Maximal Cliques in Unit Disk Graphs: Polynomial Approximation

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1 Maximal Cliqes in Unit Disk Graphs: Polynomial Approximation Rajarshi Gpta, Jean Walrand, Oliier Goldschmidt 2 Department of Electrical Engineering and Compter Science Uniersity of California, Berkeley, CA 94720, USA {gptar, wlr}@eecs.berkeley.ed 2 OPNET Technologies Inc, 2006 Delaware Street Berkeley, CA 94709, USA ogoldschmidt@opnet.com Abstract We consider the problem of generating all maximal cliqes in an nit disk graph. General algorithms to find all maximal cliqes are exponential, so we rely on a polynomial approximation. Or algorithm makes se of certain key geographic strctres of these graphs. For each edge, we limit the set of ertices that may form cliqes with this as the longest edge. We then consider seeral characteristic shapes determined by that edge, and proe that all cliqes haing this as the longest edge, are inclded in one of the sets of ertices contained in these shapes. Or algorithm works in O(m 2 ) time and generates O(m ) cliqes, where m is the nmber of edges in the graph and is its maximm degree. We also proide a modified ersion of the algorithm which improes the performance in many cases, albeit withot affecting the worst case rnning time. Keywords: Cliqes, Unit Disk Graphs, Ad-Hoc Networks. Introdction We address the problem of finding all maximal cliqes in an Unit Disk Graph (UDG) []. An nit disk graph G = (V, E) is defined on a finite set of points V in the plane. Two ertices, V are connected by an edge E if and only if their Eclidean distance is less than or eqal to. An indced sbgraph in G that is a complete graph is called a cliqe. A maximal cliqe of G is one that it is not contained in any other cliqe. UDGs hae recently attracted a lot of importance de to the adent of wireless ad-hoc networks. Modelling of ad-hoc networks reqires the nderstanding of interference between neighboring nodes and links. Two nodes/links are often modelled as interfering when they lie within an interference range of each other, making the nderlying graph a UDG [2]. UDGs also appear in channel assignment problems in broadcast and celllar networks (e.g. []). Cliqes in sch networks are of importance since only a single ertex in a cliqe may be actie at once leading to considerations of capacity and qality of serice. In this paper, we attempt to find all maximal cliqes in a UDG, in polynomial time. We begin by describing in Sec. 2 the backgrond of the problem, and the related work in the field. Or approximation algorithm and its rationale is presented in Sec. 3. We complete the paper with a discssion of simlation reslts in Sec. 4, and a conclsion in Sec Backgrond and Related Work We cite a sampling of papers in the ad-hoc networking arena that tilize the concept of cliqes for schedling, roting and QoS prposes. In [3], the athor ses these ideas for optimizing traffic flows, while the athors of [2] and [4] se cliqes to derie necessary and sfficient bonds on the maximm capacity of an ad-hoc network. In [5], the athors propose a cliqe-based pricing approach to optimize resorce allocation in an ad-hoc network. All these ideas reqire the comptation of maximal cliqes, preferably in a distribted manner. Cliqes hae been stdied in great detail in the area of graph theory. Algorithms to generate maximal cliqes from a graph were first introdced by Harary and Ross [6] in 957. Dring the 960s and 970s, the Bierstone Algorithm was deeloped [7, 8], and was frther refined in [9]. All of these algorithms work on a general graph and otpt maximal cliqes. The nmber of maximal cliqes in a graph is typically exponential, as sch algorithms are conseqently (e.g. [0]). While the aboe algorithms also work for UDGs, or specific applications often reqire a qicker approximation approach.

