st D H for H. This mans that D H has bn making up th connctions of F lost from cutting a part in H. Thrfor, th connctivity of A a follows from th connctivity of F. To stablish th tim for computing A a, w not th fact that, for a squar with dg lngth 2/2, all vrtics lying insid th squar induc a complt subgraph in which any vrtx must dominat all othr vrtics. It follows from this 204 NETWORKS 2003

4 FIG. 3. Two dgs (u, v) and ( x, y) hav crosspoint w. fact that th minimum dominating st for th subst V of vrtics lying in cll has siz ( 2m ) 2. Hnc, th MCDS for V has siz at most 3( 2m ) 2 by Lmma 2.1. Thrfor, K 3( 2m ) 2. Suppos that cll contains n vrtics of th input unit-disk graph. Thn, th numbr of candidats for ach dominator in K is at most 3 2m 2 n O m k n 2. k 0 Hnc, computing A a can b don in tim n O m 2 n O m2 n O m2. By Lmma 3.1, w may tak A a to approximat th MCDS. Th nxt lmma will hlp us stimat th approximation prformanc of A a : Lmma 3.2. Suppos that h 7 3 log 2 (4m 2 / ). Lt D* b an MCDS for input graph G. Thn, K a D* for 0 a m 1. Proof. Rcall that G a (h) is th subgraph of input graph G (V, E) inducd by its vrtics lying in th cntral ara C a (h) of th partition P(a, a). Lt D b an MCDS in G for G a (h). Thn, w must hav D D*. Now, lt G[D] dnot th subgraph of G inducd by D. W first claim that G[D] has a spanning tr T without crossing dgs in th plan. In fact, suppos that T is a spanning tr of G[D] with th minimum total dg lngth. Suppos that T contains two dgs (u, v) and ( x, y) crossing at a point w in th plan. Without loss of gnrality, assum that sgmnt (v, w) is th shortst on among th four sgmnts (u, w), (v, w), (x, w), and ( y, w) (Fig. 3). Rmoval of ( x, y) from T would brak T into two connctd componnts containing vrtics x and y, rspctivly. On of thm contains dg (u, v). Not that d( x, v) d( x, w) d(v, w) d( x, w) d(w, y) 1 and d( y, v) d( y, w) d(v, w) d( y, w) d(w, x) 1. Thrfor, w can add ithr ( x, v) or(y, v) to connct th two connctd componnts of T ( x, y) into on. [In Fig. 3, th right sid shows a cas whr (u, v) is in th FIG. 4. Opration 2. connctd componnt containing y and, hnc, (v, x) is addd to connct th two componnts into on.] This opration rducs th total dg lngth of th tr, contradicting th assumption on T. Assum that T is a spanning tr of G[D] without any crosspoint. Lt T b b th subforst of T inducd by thos vrtics not dominating any vrtx in G a (h). W nxt modify T to a forst with thr oprations: Opration 1: If, aftr dlting a vrtx u of T b, T still kps th following proprty (B1), thn dlt u. (B1) For any connctd componnt H of G a (h), T conncts vry two vrtics in H T, that is, T has a connctd componnt dominating H. Opration 2: If, aftr dlting an dg of T, T still kps th proprty (B1), thn dlt th dg (Fig. 4). Through Oprations 1 and 2, T bcoms a forst with th proprty that dlting any vrtx or dg would dstroy proprty (B1). Now, w apply th third opration to T. Opration 3: If T b has two adjacnt vrtics u and v both with dgr two, thn dlt thm and rstor th proprty (B1) as follows: Not that dlting u and v braks a connctd componnt of T into two parts, say C 1 and C 2. Sinc T alrady passd Opration 1, thr must xist a connctd componnt H of G (h) such that T H xists in both C 1 and C 2. Sinc T C 1 and T C 2 dominat H, thr must xist ithr on vrtx x in H such that x is dominatd by both T C 1 and T C 2 or two adjacnt vrtics x and y in H such that x is dominatd by T C 1 and y is dominatd by T C 2. Thrfor, adding ithr x or x and y to T would rstor th proprty (B1). Aftr Opration 3 is mployd onc, it may b possibl to apply Oprations 1 and 2 again. At any tim, if Opration 1 or 2 can b applid, thn w us it; if Oprations 1 and 2 cannot b applid but Opration 3 can b, thn w mploy Opration 3. Sinc both Oprations 1 and 3 rduc th numbr of vrtics in T b and Opration 2 rducs th NETWORKS

