Generalized Edge Coloring for Channel Assignment in Wireless Networks

Transcription

1 TR-IIS Generalize Ege Coloring for Channel Assignment in Wireless Networks Chun-Chen Hsu, Pangfeng Liu, Da-Wei Wang, Jan-Jan Wu December 2005 Technical Report No. TR-IIS

2 Generalize Ege Coloring for Channel Assignment in Wireless Networks Chun-Chen Hsu Institute of Information Science Acaemia Sinica Taipei, Taiwan Da-Wei Wang Jan-Jan Wu Institute of Information Science Acaemia Sinica Taipei, Taiwan Pangfeng Liu Department of Computer Science an Information Engineering National Taiwan University Taipei, Taiwan Abstract This paper introuces a new graph theory problem calle generalize ege coloring (g.e.c). g.e.c is similar to traitional ege coloring, with the ifference that a vertex can be ajacent to up to k eges that share the same color. g.e.c. can be use to formulate the channel assignment problem in multi-channel multi-interface wireless networks. We provie theoretical analysis for this problem. Our theoretical finings can be useful for system evelopers of wireless networks. We show that when k = 3, there is no optimal solution. When k =2anthemaximum egree of the graph is no more than 4, or is a power of 2, we erive optimal algorithms to fin a g.e.c. Furthermore, if given one extra color, we can fin a g.e.c. that uses minimum number of ege colors for each vertex. 1 Introuction Many moern wireless LAN stanars, such as IEEE b/802.11g [2] an IEEE a [1], provie multiple non-overlappe frequency channels that can be use simultaneously within a neighborhoo. Ability to utilize multiple channels substantially increases the effective banwith available to wireless network noes [3, 10, 12, 11]. One way to utilize multiple channels is to equip each noe with multiple network interface cars (NICs) [7]. For irect communication, two noes nee to be within communication range of each other, an nee to have a common channel assigne to their interfaces. Noe pairs using ifferent channels can communicate simultaneously without interference. Furthermore, since the number of interface cars per noe is limite, each noe typically uses one interface to communicate with multiple of its neighbors. The channel assignment problem is to bin each neighbor to a network interface an also bin each network interface to a raio channel with the goal to minimize interference [7, 12] Specifically, we consier channel assignment that satisfies the following constraints: (1) The total number of raio channels assigne to a wireless noe is boune by the number of network interface cars it has, (2) the capacity of a raio channel within a communication range is boune by a constant number k; the consequence is that an interface on a noe can communicate with up to k neighboring noes, an (3) two noes that nee to communicate 1

3 with each other irectly shoul share at least one common channel. Clearly, the channel assignment for each network interface affects the number of interface cars a noe must have in orer to communicate with all of its neighbors. It also affects the total number of channels that are actually use. Graph coloring seems to be a natural formulation for this problem. However, stanar vertex coloring [8, 4] (an more recently, vertex-multi-coloring) [5] cannot capture the thir constraint that communicating vertices nee to be assigne a common color. Stanar ege coloring [8] fails to capture the secon constraint that no more than k colors can be assigne to the ajacent eges of a vertex. In this paper, we introuce a new graph theory problem calle generalize ege coloring (g.e.c.). Generalize ege coloring is similar to traitional ege coloring, with the ifference that a vertex can be ajacent to up to k eges that share the same color. We show that the channel assignment problem escribe above can be formulate as a generalize ege coloring problem as follows. By picking a color for an ege, we assign the channel number on the two interfaces on two neighboring noes. By restricting the number of ajacent eges that have the same color, we limit the number of neighbors that can communicate with the same interface. In this paper, we provie theoretical analysis for the generalize ege coloring problem. Our theoretical finings are interesting an can be useful for system evelopers of multi-channel multi-interface wireless networks. There are two criteria to evaluate the quality of a generalize ege coloring. The first is the total number of colors use (which is equivalent to the total number of channels use in the wireless network), an the secon is the number of ege colors ajacent to a vertex (which is equivalent to the number of network interface cars on each noe). An optimal g.e.c. shoul use only D k channels where D is the maximum egree of the wireless mesh network, an for each noe with neighbors, it shoul use only k network interfaces. We show that when k is 3, it is impossible to fin optimal g.e.c. for some graphs. However, when k is 2, we have an optimal algorithm for several cases: (1) when D is no more than 4,(2) when D is a power of 2, or (3) the graph is bipartite. In aition, if we are given one extra channel, we can erive a g.e.c. that achieves optimal number of interfaces for each noe. This is very similar to the case of traitional ege coloring where k is 1, an fining an ege coloring with D colors is NP-complete, but it is always possible to color any graph with D +1colors. The rest of the paper is organize as follows. Section 2 formally efines the generalize ege coloring an the quality measurement criteria. Section 3 escribes our results on the generalize ege coloring problem. Finally Section 4 conclues with some interesting open problems in this research topic. 2 Problem This section efines our terminology about generalize ege coloring. Given a graph G = (V,E) we color every ege with mapping function f from E to a color set C. Inparticular, we require that every noe in V isajacenttoatmostk eges of the same color. As a result the traitional ege coloring is a special case when k is 1. We can erive some trivial lower bouns on the number of colors require. For example, let D be the maximum egree of G, then we nee at least D k colors to color G. Also if the number of neighbors of a noe v is v, the number of colors require to color the eges ajacent to v is v k. We efine the global iscrepancy of a coloring function f as ifference between the total number of colors f actually uses an the lower boun D k, i.e. C D k. Similarly we efine the local iscrepancy of a noe v to be the ifference between the actual number of colors ajacent to a noe v an the lower boun v k, i.e. n(v) v k,where n(v) is the number of colors ajacent to v uner f. The maximum local iscrepancy is the maximum among all the local iscrepancy, i.e. max v (n(v) v k ). 2

