Revisiting Connected Dominating Sets: An Optimal Local Algorithm?

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1 Reisiting Connected Dominating Sets: An Otimal Local Algorithm? Samir Khller Deartment of Comter Science Uniersity of Maryland College Park, MD 0740, USA Sheng Yang Deartment of Comter Science Uniersity of Maryland College Park, MD 0740, USA Abstract In this aer we consider the classical Connected Dominating Set (CDS) roblem. Twenty years ago, Gha and Khller deeloed two algorithms for this roblem - a centralized greedy aroach with an aroximation garantee of +, and a local greedy aroach with an aroximation garantee of ( + ) (where H() is the harmonic fnction, and is the maximm degree in the grah). A local greedy algorithm ses significantly less information abot the grah, and can be sefl in a ariety of contexts. Howeer, a fndamental qestion remained - can we get a local greedy algorithm with the same erformance garantee as the global greedy algorithm withot the enalty of the mltilicatie factor of in the aroximation factor? In this aer, we answer that qestion in the affirmatie. I. INTRODUCTION A connected dominating set (CDS) in a grah is a sbset of ertices that indces a connected sbgrah, and is also a dominating set at the same time. A dominating set is a sbset of ertices sch that eery node in the grah, is either in the dominating set, or adjacent to a node in the dominating set. Finding a minimm connected dominating set is NP-hard, and ths for the last twenty years, researchers hae exlored aroximation algorithms for this roblem starting with the work of Gha and Khller [7]. One of the original motiations for the roblem was in bilding a backbone for roting in the context of wireless ad hoc networks. Oer time many other alications hae been exlored in the context of social networks and AI [], []. See other alications and reslts in the book by D and Wan [4]. In their aer, Gha and Khller [7] deeloed two simle aroaches - the first one is a local aroach, where we start from a single ertex in the soltion, and incrementally (greedily) add neighboring nodes while maintaining a connected sbset of nodes at all times. It is temting to imagine that adding one node at a time, maintaining connectiity might work well. Howeer, this does not work and there are instances where we might end with a soltion with (n) nodes, while the otimm soltion has only O() nodes! Interestingly, a simle modification of this algorithm that exlores a -ho neighborhood making greedy choices at each ste that inoles selecting to two nodes at each ste, works mch better and This work was recently awarded the ESA Test of Time Award. they show that a ( + ) aroximation can be obtained for this roblem (H(n) = P n i= i is harmonic fnction, and is the largest degree in the grah). The main benefits of the local algorithm are that no knowledge of the entire grah is needed at each ste, and only the art of the exlored grah sffices to select the next node. This may not be a sefl for a static grah, as yo can calclate it once and foreer. Bt to get a CDS for a grah that changes dynamically, like social networks or wireless ad hoc networks, limiting the amont of information reqired can be both imortant and challenging. They also deeloed an imroed global algorithm that is again a greedy algorithm, and constrcts a soltion that does not maintain any connectiity roerty, and is only connected at the ery end. This centralized greedy algorithm yields a bond of + O(), eliminating the factor of, and gets s close to the lower bond on this roblem of ( ) de to set coer hardness [6]. Desite mch interest (see book by D and Wan [4]), the key qestion remained oen for two decades - is there a local algorithm that gies s the same bond as the global algorithm? In this aer, we answer this qestion in the affirmatie. We deelo a ery simle local algorithm, and are able to show that it matches the bond of the global algorithm. Sch a reslt is esecially srrising de to its simlicity. Connected Dominating Sets became a central toic in the context of wireless ad hoc networks, where the CDS acts as a roting backbone for acket roting. Often it is exensie to connect all the nodes, as the cost can become rohibitie, and in this case it is fine to connect most of the nodes (or a gien fraction). Li and Liang [0] formalized the roblem of artial connected dominating sets (PCDS) and roided heristics withot erformance garantees. Arachenko et al. [] defined the bdgeted connected dominating set roblem (BCDS) where we hae a bdget of k nodes and we wish to find a connected set of at most k nodes that maximizes the nmber of nodes it can coer. Insired by alications in social networks, they deeloed ractical heristics sing only local information. Khller et al. [9] deeloed the first algorithms with theoretical garantees for both these roblems with aroximation factors of O(ln ) for PCDS and 3 e for BCDS. Extensions to node weighted ersions were considered

