Algorithms for Channel Assignment in Mobile Wireless Networks using Temporal Coloring∗ Feng Yu
Amotz Bar-Noy
City University of New York Graduate Center New York, NY, 10016, USA
City University of New York Graduate Center New York, NY, 10016, USA
[email protected] Prithwish Basu
[email protected] Ram Ramanathan
Raytheon BBN Technologies Cambridge, MA, 02138, USA
Raytheon BBN Technologies Cambridge, MA, 02138, USA
[email protected]
[email protected]
ABSTRACT
Keywords
We model the problem of channel assignment in mobile networks as one of temporal coloring (T-coloring), that is, coloring a timevarying graph. In order to capture the impact of channel re-assignments due to mobility, we model the cost of coloring as C + αA, where C is the total number of colors used and A is the total number of color changes, and α is a user-selectable parameter reflecting the relative penalty of channel usage and re-assignments. Using these models, we present several novel algorithms for temporal coloring. We begin by analyzing two simple algorithms called SNAP and SMASH that take diametrically opposite positions on colors vs re-assignments, and provide theoretical results on the ranges of α in which one outperforms the other, both for arbitrary and random time-varying graphs. We then present six more algorithms that build upon each of SNAP and SMASH in different ways. Simulations on random geometric graphs with random waypoint mobility show that the relative cost of the algorithms depends upon the value of α and the transmission range, and we identify precise values at which the crossovers happen.
Channel Assignment; Mobile Wireless Networks; Graph Coloring; Random Graph; Time Varying Graph
Categories and Subject Descriptors C.2.1 [Network Architecture and Design]: Wireless Communication; G.2.2 [ Graph Theory]: Graph algorithms ∗ Research was sponsored by the Army Research Laboratory and was accomplished under Cooperative Agreement Number W911NF-09-2-0053. The views and conclusions contained in this document are those of the authors and should not be interpreted as representing the official policies, either expressed or implied, of the Army Research Laboratory or the U.S. Government. The U.S. Government is authorized to reproduce and distribute reprints for Government purposes notwithstanding any copyright notation here on.
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1.
INTRODUCTION
The assignment of channels to communicating nodes is a key part of contention-free medium access in a wireless network. Depending upon the access method, the “channel" could be time slot [19], code [11] or frequency [9]. The problem also arises in multi-channel multi-radio cognitive networks where nodes opportunistically use a subset of a large number of unused frequencies [21]. Graph coloring is a natural model for channel assignment [10,15, 19]. It can capture several access methods such as TDMA, FDMA and CDMA in a unified way [18]. In this formulation, the vertices in the graph represent nodes in a multi-hop network, edges represent links, and colors represent channels assigned to nodes. The coloring of the graph is typically done using constraints that reflect the application context. For example, for conflict-free slot assignment in TDMA one seeks to color the graphs using a minimum number of colors such that vertices within two hops (i.e., distance2) of each other receive different colors, which in turn ensures that transmissions in two slots never collide. Most of the work thus far on graph coloring has been on static graphs. This does not capture networks where due to mobility or other link fluctuations, the network topology is varying over time. Applying static graph coloring to successive graph “snapshots" does not capture properties that are valid over time, for instance, a coloring that is conflict-free for t seconds in future. Further, the mismatch between the model (static graph) and the reality (mobile networks) is far from ideal. In this paper, we use a model that inherently captures dynamism, and hence mobility. Specifically, we model a network as a timevarying graph or temporal graph (T-graph). A T-graph is more than just a sequence of snapshots – it is a structure in and of itself that is manipulable as a whole, and for which properties can be defined across time. For example, a “valid coloring" property can now be attributed to (an entire) temporal graph, e.g., a contiguous sub-sequence T-graph. One can also aggregate snapshots between arbitrary points in time to get new T-graphs, as detailed later. Figure 1 illustrates a time-varying graph with a conflict-free distance-1 time-varying color assignment.