2 It is tempting to hope that the special strctre of UDG s caps the nmber of maximal cliqes in a UDG. In fact, the nmber of cliqes cold be exponential in n, the nmber of ertices in the graph, as we hae shown. Theorem [] The total nmber of cliqes in a UDG grows exponentially with n in the worst case. Proof We illstrate this by the following example. Assme n = 2p. Draw a circle of diameter + ɛ, where 0 < ɛ <<. Place the 2p nodes niformly on the edge of the circle and label them clockwise from to 2p. If ɛ is small enogh, we hae edges from any ertex i to all other ertices except the diametrically opposite ertex (i + p) mod 2p. Ths we hae constrcted a complete p-partite graph with its ertex set being a pair of diametrically opposite nodes i.e. {i, i + p}, i =,... p. The selection of one ertex from each of the p sets will form a maximal cliqe. Clearly we hae a total of 2 p = 2 n/2 maximal cliqes. This pessimistic reslt led s to look at sitable approximations, that wold enable a polynomial-time algorithm. The essence of the approximation is to se slightly sper-maximal cliqes. When the nmber of cliqes grows large, the approximation algorithm generates the nion of seeral nearby cliqes as a single sper-maximal cliqe. In [2], we hae earlier presented an approximation algorithm to find all maximal cliqes in ad-hoc networks. We recognized two key featres abot the geographic natre of ad-hoc networks. First, two nodes that are part of a cliqe mst be within an interference range ω = of each other. Second, if a grop of nodes form a cliqe, then the maximm distance between any pair of them mst be. The heristic approximation ses a small disk of diameter (i.e. radis = /2) to scan a larger disk of radis arond a node. Alternatiely, the scanning disk is sed to scan the entire region in which the nodes are placed. Each position of the scanning disk generates a cliqe. The generated set of cliqes is then shrnk to reslt in the approximate set of maximal cliqes arond the node. There are howeer a few problems with the scanning disk approach presented in [2]. The foremost of which is that the rnning time of the algorithm depends on the step size and the size of the field. Frther, the scanning disk of diameter fails to catch all cliqes, e.g., three nodes located at the corners of an eqilateral triangle of side. We wold like to address these isses as we moe beyond the realm of ad-hoc networks, to more generic UDGs. 3 Algorithm We order the edges in G in decreasing order of length. Then, for each edge, we find all maximal cliqes with as the longest edge. This method will generate all maximal cliqes in G, together with some extra cliqes that may be sbsets of other larger maximal cliqes. The set of cliqes generated may be processed at a later stage to prne the non-maximal cliqes. 3. Important Shapes in UDG We begin by obsering seeral important geometric strctres determined by an edge, and their characteristics w.r.t. maximal cliqes in UDGs. We consider three shapes in particlar we call these the football, the disk and the cred triangle. 3.. Football Gien an edge, let d be the Eclidean distance between and. Obiosly, d, since or graph is an UDG. We draw two circles of radis d centered at and. Denote the set of ertices in the football-shaped (American Football, not Soccer) intersection of the two circles as F, shown in Fig. (a). Then we proe the following. Theorem 2 [] A cliqe whose maximm edge is mst be contained in F. Proof 2 Sppose the cliqe with as the longest edge contains a ertex w V \ F. Then, either d w > d or d w > d. In either case, is not the longest edge. (Contradiction) Howeer, F in itself is not a maximal cliqe as defined. An example is shown in Fig. (a), where nodes i, j F. Yet d ij > d and hence not part of a maximal cliqe with as the longest edge. In fact, we cold een hae d ij >, in which case, ij / E. Ths F is a sperset of the reqisite maximal cliqes. By simple geometry, the height of the football F is d 3. A special sitation occrs when / 3. In this case, the height of the football is ; so eery ertex in F is connected to each other. Then there is in fact a single cliqe F, which incldes all maximal cliqes with as the longest edge.