4 We now use the global an the maximum local iscrepancy to evaluate the quality of a coloring function. A coloring function is a (k, g, l) generalize ege coloring if every noe in V is ajacent to at most k eges of the same color, the global iscrepancy is boune by g, an the maximum local iscrepancy is boune by l. For example, we know that the problem of etermining whether a graph has a (1, 0, 0) g.e.c. is NP-complete, an the Vizing s theorem says that it is always possible to fin a (1, 1, 0) g.e.c. for any graph. A generalize ege coloring is optimal if an only if it is a (k, 0, 0) coloring. 3 Results We first give an impossibility result on the case of k 3, an show that there are cases that we cannot fin (k, 0, 0) g.e.c. for them. The construction is as follows. First we construct aringof2k noes, an each noe is connecte to its two neighbors with two eges. This leaves k 2 eges for each noes along the ring. Now we place k 2 noes in the mile of the ring, an connect each one of them to every noe along the ring. Now each noe in the mile has egree 2k. Suppose we can fin a (k, 0, 0) g.e.c. for this graph, the eges along the ring must be colore with the same color, since each noe along the ring is of egree k, an from the 0 local iscrepancy requirement, it can have at most one color. This forces all the eges going to the noes in the mile to be colore with the same color, which violates the requirement that a noe can be ajacent to at most k eges of the same color. Figure 1 illustrates the constructe graph when k is 3. Figure 1: A graph without (3, 0, 0) g.e.c. From now on we focus on the case where k is 2. We will show that when the maximum egree is boune by 4, we can always fin (2, 0, 0) g.e.c. using Euler cycle. 3.1 Euler Cycle It is well known that a graph has a Euler cycle if an only if every noe is of even egree. We will construct a (2, 0, 0) g.e.c. base on the Euler cycle. The first step of our algorithm is to pair up all the noes with egree 1 or 3, so that every noe is now of egree 2 or 4. Since the number of o-egree noes in a graph is always an even number, the step will not leave any o-egree noes. We use G to enote the graph after transformation. The secon step is to remove some egree 2 noe to simplify the later coloring process. Consier the noes with egree 2 these noes are all on paths that connect egree 4 noes. If the path connect two ifferent egree 4 noes, as in Figure 2 (a), we remove all of them an place a single ege. If the path goes back to the same egree 4 noe an forms a self loop, as in Figure 2 (b), we remove all but two noes from the path. We enote the transforme graph as G. 3

5 (a) (b) Figure 2: Two cases to remove some egree 2 noes. Now we construct a Euler cycle for the transforme graph. Since every noe is of egree 2 or 4, the construction is possible. We then inex each ege with a sequence number accoring to the orer it appears in the cycle. For all eges that have even inices we color them with 0, an the other eges are colore with 1. Lemma 1 The Euler cycle constructe from G has even length, an every noe has the same number of ajacent eges that are colore with 0 an 1. Proof. The length of the Euler cycle is equal to the number of eges in G, which is equal to the sum of all egrees of noes in G ivie by 2. Since there are only egree 4 noes an pairs of egree 2 noes in G, the Euler cycle has even length. In aition, the color are given in alternative manner, each egree 4 noe has two eges of 0 an two eges of 1, an each egree 2 noe has one 0 ege an one 1 ege. Now we nee to erive the actual coloring function for G. If a set of noes is replace by a single ege since the path they form connects two ifferent egree 4 noes, the entire path is colore with the same color from the G coloring. This is feasible since k is 2. On the other han, if a path form a self loop an is replace by a path of length 3 (with two egree 2 noes), the first an the thir ege is colore the same color ue to the alternating coloring. As a result we can color all the noes in that path with the same color. Note that this special treatment is necessary, otherwise the alternating coloring process will be complicate. Finally we nee to remove the ae eges from G. We only ae eges to those noes in G that have egree 1 or 3. These noes in G now has the same number of eges colore by 0or1,sonomatterwhichegeweremove,thelocal iscrepancy will not increase. Formally we have the following theorem. Theorem 2 There exists a (2, 0, 0) generalize ege coloring for every graph with maximum egree boune by 4. The pseuo coe of the alternating coloring process is as follows. 3.2 One Extra Color We now escribe an algorithm that fins (2, 1, 0) g.e.c. for every graph. It is well known that although to etermine if a graph allows a (1, 0, 0) g.e.c. is NP-complete, to fin a (1, 1, 0) 4