SMASH w.h.p. if α ≤
Figure 1: Temporal Graphlets for t = 1,2,3, and a conflict-free temporal coloring. In static networks (graphs), the minimization criterion is typically just the number of channels (colors) used. In a mobile network (temporal graph), however, there are two parameters of interest: the total number of colors used, and the number of times a node has to be re-assigned a color due to conflicts caused by mobility. In most real-life networks, a re-assignment of time slot or frequency requires the exchange of control packets which reduces data capacity. More crucially, it may cause packet drops since the distributed algorithms that implement a re-assignment of the channel take time to converge. Note that even the addition of a single edge can cause a network-wide ripple effect of re-coloring. In general, the relative “cost" of total colors used and the effect of re-coloring depends on the network system and its protocols. Accordingly, we use a generalized cost model where the cost of a dynamic coloring of a temporal graph is given by C + α.A where C is the total number of colors used, A is the number of re-assignments over a period of time T and α is a user-defined parameter that “weights" the penalty of re-assignment relative to that of the total number of colors. For example, in a cognitive radio network where spectrum holes are plentiful, α would be smaller than in a TDMA network where slots are precious. In Figure 1, for example, C = 2, A = 2. We present several novel algorithms for coloring T-graphs (we call this T-coloring). We begin by taking two simple baseline algorithms that take extreme positions on the optimization function: SNAP, which colors each snapshot independently, and thus does not care about re-assignments at all; and SMASH, which takes the entire T-graph, “smashes" it into a single graph by taking the union of all edges, colors this graph, and uses the colors for the entire duration [1, T ]. Thus, SMASH never re-assigns but makes little attempt to minimize total colors used. We then present six more sophisticated algorithms that enhance each of SNAP and SMASH in various ways, for example, by incrementally coloring or partially smashing. We evaluate the performance of several T-coloring algorithms using random geometric graph (RGG) and random waypoint mobility (RWP) models. Our main contributions are as follows: Temporal coloring and cost model. We present a new coloring framework that captures mobility inherently. Our cost model C + αA captures the relative impact of channel re-assignments and channels used in a user-selectable manner. Together, our framework better reflects real-world considerations. Temporal coloring algorithms and analysis. We present a total of eight algorithms that cover a spectrum of strategies, including a hybrid one. We prove a number of properties on the relative performance of SNAP and SMASH. For example, we show that if α > 1 SMASH is always better than SNAP, and if α = 0 SNAP is never worse than SMASH, and for random graphs SNAP outperforms
1 1 n log 1−p
for distance-1 coloring (we also
give an analogous result for distance-2 coloring). We also derive a network dynamics based threshold for a hybrid algorithm, based on the RGG-RWP model, that we then use for switching between two algorithms. Insights from experiments. We present several experiments comparing the algorithms and identify “pivot" points where pairs or triplets of algorithms reverse their relative performance. While no single algorithm is superior over the entire range of α, our work provides insights to practitioners to choose the best algorithm for a given scenario. Four of our eight algorithms require the entire T-graph in advance and four do not. The former captures mobility or dynamism that are pre-defined (e.g. networks of aircraft or underwater nodes with pre-defined trajectories, scheduled sleep/wake sensor networks) or can be reasonably predicted somewhat into the future [24]. It is also applicable to dynamic job scheduling modeled as a coloring problem [1]. However, for simplicity and clarity, we assume in the description and in the experimental evaluation that all algorithms have the same input, namely the entire T-graph. This paper follows the so-called "protocol model" for interference vice the SINR-based "physical model" [8]. While we recognize that the physical model is a better reflection of radio characteristics, our focus is on temporal aspects and a novel cost model. These are first better investigated in the simpler protocol model, with physical model extensions a topic for future work. Further, recent work [23] has shown that one can narrow the gap between the two models by appropriately setting the interference range. Finally, while we present our algorithms as using the entire graph information (i.e., as a centralized algorithm), four of our algorithms work in a way that only uses local information at each step, and can be implemented as a distributed algorithm along the lines of [14, 27]. That is, the same techniques can be applied per (smashed) snapshot to make a distributed version of the proposed algorithms. The rest of the paper is organized as follows. After discussing relevant background work in the next section, we provide in section 3 definitions of temporal graph and coloring. In section 4 we provide theoretical results for SNAP and SMASH. Section 5 describes six more algorithms, and section 6 compares the performance of algorithms using simulation experiments.
