Packet Scheduling with Interference

Transcription

1 Computer Science Computer Science Department Thomas Keßelheim Packet Scheduling with Interference Diploma Thesis January 3, 2009 Thesis advisor: Second advisor: Prof. Dr. Berthold Vöcking Prof. Dr. Petri Mähönen

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3 Erklärung Hiermit versichere ich, dass ich die Arbeit selbstständig verfasst und keine anderen als die angegebenen Quellen und Hilfsmittel benutzt sowie Zitate kenntlich gemacht habe. Aachen, 3. Januar 2009 (Thomas Keßelheim) i

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5 Abstract Routing and scheduling in wireless networks is a popular field of research these days. There are many parallels to wired networks but also new aspects such as interference and power control. A common model for these interference constraints is the so-called physical model that is based on the signal to interference plus noise ratio (SINR): For a successful transmission the received signal has to be β-times higher than the signal of other senders plus ambient noise. A typical problem is to route and schedule packets via multi-hop networks in the shortest possible time. This task is formalized in the cross-layer latency minimization (CLM) problem. It affects three major fields, namely routing, power control and scheduling. Given N source-destination pairs (s i, t i ), the aim is to find suitable paths and powers for each packet, and a schedule. We study approximation algorithms for the CLM problem, which we prove to be NP-hard before. When powers are fixed such that the received signal is the same at each receiver we are able to find a randomized O(log 2 n) whp-approximation. However, we get to know that it is crucial in some instances to choose the right power levels rather than just leaving them fixed. But these instances have a large length diversity compared to the number of packets and networks nodes. Expressed in terms of this length diversity, we find out that the loss is at most O(log log 2 n) resulting in a total approximation factor of O(log log 4 n) whp. Thomas Keßelheim iii

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7 Contents Introduction. Formal Description of the Interference Model Algorithmic Problems Comparison to the Protocol Model Related Work Outline of this Thesis NP-hardness 7 3 Scheduling in General Metrics 3 3. Estimating the Optimal Schedule Length A Randomized Algorithm for Single-hop Scheduling Extensions for Multi-hop Scheduling Multi-hop Scheduling with Fixed Paths Finding Optimal Paths Summary Different Power Assignments Lower Bounds Upper Bounds Summary Scheduling in the Euclidean Plane A Scheduling Algorithm Equivalence of the Protocol Model and the SINR Model Conclusions and Further Work 39 v

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9 Chapter Introduction Communication networks have been subject to research for quite a long time up to now. There have also been many interesting results for wired networks such that they are well understood (see e. g. [Sch98]). These days wireless communication becomes more and more important where constraints are different. For example, in wired networks each transmission may use its own link. In contrast, transmissions are not that separated in wireless networks. Instead every transmission is exposed to some interference by other transmissions or ambient noise. In this thesis we study the algorithmic aspects of these networks. This means our analyses give worst-case guarantees for arbitrary networks in comparison to what theoretically can be achieved. In literature, many models for the interference constraints have been proposed. Many approaches are based on graphs. This means apart from the graph modelling the network topology there is a second one that defines the interference constraint. An edge between two senders indicates that not both of them can successfully transmit in a single time slot. In contrast, we study a model closer to reality called SINR model or physical model as stated by Gupta and Kumar [GK00]. It is based on the signal to interference plus noise ratio (SINR): The received signal has to be β-times larger than the interfering signals plus ambient noise, for some constant β >. The fundamental difference is that interference from all senders has to be summed up in the SINR model. This means in general all other senders have to be considered to state whether a transmission is successful. In contrast, in the graph-based models there is only a binary condition based on the local environment. We study the traditional end-to-end routing and scheduling problem: Given N pairs (s i, t i ), the aim is to route one packet for each pair from s i to t i in the shortest possible time. It is considered as static which means all requests arise at a certain starting point. In addition, our algorithms are centralized and have full knowledge over the network. Although these limitations narrow the practical use of our results, we get some interesting insight on what future protocol design has to deal with. Our results are applicable to many different kinds of wireless networks. We do not need to distinguish if there is infrastructure such as access points or not. Furthermore we find our

10 CHAPTER. INTRODUCTION results for single-hop networks (e. g. cellular networks) first, which we extend to multi-hop networks (e. g. ad-hoc networks) later on. The networks we consider have in common that nodes are able to make use of power control. This means there may be chosen a different power level for each transmission. We keep the view of slotted protocols. This means time is divided into slots and any direct transmission from one node to another one needs exactly one time slot.. Formal Description of the Interference Model We model the network as a metric space (V, d). The set V represents the nodes of the network. The metric d defines the distance d(u, v) between any two nodes u, v V. Note that in general we do not require the nodes to be located in the plane. However, in Chapter 5 we will study the impact of restricting to this special case. The metric enables us to model the decay of signals as follows. If a node u transmits a p signal with power level p then node v receives it with power. The constant α > 0 d(u,v) α is called path-loss exponent. A signal can be successfully received if its received strength exceeds the interference plus ambient noise at least by a factor β. The interference is defined as the received signal strength caused by all other senders transmitting in the same time slot. The constant β > is called antenna gain. This SINR constraint can formally be defined as follows. Definition.. (SINR condition). Consider a single time slot in which k packets are transmitted: u i transmits to v i using power level p i > 0 for all i [k] := {,..., k}. Then v j receives the packet iff the SINR condition SINR(v j ) = p j d(u j,v j ) α Noise + i k p i β (.) d(u i,v j ) α is satisfied. By this definition it is implicitly guaranteed that a node can either transmit or receive one packet in one time slot. Furthermore, by scaling powers up, the noise term gets less and less important. It can even be neglected by slightly changing β. Since we do not aim at minimizing energy, noise will be set to 0 in our theoretic analyses throughout this thesis..2 Algorithmic Problems In our end-to-end scheduling problem, we are given N source-destination pairs (s i, t i ). The aim is to route a packet from each s i to t i such that the latency is minimal, which is the overall number of time slots used. In each of these time slots the SINR constraint has to be satisfied. 2 Thomas Keßelheim