2 by Gha and Khller [] as well. Extensie research was sbseqently done on this toic with the deeloment of distribted algorithms [], as well as for many secial classes of grahs [4]. A. Or Contribtions Or reslts can be smmarized as follows - In Section III, for the Connected Dominating Set roblem, we obtain the first local information algorithm whose aroximation ratio is within additie constant to global algorithm, i.e. + O(). To be recise, or aroximation garantee is H( + ) +. This algorithm reqires -ho local information (see Section II for definition). - In Section IV, with -ho local information, we obtain an + + aroximation algorithm. In addition to better aroximation ratio, it also rns faster than the Randomized CDS algorithm [7] (Section II-C), becase it exlores fewer nodes. II. BACKGROUND We first reiew existing aroaches [7], []. A. Global Algorithm for CDS The global algorithm rns in two hases. Initially, all nodes are colored white. In the first hase, the algorithm iteratiely adds a node to the soltion, colors it black and all its adjacent white nodes gray. A iece is defined as a white node or a black connected comonent. A new node is chosen to be colored black to get maximm redction in the nmber of ieces. This hase ends when no sch node exists that can gie non-zero redctions. At this time, there are no white nodes left. Intitiely seaking, black nodes are selected nodes, gray nodes are nodes that are dominated, i.e. adjacent to black nodes. In the second hase, we start with a dominating set that consists of seeral black comonents that we need to connect. The connection is done by recrsiely connecting airs of black comonents with a chain of ertices, ntil there is only one black comonent, which will be or final soltion. The aroximation ratio for this algorithm is +, where H(n) is harmonic fnction. B. -ho Local Information Algorithm for CDS The same aer roosed another algorithm, sing only local information. Instead of sing information abot the entire grah, it only relies on information within -hos to the nodes chosen in the soltion. The formal definition of local information is as follows. Before we define what local information is, we first define the distance between a node and a set of nodes. Definition (Distance): In an ndirected grah, denote the distance between and in a grah as d(, ). It is the length of shortest ath from to. d(, S) is defined to be min S d(, ) We now define the local neighborhood of some node, or a set of nodes. Definition (Local Neighborhood): Gien a set of nodes S in grah G, the r-ho neighborhood arond S is the indced sbgrah of G containing all nodes sch that d(, S) ale r. We denote the r-ho neighborhood as N r (S). When there is no confsion, we se the same notation to denote the set N r (S) ={ d(, S) ale r}. An algorithm with local information ses information only within the local neighborhood of the nodes it has chosen. To be secific, if at some ste, the set of nodes that an algorithm has chosen is S, and we hae r-ho local information, then we know the indced grah of N r (S), as well as the degree of nodes in N r (S)/N r (S). From an arbitrary starting node, nodes are added iteratiely. For each loo in this algorithm, one chooses a node, or a node and one of its neighbors. This means we need knowledge of - ho neighborhood to maximize the nmber of newly coered nodes, which exlains why it ses -hos of local information. Algorithm : CDS with -ho Local Information Data: Grah G =(V,E) Reslt: A connected dominating set S s an arbitrary node; S {s}; 3 while S is not a dominating set do 4 S a node in N (S) or a node in N (S) and one of its neighbors in N () (which also lies in N (S)) that maximize the nmber of newly coered nodes; S S [ S; The aroximation ratio for this algorithm is ( + ). We can imroe it in ractice by maximizing the nmber of newly coered nodes for each new node selected, which is sed in [], bt the theoretical worst case bond is the same. C. -ho Local Information Algorithm for CDS Borgs et al. [] first came with this algorithm, which is based on the reios one. Instead of choosing a node or a air of adjacent nodes