2.
RELATED WORK
Prior work related to this paper can be broadly classified along two lines: channel assignment using a static coloring model, and recent work on time-varying graphs. Channel assignment in multi hop wireless networks using coloring has been studied in various contexts such as FDMA [9], TDMA [5, 19] and CDMA [11]. A framework that unifies the above coloring problems is presented in [18] along with an algorithm. Most variants are NP-complete, and complexity issues have been discussed in [2]. Distributed coloring algorithms are the subject of [17, 20], while an incremental coloring algorithm is given in [15]. Graph-theoretic models of mobile or dynamic networks have been considered in [7, 12, 22, 25]. Each uses a different way of representing the time domain – labels are attached to edges in [7], while temporal reachability graphs (TVGs) are proposed in [25]. The evolution of random graphs [4] has been studied in [6]. Our work is unique in that it is the first that considers the coloring problem in a temporal graph setting with a cost model that incorporates re-assignments of colors. In achieving our advances, we have leveraged considerably from previous work, in particular, that on greedy coloring and incremental coloring.
on a time varying graph such that these constraints are satisfied for every snapshot in the T -graph. The distance-1 constraint means that every pair of adjacent nodes must receive different colors; the distance-2 constraint means that every pair of nodes with shortest distance less than or equal to 2 must receive different colors. D EFINITION 4. Given a coloring constraint, a T -Coloring is a labeling function C : [1, T ] × V → N of the vertices of a temporal graph G[1, T ] such that C(t, v) satisfies the coloring constraint for each t ∈ [1, T ] and2 v ∈ V . Figure 2: Various representations of temporal graphlets for the TGS in Fig. 1. A stacked graph is constructed by drawing directed edges in the direction of time between successive temporal graphlets in a TGS; a smashed graph is a “collapsed” version of the stacked graph. Alternatively, it is union of the graphlets. Both smashed and m-smashed graphs are “lossy” representations.
3.
PRELIMINARIES
3.1
Temporal Graph
Assume slotted time starting at time 0. Slot t > 0 starts just after time t − 1 and ends at time t. For 0 ≤ T1 ≤ T2 , a Temporal Graphlet Sequence (TGS) is defined as
For example, a T -Coloring with the dist-2 constraint is a labeling function C : T ×V → N such that for any t ∈ [1, T ] and u, v ∈ V , if dist(u, v) ≤ 2, then C(t, u) 6= C(t, v). In T -Coloring, we count the total number of colors used (denoted by C) and the total number of times any vertex changes its color between graphlets (denoted by A). The total number of colors C is the maximum label used if the positive integers 1, 2, . . . are the labels. Formally, D EFINITION 5. Given a T-graph G[T1 , T2 ], and a coloring algorithm, graphlet Gi has its color set C(Gi ) = {ci (vj )|vj ∈ V (G)}, ci ∈ Z+ . Then the total number of color used is C = max(ci (vj )), and the total number of times a color is changed is T2 −1
A=
T GS(T1 , T2 ) = {G(t) = (V (t), E(t))}, T1 ≤ t ≤ T2 This model attempts to capture its space-time trajectory (see Figure 1). Each G(t) is referred to as a temporal graphlet, a graphlet, or a snapshot. Alternate notations that we use, depending on the emphasis, include G(T1 , T2 ), G[1, T ] (shifting the frame of reference maintains properties), G[T ] (reference shifting is implied). A temporal graph, called T -graph, is a single-graph representation of the TGS. Such representations include the stacked graph (StG), the smashed graph (SmG) and its generalization the m-smashed graph (m-SmG), and the evolving graph as shown in Figure 2. Formally, D EFINITION 1. Given a TGS G[1, T ], the stacked graph (StG) of G[1, T ] is StG = (VS , ES ), where VS = ∪t V (t), ES = ∪t E(t) ∪ EC where EC = ∪t,i (ui (t), ui (t + 1)). D EFINITION 2. Given a TGS G[1, T ], the smashed graph (SmG) of G[1, T ] is SmG = (VM , EM ), where each sequence of u(t), u(t+ 1), . . . is replaced by a single vertex u ∈ VM , and EM = ∪t E(t) with endpoints of edges mapped to the replaced vertices in VM . D EFINITION 3. Given a TGS G[1, T ], the m-smashed graph (m-SmG) of G[1, T ] is m-SmG = (VM , EM ), where the smashing operation is not performed on the entire G[1, T ] but on each of G[1, m], G[m + 1, 2m], G[2m + 1, 3m], . . . instead. We note that the "evolving graph” representation proposed in [7] which labels edges with the times at which they are active is equivalent to the stacked graph1 but an evolving graph is not a traditional graph. Hence reducing to an evolving graph does not allow us to easily leverage existing algorithms or code.