11 .2. Algorithmic Problems The most general problem formulation is the cross-layer latency minimization (CLM) problem as stated by Chafekar et al. Definition.2. (Cross-layer latency minimization problem [CKM + 07]). Given N sourcedestination pairs (s i, t i ) the aim is to find. a routing path P i for each packet i from s i to t i defining which hops to take, 2. a power assignment p defining a power for each hop, and 3. a schedule S stating which hop has to be done in which time slot such that the schedule length T (S) is minimal. We distinguish two variants: In the single-hop case all transmissions are made directly from s i to t i. In the multi-hop case intermediate nodes may be used to store and forward the packets. In fact, the single-hop case is a special case of the multi-hop case where additionally paths are fixed to length. We will deal with the problem as follows. First, we will consider the single-hop case where the power assignment is fixed. Afterwards we will extend our results to multi-hop schedules with fixed routing paths and power assignments. As a last step we will discuss how to find optimal routing paths and power assignments. Let us now introduce some more notation on paths. Having a set P of N paths P,..., P N we denote the number of involved edges on a path P i by l(p i ). Its nodes are referred to as P i,0, P i,, P i,2,..., P i,l(pi ). Furthermore we set D as the maximum path length by D := max i [N] l(p i ). To simplify notation let us define furthermore the set of indices of receivers as dom(p) = { (i, j) N N i N, j l(p i ) }. In case D = we identify dom(p) with [N]. Using these definitions we can formally define a power assignment setting a power for each hop. Definition.2.2 (Power Assignment). Given a set of paths P, a power assignment p is a function p: dom(p) R >0. There are two generic possibilities to assign powers that are of much relevance. On the one hand, we can make all senders use the same transmission power: p(i, j) = for all (i, j) dom(p). This uniform power assignment is the only possible choice when we have identical devices without power control. On the other hand, we can set the powers such that the received power is the same at all receivers: p(i, j) = d(p i,j, P i,j ) α for all (i, j) dom(p). Thomas Keßelheim 3

12 CHAPTER. INTRODUCTION This power assignment is of special relevance when considering the problem to minimize energy rather than latency: The power used is proportional to the minimum power needed to deal with the ambient noise, which must not be neglected in this case. This is why it is called linear power assignment. Now a schedule stating which hop will be done in which time slot can be formally defined. Definition.2.3 (Schedule). Given a set of paths P and a power assignment p a schedule S of length T (S) is a function S : dom(p) [T (S)] such that. for all (i, j) dom(p) with j > 2 it holds and S(i, j) > S(i, j ) (.2) 2. for all (k, l) dom(p) p k,l d(p k,l, P k,l ) α β (i,j) S {(k,l)} (i,j) (k,l) p i,j d(p i,j, P i,j ) α. (.3) In words, Condition.2 means each hop has to take place after the packet has reached the starting point of this hop. Of course, this is of no importance in single-hop schedules. Condition.3 simply states the SINR condition (.) has to be satisfied for all hops. By transmitting each hop in its own time slot, a trivial schedule of length N D can be constructed..3 Comparison to the Protocol Model As outlined before, there are a number of graph-based interference models. One typical representative is the protocol model introduced by Gupta and Kumar in [GK00] and studied further in [KMPS04]. Definition.3. (Protocol model [GK00]). In the protocol model a transmission from u to v is successful if for all transmissions from u to v scheduled in the same time slot it holds that d(u, v) ( + ε) d(u, v) where ε > 0 is a model constant. In general, an optimal schedule can be much shorter when using the SINR condition instead of the protocol model. This was previously shown by Moscibroda et al. [MWW06] in theory and also in simulations. Nevertheless in these examples the SINR schedule is not asymptotically shorter but only within a constant. The following example shows that even asymptotically large differences in schedule lengths can occur. 4 Thomas Keßelheim

13 .4. Related Work In our instance we have N pairs of senders and receivers (s i, t i ) which are placed on a line, i. e. V R, d(u, v) = v u. To simplify notation we set the constant a = 2 α+ β +. The i-th sender s i is located at a 2i α, its receiver t i is located at a 2i α. First observe that for all i [N] receivers t j with j < i are located between s i and t i. In the protocol model this means no two packets can be transmitted together in a single time slot. In contrast, in the SINR model only a single time slot is needed for all transmissions when the packet from s i to t i is transmitted with power a i. To show this we fix a receiver t j, j [N]. We can now bound the interference caused by senders that are located between s j and t j : i<j p i d(s i, t j ) α i<j a i a 2j a a j a 2j. The interference caused by the outside senders can be bounded similarly: i>j p i d(s i, t j ) α i>j a i a a j+ 2i a Thus the total interference is at most 2 a j a a = 2j 2 α β a j a 2j = β p j d(s j, t j ) α. = a j. a a 2j This means the SINR constraint is satisfied, all transmissions can be done in the same time slot. For this example it is crucial to choose the right power assignment in the SINR model. Similar examples demonstrating the impact of this choice can be found in Section 4.. Another important factor is the exponential growth of this instances. We will show in Section 5.2 that when distances are not too different the optimal lengths only differ by a constant factor..4 Related Work Networks under SINR constraints have been studied widely up to now from several points of view. The first study of SINR scheduling problems on nodes that are arbitrarily and not randomly distributed has been presented by Moscibroda and Wattenhofer [MW06]. In contrast to our problem, they aim to schedule all links on a spanning tree. The idea is to have a protocol such that packets between any two nodes are transmitted via a virtual backbone. This result has been extended by Moscibroda et al. [MWZ06] to arbitrary demands, which is in fact the single-hop case of our problem. Their result is an O(log 2 N I in ) algorithm, where I in is a measure of interference in the instance. Unfortunately, I in is no lower bound for the optimal schedule length. Thomas Keßelheim 5