3 t T F y k i Figre (b) x > <= <= j D Figre (c) Figre (a) Figre (d) Figre : Characteristic Shapes in UDG: (a) Football F (b) Cred Top Triangle T (c) Proof that T is a cliqe (d) Conter-example showing T and T 2 not enogh to coer all cliqes 3..2 Disk The next shape of interest is the disk D which is the set of all ertices in the circle with as its diameter, also shown in Fig. (a). Clearly, the maximm distance between any two nodes in D is d. Ths D is a cliqe with as its longest edge. Note howeer that D may not always be maximal, e.g., in Fig. (a), D = {,, k}, yet the cliqe {,, k, i} contains D Cred Triangle Finally, we look at the cred triangle formed by,, and t the pper intersection point of the two circles of radis d centered at and, as denoted in Fig. (b). The three points form an eqilateral triangle. We expand this area by drawing cres with radis d from to centered at t and so on. All ertices in the reslting top cred triangle is denoted by T. A similar bottom cred triangle T 2 may also be drawn. The ertices in these form cliqes, as we proe below. Theorem 3 The ertices in T or T 2 form a cliqe with as the longest edge. Proof 3 We show that any two ertices in T are separated by a distance d. Then, T forms a cliqe with as the longest edge. Sppose x, y T and d xy > d. We choose two points ot of the eqidistant points,, t sch that both x and y lie on the same side as the edge connecting these two points. W alog, we choose and. The reslting qadrilateral formed by joining them to x and y is shown in Fig. (c). Note that all edges except xy are d. Ths we hae a contradiction. In x, x, x. Then x x xy. () Bt in xy, xy > x, y. Then xy > xy x. (2) Howeer, the cred triangle also does not necessarily coer all cliqes. For instance, the for ertices in a cross shown in Fig. (d) form a cliqe, bt can not be coered by any cred triangle as defined. To smmarize, we hae otlined three characteristic shapes determined by the edge : Football F is a sperset of all maximal cliqes with as the longest edge. Disk D is a cliqe with as the longest edge, bt may not be maximal. Cred Triangles T and T 2 are cliqes with as the longest edge. Bt these too may not be maximal. If eery node in F is contained in any of D, T or T 2, then we hae fond the single maximal cliqe with as the longest edge. Since that may not always happen, we need an alternatie method.

4 k i j F Band B y > B2 z y l x Figre (a) B Figre (b) x B Figre (c) Figre 2: Moing band in football: (a) Band B in Football (b) B may inclde some extra ertices (c) Positioning bands with bondaries on ertices in F 3.2 Moing Band in Football Consider the football shaped area as before, and look at a band of height d within this, as shown in Fig. 2(a). Let B be the set of nodes within the band. Since the height of the band is greater than half of the height d 3 of the football, any sch band always contains and. As we know from Theorem 2, all maximal cliqes with as the longest edge mst lie in F. Then we hae the following theorem. Theorem 4 Any cliqe with as the longest edge mst lie in some band B within football F. Proof 4 Any ertex x F is d from both and. So the only way we cold iolate the theorem is by haing two ertices x, y F that form a cliqe with and, bt do not lie in any band. Bt then d xy > d since the height of the band itself is d. (Contradiction) Ths we will coer eery possible maximal cliqe with as the longest edge, if we select all sch bands B. Note that the band B may inclde some extra ertices that are not part of the maximal cliqe as defined. As shown in Fig. 2(b), ertices x and y are inclded in the band, yet their distance is greater than d. When the ertices in F are not all inclded in either of D, T or T 2, we need to se the bands to generate the maximal cliqes. The ertices in F can be diided into two disjoint grops those to the north of and those to the soth (assign bondary nodes to any one). For each ertex x F, depending on whether it is to the north (or soth) of, we position a band sch that its north (or soth) bondary lies on x. If the distance of x from is less than 3 2, we simply position a band at its sothernmost (or northernmost) position. There are at most sch bands, since or has no more than neighbors. Fig. 2(c) shows two sch bands B and B2. In this case we hae the following theorem: Theorem 5 Choose a set of bands B i, i =,..., corresponding to each ertex in F, as described aboe. Then, eery maximal cliqe with as the longest edge, is contained in some B i in this set. Proof 5 Consider any maximal cliqe q F. Amongst all its ertices, let x be the ertex farthest from the line. Then, some band B j is positioned with x on one of its bondaries, and also contains and. Since x is the farthest ertex in the cliqe, all other ertices mst lie on the same side of x as. Bt they are all at a distance d from x. So all the other members mst lie within the band, whose width is d. Hence q B j. 3.3 Basic Algorithm findallmaximalcliqesinudg:. order edges in decreasing order of length; 2. for each edge 3. if d / 3 4. otpt maximal cliqe F ; 5. else 6. otpt three maximal cliqes: D, T, and T 2 ;