5 FIG. 5. Tr T. numbr of dgs of T without incrasing th numbr of vrtics of T b, this procss has to nd in finitly many stps. At th nd, forst T would still hav proprty (B1) and in addition hav th following proprtis: (B2) T b has no adjacnt two vrtics both with dgr two. (B3) T has at most D* vrtics. (Not that initially T is a spanning tr for G[D] and D D*. Sinc Oprations 1, 2, and 3 do not incras th numbr of vrtics in T, th final T has at most D* vrtics.) Nxt, w mov som dgs of T to th insid of C a (h 1) by th following opration: Opration 4: If T has two vrtics u and v lying in C a (h 1) such that d(u, v) 1 and T has a path btwn u and v which contains an dg not lying in C a (h 1), thn dlt th dg and add dg (u, v). Aftr Opration 4 is mployd onc, it may b possibl to apply Oprations 1, 2, and 3 again. At any tim, if Opration 1, 2, or 3 can b applid, thn w us it; if Oprations 1, 2, and 3 cannot b applid but Opration 4 can b, thn w mploy Opration 4. Sinc Opration 4 rducs th numbr of dgs of T not lying in C a (h 1) and Oprations 1, 2, and 3 do not incras th numbr of dgs of T not lying in C a (h 1), this procss has to nd in finitly many stps. At th nd, forst T would still hav proprtis (B1), (B2), and (B3) and, in addition, hav th following proprty: (B4) Opration 4 cannot b applid. Sinc any vrtx dominating som vrtx in th cntral ara of a cll must li in C (h 1), vry vrtx of T lying in B (h 1) must blong to T b. Now, w considr a maximal subtr T of T such that (C) T has all lavs in C (h 1) and all othr vrtics not in C (h 1) (Fig. 5). For simplicity, w call such a maximal subtr satisfying (C) an ar of th cntral ara of cll. W claim that T lis in cll. To show our claim, suppos that T has k lavs. Sinc h 7, vry dg of T incidnt to a laf dos not li in C a (h 1). Thus, vry path in T conncting two lavs must contain an dg not lying in C a (h 1). By (B4), vry two lavs of T hav distanc mor than on. By Lmma 2.2, k 4m 2 /. Plas rcall th notation that T (h 1) is th subgraph of T inducd by its vrtics lying in th ara C (h 1) and T is a forst obtaind in th abov proof. Not that T (h 1) consists of k lavs of T and, hnc, has k connctd componnts. Th outr path p of T is a path btwn two lavs such that T lis in th ara btwn th path p and th boundary of C (h). Sinc T has no crosspoint and T is a maximal subtr satisfying (C), only vrtics in path p may mt an dg in T but not in T. For contradiction, suppos that T has a vrtx r lying outsid of cll. Without loss of gnrality, w may assum that r is on th path p. W considr r as a root for T and study th k paths from lavs to r. Th path p is brokn at r into two such paths. Not that any path passing through ara C (h 4) C (h 1) must mt an dg not on th path. (Othrwis, th path would contain two vrtics in T b both with dgr two.) It follows that, xcpt for th two paths obtaind from path p, vry path has to b mrgd into anothr on in ara C (h 4) C (h 1). This mans that ths k paths bcom at most 2 k/ 2 paths whn thy go out from C (h 4), namly, T (h 4) contains at most 2 (k 2)/ 2 connctd componnts. Similarly, T (h 1 3( log 2 (k 2) ))( T (6)) contains at most thr connctd componnts and T (3) contains at most two connctd componnts, that is, all k paths in C (3) hav mrgd into two paths. Not that ths two paths will mrg into on at r lying outsid cll. Thrfor, ach of thm has a vrtx u in ara C (0) C (3) incidnt to an dg in T T. This mans that thr must xist anothr cll whos cntral ara has an ar T touching T at a point in cll. (Not that aftr Oprations 1, 2, and 3 T b is containd in th union of ars of all cntral aras of clls.) So, T cannot li in cll. Similarly, this implis th xistnc of anothr cll whos cntral ara has an ar T touching T at a point in cll. Sinc T contains no cycl, this procss may go on indfinitly, so that a path of infinit lngth is found in T, a contradiction. This contraction complts th proof of our claim that T lis in cll. By (B1) and our claim, T (0) is a union of connctd dominating sts for connctd componnts of G (h). It follows that th numbr of vrtics in T (0) is at last K sinc K is a minimum on. Thus, K a K T 0 T D*, whr T dnots th numbr of vrtics in T. W ar rady to prsnt th following main thorm: 206 NETWORKS 2003