6 proceure AlternatingColoring 1. Pair up o-egree noes an a eges. 2. Remove some egree 2-noes accoring to Figure Fin a Euler cycle. 4. Color the eges alternatively with 0 or Color the eges along the path in Figure 2. with the same color. 6. Remove eges ae in step 1. Figure 3: The pseuo coe of fining a (2, 0, 0) g.e.c. for graph with maximum egree 4. g.e.c. in polynomial time is always possible from Vizing s theorem [13]. Therefore the first step of our algorithm is fin a (1, 1, 0) g.e.c. Now we will reuce the number of color by half. Let D be the maximum egree of the graph G. From Vizing s theorem we know that we nee at most D+1 colors to come up with a(1.1.0) g.e.c. By grouping two colors into a new color, we will have at most D+1 2 new colors. Since the original coloring is a (1, 1, 0) g.e.c. The new coloring is a (2, 1,L)where L is about D 4. The reason is that we might use one more color than the D 2 lower boun, an a noe with D 2 eges may still have D 2 new colors ajacent to it after we combine colors, which is about D 4 higher than the D 4 lower boun, hence the maximum local iscrepancy can go up to about D 4. Now we want to reuce the maximum local iscrepancy to 0. The iea is to fin a noe v an two colors c an so that v is ajacent to exactly one ege (enote by (v, w)) colore by c, an one ege (enote by (v, u)) colore by. If we can change the color of (v, w) from c to without increasing the local iscrepancy of w, we can reuce the local iscrepancy of v. For ease of notation we use N(v, c) to enote the number of eges ajacent to v that are colore c. If we can o this for every noe v that has N(v, c) =N(v, ) = 1 for two colors c an, we can reuce the maximum local iscrepancy to 0 by repeately changing the c to for every noe v that has N(v, c) =N(v, ) =1. The key operation for changing color is to fin a c path. The iea about c path is inspire by [9]. Without lose of generality we assume that we want to change color c to. A c path is efine as follows: A c path starts from v, goes through the unique ege (v, w) that is colore c, travels along only eges colore with c or, an ens at a noe other than v. If we exchange the colors of the eges between c an along the c path, we will not increasing the local iscrepancy of any noe along the path. Suppose we can alway fin a c path from v, we can reuce the maximum local iscrepancy to 0. The c path construction is as follows: We always check for whether the current path uner consieration is alreay a c path. If so, we stop an eclare that a path is foun. If not we exten the current path an hope that we can stop at the next ege. Initially the path uner consieration is from v to w, i.e. the unique ege colore c. There are several case to consier while etermining whether we shoul stop or exten. Withoutloseofgeneralityweassumethewejustextentoanoexthrough an ege colore c. Similar argument can be mae for an ege colore, since we are extening a path that coul have both color c an. If N(x, c) =1anN(x, ) = 0, we stop at x since x is ajacent to one c, an changing that only c to will not increase the local iscrepancy of w. In the secon case we have N(x, c) =2anN(x, ) = 0. We cannot stop at x in this case since that will increase the number of colors ajacent to x by one. As a result we exten the c path through the other ege colore by c. Note that changing both c will not increase 5