14 CHAPTER. INTRODUCTION A further extension [MOW07] introduces a new measure of interference χ ρ called disturbance where ρ > 0 is some parameter. The algorithm described achieves a schedule length of O(χ ρ ρ 2 log N (log N + ρ)). There is still no comparison to the optimal schedule length. This is why both results lack of an approximation factor. Goussevskaia et al. [GMW08] also examine a related problem. In the local broadcasting problem any node in the network intends to transmit a packet to all nodes within its socalled local broadcasting range. They describe two distributed algorithms which have a polylogarithmic approximation factor. Chafekar et al. study similar problems. In [CKM + 08] they deal with maximizing the throughput in an SINR network. In [CKM + 07] they give an approximation algorithm for the CLM problem. It is crucial for their analysis that the instance is located in the plane. This allows to use graph coloring in a similar way to the approaches used in the protocol model. Our algorithms instead work in general metrics while the approximation factor is still better. Apart from this, Goussevskaia et al. [GOW07] prove the scheduling problem in the Euclidean plane with fixed power assignment to be NP-hard. The idea is quite a simple reduction of the Partition problem. However, this result still does not prove the scheduling problem when powers are not fixed to be NP-hard. We will give this result for general metrics..5 Outline of this Thesis As outlined before, we will prove the NP-hardness of the CLM problem. This will be done in Chapter 2. Afterwards, in Chapter 3, we will find an approximation algorithm for the CLM problem where powers are fixed to the linear power assignment. The effects of this restriction are analyzed in Chapter 4. This allows us to give an approximation factor for our algorithm for the CLM problem. As a last point, we will study a different approach that can be applied when restricting to the Euclidean plane in Chapter 5. 6 Thomas Keßelheim

15 Chapter 2 NP-hardness As a first step, let us prove the defined problem is NP-hard. Goussevskaia et al. [GOW07] already proved the scheduling problem with fixed paths and a fixed power assignment to be NP-hard. Thus we will only have to deal with the CLM problem where powers are subject to optimization. First, we need to prove two simple bounds. Lemma For x, y 0 with x + y 2 it holds that [( ) α ( ) α ] 2 α + 2 α α (x + y) + 2 α + 2 α (x + y) + ρ(x + y) 2 x y where ρ is a constant defined as follows ρ = (2 α + ) α 2 α+2 ( 2 α+ + α + ). Proof. Let us fix the sum s of x and y, i. e. 0 < s and s = x + y, and analyze the 2 functions f, g : [0, s] R defined by ( ) α ( ) α g(x) = + f(x) = g(x) α. x (s x) The extremal points of f and g are obviously the same. Having a look at the derivative of g ( ) α+ ( ) α+ dg dx (x) = α α x (s x) we remark that g must have its extremal points at the critical point x = s resp. at the 2 boundary points x = 0 and x = s. Thus it suffices to have a closer look at the functions at these points. s we have: For the critical point x = 2 ( ) f 2 s = 2 α 2 s = 2 α k=0 ( ) k 2 s. 7

16 CHAPTER 2. NP-HARDNESS So: ( ) 2 α + 2 α s f 2 s 2 α + 2 α s + s 2. (2.) We now consider the boundary points x = 0 and x = s: Let us define another function h: [ 0, 2] R by h(s) = [( ) α α + ]. s The first and the second derivative of h are: dh ds (s) = [( ) α α + ] ( ) α+ α α s s [( ) d 2 α h ds (s) = α + ] 2 ( ) α+2 [ ( ) α 2 + α + ] 2 s s s By Taylor s theorem it holds that. h(s) = h(0) + dh ds (0) s + R (s) = 2 α + 2 α s + R (s). For the remainder term R (s) the Lagrange form states there is a ξ, 0 < ξ < s, such that R (s) = d 2 h 2 ds (ξ) 2 s2. Since s 2 we can conclude 0 R (s) ρs 2. In total, this means 2 α + 2 α s h(s) 2 α + 2 α s + ρs 2. (2.2) From Equations 2. and 2.2 we can conclude the claim holds. We will present a reduction of 3-Partition, which can be regarded as a variant of the Bin Packing Problem in which the weights are much more restricted. Definition (3-Partition [GJ79]). Given B N, m N, a,..., a 3m N, B < a 4 i < B, 3m 2 i= a i = m B Question: Is there S,..., S m [3m] such that j S j = [3m] and i S j a i = B for all j [m]? 8 Thomas Keßelheim

17 v d v 2 l d c d 2 l 2 u d2 u 2 Figure 2.: A visualization of the scheduling instance network. For a feasible solution it obviously suffices to have i S j a i B for all j [m] since 3m i= a i = m B. To show the CLM problem is strongly NP-hard we reduce it to 3-Partition, which is known to be strongly NP-hard [GJ79]. Theorem The single-hop and the multi-hop case of the CLM problem are strongly NP-hard. Proof. We will define a reduction of a 3-Partition instance as defined in Definition to a scheduling instance in which all numbers are bounded polynomially in m. First define two shortcuts depending on B: ( ) A := α ρ + B 2 4 and χ := 2 α + 2 α 3 B A + ρ ( B A ) 2 = 4A2 + BA α ρb α A 2. The scheduling instance network is visualized in Figure 2.. It consists of 2 3m + nodes called c, u,..., u 3m, v,..., v 3m. There is an edge between u i and v i, between u i and c, and between v i and c for all i [3m]. The distances are defined as d(u i, c) = d(v i, c) = A a i =: d i and d(u i, v i ) = χβ α ( A a i 4 ) =: l i. There is a communication request between the nodes u i and v i for all i [3m]. We now claim: There is a schedule of length m There is a solution for 3-Partition. Note that it does not make sense to do multi-hop scheduling. This is why we only have to consider the single-hop case. We have to show that if there is a schedule of length m there is a solution for 3-Partition. Obviously, in any schedule of length m, in each step exactly three transmission requests are operated. Consider such a single step. For the ease of Thomas Keßelheim 9

18 CHAPTER 2. NP-HARDNESS notation let, w. l. o. g., the transmissions take place between u i and v i for i {, 2, 3}. Let furthermore be p p 2 p 3. By SINR constraint it holds ( ) p 2 p 3 β (d + d 2 ) α + (d + d 3 ) α p l α Thus we can conclude ( ) β (d + d 2 ) α + (d + d 3 ) α l α ( ) α ( ) α ( ) α + 2A a a 2 2A a a 3 A a χ α 4 [( ) α ( ) α ] α ( a a + 2 a a A a ) 2Aχ A 2A 2A 2A By using Lemma 2.0. we have [ ( 2 α + 2 α a A + a 2 2A + a )] ( 3 A a ) 2Aχ ( 2A 4 (4A + 2a + a 2 + a 3 ) A a ) 2 3 α A 2 χ 4 4A 2 + (a + a 2 + a 3 )A B α A 2 χ = 4A 2 + BA α ρb 2 ( ) (a + a 2 + a 3 ) 2 3 B 2 α ρ + B. 4 A }{{} = 2. Since a, a 2, a 3 and B are integer, it has to be a + a 2 + a 3 B. This means the schedule can be transformed to a valid solution of 3-Partition. Let there be a solution S,..., S m of 3-Partition. We claim that each set S i can be regarded as a valid schedule step using the uniform power assignment. For the ease of notation, we we write S i = {, 2, 3} as well. We have (d + d 2 ) α + (d + d 3 ) α ( ) α ( ) α = + 2A a a 2 2A a a 3 ( ) α [( ) α ( ) α ] = 2A a a + 2 a a 3 2A 2A 2A 2A 0 Thomas Keßelheim.

