On Power Efficient Communication over Multi-hop Wireless
Networks: Joint Routing, Scheduling and Power Control
Randeep Bhatia
Murali Kodialam
Bell Laboratories
Lucent Technologies
101 Crawfords Corner Road
Holmdel, NJ 07733, USA
{
randeep, muralik
}
@bell-labs.com
Abstract— With increasing interest in energy constrained multi-hop wireless networks [2], a fundamental problem is one of determining energy efficient communication strategies over these multi-hop networks. The simplest problem is one where a given source node wants to communicate with a given destination, with a given rate over a multi-hop wireless network, using minimum power. Here the power refers to the total amount of power consumed over the entire network in order to achieve this rate between the source and the destination. There are three decisions that have to be made (jointly) in order to minimize the power requirement.
• The path(s) that the data has to take between the source and the destination. (Routing)
• The power with with each link transmission is done. (Power Control).
• Depending on the interference or the MAC characteristics, the time slots in which specific link transmissions have to take place. (Scheduling)
To the best of our knowledge, ours is the first attempt to derive a performance guaranteed polynomial time approximation algorithm for jointly solving these three problems. We formulate the overall problem as an optimization problem with non-linear objective function and non-linear constraints. We then derive a polynomial time3-approximation algorithm to solve this problem. We also present a simple version of the algorithm, with the same performance bound, which involves solving only shortest path problems and which is quite efficient in practice. Our approach readily extends to the case where there are multiple source-destination pairs that have to communicate simultaneously over the multi-hop network.
I. INTRODUCTION
The problem of determining the achievable rate for a given power is a well studied problem in information theory [3]. For example, in the case of an Additive White Gaussian Noise (AWGN) channel the rate R, for a channel with bandwidth W, transmission powerP and noise spectrum densityN0 is given by
R=Wlog2(1 + P
N0W)bits/second.
This formula gives a theoretical upper bound on the rate that can be achieved for a given power level or equivalently a lower bound on the amount of power required to achieve a given rate. One generalization of this problem is to determine the capacity of a system where there are additional (relay) nodes that help transmission between the sender and the receiver. The capacity
region even for one relay node is still an open problem [3]. In this paper, we consider a simpler variant of the problem, where information has to be sent from the source to the destination over a multi-hop network where the transmissions take place on point to point links. We do not consider the case where a node overhears the communication taking place between two other nodes in the network and uses this information to increase the capacity of the channel between the sender and the receiver. The fact that communication takes place on point to point links is generally true in networks where a routing algorithm is used to transmit information from the source to the destination. In this case, when an intermediate node receives traffic, the routing algorithm decides the next hop on the path to the destination and this node forwards traffic along to the next hop node. This is the mechanism that is generally used in ad-hoc and as well as sensor networks. Therefore, one can view this as communication along links in the network. Though the routing algorithm eliminates the broadcast aspect of the wireless channel, the multiple access aspect of the communication channel still remains. Each node along the path from the source to the destination has to ensure that when it is transmitting to the downstream node, that the downstream node is not transmitting to or receiving data from any other node.
Given a particular source node, that wants to communicate at a given rate with a given destination node, the objective of this paper is to determine how this communication takes place across the multi-hop network in order to minimize the overall network power consumption. Given a particular node configuration, is it better to take many short hops on the way to the destination, or is it better to take fewer long hops? Is it more energy efficient to send all the data along a single path or is it preferable to split the data along multiple paths? result to derive a feasible solution to the problem with b paths? The problem becomes more complex when there are many source-destination pairs that have to communicate across the same multi-hop network. In general, there are three components that determine the power required to achieve a given rate.
• The routing problem of determining the paths along
which flow is routed from the source to the destination.
which communication takes place between the nodes in the network.
• Thepower control problemof determining the power with
which the nodes communicate in a given time slot. A more general version of this problem is where given data rates have to be simultaneously achieved between many different source-destination pairs. The objective in this case is to determine the power required to achieve this given rate vector.