greedily, it chooses only one gray node to maximize the nmber of newly coered nodes, and in addition selects one of the newly coered nodes niformly at random. The maximization rocess only reqires -ho neighborhood information, and we do not worry abot the random node we are abot to choose. Not only does this algorithm reqire less information, it rns mch faster. The aroximation ratio is nchanged. D. Insiration Comaring the two algorithms sing global information and local information, we find a ga of factor in the aroximation ratio: () + O() for a local algorithm erss + O() for global algorithm. This looks reasonable: we are always better off when offered more information. Bt is it really the case? Is this ga the rice we aid for lack of information essential, or the same aroximation

3 Algorithm : Randomized CDS with -ho Local Information Data: Grah G =(V,E) Reslt: A connected dominating set S s an arbitrary node; S {s}; 3 while S is not a dominating set do 4 a node in N (S) that maximize the nmber of newly coered nodes; a niformly randomly chosen node from N () N (S), i.e. the newly coered nodes; 6 S S [{, }; Algorithm 3: CDS with -ho Local Information Data: Grah G =(V,E) Reslt: A connected dominating set S s an arbitrary node; S {s}; 3 V N (S); 4 while 9 V, s.t. w + c > 0 do argmax V w + c ; 6 S S [{}; 7 V N (N ()); for all nodes in V, date c,w ; ratio can be achieed regardless of limited information. Or answer is affirmatie: with local information, we can still get +O() aroximation. To be secific, an H( +)+ aroximation. III. IMPROVED -HOP LOCAL INFORMATION ALGORITHM A. Algorithm S is initially any node. Iteratiely add nodes within N (S) to S. When new nodes are added to S, the -ho neighborhood of S exands, and we reeat this rocess. By the end of the rocess, we hae colored all the nodes. Initially, all nodes are white. When a node is added to S, we color it black, and all its white neighbors gray. The selected black nodes will form comonents, and these comonents may merge when additional nodes are selected. At each ste, we look within a -ho neighborhood of S, i.e., among the nodes that are either gray or adjacent to gray nodes. Add the node that maximizes w +c, where c is the nmber of different black comonents connected to, and w is the nmber of white nodes in N (). Or knowledge of the grah is enriched when we add nodes to or soltion. The algorithm ends when there is no sch that w +c > 0. Note that if was gray before the selection, c is garanteed to be greater or eqal to. If it was white, c =0. Fig.. Consider and as otential choices. For node, c =, as there is only adjacent comonent. Note that w =4, it has for white neighbors, and itself is not white, hence w + c =. For node, since it is white, there is no adjacent black comonent, so c =0. Note w eqals 3, since both node itself and two of its neighbors are white. So the ale is w + c =6+0 =. For comleteness, here is the sedo code. ) Correctness: First we roe that what we get is a dominating set. If not, then there exists a node that is not dominated, i.e. a white node. Bt w +c =w +0 > 0, we wold hae added this node to or soltion, a contradiction. Next, we roe the following theorem that this dominating set is indeed connected. Theorem 3: The soltion retrned by Algorithm 3 is connected. We hae the following obseration: Obseration 4: When a node is added to or soltion, it is within -hos of some black node. It is tre becase otherwise, this node wold be beyond or knowledge and wold not be considered at this ste. This ensres the following corollary: Corollary : Each newly added node can be connected to existing comonents at the cost of at most one additional node. Proof: [Proof of Theorem 3] We roe this by contradiction. Sose the algorithm gies a dominating set that is not connected, i.e. has more than one comonent. Exactly one of these comonents will contain the starting node, call it starting comonent. Consider the first time when a node otside the starting comonent joins or soltion. According to Corollary, there mst exist another node that can join to the starting comonent. Since is disconnected from the starting comonent, is not in or soltion. Bt at the end of the algorithm, w +c 0+ > 0, which means that can be added to or algorithm. This gies a contradiction. So or algorithm will indeed gie a connected dominating set. B. Analysis We are going to se a charging scheme to charge the cost of each selected node and bond the total charge, which in trn bonds the nmber of nodes we select. Before that, we define an ncharged black node as: Definition 6: An ncharged black node is a node that was charged once when it trned directly from white to black, and has not been charged afterwards. For ncharged black nodes, the following lemma holds. Lemma 7: Each comonent contains exactly one ncharged black node. The roof of this lemma will come later. Recall in the algorithm, we select a node that maximizes w + c, where w is the nmber of white nodes in