3.2
Temporal Coloring
Temporal coloring (T -coloring) is the process/task of coloring a given temporal graph in a conflict free manner. Specifically, it is a vertex coloring problem with constraint distance-1 and distance-2 1
And deleting the labels yields a smashed graph.
X
X
ci (vj ) 6= ci+1 (vj ) .
i=T1 vj ∈V (G)
As mentioned in section 1, practical applications require minimizing both the number of colors and the number of color reassignments. Accordingly, we define a simple and natural cost function to evaluate the quality of a T-coloring. D EFINITION 6. The cost of a T -coloring X is given by COST (X) = C(X) + α · A(X) where α ≥ 0 is a user defined parameter, and C and A are, respectively, the total number of colors used and the total number of re-assignments as defined above.
3.3
Greedy Static Coloring
Our temporal coloring algorithms use as a key component a coloring of a graphlet. Coloring a graph is NP-hard to even approximate within a constant factor for both distance-1 and distance-2 constraints [13, 26]. While our temporal coloring algorithms are agnostic to the particular choice of static coloring method, including exponential-time optimal, we use a greedy heuristic from [18] called UxDMA with Progressive Minimum Neighbor First (we call it simply PMNF) ordering. Our implementation of PMNF has a running time of O(|V ||E|). However, one may replace PMNF with any other algorithm without impact on fundamental contributions, and therefore we shall refer to the procedure static-coloring.
4.
BASELINE ALGORITHMS: SNAP AND SMASH
The set of possible T -coloring algorithms for the cost function COST = C + αA can be thought of as lying along a spectrum that balances the number of colors and number of re-assignments in different ways. We begin by describing and theoretically analyzing two simple and naive algorithms at the two ends of that spectrum. These algorithms are oblivious to the value of α. 2
We note that color conflicts can occur across the directed “time" edges. T-coloring is not the same as (traditional) coloring of a Tgraph.
Algorithm SNAP colors every snapshot G[Ti ] in the stacked graph independently and afresh using a static coloring algorithm (e.g. PMNF discussed in section 3.3). In this section, SNAP “ignores” the coloring of G[Ti−1 ] when it colors G[Ti ]. In later sections, we show that a greedy version of SNAP could do better if, for example, it considers the vertices in the same order for all the T graphlets. Algorithm 1 SNAP for T-Coloring Input: T-Graph G[1, T ], constraint ∈ {dist-1,dist-2} Output: Color assignment of each graphlet: Color[1, T ] 1: for i from 1 to T do 2: Color(i)=static-color(Gi , constraint) 3: end for 4: return Color[1, T ]
Algorithm SMASH colors the smashed graph (refer definition in section 3.1) and the colors stay the same for all T graphlets. In other words, we simply run the static coloring algorithm on SmG(G) with, say, C colors. We observe that A = 0 and Cost = C. The main drawback of SMASH is that it smashes all the T graphs, thus creating a graph whose chromatic number is too high. In later sections, we show that partial smashing could help reduce the chromatic numbers at the cost of having some color changes. Algorithm 2 SMASH for T-Coloring Input: T-Graph G[1, T ], constraint ∈ {dist-1,dist-2} Output: Color assignment: Color (same for all graphlets) 1: G = SmG(G[1, t])) 2: Color=static-color(G, constraint) 3: return Color
For the simple implementation of PMNF that was sufficient for our experiments, the worst-case running time of SNAP is O(|V |3 T ) and that of SMASH is O(|V |3 +|V |2 T ). Thus, SMASH is superior in running time. We now consider the relative theoretical performance of SMASH and SNAP with respect to α, first with a worst-case viewpoint, then for Erdos-Renyi random graphs. First, we introduce some additional notation for dist-2 coloring. Let G2 denote the dist-2 graphlet that results from adding an edge between any two nodes in G that are within two hops away from each other. Also, let SmG(G2 ) denote the smashed dist-2 T-Graph = SM ASH(G21 , . . . , G2T ) and SmG2 (G) denote the dist-2 smashed T-Graph = (SM ASH(G1 , ..., GT ))2 . It is easy to show that SmG(G2 ) is a subgraph of SmG2 (G).