19 Lemma 2.0. states this is at most ( ) α [ ( 2 α + 2 α a 2A 2A + a 2 2A + a 3 2A + a ) ( a + ρ 2A 2A + a 2 2A + a 2A + a ) ] 2 α 3. 2A Since a + a 2 + a 3 B this is at most ( ) [ α ( 2 B α + 2 α 2A 2A + a ) ( ) ] 2 α B + ρ. 2A A Thus we can bound the interference divided by the signal strength by [ ] / (d + d 2 ) + α (d + d 3 ) α l α ( ) [ α ( 2 B α + 2 α 2A 2A + a ) ( ) ] 2 α B ( + ρ A a ) α 2A A 4 χ α β ( = ( 2 B 2 α α + 2 α 2A + a ) ( ) 2 B + ρ 2A A ( 2 a α 4A 2 α Ba 8A + a ) ) α ρb2 a 2 8A 2 4A 3 χ α β ( 2 α + 2 α 3 B ( ) ) 2 α B A + ρ A χ α β = β. So the SINR condition is met and the solution for 3-Partition can be regarded as a valid schedule consisting of m steps. All in all, we can conclude that the CLM problem is NP-hard. It is even strongly NP-hard since 3-Partition is and all numbers are bounded polynomially in m which is a lower bound of the input length. We have seen that under the assumption P NP there is no polynomial time algorithm. This is why we will design and analyze approximation algorithms in the next chapters. Thomas Keßelheim

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21 Chapter 3 Scheduling in General Metrics Having proven NP-hardness, we will analyze a randomized approximation algorithm for the CLM problem in this chapter. We will not try to find optimal powers but leave them fixed to the linear power assignment and analyze the effects of this choice in the next chapter. Under these assumptions we will find a way to estimate the optimal schedule length. Afterwards it is a simple task to find an algorithm for single-hop scheduling. Finally, we will extend this algorithm to multi-hop scheduling and add a path selection algorithm. 3. Estimating the Optimal Schedule Length Given a set of paths and a power assignment, an interesting question is to get upper and lower bounds indicating how long an optimal schedule has to be. In wired networks there is quite a simple, yet surprising answer that was first given by Leighton et al. [LMR94]: This optimal schedule has length Θ(C + D) where C is the congestion, which is the maximum number of paths using a certain edge, and D is the dilation, which denotes the maximum length of a path. In wireless networks under SINR constraints it is unfortunately not that easy. Nevertheless, we will find an analogon to the congestion C measuring the interference. It is a lower bound on the optimal schedule length in the linear power assignment and we will use it to give approximation ratios. Definition 3.. (Measure of Interference I). For (k, l) dom(p) define { I k,l = min, p } i,j d(p k,l, P k,l ) α. p k,l d(p i,j, P k,l ) α (i,j) dom(p) Using this we define the measure of interference: I = max I k,l. (k,l) dom(p) 3

22 CHAPTER 3. SCHEDULING IN GENERAL METRICS This measure of interference allows us to make quite a good estimation for the optimal schedule length in the linear power assignment. { } ( ) I q Theorem Every schedule has length at least max min κ q max, D = Ω I q min q max + D, where κ is a constant and q max = max (i,j) dom(p) (i,j) dom(p) p i,j d(p i,j, P i,j ) α and q min = min (i,j) dom(p) p i,j d(p i,j, P i,j ) α. Proof. Any schedule has length at least D since there is a path consisting of D hops which all need a single time slot. It remains to show any schedule has length at least I q min κ q max. For w V we define { } qmin p i,j J w = min,. q max d(p i,j, w) α q max It can be easily seen that qmax q min J w I k,l for w = P k,l. Thus it obviously suffices to show that T (S) Jw for all w V. To prove this it suffices to show that in any step J κ w decreases by κ at most. Let us now consider such a single step where transmissions are made from u i to v i for i [k]. Let furthermore be w V. We have to show: i [k] { qmin min, q max q max } p i κ. d(u i, w) α We define j arg min i [k] d(v i, w), i. e. v j is the closest (active) receiver from w. This might also be w itself. We define a set U of indices of near senders from w by U = {i [k] d(u i, w) 2 d(v j, w)} (cf. Figure 3.). Using the triangle inequality we can conclude for all i U: d(u i, v j ) d(u i, w) + d(w, v j ) 3 2 d(v j, w). (3.) In addition, we have d(v j, w) d(v i, w) d(v i, u i ) + d(u i, w) since v j is the closest receiver by triangle inequality d(v i, u i ) + 2 d(v j, w) by definition of U. This implies d(v j, v) 2d(u i, v i ). (3.2) 4 Thomas Keßelheim

23 3.. Estimating the Optimal Schedule Length w v j U Figure 3.: The definition of v j and U. There is no receiver inside the outer circle. Putting together 3. and 3.2 we get d(u i, v j ) 3d(u i, v i ). Thus it holds q min U\{j} p i q max q max d(u i, v i ) α q max i U i j For all i [k]\u it holds that i U i j p i d(u 3 α i, v j ) 3α α q max β p j d(u j, v j ) α 3α β. d(u i, v j ) d(u i, w) + d(w, v j ) d(u i, w) + 2d(u i, w) = 3d(u i, w). by triangle inequality by definition of U Now, we can sum up all i [k]\u: q max i [k]\u i j p i d(u i, w) α q max i [k]\u i j p i d(u 3 α i, v j ) 3α α β. Summing up all i [k] gives { } qmin p i min, q max d(u i, w) α i [k] q max q min q max U\{j} + 3α β + 3α β + i [k]\u i j q max p i d(u i, w) α + Thomas Keßelheim 5