7 the local iscrepancy of x, an we only exten the path by one more noe. In the thir case we have N(x, ) = 1. In this case we can stop at w since both (x, c) an N(x, ) are greater than 0. Changing the incoming c ege will not increase the number of colors ajacent to x, an since there is only one ege before the change, aing another one will not violate the k = 2 constraint either. In the final case we have N(x, ) = 2. In this case we cannot stop at x, otherwise the number of eges ajacent to x will be 3, violating the k =2constraint. Asaresultwe pick an ege colore by an extent the path. Since each ege can only be use once in the process, eventually the process must stop an we fin a c path. The only complication is that the en noe might be v, therefore we will not be able to reuce the local iscrepancy of v. The following lemma says that we can always fin a c path that stops at a noe other than v. v c w c n j i c h m Figure 4: There exists a c path that starts from v but oes not en at v. Lemma 3 There exists a c path that stops at a noe other than v. Proof. Assume that we construct a c path an eventually go back to v by a cycle C. Since the path starts with a c ege an ens with a ege, let h enote the last noe that extens a c ege, an this ege leas to noe i. Since uring the construction we exten through noe i, therefore N(i, ) = 2 an there exists another ege (i, m) that is colore. See Figure 4 for an illustration. If we pick (i, m) to exten (instea of (i, j)) the c path, it will be impossible to get back to v. The reason is that both N(v, c) ann(v, ) are 1, so the only way back to v is through the ege. Please refer to Figure 4 for an illustration. If the c path oes reach v, wetracebackfromv to the noe where it branches off the cycle C at noe n. By efinition we know that we will see only eges before n, an we branch off C via a c ege, ue to the k = 2 constraint. Recall that uring the construction of C, whenweentern we will leave through another ege only if there is no c ege ajacent to n (from the secon case above). However, we o have N(n, c) > 0 an this contraicts to the formation of C. As a result we can be assure that there exists a c path that will not go back to v. Theorem 4 There exists a (2, 1, 0) generalize ege coloring for every graph. 3.3 Power of 2 We now escribe an algorithm that construct a (2, 0, 0) for every graph with maximum egree as a power of 2, i.e., the graph G =(V,E) has the maximum egree D =2 for an positive integer. 6

8 The basic iea of the construction is to ivie the original graph G into two subgraphs, so that the maximum egrees of both subgraphs are equal. Recall that uring the construction of (2, 0, 0) g.e.. for D 4, we use a Euler cycle to color every ege so that that the number of 0-eges an 1-eges ajacent to a noe iffer by at most 1. Now we apply the alternating coloring process (Figure 3) to G, then ivie the eges accoring to their colors. We have two inuce subgraphs G 0 =(V,E 0 )ang 1 =(V,E 1 ), where E 0 are those eges in G that are colore 0, an E 1 are those colore by 1. Both the maximum egree of G 0 an G 1 are 2 1. We can recursive apply this coloring process until the maximum egree is own to 4, an erive a (2, 0, 0) for each subgraph. Now when we put all these g.e.c. together an view those colors in ifferent g.e.c. s as ifferent colors, we have a (2, 0, l) g.e.c. (enote by C),wherel is the maximum local iscrepancy. Note that the key point of this construction is that we use only D colors to color the entire graph, so the global iscrepancy is 0. Next we nee to reuce the local iscrepancy of C for every noe. Recall that uring the construction for the (2, 1, 0) in Section 3.2, we are able to convert a color c ege into a ege, as long as they are ajacent to the same noe v an there is no other eges colore by c or ajacent to v. We now apply the same technique to the coloring C obtaine in the previous step. As long as there exists a noe v an two colors c an so that N(v, c) =N(v, ) =1, we convert the c ege into an ege, without increasing the local iscrepancy of other noes. We repeat this step, just as we i in the construction of (2, 1, 0), an eventually will convert C into a (2, 0, 0) g.e.c. Theorem 5 There exists a (2, 0, 0) generalize ege coloring for every graph with maximum egree as a power of Bipartite graph It is well known that given a bipartite graph with maximum egree D, we can fin an ege coloring with D colors in polynomial time [6]. In our terminology, it is easy to compute a (1, 0, 0) g.e.c. for bipartite graphs. By combining this (1, 0, 0) g.e.c. with the concept of c path, we are able to fin (2, 0, 0) g.e.c. for every bipartite graph. Given a bipartite graph, the algorithm first fins an ege coloring with D colors. We then group the colors into D 2 new colors. This results in a (2, 0,l) g.e.c. where l is the new local iscrepancy. We then examine every noe v. If there are two colors c an, sothat N(v, c) =N(v, ) = 1, we fin a c path for them. Eventually we have a (2, 0, 0) g.e.c. Theorem 6 There exists a (2, 0, 0) generalize ege coloring for every bipartite graph. 4 Conclusion This paper introuces a new graph theory problem calle generalize ege coloring. We show that when the number of eges of same color is limite to 3, it is impossible to fin optimal g.e.c. for some graphs. However, when the limitation is 2, we have an optimal algorithm when D is no more than 4, or is a power of 2, or the graph is bipartite. In aition, if we are given one extra color, we can erive an generalize ege coloring that achieves optimal number of colors for each noe. This is similar to the case of traitional ege coloring where k is 1 fining an ege coloring with D colors is NP-complete, but it is always possible to color any graph with D +1colors. There are several interesting open problems along this line of research. For example, although it is impossible to fin (k, 0, 0) g.e.c. for k 3, is it possible to fin a (k, 1, 0) solution, just like the case when k is 2? Also we can erive optimal g.e.c. for k =2an some special values of maximum egree D, is it true that we can always fin optimal g.e.c. for any maximum egree D? The authors will continue the investigation on these interesting problems. 7

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