With increasing interest in sensor networks where nodes are energy constrained, there has been a recent emphasis on energy efficient routing and scheduling in multi-hop networks [2]. Cruz and Santhanam [4] study the problem of joint routing, link scheduling and power control for wireless multi-hop networks. They present an algorithm to compute an optimal link scheduling and power control policy that minimizes the total average transmission power in the wireless multi-hop network, subject to given constraints regarding the minimum average data rate per link, as well as peak transmission power constraints per node, and multi-access signal interference constraints. In addition, the authors use the link costs obtained from the algorithm to design a joint routing, link scheduling and power control algorithm. Their algorithm although quite efficient has a worst case exponential complexity. Neely et al [8] consider dynamic routing and power allocation for a wireless network with time varying channels and power constrained nodes. They establish the capacity region of all rate matrices that the system can support and develop a joint routing and power allocation policy, for general arrival and channel state processes, which stabilizes the system and provides bounded average delay guarantees whenever the input rates are within this capacity region. The policy is developed by solving a hard optimization problem which can be nonlinear and non-convex in its most general formulation. Heuristics are presented to solve this problem for special cases. The work of ElBatt and Ephremides [5] considers the problem of joint scheduling and power control in multi-hop networks. They consider several general interference models and they develop algorithms for joint scheduling and power control. They do not consider the routing problem. In our work we consider the joint routing, scheduling and power control problem albeit on a simple interference model. There has also been some recent work on joint routing and power allocation in CDMA networks [13]. They approximate the problem as a convex optimization problem and develop techniques for solving this problem. To the best of our knowledge, this is the first performance guaranteed approximation algorithm for a joint routing-scheduling-power control problem.
Though the algorithms developed in this paper are centralized, we believe that with some modifications it is possible to develop a distributed version of the algorithm. The main step in the algorithm is the solution of shortest path problems and there are standard ways of implementing Bellman-Ford algorithm in a distributed fashion. The scheduling algorithm can be solved as a greedy coloring problem. The power control
portion of our algorithm is also a distributed algorithm. There are still non-trivial issues that have to be resolved before this algorithm can be implemented in a truly distributed fashion but the approach shows a lot of potential for distributed implementation.
The rest of the paper is organized as follows: We outline the model and assumptions in Section II. In Section III we illustrate the technical challenges with the help of a simple example. In Section VI we consider a simpler problem of determining the minimum power schedule in the case the link flows are specified. This problem only considers the scheduling and the power control problem. Since there is no end-to-end flows to be achieved, there is no routing component to the problem. Instead of solving this problem exactly, we develop a 2-approximation algorithm for the problem. The structure of the (near) optimal solution to this problem is used to formulate and efficiently solve the general problem in Section VII.
II. MODEL ANDASSUMPTIONS
We consider a multi-hop wireless network with n nodes. The nodes communicate with each other via wireless links. Each node in the network can communicate directly with a subset of the other nodes in a network. Any time a nodeucan transmitdirectlyto nodev, we represent this fact by a directed edge (link) from nodeuto nodev. We represent the nodes in the network and the possible communications between nodes by a directed graphG= (V, E). HereV represents the set of nodes in the network andEthe set of directed edges (links) in the network. We assume that there aremlinks in the network. We do not require that the links be bidirectional. Given a link e∈E, we uset(e)to represent the transmission end of the link andr(e)to be the receiving end of the linke. We say that a link
eis active to mean that there is a transmission fromt(e)to
r(e). We assume that system operates in a synchronous time-slotted mode. For a system that operates in an asynchronous mode, the results in this paper can provide upper bounds on its performance. Given a nodev ∈V, we use Nin(v) to denote the set of links that terminate at nodev. In other words,
Nin(v) ={e∈E:r(e) =v}.
Similarly, for a given nodev∈V, we useNout(v)to denote the set of edges that emanate fromv, i.e,
Nout(v) ={e∈E:t(e) =v}.
We use N(v) to denote Nin(v)∪Nout(v). Note that N(v) is the set of links incident on v. Let δ denote the maximum degree in the network, i.e.,
δ= max
v∈V |N(v)|.
We assume that the length of each time slot isτunits. Letσ(e) represent the path loss on linke. We useP(e, j)to denote the transmit power on link e in time slot j. In other words, we assume that node t(e) transmits with power P(e, j) in time slotj. Thus the received power at node r(e), in time slot j,
1
2
3
4
Fig. 1. An example multi-hop wireless network.
is P(e, j)σ(e). Note that our model includes the special case where the underlying graph is a complete graph (every pair of nodes can directly communicate with each other) in which the path loss for nodes not in the range is set to zero.
A. Interference Model
The interference model that we consider is a synchronous time-slotted joint TDMA/CDMA system where the nodes use unique signal sequences. This is one of the systems considered in [5]. This has also been studied in [12] [6]. Another view of this system is to view the communication as taking place in a spread spectrum mode. We assume that the hopping sequences for different link transmissions in the same neighborhood is designed so that there is no interference between different link transmissions in the same time slot. However, we require that a given node can interact (either send or receive) with at most one node in any time slot. If a solution requires a node to interact with multiple nodes in the same time slot then we say that the solution has aprimary conflict. Therefore we are required to construct primary conflict-free schedules. We assume that a total bandwidth of W Hertz is available for use in this multi-hop network. Since we assume that two links primary conflict-free link transmissions do not interfere with each other, we assume that each link transmission uses the entire bandwidth W. We also assume that each link can be modeled as an Additive White Gaussian Noise (AWGN) channel with a noise spectral density N0. Therefore, if edge e is active in time slotj then
τ Wlog2
1 + σ(e)P(e, j)
N0W
bits of data will be transferred from t(e)tor(e)in time slot j.