4 N (), and c is the nmber of black comonents adjacent to. We charge for selecting this node, and slit this charge into w + c shares. For eery white neighbor of, it takes two shares, i.e. gets /(w + c ) charge. For all bt one adjacent comonents, a share of charge is laced on the ncharged black node of this comonent (assming the correctness of Lemma 7). Notice may be white or gray before the selection. If it was white, it means that is not adjacent to any black nodes, so c =0, and it is charged one share, i.e. /(w +c ). If it was gray, then nothing needs to be done. In conclsion, if the node was white, the nmber of shares eqals (w )+0+ = w +c ; if the node was gray, the nmber of shares eqals w +(c ) = w +c. This means we are not oer or nder conting charges. A isal exlanation is in Figre. Fig.. Consider the charging for and if they were selected at this ste. For, w + c =, each white neighbor of gets shares. For node, w + c =6+0 =. Each white neighbor gets shares, bt since node goes from white directly to black, it only gets one share, and become an ncharged black node. We now roe the correctness of Lemma 7. Proof: [Proof of Lemma 7] An ncharged black node comes into existence when a white node is chosen and added to or soltion. According to the charging scheme, this node itself forms a comonent, and it has exactly one ncharged black node. Comonents are connected when a gray node is chosen which connects seeral comonents. Since all bt one comonent was charged, assming all existing comonents hae exactly one ncharged black node, the reslting comonent also has exactly one ncharged black node. A isal exlanation is in Figre 3. w Fig. 3. Sose the black nodes are already chosen, and we are adding node to the soltion. There are two black comonents adjacent to, so c =. Ths w +c =4+ =. Each white neighbor of gets shares. For the two comonents, all bt one comonent will get a share. This share is charged against the ncharged black node (node w for the left comonent and node for the right comonent) of the comonent, which may not be the node adjacent to. After charging, the two comonents are merged, and the merged comonent still has exactly one ncharged black node, i.e. node. Note there is no charge against node. Eerything reared, we state the main theorem and roe it by bonding the total charge. Theorem (Main Theorem for -ho Local Information Algorithm): The imroed -ho local information greedy algorithm gies H( + ) + aroximation. Proof: [Proof of Theorem ] Sose the otimal soltion is OPT, OPT = k, with ertices OPT, OPT,..., OPTk. We artition all nodes into S OPT, S OPT,...,S OPTk. Withot loss of generality, we reorder the nodes, and se i to denote ertex OPTi. So S i contains ertex i and its neighbors. Ties are broken arbitrarily as long as i S i. Withot loss of generality, we assme that the charge in S i changes at each ste. To describe the total nmber of charges that nodes in S i can receie, we define i as the following ale for S i in ste: w j i + bj i where w j i is the nmber of white nodes in S i at ste, b j i is the nmber of ncharged black nodes at ste in S i. A symbol (e.g. w i,b i ) with an er scrit j, i.e. w j i,bj i,cj i, denotes the corresonding ale at ste. Then i = (i)+ ( (i) is the degree of node i). A white node can receie two charges (in fact all white nodes will receie two charges, excet for one, which is the only ncharged black node when the algorithm ends). Either it will receie two charges when it first becomes gray, and neer receie any more charge. Or it receies one charge when it becomes a ncharged black node, and receies another one to become a charged black node. We hae the following lemma: Lemma 9: When the total charge inside S i changes, i decreases by at least one. Proof: [Proof of Lemma 9] Total charge changes when at least one node in S i receies charges. There can be three cases: this node was white, gray, or black. If this node was white before charging, either it is now gray, which means it receies two charges, or it is now black, meaning one charge. Therefore this node will case i to decrease by or (or more since there can be