24 CHAPTER 3. SCHEDULING IN GENERAL METRICS 2 3α β +. Thus we have shown the claim for κ = 2 3α β +. Since having q max = q min in the linear power assignment, we have proven a lower bound of Ω(I + D) for the linear power assignment. In the remaining part of this chapter we will complete this result by an algorithm generating a schedule of length O(I + D log 2 ˆn). 3.2 A Randomized Algorithm for Single-hop Scheduling Using the measure of interference defined in the previous section, we can construct quite simple randomized algorithms for the single-hop case and also for the multi-hop case. Please note that these algorithms work with any power assignment. However, their performance guarantees are always given in terms of I, which is a lower bound on the optimal schedule length for the linear power assignment only. Algorithm A simple single-hop algorithm while packet has not been successfully transmitted do try transmitting with probability 2βI end while Algorithm is a simple waiting-for-success algorithm where all packets use the same probability of transmission at any time. Lemma Algorithm generates to a schedule of length at most O(I log N) whp. Proof. First consider a fixed packet k in a single step of the algorithm. Let X i, i [N], be the 0/ random variable that packet i is chosen to be transmitted in this step. Let Z be the interference at receiver t k divided by its signal strength. Since we have β > the transmission cannot be successful anymore when Z. Thus we may cut off the influence by a single sender at. Z = N i= i k min {, d(s k, t k ) α p k Its expectation value is: E Z = N i= i k min {, d(s k, t k ) α p k } p i X d(s i, t k ) α i } p i E X d(s i, t k ) α i 6 Thomas Keßelheim

25 3.2. A Randomized Algorithm for Single-hop Scheduling = N i= i k min {, d(s k, t k ) α p k } p i d(s i, t k ) α 2βI 2β So by Markov s inequality the probability that this packet cannot be transmitted successfully given X k = is [ Pr Z ] Pr[ X 2 E X ] β 2. To make the transmission successful the two events X k = and Z β Since they are independent it holds that [ Pr X k =, Z ] [ = Pr[ X k = ] Pr Z ] β β have to occur. ( ) = 2βI 2 4βI. The probability for packet k not to be successfully transmitted in (k 0 + )4βI ln N independent repeats of such a step is therefore at most ( ) (k0 +)4βI ln N e (k 0+) ln N = N (k0+). 4βI Applying a union bound leads to an overall bound that one of N packets is not successfully transmitted in (k 0 + )4βI ln N = O(I log N) independent repeats: Pr[ T (S) > (k 0 + )4βI ln N ] N Pr[ Packet i needs more that (k 0 + )4βI ln N attempts ] i= N N (k 0+) = N k 0 This means all senders are successful within O(I log N) steps whp. For the analysis of a more sophisticated algorithm, we need a Chernoff bound in which random variables do not have to be independent. Lemma Let X,..., X n be 0/ random variables for which hold that there is a p [0, ] such that for all k [n] and all I [k ] Pr X k = X i =, X i = 0 p. (3.3) i I i [k ]\I Let furthermore w,..., w n be reals in (0, ] and µ p w i. Then the weighted Chernoff bound ] ( ) e Pr[ δ µ wi X i ( + δ)µ ( + δ) (+δ) holds. Thomas Keßelheim 7

26 CHAPTER 3. SCHEDULING IN GENERAL METRICS Setting + δ = γ 2e this implies ] Pr[ wi X i γµ 2 γµ. Proof. To show this Chernoff bound, a standard proof [Rag88] can be adapted. The only step that needs to be changed is that still E[e tx ] n ( pe tw i + p ) i= although random variables are no more independent. No other step makes use of the independence. Let us first prove that for all k {0,... n } and for all J [k] I [n]\[k] ( i I e tw i ) Pr X i =, i I X i = 0 X i =, X i = 0 i J i [k]\j ( ) pe tw i ( p). (3.4) i ([n]\[k])\i I [n]\[k] i I i ([n]\[k])\i by reverse induction. For k = n this is a direct conclusion from Condition 3.3, which is satisfied by assumption. Now let us assume Equation 3.4 has already been shown for k + and we have to prove it for k. We split up the sum on the left side depending on whether k + I or not: ( ) e tw k+ I [n]\[k+] i I e tw i Pr X k+ =, X i =, i I + I [n]\[k+] ( i I e tw i ) Pr X k+ = 0, X i =, i I i ([n]\[k+])\i i ([n]\[k+])\i X i = 0 X i =, i J X i = 0 X i =, Using the definition of conditional probability is the same as ( ) e tw k+ Pr[ X k+ = ] e tw i I [n]\[k+] i I i J i [k]\j i [k]\j X i = 0 X i = 0. 8 Thomas Keßelheim

27 3.2. A Randomized Algorithm for Single-hop Scheduling Pr X i =, X i = 0 i I i ([n]\[k+])\i X k+ =, X i =, i J ( ) + Pr[ X k+ = 0 ] Pr X i =, i I I [n]\[k+] i I e tw i X i = 0 X k+ = 0, X i =, i J i ([n]\[k+])\i X i = 0 i [k]\j X i = 0. i [k]\j Applying the induction hypothesis, we can bound this by ( e tw k+ Pr[ X k+ = ] + Pr[ X k+ = 0 ] ) ( pe i) tw ( p). I [n]\[k+] i I I [n]\[k+] i I i ([n]\[k+])\i As we have e tw k+ Pr[ Xk+ = ] + Pr[ X k+ = 0 ] e tw k+ p + ( p), this is at most: ( e tw k+ p + ( p) ) ( pe i) tw ( p) = ( ) pe tw i I [n]\[k] i I i ([n]\[k])\i ( p). i ([n]\[k+])\i Thus we have successfully proven Equation 3.4 for all k {0,..., n }. Obviously, the expectation of e tx can be calculated as follows: E[e tx ] = ( ) e tw i Pr X i =, X i = 0. I [n] i I i I i [n]\i Using the above result with k = 0 we can bound it by ( ) pe tw i ( p). I [n] i I i [n]\i It can easily be seen that this is exactly n ( pe tw i + p ). i= Thus we have filled the gap. Thomas Keßelheim 9