4.1
Properties of SNAP and SMASH
For clarity, consider an optimal static coloring usage for SMASH and SNAP. However, the following discussion is valid for any “reasonable” greedy alternatives as well. For the rest of this section, we assume the dist-1 constraint, unless otherwise specified. The next two propositions give thresholds for which one strategy outperforms the other for any T-graph. P ROPOSITION 1. When α > 1, SMASH is always better than SNAP. P ROOF. Suppose we color the T-graph with a SNAP algorithm X, then we could define a new coloring: each time a vertex changes its color between Gi and Gi+1 , we color this vertex with a new unique color in G1 ...Gi+1 . At the end of this process, the coloring is a legal T-coloring of the smashed graph that uses C = C(X) + A(X) colors. Obviously, C ≥ C(SM ASH), therefore C(SM ASH) ≤ C(X) + A(X) < C(X) + αA(X) since α > 1.
We remark that the above proposition is not true for dist-2 Tcoloring on the original T-graph (G1 , G2 , . . . , GT ). However, it is true if both SMASH and SNAP were operating in dist-1 mode on the T-graph (G21 , . . . , G2T ). P ROPOSITION 2. When α = 0, SNAP is at least as good as SMASH. P ROOF. When α = 0, there is no penalty for changing colors and so cost of SNAP is the chromatic number under the optimal static coloring assumption. The chromatic number is monotonic in the sense that its value for a graph G is at least its value for any sub-graph of G. Therefore, C(Gi ) ≤ C(G). Thus, the cost of SNAP is no larger than the cost of SMASH. Given the above, a natural question is: in the intermediate values of 0 < α < 1, which one is better? In particular, as α is increased above 0, at what point does SNAP start to be worse than SMASH? We show that for an arbitrary sequence of graphlets there is no such threshold. However, for random graphlets such a threshold exists. For the next two propositions we assume a “reasonable” implementation of SNAP that does not try hard to avoid changes but also does not try to generate too many changes. P ROPOSITION 3. For any α < 1, there exists a T-graph for which SNAP is better than SMASH. P ROOF. Let n = 3 and call the vertices A, B, C. The edges E = {(A, B), (A, C)} exist in G1 while E = {(A, B), (B, C)} for G2 , . . . , GT . The cost of SMASH is 3 since SMASH colors the smashed graph with 3 colors. A “reasonable” SNAP uses only 2 colors and changes the color of C once for a total cost less than 3. Note, that any greedy SNAP would use at most 2 colors and at most 3 changes. For an “adversarial” implementation of SNAP, the above proposition holds for α < 1/3. P ROPOSITION 4. For any α > 0, there exists a T-graph for which SMASH is better than SNAP. P ROOF. Let n = 4 and call the vertices A, B, C, D. The edges E = {(A, B), (C, D)} exist in G1 . A greedy or an optimal SNAP will color G1 with 2 colors. Without loss of generality, assume that A and C are colored with the color 1. Then E = {(A, B), (A, C), (C, D)} for G2 , . . . , GT . A greedy or an optimal SNAP will color G2 , . . . , GT with 2 colors but will have to change the color of at least 2 vertices. SMASH on the other hand will color the smashed graph with 2 colors in a way that is good for G1 as well. Since α > 0, the cost of SNAP is larger than the cost of SMASH for this T-graph. Note that in the above example SNAP is paying the cost of not taking into account the future state of graphlets that SMASH is naturally able to do. Also note that G2 is defined based on the particular implementation of SNAP.