III. TECHNICAL CHALLENGES
In this section we identify the technical challenges, for solving the minimum energy flow routing problem for multi-hop wireless network, with the help of a simple example. In this example which is shown in Fig 1 the nodes 2 and3 are each at distance4dfrom both nodes1and node4while nodes
1 and4are distance6dapart. In this example flow at the rate T needs to be routed from node 1 to 4. We assume a path loss model in which received power is inversely proportional to the square of the distance from the transmitter. In other
words for a linkeof length (distance)d, the path loss function σ(e) = 1/d2.
The first issue is of determining the paths and their associ-ated flows over which to route the flow from node1to4. Here the constraints are flow conservation at intermediate nodes and the requirement that flow at the rateTneeds to be routed from node 1 to 4. There are 3 possible 1 to 4 paths P1 = 1,4, P2 = 1,2,4 and P3 = 1,3,4. We consider 3 solutions. SolutionS1routes flow at the rateT on the pathP1. Solution S2 routes flow at the rate T on the pathP2 and solution S3 routes flows at the rate of T /2 on paths P2 and P3 each. The second issue is of determining the schedule for the links that support non zero flow. Here the constraints are that no two links that are scheduled simultaneously can belong to the same node, since a node can only be participating in a single operation (send/receive) with at most one other node at any given time slot. In solutionS1link(1,4)is scheduled in every time slot, with a frequency of 1. In solution S2 links (1,2) and(2,4)cannot be scheduled simultaneously and hence they are scheduled in alternate time slots, each with a frequency of
1/2. In solutionS3link(1,2)can only be scheduled with link
(3,4) and link(1,3) can only be scheduled with link (2,4). Thus in S3 links (1,2) and (3,4) are scheduled together in slots that alternate with the slots in which links(1,3)and(2,4) are scheduled together. Thus all links in schedule S3 have a frequency of1/2. Note that in every solution for this example the sum of the frequencies of all the links that belong to a node is at most1. This is not a coincidence but is required by the scheduling constraints. However in general the scheduling constraint are not necessarily satisfied even when the sum of the frequencies of all the links that belong to a node is at most
1. The rateR(e)of a linke(while it is active) is determined by its flow f(e) and its frequency h(e). More specifically h(e) = f(e)/R(e). Given the rate R(e)of link e the rate of energy consumption (power) for the nodet(e)while link eis active is given byP(e) =N0Wσ(e)
2RW(e) −1
.Thus the rate of energy consumption for the nodet(e)for linkeish(e)P(e). Getting back to the example the rate of link(1,4)inS1isT. Rate of both links (1,2)and(2,4) inS2 is2T each. Finally in solutionS3 each of the links(1,2),(2,4) (1,3) and(3,4) has a rate ofT each. We are now ready to compute the energy consumption for each solution. In the following we set total bandwidthW = 4T. Thus the total rate of energy consumption (power) for solutionS1 is
T T ·N04T·36d 22T W −1 = 27.25N0T d2, for solutionS2 is 2·2T T ·N04T·16d 222T W −1 = 26.51N0T d2, and for solutionS3 is
4· T /2 T ·N04T·16d 22T W −1 = 24.224N0T d2. Thus solutionS3is the best solution among the three solutions (for W = 4T) for minimizing the energy consumption in
routing the given flow. Note that had we chosen W = 2T then solution S1, where no intermediate hops are used, is a lower energy solution than solution S2 which requires going through a relay node. This example shows that in a multi-hop network lower energy may be consumed on longer paths if intermediate relay nodes are used. Also sometimes the lower energy solution requires spreading the flow over multiple paths rather than sending all the flow on a single path. The set of paths from the source to the destination will in general depend on the desired rate.
We will show later that the problem for determining the minimum energy solution can be formulated as a non-linear optimization problem over non-linear constraints. One option is to attempt to solve this using a general purpose solver. (Though it is not clear these solvers can handle non-linear constraints.) The approach that we take is to exploit the combi-natorial structure of the problem to develop an approximation algorithm to solve this problem.
IV. OVERALLSTRATEGY
We now outline the overall strategy to obtain the minimum energy schedule. We obtain an approximation algorithm to this problem in three steps. In the following T denotes flow constraints, S denotes the scheduling constraints and g() is the energy function.
1) We first formulate the minimum energy problem over two sets of vectors f and R (both defined on the set of links). The vectorf represents the achieved data rate flow on links and the vector R represents the rate at which data is transmitted in a given time slot on link e, when it is active. The minimum energy problem is formulated as the following optimization problemP1:
ming(f, R)
(f, R)∈ S f ∈ T.