28 CHAPTER 3. SCHEDULING IN GENERAL METRICS Algorithm 2 An O(I + log 2 N) whp algorithm while I curr log N do J := I curr while I curr J 2 do if packet i has not been successfully transmitted then assign a delay δ i 6eβJ i. u. r. try transmission after waiting the delay end if end while end while execute algorithm We can now use this bound to analyze Algorithm 2. This algorithm assigns random delays to all packets. The maximum delay is decreased depending on how many transmissions have already been successful. Let I curr denote the measure of interference that is only caused by the remaining transmissions to be made. The delay maximum is reduced to its half when I curr = I. 2 We first observe that the inner while loop, which reduces I to I generates O(I) steps whp. 2 Lemma During one iteration of the outer while loop, the inner while loop of Algorithm 2 is executed at most k times with probability at least N k 0. Proof. We first consider a single iteration of this loop. All senders are taking part as if they had not yet been successful. We observe if the senders of a set S are transmitting and there is a collision for packet i we have j S j<i min {, d(s i, t i ) α p i } p j > d(s j, t i ) α 2β or j S j>i min {, d(s i, t i ) α p i } p j > d(s j, t i ) α 2β. In the first case let be Y i < =, in the second one Y i > =. We now show that the random variables Y <,..., Y N < fulfill Equation 3.3 with p =. Let 8e us fix k [N] and I [k ]. Since the delays δ i are drawn independently they can be considered as if they were drawn one after the other in order δ, δ 2,.... Then the value of Y i < would already be determined after drawing δ i by definition. In other words: The values of δ,..., δ k already determine the values of Y <,..., Y k <. This means there is a subset M [6eβJ] k of delay values such that i I Y i < =, i [k ]\I Y i < = 0 iff (δ,..., δ k ) M. Let now be X i be a 0/ random variable for i [k ] such that X i = iff δ i = δ k. We can observe that we have for all (a,... a k ) [6eβJ] k : E[X i δ = a,..., δ k = a k ] = 6eβJ. 20 Thomas Keßelheim

29 3.2. A Randomized Algorithm for Single-hop Scheduling Define furthermore Z < k = k i= min {, d(s } k, t k ) α p i X p k d(s i, t j ) α i with E[Z < k δ = a,..., δ k = a k ] 6eβ. Now it holds that Pr[ Y < k = δ = a,..., δ j = a k ] = Pr [ Z k < > ] 2β δ = a,..., δ k = a k 2β E[Z k < δ = a,..., δ k = a k ] = 8e = p. We can now apply the so-called law of alternatives: Pr Y k < = Y i < =, Y i < = 0 i I i [k ]\I = Pr δ = a,... δ k = a k p. (a,...,a k ) M i I Pr[ Y < k = δ = a,..., δ k = a k ] Thus we may apply Lemma on I < j I < j = N i= i j min {, d(s j, t j ) α p j defined as follows: } p i Y < d(s i, t j ) α i. Y < i =, i [k ]\I Y < i = 0 This random variable indicates the remaining measure of interference that is caused by these collisions. Setting γ = 2e, µ = J Lemma states: 8e [ Pr I j < J ] 2 J 4 N. 4 Let us now consider the situation after k 0 +2 iterations of the inner while loop. Since these are independent repeats we have [ Pr I j < J ] N (k0+2). 4 For reasons of symmetry this also applies to I > j. Thomas Keßelheim 2

30 CHAPTER 3. SCHEDULING IN GENERAL METRICS For a sender that has not been successful we have Z j < + Z j >. This means we have the bound Ij curr I j < + I> j. For the remaining measure of interference Icurr = max j [N] Ij curr we can conclude: [ Pr I curr J ] [ Pr Ij curr J ] 2 2 j [N] [ Pr I j < J 4 or I< j J ] 4 This proves the claim. j [N] N ( N (k 0+2) + N (k 0+2) ) N k 0. Using this lemma, we can add up all numbers of steps that are generated in the while loops. Theorem Algorithm 2 generates a schedule of length at most O(I + log 2 N) steps whp. Proof. Let T k be the number of scheduling steps generated in the k-th execution of the outer while loop. As shown in the previous lemma, it holds that [ Pr T k (k 0 + 3) 6eβ ] 2 I k N. k 0+ Let furthermore U denote the number of scheduling steps generated in the execution of Algorithm. As shown in Lemma 3.2., it holds that Pr[ U (k 0 + 2) 4β ln N log N ] N k 0+. Thus the total number of steps generated in the while loops k T k + U can be estimated by [ ] Pr T k + U (k 0 + 3) 32eβI + (k 0 + 2) 4β ln N log N k [ ] Pr T k (k 0 + 3) 6eβ 2 I U (k k 0 + 2) 4β ln N log N k [ Pr T k (k 0 + 3) 6eβ ] 2 I + Pr[ U (k k 0 + 2) 4β ln N log N ] k N + k 0+ N k 0+ k (log N + ) N k Thomas Keßelheim

31 3.3. Extensions for Multi-hop Scheduling N k 0. This means the total number of steps is at most (k 0 + 3) 32eβI + (k 0 + 2) 4β ln N log N = O(I + log 2 N) with probability at least N k Extensions for Multi-hop Scheduling Up to now, the described algorithms only take care of single-hop scheduling with a fixed power assignment. As a first generalization, we extend the results to multi-hop schedules in this section. This means we route along fixed paths first and approximate optimal paths afterwards. To simplify the proofs we introduce ˆn = V 2 N as an upper bound for the maximum number of reasonable hops. Obviously, it is still polynomially bounded in the input size Multi-hop Scheduling with Fixed Paths In our first approach, we leave paths fixed. Thus the considered scheduling problem is more or less the same as considered before apart from some dependencies such that one communication hop has to be done after another one has already taken place. With Algorithm 3 we succeed in re-using single-hop algorithms for this problem. The idea is to simply assign a delay to each packet. By this shift, a number of time frames is created. The measure of interference I is sufficiently balanced between these time frames and thus the single-hop algorithm A can be used to schedule the time frame to generate feasible time slots since there is at most one hop of a packet assigned to each time frame. Algorithm 3 Fixed path multi-hop scheduling for all i [N] do assign a delay δ i 2eI i. u. r. log ψ ˆn end for for all t 2eI + D do log ψ ˆn execute A on all hops (i, j) with δ i + j = t end for Theorem Let A be a single-hop algorithm generating a schedule of length O(I + log ψ N), ψ, whp. Then the schedule generated by Algorithm 3 has length O(I+D log ψ ˆn) whp. Proof. For (k, l) dom(p) let I (t) k,l be the random variable of I caused by all hops assigned to time frame t. Let T t denote the schedule length that is used to schedule the t-th time frame. Thomas Keßelheim 23