4.2
Analysis of T-coloring in Random T-graphs
In this section, we study properties of T-coloring in the simplest model of random T-graphs – each graphlet is an independently generated graph obeying the Erdos-Renyi random graph model ER(n, p), in which, each of the possible n2 edges exists with an independent probability p. We study the scaling of the number of colors necessary to color such graphs under both dist-1 and dist-2 constraints as a function of n and p. We note that theoretical results for coloring in RGGs, the more relevant model for MANETs, have been more elusive even for static graphs, with the exception of the recent results by McDiarmid [16]. A theoretical study of T-coloring of RGGs is a topic of future work for us.
4.2.1
Erdos-Renyi graphlets: dist-1 constraints
T HEOREM 5 (B OLLOBAS [3]). Let χg denote the chromatic number under the greedy coloring algorithm, Gn,p is an ER(n, p) graph. Then χg (Gn,p ) = log n1 n , almost surely for any greedy 1−p
algorithm and any vertex ordering. L EMMA 1. CSM ASH = χg (Gn,p0 ) = T ∗
n log 1
n
, where
1−p
p0 = 1 − (1 − p)T . P ROOF. The smashed graph is ER(n, p0 ) where p0 = 1 − (1 − p) . Applying Bollobas’s theorem, we have: CSM ASH = χg (Gn,p0 ) = n n = T ∗ log n1 n . The last equality = log log 1 n n 1 T
1−p0
1−(1−(1−p)T )
1−p
is true since logxT n = (1/T ) logx n. L EMMA 2. Let A0 denote the expected number of color changes between two consecutive ER(n, p) graphlets. Then COSTSN AP = CSN AP + αASN AP = log n1 n + α(T − 1)A0 . 1−p
P ROOF. We rely on the assumption that T is fixed while n tends to infinity, therefore, because of the concentration property in Bollabas’ theorem mentioned above, CSN AP = max{C1 , . . . , CT } = n . log 1 n 1−p
The next corollary follows from the above two lemmas. C OROLLARY 6. Let C0 =
n log 1
. Then with high probabiln
1−p
ity SNAP outperforms SMASH iff α
θ(∆∗ , T, α) then 4: Color(G0 ) = Algorithm INC (G, G0 ,constraint) 5: else 6: Color(G0 ) = Algorithm P-SNAP (G0 , P M N F − labeling(G), constraint) 7: end if 8: return Color(G0 )
6.
EXPERIMENTAL ANALYSIS
In this section, we evaluate conflict-free channel assignment (over a period of time) in a MANET modeled as temporal graph coloring. We evaluate by simulation the performance of the various temporal coloring schemes described in sections 4 and 5 on a random MANET whose nodes follow the random waypoint mobility model (RWP). The parameters used in the simulation study are given in the table below. Note that all possible combinations of parameters in the table were not simulated. Parameter and Description n (Number of nodes) D (Side of square area) s (Node speed) r (Transmission range) T (Time window of evaluation) α (Significance of reassignment) K (Budget for smashing algorithms) m (Number of graphlets compressed for m-SMASH)
Range 50, 100 1000 5, 10, 15, 20 [25, 200] 100, 200 [0.0001, 1] 10,20 T ∼ [K ]
The specific combinations that were simulated have been reported in the following text.
Figure 5: For most values of α P-SNAP outperforms SNAP, especially at lower speeds and higher range.