5 7. if F = D or F = T or F = T 2 8. we are done; 9. else 0. for each ertex x F. Position band B with bondary on x; 2. Vertices in band form cliqe; 3.4 Complexity We assme that the description of V also contains the geometric locations of the ertices. Let m be the nmber of edges, and be the maximm degree of the graph. Then, step 2 is ealated m times. There may be at most ertices in F, so step 0 is ealated times. Finally, in order to generate a cliqe, we need to look at eery neighbor of and, and check if they lie within the prescribed region. This may be achieed in at most 2 operations. Ths the oerall algorithm is O(m..2 ) = O(m 2 ). For each link, the algorithm generates p to cliqes. Ths, the nmber of cliqes generated is O(m ). 3.5 Modified Algorithm Each cliqe fond in Sec. 3.3 is maximal with as its longest edge. Howeer, recall that or motiation is in finding maximal cliqes in the entire graph; so any generated cliqe that is a sbset of some other cliqe is redndant. Sppose that instead of taking the disk D with diameter d, we consider a potentially larger disk of diameter, lying on (both disks hae the same center). Let D be the set of ertices in this larger disk, as shown in Fig. 3(a). D may inclde edges longer than, bt these edges are. So D is a maximal cliqe in G, and D D. Ths we in fact do better by looking at D instead of D. Similarly, ertices in the pper cred triangle T as shown in Fig. 3(b) (as also the lower cred triangle T 2 ), with sides, also forms a maximal cliqe. T shares the same ppermost point with T, and so T T. We then hae the following theorem. Theorem 6 If d 3, eery band B is contained in either F T or F T 2. Proof 6 We note that T and T 2 oerlap, as shown in Fig. 3(c). For a small enogh d, the oerlap is sch that any band B is wholly contained in either T or T 2. This happens when d d 3 2 ( 3 + ) d = 3 (3) Also, B F. Hence for d 3, eery possible B is contained in either F T or F T 2. Bt T and T 2 are both maximal cliqes. Hence it is enogh to consider only T and T 2 instead of looking at all the bands. We se this knowledge to modify the algorithm. findallmaximalcliqesinudgmodified:. order edges in decreasing order of length; 2. for each edge 3. if d / 3 4. otpt maximal cliqe F ; 5. else 6. otpt three maximal cliqes: 7. if d 3 F D, F T, and F T 2 ; 8. we are done; 9. else if no ertex in F \ D or F \ T or F \ T 2 0. we are done;. else 2. for each ertex x F 3. Position band B with bondary on x; 4. Vertices in band form cliqe;

6 D T T D oerlap T2 Figre (a) Figre (b) Figre (c) Figre 3: Modified algorithm coering more nodes in maximal cliqe: (a) Larger band D (b) Larger cred triangle T (c) Oerlapping T and T 2 when d is small The modified algorithm has the same rnning time O(m 2 ) in the worst case as the basic algorithm (Sec. 3.4), and generates the same order of cliqes. Howeer, we are more likely to aoid the scanning band in steps 2-4 by checking in steps 7-0 if we hae in fact already generated all the maximal cliqes we need. 3.6 Analyzing Extra Nodes Coered by Band In a general graph, it is difficlt to assess how many extra nodes the band captres in its cliqes. Howeer, it is interesting to analyze the effect in a niform random graph. In sch a case, the nmber of ertices captred by each coering shape is proportional to its area. We hae calclated the areas coered by each of the shapes that we hae sed. Note that the band has ariable area it is largest when at the middle of the football, and smallest when it is at the extremities. Shape Football Disk Cred Triangle Band (max) Band (min) Area [ 2π ]d2 =.228d Handling Changes in the Network π 4 d2 = 0.785d 2 [ π ]d2 = 0.705d 2.008d d 2 It is important to analyze the ealation of the algorithm when changes occr in the network. We wold like the reaction to these changes to be limited to the locality of the change. We identify fie types of changes New Vertex When a new ertex is added to the network, we need to ealate cliqes at all the new edges. This takes O(. 2 ) = O( 3 ), since there may be p to edges. The new ertex may add to any of the cliqes inoling its neighbors; so we need to re-ealate cliqes in the neighborhood of the new ertex. With nodes in the neighborhood, there may be p to 2 edges hence the oerall algorithm is O( 2. 2 ) = O( 4 ) Delete Vertex Let the maximal cliqes generated be stored in a q n cliqe-node incidence matrix, where q is the nmber of cliqes. From Sec. 3.4, we know that q is O(m ). To delete a ertex i, it sffices to delete the i th colmn from the matrix. This may be done in O(m ) New Edge We need to re-ealate cliqes in the neighborhood of both ertices forming the new edge. There may be p to 2 2 edges in the neighborhood, so the oerall algorithm is O( ) = O( 4 ).