32 CHAPTER 3. SCHEDULING IN GENERAL METRICS Let be X i,j,t = iff δ i + j = t. Then we have I (t) k,l = (i,j) dom(p) min {, d(p } i,j, P i,j ) α X d(p i,j, P k,l ) α i,j,t. This means E I (t) k,l = (i,j) dom(p) { min, p } i,j d(p i,j, P i,j ) α logψ ˆn p k,l d(p i,j, P k,l ) α 2eI logψ ˆn 2e. For fixed t the random variables X i,j,t are negatively associated as defined by Dubhashi and Ranjan [DR98]. So a Chernoff bound is applicable: for all k 2 it holds that [ ] Pr I (t) k,l k 2 log ψ ˆn 2 k 2 log ψ ˆn 2 k 2 log ˆn = ˆn k 2. By definition of A for all constants k and k 2 there is a constant k 0 such that: [ Pr T t k 0 k 2 log ψ ˆn ] max (k,l) dom(p) I(t) k,l k 2 log ψ ˆn ˆn k We can now estimate the overall schedule length: [ ] Pr T t k 0 k 2 (I + D log ψ ˆn) t t Pr [ T t k 0 k 2 log ψ ˆn ] ˆn k + t ( 2eI log ψ ˆn + D ˆn k 3 (k,l) dom(p) ) ( [ ] Pr I (t) k,l k 2 log ψ ˆn ˆn k + ˆn ˆn k 2 ) for some constant k 3 depending on k and k 2. This means we have a schedule length of O(I + D log ψ ˆn) whp for fixed path scheduling in the linear power assignment.. In total, this means by combining Algorithms 2 and 3 we get an O(I + D log 2 ˆn) whp algorithm. 24 Thomas Keßelheim

33 3.3. Extensions for Multi-hop Scheduling Finding Optimal Paths So far the paths taken by the packets are fixed. In this section we will find a way to approximate optimal paths when using the linear power assignment. For this purpose we will adapt an approach first used by Srinivasan and Teo [ST97] for wired networks. Chafekar et al. [CKM + 07] also use it as a part of their CLM algorithm. We solve an Integer Linear Program (ILP) approximatively by using relaxation and randomized rounding. First, let us formalize the problem of finding paths such that max{i, D} is minimal as ILP. We introduce a set of edges E V V which describes the set of links that may be used. Let furthermore N in (v) resp. N out (v) denote the incoming resp. outgoing edges from v. Minimize w subject to: i [N] y(i, e) y(i, e) = (3.5a) i [N], v V \{s i, t i } i [N] i [N], v V e N out(s i ) e N out(v) y(i, e) y(i, e) w e E e =(u,v ) e N in (s i ) e N in (v) y(i, e) = 0 } y(i, e ) min {, d(u, v ) α w d(u, v) α (3.5b) (3.5c) (3.5d) i [N], e E y(i, e) {0, } (3.5e) This ILP is designed to minimize w = max{i, D} as follows. Condition 3.5d ensures that I w whereas condition 3.5c ensures D w. By leaving out condition 3.5e, this ILP can be relaxed to an LP which then describes a multi-commodity flow problem. Algorithm 4 solves this LP and uses the result to approximate a solution of the ILP applying the technique of randomized rounding [RT87]. Algorithm 4 Finding optimal paths solve the relaxed ILP 3.5 for all i [N] do decompose the resulting flow for commodity i to paths P,..., P k with flows f,..., f k remove paths longer that 2w and scale the flows along the other paths proportionally select one of the remaining paths at random with probability f i end for Theorem Let I and D be the values such that I+D is minimal. Then Algorithm 4 calculates paths such that I = O(I log ˆn) whp and D 2D. Proof. We fix i [N]. Observe that f j 2 j:l(p j ) 2w Thomas Keßelheim 25

34 CHAPTER 3. SCHEDULING IN GENERAL METRICS because otherwise condition 3.5c could not be satisfied. This means by leaving out all paths longer than 2w and scaling the flows along the other paths proportionally all flows are doubled at most. Therefore conditions 3.5c and 3.5d hold after substituting w by 2w. Since all remaining paths are not longer than 2w, it is ensured that D 2w. Thus only I will increase depending on which paths are selected at random. Let I k,l denote the random variable of measure of interference at link (k, l). Its expectation value is obviously E[I k,l ] 2w. By applying a Chernoff bound, we get Pr[ I k,l (k 0 + )2e log ˆn 2w ] 2 (k 0+)2e log ˆn 2w ˆn (k 0+). For the maximum value I we have Pr[ I (k 0 + )2e log ˆn 2w ] ˆn k 0. In total this means: if I and D are the values such that I + D is minimal which is the optimal solution for the ILP 3.5 we found out how to calculate paths such that I = O(I log ˆn) whp and D 2D. Thus we have found a path selection algorithm for the linear power assignment. Using the scheduling algorithms we can now compose an approximation algorithm for the CLM problem restricted to the linear power assignment. 3.4 Summary Let us summarize what we have found in this chapter for the CLM problem. We left the powers fixed to the ones given by the linear power assignment. On the one hand, we found a way to estimate the optimal schedule length when paths are fixed by Ω(I + D) where I is an analogon to the congestion C measuring the interference. On the other hand, using Algorithm 3 together with Algorithm 2 we get an O(I + D log 2 ˆn) whp schedule. In combination with our path selection, Algorithm 4, we get a schedule of length O(I log ˆn + D log 2 ˆn) whp. In other words, we have found an O(log 2 ˆn) whp approximation algorithm. 26 Thomas Keßelheim