6 Thorm 3.3. Suppos that h 7 3 log 2 (4m 2 / ) and m/(h 1) 32s. Thn, thr is at last half of i 0, 1,..., m/(h 1) 1 such that A i(h 1) is a (1 1/s)- approximation for th minimum connctd dominating st. Proof. By Lmma 3.2, for vry i 0, 1,..., m/(h 1) 1, K i(h 1) D*, whr D* is an MCDS for G. Rcall that F is a connctd dominating st for G such that F 8 D* and F a(h 1) F B a (h 1). Morovr, lt F a H (F a V ) dnot th subst of vrtics in F a(h 1) ach with distanc h 1 from th horizontal (vrtical) boundary of som cll in P(a, a). Thn, F a(h 1) F a H F a V. Morovr, all F i(h 1) H for i 0, 1,..., m/(h 1) 1 ar disjoint. Hnc, i h F 1 H F 8 D*. Similarly, all F V i(h 1) for i 0, 1,..., m/(h 1) 1 ar disjoint and Thus, i h F 1 V F 8 D*. F i h 1 h 1 F i h 1 H F i h 1 V Thrfor, 16 D*. A i h 1 K i h 1 F i h 1 h 1 that is, m/ h 1 16 D*, 1 A m/ h 1 i h 1 1 1/ 2s D*. This mans that thr ar at last half of A i(h 1) for i 0, 1, m/(h 1) 1 satisfying A i h 1 1 1/s D*. Th following corollary follows immdiatly from th thorm: Corollary 3.4. Thr is a (1 1/s)-approximation for an MCDS in connctd unit-disk graphs, running in tim n O((s log s)2). Proof. Not that computing ach A a nds tim n O(m2). By Thorm 3.3, a (1 1)/s)-approximation can b obtaind by computing all m/(h 1) A a s and choosing th bst on. Thus, th total running tim is mn O(m2) n O(m2). Choos m to b th last intgr satisfying m/(h 1) 32s, whr h 7 3 log 2 (4m 2 / ). Thn, m O(s log s). This complts th proof. 4. CONCLUSIONS W hav dsignd a PTAS for th MCDS in unit-disk graphs. Thr is vidnc to show that currntly xisting implmntd approximations hav a larg spac for improvmnt. Acknowldgmnts Th authors wish to thank Dr. Pnjun Wan for his hlpful suggstions and corrctions and also wish to thank a rfr for insightful commnts. REFERENCES [1] K.M. Alzoubi, P.-J. Wan, and O. Fridr, Nw distributd algorithm for connctd dominating st in wirlss ad hoc ntworks, Proc HICSS, Hawaii, 2002, pp [2] A.D. Amis and R. Prakash, Load-balancing clustrs in wirlss ad hoc ntworks, Proc 3rd IEEE Symp on Application- Spcific Systms and Softwar Enginring Tchnology, 2000, pp [3] X. Chng and D.-Z. Du, Virtual backbon-basd routing in ad hoc wirlss ntworks, Tchnical rport , Dpartmnt of Computr Scinc and Enginring, Univrsity of Minnsota. [4] B.N. Clark, C.J. Colbourn, and D.S. Johnson, Unit disk graphs, Discr Math 86 (1990), [5] T. Clausn, P. Jacqut, A. Laouiti, P. Mint, P. Muhlthalr, and L. Vinnot, Optimizd link stat routing protocol, IETF Intrnt Draft, draft-itf-mant-olsr-05.txt, Octobr [6] B. Das and V. Bharghavan, Routing in ad hoc ntworks using minimum connctd dominating sts, ICC 97, Montral, Canada, Jun 1997, pp [7] S. Guha and S. Khullr, Approximation algorithms for connctd dominating sts, Algorithmica 20 (1998), [8] P. Johansson, T. Larsson, N. Hdman, B. Milczark, and M. Dgrmark, Scnario-basd prformanc analysis of routing protocols for mobil ad hoc ntworks, Proc IEEE MOBICOM, Sattl, August 1999, pp [9] D.B. Johnson and D.A. Maltz, Dynamic sourc routing in ad hoc wirlss ntworks, Mobil computing, Tomasz Imi- NETWORKS

7 linski and Hank Korth (Editors), Kluwr, Boston, 1996, pp [10] C. Lund and M. Yannakakis, On th hardnss of approximating minimization problms, J ACM 41 (1994), [11] M.V. Marath, H. Bru, H.B. Hunt III, S.S. Ravi, and D.J. Rosnkrantz, Simpl huristics for unit-disk graphs, Ntworks 25 (1995), [12] S.-Y. Ni, Y.-C. Tsng, Y.-S. Chn, and J.-P. Shu, Th broadcast storm problm in a mobil ad hoc ntwork, Proc MOBICOM, Sattl, 1999, pp [13] C.E. Prkins and E.M. Royr, Ad hoc on-dmand distanc vctor routing, Proc 2nd IEEE Workshop on Mobil Computing Systms and Applications, Nw Orlans, LA, Fbruary 1999, pp [14] P. Sinha, R. Sivakumar, and V. Bharghavan, Enhancing ad hoc routing with dynamic virtual infrastructurs, INFO- COM 3 (2001), [15] R. Sivakumar, P. Sinha, and V. Bharghavan, CEDAR: A cor-xtraction distributd ad hoc routing algorithm, IEEE J Sl Aras Commun 17 (1999), [16] J. Wu and H. Li, On calculating connctd dominating st for fficint routing in ad hoc wirlss ntworks, Proc 3rd Int Workshop on Discrt Algorithms and Mthods for MOBILE Computing and Communications, Sattl, WA, 1999, pp NETWORKS 2003

Week 3: Connected Subgraphs

Week 3: Connected Subgraphs Wk 3: Connctd Subgraphs Sptmbr 19, 2016 1 Connctd Graphs Path, Distanc: A path from a vrtx x to a vrtx y in a graph G is rfrrd to an xy-path. Lt X, Y V (G). An (X, Y )-path is an xy-path with x X and y