7 2 Bands Aerage nmber of Cliqes Cliqes/Edge Cliqes/Edge(Modified) maxdegree/ Node Density (a) Nmber of cliqes generated by the approximation algorithms Fraction of edges for each cliqe generation method D, T and T2 d < /sqrt(3) Bands D, T and T2 d < sqrt(3) d < /sqrt(3) Node Density (b) Comparing the two ersions of the algorithm. Left bar = Basic Algorithm. Right bar = Modified Algorithm Figre 4: Approximating Maximal Cliqes in a Random UDG Delete Edge Again, we need to re-ealate cliqes at both affected ertices. As aboe, it takes O( 4 ) Moe Vertex If no new edges are formed or lost, we need not take any steps. Howeer, a change in the network needs to be handled as a Delete Vertex (Sec ), followed by a New Vertex (Sec. 3.7.). The cmlatie algorithm is O( 4 ). For all of the modification operations described aboe, note that O( 2 ) is no greater than O(m), so O( 4 ) is at least as good as O(m 2 ). Typically << m, so O( 4 ) presents a sbstantial gain oer O(m 2 ). 3.8 Distribted Algorithm The maximal cliqe algorithms presented in Sec. 3.3, 3.5, and 3.7 rely only on localized information. For instance, in an ad-hoc network, we wold only reqire each node to know abot location and topology in its 2-hop neighborhood. In a practical scenario, the algorithms may ths be implemented in a distribted fashion with each node compting all maximal cliqes arond it. This wold take O( 2 ) time. 4 Simlation Reslts We ealate the performance of or algorithms in a UDG formed by nodes distribted randomly in the plane. We consider a field of dimension 0 0 and niformly distribte between 00 to 2000 nodes on it, yielding a node density ranging from to 20 per nit sqare. We then execte the algorithms (both basic and modified) on the generated topologies, the reslts of which are presented in Fig. 4. Note that each data point on the plot is an aerage oer 0 separate randomly generated topologies, haing the same node density. Fig. 4(a) shows the nmber of cliqes per edge, as the node density increases. We also plot the ale of the maximm degree on the same graph, sing the dotted line (note that this ale is scaled down by a factor of 0, to improe isibility). In a random UDG, increases linearly with node density. Recall that the nmber of cliqes generated by or approximation algorithm is O(m ), ths the nmber of cliqes per edge is O( ). As expected, cliqes/edge shows the same shape as. Howeer, the optimizations achieed by sing the disks and the cred triangles ensre that the actal nmber of cliqes is significantly less. In fact, the actal cliqes/edge generated by the basic algorithm is only arond /8. The modified algorithm presented in Sec. 3.5 proides a frther redction in the nmber of cliqes, as shown in the figre. Fig. 4(b) compares the detailed workings of the two flaors of the algorithm. Using a bar graph, we look at the fraction of edges that are ealated sing each block in the logic of the algorithms, as gien in sections 3.3 and 3.5. We distingish