35 Chapter 4 Different Power Assignments Up to now, we did not take the power assignment into consideration but left it fixed such that all schedules had to use the linear power assignment. Although it is quite common to use the linear power assignment (see e. g. [CKM + 07]) there has been no study on the direct effect of this choice. Even if the linear power assignment has the advantage of using the minimum power needed to deal with ambient noise, we have to analyze the impact of this restriction on the optimal schedule length. 4. Lower Bounds Both of the generic power assignments considered so far, the linear as well as the uniform power assignment, have in common that each transmission power only depends on the distance between the sender and the receiver and not on the other transmissions to be done. We call this behaviour oblivious. Definition 4.. (Oblivious Power Assignment). A power assignment p is called oblivious iff there is a function f : R >0 R >0 such that p i,j = f(d(p i,j, P i,j )), i. e. the assigned power only depends on the distance between sender and receiver. Of course, one could imagine to use more complicated functions than the linear or the constant function. In this section, we will see that in general any choice of a function can have a very bad effect on the optimal schedule length. Theorem Let f : R >0 R >0 be an oblivious power assignment function. Then there exists a family of instances on a line that needs Ω(N) steps when using the powers defined by f whereas all senders can transmit in a single step using a different power assignment. Proof. We consider the family of instances illustrated in Figure 4.. Formally, this kind of instance can be defined by s, t,..., s N, t N R such that { 0 if i = s i = and t i = s i + x i. t i + y i otherwise 27

36 CHAPTER 4. DIFFERENT POWER ASSIGNMENTS s t s 2 t 2 s N t N x y 2 x 2 x N Figure 4.: A visualization of the instances. x i and y i are chosen depending on f. We fix the distance x to. To simplify notation, let us furthermore define κ := 2β + as a constant. We now define the distances x i and y i between the nodes recursively depending on the function f. We can distinguish three cases as an arbitrary function f : R >0 R >0 obviously satisfies at least one of these conditions: (a) For every x 0 > 0 and every c > 0 there is a value x > x 0 such that f(x) > c. (b) There is a value b > 0 such that for all x 0 > 0 there is an x > x 0 with b 2 f(x) b. (c) For every x 0 > 0 and every c > 0 there is a value x > x 0 such that f(x) < c. In case (a) we set: y i = κ 2/α (x i + y i ). Given x,..., x i and y i, we choose x i such that x i y i and f(x i ) y α i f(x j ) x α j for all j < i. By this construction it is ensured that a link k is exposed to high interference by links with larger indices. To show this, let S be a set of indices of packets that can be transmitted together in one step; k = min S. For i S\{k} it holds that d(s i, t k ) = i j=k+ x j + i j=k+ y j 2 i y j 2 j=k i j=k κ 2(i j)/α y i 2 κ 2/α y i. Since all signals in S can be transmitted in one step the SINR condition is satisfied for packet k: β i S\{k} p i d(s i, t k ) α Putting this together, we get f(x k ) β x α k i S\{k} p k d(s k, t k ) = f(x k) α x α k p i d(s i, t k ) α. y α i ( i S\{k} f(x k ) x α k y κ 2/α i ) = α 2 ( S )( α κ 2/α ) α f(x k). x α k 28 Thomas Keßelheim

37 4.. Lower Bounds In case (b) we choose x i = y i such that x i+ κ 2/α x i and b 2 f(x i) b for all i [N]. Again, let S be a set of packets that can be transmitted together in one step; this time let be k = max S. For i S it holds that d(s i, t k ) = k x j + j=i k j=i+ y j 2 k x j 2 j=i k j=i κ 2(k j)/α x k 2 κ 2/α x k. The SINR condition states for packet k β i S\{k} p i d(s i, t k ) α p k d(s k, t k ) b. α x α k Thus: β b x α k i S\{k} p i d(s i, t k ) α i S\{k} b 2 ( x κ 2/α k ) = α 2 ( S )( α κ 2/α ) α b 2 (x k ) α. In case (c) we choose x i = y i such that x i+ κ 2/α x i and f(x i+ ) f(x i ) for all i [N]. Let S be a set of packets that can be transmitted together in one step; k = max S. The SINR condition is satisfied for packet k: β i S\{k} Now, we have: f(x k ) β x α k p i d(s i, t k ) α i S\{k} p k d(s k, t k ) f(x k) α x α k p i d(s i, t k ) α i S\{k} 2 2 α β( κ 2/α ) α f(x k ) 2 ( x κ 2/α k ) = α 2 ( S )( α κ 2/α ) α f(x k) (x k ) α In all three cases, we have S +. This means only a constant number of packets can be transmitted in one step and therefore Ω(N) steps are needed when using p i = f(d(s i, t i )). In contrast, there is a power assignment, p i = κ i, which needs only one step in each of these instances. Note that in all instances it holds that y i x i and y i+ κ 2/α x i. Let us first bound the interference on a receiver t j by senders s i with index i < j: i<j p i d(s i, t j ) α i<j p i x α j = x α j κ i κ i<j κ j x α j. The interference by the remaining senders: i>j p i d(s i, t j ) α i>j p i κ 2(i j) x α i κ2j x α i i>j κ κ2j κ j+ i x α i κ = κ κ j x α j. Thomas Keßelheim 29

38 CHAPTER 4. DIFFERENT POWER ASSIGNMENTS Summed up we get the total interference on a receiver t j : i j p i d(s i, t j ) 2 α κ κ j x α j = β p j d(s j, t j ) α. We can see that the SINR condition is satisfied. This means all packets can be transmitted in a single time slot using this power assignment. In total, we see that particularly the linear and the uniform power assignment can have a the worst possible performance that is even fulfilled by trivial schedules. 4.2 Upper Bounds Having a closer look at the instances used as counter-examples in the last proof, it shows that the distances differ very much. The smallest value is whereas the largest is Ω(2 N ). In this section we will show that this large growth has to occur in worst-case examples. To express this difference, we introduce the ratio between the longest and the shortest distance between two nodes which act as a pair of sender and receiver for a hop: = max (u,v) E d(u, v) min (u,v) E d(u, v). We furthermore define the set of indices of senders around a node with distance at most r K r (v) = { i [N] d(s i, v) r }. Our analysis will be non-constructive in the following way: We assume to know an optimal schedule S (using any power assignment) and to split it up. Then we consider the single time slots separately. Each of these time slots corresponds to an instance for which there is a schedule of length. We only need to bound the number of time slots needed for scheduling such a one-step instance. A central result is that we can bound K r (v) in a one-step instance for every node v when 2. Lemma Given a single-hop instance such that all packets may be transmitted in a single time slot using any power assignment. If there is an L > 0 such that L d(s i, t i ) 2L for all i [N], then it holds for all v V, l L: K l (v) β ( ) α 4l +. L 30 Thomas Keßelheim