Figure 6: INC outperforms both P-SNAP and SMASH. The relative performance of P-SNAP and SMASH depends upon the parameters. We focus on “pivot" or “crossover values", that is, when the relative performance (cost) of two or more algorithms reverse themselves. Recall that cost is C + αA where C is the total #colors, A total #re-assignments, and α is a user-defined parameter. We note that SNAP, P-SNAP, INC and hybrid strictly require only one snapshot at a time whereas SMASH, m-SMASH, BCSMASH, AE-SMASH require the entire T-graph. In order to compare all algorithms with the lowest common denominator of requirement, we assume all algorithms receive the full T-graph. In all experiments below, we generate a T-graph based on the random geometric graph (RGG) with random waypoint mobility model (RWP). In the RGG(n; D; r) model, n nodes are placed uniformly randomly in a D × D square, and two nodes have an edge between them iff they are within a transmission range r. In the RW P (n; D; s) mobility model, each node picks a destination uniformly randomly in the square and moves to it at a constant given speed s, then picks another destination, and so on. There is no “pause time" at the destination. A T-graph is generated as follows. At each time slot t, graphlet G(t) is the RGG based on vertex positions at time t, and we store each graph let G(t) as a stacked graph for T-coloring. In the generation of RGG and RWP, we use generic “units" – one may assign any (the same) transformation from “units" to actual metrics (e.g. 1 unit could be 10 meters). All experiments below are done using an RGG with n = 50 and D = 1000. The speed, range and the cost bias α are varied differently for each of the experiments. All of the colorings done use the dist-2 constraint which
represents broadcast scheduling for TDMA and is the most applicable vertex coloring constraint in practice. Each point in the plot represents an average over 50 random T-graph samples for all algorithms except Adaptive Hybrid for which we used 20 samples to accommodate its higher running time. We first compare SNAP and P-SNAP. For this set of experiments, the radius r = 75. Figure 5 shows the percentage improvement of P-SNAP over SNAP as a function of α for varying RWP speeds and ranges. We observe that for all but very small values of α P-SNAP outperforms SNAP. As discussed earlier in section 5, for dynamism at the rates studied, two consecutive graphlets are nearly the same, and using the same ordering obviously results in better performance. At very low α, the cost is essentially the number of colors used. PMNF labeling is very effective in reducing the colors used, and since SNAP always uses PMNF while P-SNAP uses it only once at the beginning, when α is very small, SNAP uses less colors than P-SNAP. We also observe that P-SNAP outperforms SNAP to a greater extent at lower s and higher r. Lower speeds result in similar consecutive graphlets, thus increasing P-SNAP’s advantage of “sticky" labeling. At higher ranges, the graphlets are more connected and have a higher chromatic number. The higher number of colors results in a larger gap between re-assignments in SNAP and P-SNAP. Given that P-SNAP outperforms SNAP for most of the parameter space of interest, we use P-SNAP going forward, and compare it with SMASH and INC. For this set of experiments, r = 75 and s = 5. Figure 6 shows the dependence of the cost of each algo-
Figure 7: SMASH, m-SMASH, Big-Change-SMASH exhibit a "pivot" or "crossover" point, at approximately α=0.05 rithm as a function of α and r. INC outperforms both SMASH and P-SNAP. Thus, although we showed in section 5 that there are instances where INC incurs greater cost than both SMASH and PSNAP, it appears that this is not evident in practice, at least in the RGG-RWP model. However, there is no clear "winner" between SMASH and PSNAP. There is a value of α, approximately 0.05, below which P-SNAP is better, and above which SMASH is better. At higher α, re-assignments are more costly and SMASH does not re-assign. Further, as the plot on the right in Figure 6 shows, for α = 0.05, the gap between SMASH and P-SNAP widens with increasing range. A denser graph is more suitable for SMASH since then the difference between a single graphlet and a smashed graph narrows. We now compare the different kinds of smashing algorithms, namely m-SMASH, Big-Change-SMASH (BC-SMASH), and regular (full) SMASH. For this set of experiments, Figures 7(a,b) show the results as a function of α for budgets K = 10 and K = 20, for different values of r. We observe an interesting phenomenon around a “pivot" value of α: when α is less than the pivot value, the cost is in the order B-C-SMASH < m-SMASH < SMASH, and when α is greater than this pivot value the cost is reversed as SMASH < m-SMASH < BC-SMASH. It is particularly interesting that the crossovers between the three algorithms occur at the same point, and for different combinations of range and K (we have observed this in other combinations not shown here due to lack of space). We observe that the pivot point moves to the right with decreasing K. Figure 7(c) shows the cost versus r at α = 0.05. A crossover can be seen between BC-SMASH and SMASH (with m-SMASH outperforming both). Finally, we evaluate the performance of the adaptive hybrid algorithm (Algorithm 8) that switches between two online schemes P-SNAP and INC based on the current network dynamics and the value of α. Figure 8 illustrates that Algorithm 8 adaptively switches from P-SNAP (at low α) to INC (at higher α). Note that the threshold function θ(·) given in Algorithm 8 is only illustrative, and other more complex functions can indeed be used instead. Our experiments provide a way by which a user can select the right algorithm based on the kind of network, i.e„ the cost bias between colors and re-assignments, and range, among other things.