8 these as d < / 3, d < 3, D, T, T 2, and Bands. For each ale of node density, the left bar denotes the basic algorithm and the right bar denotes the modified algorithm. Consider the left bar for each node density, docmenting the fnctioning of the basic algorithm. In all the simlations, /3 of the edges (lowest bar, colored cyan) directly yield a single maximal cliqe F, when d < / 3. This is to be expected, since the probability of this is gien by ( / 3 ) 2 = /3. Seeral more edges are handled by one of the special cliqes D, T or T 2, as indicated by the red bar in the middle. The fraction of edges where we hae to resort to the Bands, is shown by the black bar at the ery top. As seen from the figre, ery few edges reqire this method when the node density is small, bt the fraction of edges increases with node density. At the maximm node density of 20, nearly 65% of the edges reqire the comptation of the Bands. The modified ersion of the algorithm introdces another case, whereby the sets T and T 2 sffice when d < 3. This is denoted by the second bar from the bottom (colored green) for the modified algorithm. This cases a redction in the fraction of edges reqiring the ealation of Bands (down to abot 40% when the node density is 20), and therefore a redction in the total nmber of cliqes. 5 Conclsions Wireless and ad-hoc networks are often modelled as nit disk graphs, and cliqe strctres in them are sed in seeral applications. We consider the problem of generating all maximal cliqes in a UDG. General algorithms to find cliqes in a graph are exponential, so we rely on a polynomial approximation. We consider each edge, and find all maximal cliqes with this as the longest edge. Or algorithm works by making certain key obserations abot the geometric strctre of these graphs. We first limit the possible cliqe-forming ertices into an area shaped like a football. Then we se two other shapes the disk and the cred triangle which we proe to generate cliqes. If all cliqes hae not yet been fond, we se a band shape to scan the football to generate all cliqes. Or algorithm works in O(m 2 ) time and generates O(m ) cliqes. References [] A. Graf, M. Stmpf, and G. Weisenfels, On Coloring Unit Disk Graphs, Algorithmica, ol. 20 (998), pp [2] R. Gpta, J. Msacchio, and J. Walrand, Sfficient Rate Constraints for QoS Flows in Ad-Hoc Networks, UCB/ERL Technical Menorandm M04/42, Fall [3] A. Pri, Optimizing Traffic Flow in Fixed Wireless Networks, Proc. WCNC [4] K. Jain, J. Padhye, V. N. Padmanabhan, and L. Qi, Impact of Interference on Mlti-hop Wireless Network Performance, ACM Mobicom 2003, San Diego, CA, USA, September [5] Y. Xe, B. Li, and K. Nahrstedt, Price-based Resorce Allocation in Wireless Ad-Hoc Networks, in Proc. IWQoS 2003, Monterey, California, Jne [6] F. Harary, and I. C. Ross, A Procedre for Cliqe Detection Using the Grop Matrix, Sociometry, ol. 20, pp , 957. [7] E. Bierstone, Cliqes and Generalized Cliqes in a Finite Linear Graph, Unpblished Report, 960s. [8] J. G. Agstson, and J. Minker, An Analysis of Some Graph Theoretical Clster Techniqes, Jornal of the ACM (JACM), ol. 7, no. 4, pp , October 970. [9] C. Bron and J. Kerbosch, Finding All Cliqes in an Undirected Graph, Commnications of the ACM, ol. 6, pp , 973. [0] S. Tskiyama, M. Ide, H. Ariyoshi, and I. Shirakawa, A New Algorithm for Generating all the Maximal Independent Sets, SIAM Jornal of Compting, ol. 6, pp , 977. [] G. Y, O. Goldschmidt, and H. Chen, Cliqe, Independent Set, and Vertex Coer in Geometric Graphs, npblished report, 992. [2] R. Gpta and J. Walrand, Approximating Maximal Cliqes in Ad-Hoc Networks, Proc. PIMRC 2004, Barcelona, Spain, September 2004.