more than one sch node, i may decreases more than that). If it was gray, it mst hae been flly charged and will not receie any more charge. If it was black, it mst hae been an ncharged black node to receies one charge. In this case, i also decrease one. For all three cases, either there is no charge change in S i, or there is a decrease on i of at least. We se instead of i when there is no confsion. Notice b j i ale cj i, since eery ncharged black node corresonds to a comonent, bt the ncharged black node may not be in S i. Consider each ste. Sose the selected node is. Since node i is also a alid choice, we hae w j + c j w j i + cj i w j i + bj i = The nmber of charges that S i receies at ste is +, so the total charge in this ste is: + w j + c j ale j +

5 This holds when w j i + cj i > 0, which is garanteed to be tre when i > 0. The ineqality will break down when w j =0and c j =, i.e. when w j i + cj i =0. To fix it, we notice S i, 9k i,s.t. ki i > 0, t >k i, t i ale 0. Ths we can take ot the last term and bond it searately. So the total charge ntil ste k i (inclding ste k i ) is er bonded by: k i j= k i = + j= t=+ + k i ale j= t=+ + 0 k i j= j= t=+ + 0 t t + max{ k i + +,} = j + =H( )+(0 k i + + ki+ i ) max{ k i + +,} k i + + k i t=max{ k i + +,} + A + t k i t=max{ k i + +,} max{ k i + +,} k i + + where H(n) = P n i= i is harmonic sm. The last eqality ses the fact that w + c. Using ki+ i ale 0, t i, combined with or assmtion that i decreases at each ste, there is at most one more ste before the whole algorithm stos. So the total amont of charge is er bonded by: ( ki+ i Adding together, we hae, H( )+(0 ki+ i ) ale ( ki+ i + ) ki+ i )+( ki+ i + ) = H( )+ As =w i +b i =w i ale +, the total charge in S i is bonded by H( +)+. Since there are OPT different S i, the total charge is bonded by OPT (H( + ) + ), which is also the er bond for the nmber of nodes we choose. IV. IMPROVED -HOP LOCAL INFORMATION ALGORITHM A. Intition Recall that the algorithm by Borgs et al. [], which selects a random neighbor when a gray node is chosen. It is too exensie to select a random node eery time. If instead of icking a random neighbor eery time, we can ick it with some robability. The total aroximation ratio will be (this is only an intition, detailed roof is in Section IV-C) ( + )( + ) that can be minimized when =. This works gien the assmtion that which is not always the case. is known before hand, B. Algorithm Instead of sing the largest degree in the grah, eery time when calclating, we se the largest degree in the exlored grah. In another world, the largest degree in N (S). Below is the sedocode. Algorithm 4: Imroed Algorithm for CDS with -ho Local Information Data: Grah G =(V,E) Reslt: A connected dominating set S s an arbitrary node; S {s}; 3 while S is not a dominating set do 4 A d the largest degree in N (S); t ; 6 a node in N (S) that maximizes the nmber of newly coered nodes; 7 S S [{}; if with robability then 9 a node from the newly coered nodes niformly at random; 0 S S [{}; C. Analysis Like the reios analysis, we do charging, and artition all the nodes into S,S,...,S OPT in the same manner. We bond the total charge inside S i, and the total nmber of nodes chosen in or soltion that corresonds to these charges. Theorem 0 (Main Theorem for -ho Local Information Algorithm): The imroed -ho local information greedy algorithm gies + + aroximation. Proof: [Proof of Theorem 0] When a node is selected in line 6 in Algorithm 4, we charge, and the charge is niformly diided among all the newly coered nodes. In line 9, another node is chosen with robability =, where d is the largest degree we crrently know. Ths wheneer S i receies charge c at some ste, the exected nmber of nodes in or soltion increases by c( + ). Note that this changes oer time. The analysis for each S i is diided into two hases. The first hase ends when one of the nodes in S i gets chosen. In other words, we come to the second hase when i is within the - ho neighborhood of some chosen node. We first consider the first hase. Let j be a random ariable indicating whether a node in S i is chosen at ste ( ale j ale n). If we denote the total charge S i receied at this ste as, and the corresonding robability of choosing another random neighbor as 0 j, then E[ j] = ( + 0 j ). The first hase will end at ste if j is the first to be. Ths we need to define a stoing time T to bond the total increase in exected nmber of nodes. Definition : Let T be the random ariable denoting the smallest j sch that j =(or n if j =0for all j).