cost model C + αA that allows a user-selectable tradeoff between number of colors (channels), and their changes over time. Using this model, we have presented a suite of algorithms ranging from simple ones such as SNAP, P-SNAP and SMASH, to more sophisticated ones such as INC, m-SMASH, Big-Change-SMASH, Accumulated-Error-SMASH, and an adaptive hybrid algorithm. For SNAP and SMASH, we have derived bounds for when one outperforms the other, for arbitrary graphs as well as Erdos-Renyi random graphs. For all algorithms, we have presented experimental results with typically used placement and mobility models. Our study shows that there exist crossover or pivot α points between pairs or triplets of algorithms, with α = 0.05 figuring as a kind of “magic number". Our work, with suitable adaptations, can be applied to time-, frequency- or code-based medium access in a reallife multi hop wireless network, and guide a designer to select the right algorithm for her preferred α. Many exciting avenues for future work exist. These include further extending the theoretical bounds for distance-2 coloring and for other algorithms, especially budget-based ones, improving the performance of the online algorithms, extending to a physical propagation model and distributed implementations.
7.
8.
CONCLUDING REMARKS
We have considered the problem of channel assignment in mobile or otherwise dynamic networks as a graph coloring problem. We have presented a model with two novel components: a temporal graph and an associated temporal coloring model; and a
Figure 8: Hybrid online algorithm switches between P-SNAP and INC as α is varied
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APPENDIX A.
COMPUTING THRESHOLD ∆∗ FOR HYBRID ONLINE T-COLORING
We calculate the expected number of edge-changes between two RGG-RWP graphlets G, G0 . The model parameters are: area (D × D), number of nodes (n), connectivity radius (r), and speed (s). We assume that G is a uniformly distributed RGG, and the future direction of each node is chosen randomly. Consider an edge (A, B) in G. Due to RWP mobility, A → A0 and B → B 0 (change of locations in G0 ). Define events Y = 1(A,B)∈G and Z = 1(A0 ,B 0 )∈G0 . ¯ ) Pr(Y )}. {Pr(Z|Y¯ ) Pr(Y¯ )+Pr(Z|Y Then, we have, E[∆(G, G0 )] = n(n−1) 2 πr 2 By assuming D >> r, we have Pr(Y ) = D2 and Pr(Y¯ ) = 2
1 − πr . For fixed parameters r, s, define the following function: D2 f (x) = Pr(|AB| = x, |AA0 | = s, |BB 0 | = s, |A0 B 0 | < r). After geometric probability calculations (omitted due to paucity of Rπ R1 space), it can be shown that f (x) = π1 Pr(|AB| = x, B 0 = 0 0
B + 2aseiθ , |A0 B 0 | < r)dθda. For x ≤ r − 2s, f (x) = 1. For 2 +x2 −r 2 x ≥ r − 2s, let θ∗ (x) = arccos( (2s) 2sx ). So 1 > f (x) > ∗ 1 − 21 (1 − θ π(x) )(1 − r−x ). Therefore, we have: 2s Z r s(3r − 2s) 2x ¯ )=1− (1 − f (x))dx < Pr(Z|Y 2 3r2 0 r Z D πs(3r + 2s) 2πx Pr(Z|Y¯ ) = 1 − (1 − f (x))dx < 2 − πr 2 ) D2 − πr2 3(D r 2πrs Therefore, it follows that E[∆(G, G0 )] < n(n−1) . 2 2 D O BSERVATION 14. It is obvious that if |AB| < r − 2s, then |A0 B 0 | < r; and if |AB| > r + 2s, then |A0 B 0 | > r, therefore, a naive upper bound is given by π(r + 2s)2 − π(r − 2s)2 8πrs = D2 D2 Hence, our bounding technique yields a fourfold improvement over the naive upper bound. Pr((A, B) changing)