6 With the stoing time T in hand, we can write ot the total increase h in exected nmber of nodes in the first hase PT as E T j= ( + 0 j i, ) which is bonded by the following lemma, Lemma : T E T [ ( + 0 j)] ale + j= Proof: [Proof] We roe it by indction on n. If n =, the left side is + 0 ale + = + H() < +, which is triial. Sose it holds for n = k, we roe it holds for n = k 3 T 4 ( + 0 j) E T j= 3 T = ( + 0 )+Pr[ = 0]E T 4 ( + 0 j) = j= 3 T = ( + 0 )+( 0 )E T 4 ( + 0 j) = j= ale ( + 0 )+( 0 ) + = ale+ + = + + = + The first ineqality is alying indction hyothesis on,, n, and the second ineqality ses the fact that 0 j = for some node, and () ale. So 0 H( ()) j. As for the second hase, wheneer S i receies charge c, the exected nmber of nodes increases by c( + ), =, where d is the largest degree we crrently know. Since d i = ( i ) is already known, we hae d d i. So = ale H(di), which imlies that the increase in exected nmber of nodes in S i is bonded by c. H(di) The total charge is bonded by H(d i ). This can be done sing the standard techniqe in set coer roof [3]. Ths the total nmber of nodes chosen in hase is bonded by H(d i )(+) aleh(d i ) + H(di )! This directly means that exected the size of soltion is bonded by OPT ( + + ), imlying + + aroximation. V. FUTURE WORK With only local information, we get the same aroximation ratio as when we hae global information, for connected dominating set roblem. Is it also the case for other roblems? Or does the lack of information roe to be a hge obstacle for designing algorithms? Or first algorithm reqires information within -hos. When only -ho local information is aailable, we cannot get the same reslt. Comared with reios reslt, the seed and aroximation ratio is imroed, i.e. (+ +). Bt a ga still ersists. Can we do better? Or is this the rice we ay for lack of information? REFERENCES [] Konstantin Arachenko, Prithwish Bas, Gioanni Neglia, Bernardete Ribeiro, and Don Towsley. Pay few, inflence most: Online myoic network coering. In Comter Commnications Workshos (INFOCOM WKSHPS), 04 IEEE Conference on, ages 3. IEEE, 04. [] Christian Borgs, Michael Bratbar, Jennifer Chayes, Sanjee Khanna, and Brendan Lcier. The ower of local information in social networks. In Internet and Network Economics, ages Sringer, 0. [3] Thomas H Cormen. Introdction to algorithms. MIT ress, 009. [4] Ding-Zh D and Peng-Jn Wan. Connected dominating set: theory and alications, olme 77. Sringer Science & Bsiness Media, 0. [] Dedatt Dbhashi, Alessandro Mei, Alessandro Panconesi, Jaikmar Radhakrishnan, and Araind Sriniasan. Fast distribted algorithms for (weakly) connected dominating sets and linear-size skeletons. In Proceedings of the forteenth annal ACM-SIAM symosim on Discrete algorithms, ages Society for Indstrial and Alied Mathematics, 003. [6] Uriel Feige. A threshold of ln n for aroximating set coer. Jornal of the ACM (JACM), 4(4):634 6, 99. [7] Sdito Gha and Samir Khller. Aroximation algorithms for connected dominating sets. Algorithmica, 0(4):374 37, 99. [] Sdito Gha and Samir Khller. Imroed methods for aroximating node weighted steiner trees and connected dominating sets. Information and comtation, 0():7 74, 999. [9] Samir Khller, Manish Prohit, and Kanthi K Saratwar. Analyzing the otimal neighborhood: algorithms for bdgeted and artial connected dominating set roblems. In Proceedings of the Twenty-Fifth Annal ACM-SIAM Symosim on Discrete Algorithms, ages SIAM, 04. [0] Yzhen Li and Weifa Liang. Aroximate coerage in wireless sensor networks. In Local Comter Networks, th Anniersary. The IEEE Conference on, ages 6 7. IEEE, 00. [] Adish Singla, Eric Horitz, Pshmeet Kohli, Ryen White, and Andreas Krase. Information gathering in networks ia actie exloration. In Proceedings of the 4th International Conference on Artificial Intelligence, ages 9 9. AAAI Press, 0. =H(d i )+ H(d i) H(di ) =H(d i )+ H(d i ) ale + Combining the charge from both hases, the exected nmber of nodes chosen in S i is bonded by + +.