Let(f∗, R∗)be the optimal solution to this problem and letu∗ denote the optimal solution value.
2) We now consider a restriction of the original problem where we fix anyf = ˜f ∈ T and consider the following optimization problemP2 in the Rvector:
ming( ˜f , R)
( ˜f , R)∈ S
Letθ( ˜f)be the optimal solution value to this problem attained atR˜=γ( ˜f). Note that if we setf =f∗, then
u∗=θ(f∗).
Assume that we can determine a solution to P2 with total energy consumption given by∆( ˜f)such that
∆( ˜f)≤ηφ( ˜f)≤αθ( ˜f),
for some fixed (closed form) function φ(). In other words ∆()is an α-approximate solution to P2. There-fore,
∆(f∗)≤ηφ(f∗)≤αθ(f∗) =αg(f∗, R∗) =αu∗. Therefore ∆(f∗) is an α-approximate solution for P1. Thus if we knew f∗ and if we are able to compute an α-approximate solution to P2 then we would have an α-approximate solution toP1.
3) Assume that φ()defined in the previous step is a fixed function that can be computed in closed form. We now formulate the following optimization problem P3.
minφ(f) f ∈ T.
Letfdenote the optimal solution to this problem. Then, φ(f)≤φ(f∗)and therefore
∆(f)≤ηφ(f)≤ηφ(f∗), and thus
∆(f)≤αθ(f∗) =αg(f∗, R∗) =αu∗. Thus by solving problem P3 and using its optimal solution f to compute an α-approximate solution to P2, where f˜is taken to bef, we get anα-approximate solution toP1. This is the approach that we follow to determine an approximation algorithm for the problem. In Section V, we formulate the optimization problem P1 and we define the constraint set S and the objective function. We formulate and solve problem P2in Section VI.
V. FORMULATION OF THEMIN-POWERPROBLEM In this section we formulate the minimum power problem. Assume that we are given a source node sand a destination nodetand the objective is to determine the routing, scheduling and power control to minimize the total amount of power consumed to achieve a given information rate from s to t. Let P(e, j)denote the transmission power for link e in time slot j. The following lemma (whose proof is given in the Appendix) establishes the fact that it is optimal for a given link to transmit at the same power in every time slot in which it is active.
Lemma 1: Let P(e, j) > 0 represent the transmission
power for linkein some time slotj when linkeis active. In any minimum power schedule P(e, j)is independent ofj.
Therefore we useP(e)to represent the transmission power on link eand use
R(e) =Wlog2
1 + σ(e)P(e)
N0W
to represent the data rate that is achieved over linkein each time slot where link e is active. We now give the set of constraints that the rate vector has to satisfy. We classify these constraints as those imposed by routing and those imposed by scheduling.
A. Routing Constraints
We are given a source nodesand a destination nodetand the objective is to route flow at rateT from the source node to the destination node. Let f(e)denote the flow on linke that results from routing this flow of T units froms tot. Thenf has to satisfy the following constraints:
e∈Nin(v) f(e) = e∈Nout(v) f(e) ∀v=s, t. e∈Nout(s) f(e) =T.
The first set of constraints enforces flow balance at the nodes in the network (exceptsandt). The last constraint ensures that a flow ofTunits is routed from the source to the destination. We let T denote these set of constraints. Thus, from the routing point of view, a flow vectorf is feasible only iff ∈ T.
B. Scheduling Constraints
We now give conditions on R(e) (and hence P(e)) such that the actual flows are schedulable. The results are shown in [6].
Lemma 2: ([6]) Given a set of link flowsf(e) and the set
of R(e)a necessary condition for schedulability is
e∈N(v)
f(e)
R(e) ≤1 ∀v∈V and the sufficient condition for schedulability is
e∈N(v) f(e) R(e) ≤ 2 3 ∀v∈V.
Note that there is a gap between the necessary and sufficient conditions. We can take one of two approaches. We can solve the problem with the necessary conditions and then check if this solution is feasible (schedulable). The schedulability can be checked by solving an edge coloring problem. Edge coloring is NP-hard, however, there are several well known approximations to this problem. In order to check if a solution is schedulable, assume that we use one such heuristic. If solution is schedulable then we are done. If not, we can solve the problem using the sufficient conditions that guarantee that the solution obtained is feasible. The alternate conservative approach is to bypass the first step and solve the problem using the sufficiency conditions. In order to keep our approach general, we assume that the constraint on the flow-rate vector
satisfies
e∈N(v)
f(e)
R(e) ≤β ∀v∈V
for some constant β. Note that setting β = 1 gives the necessary conditions andβ =23 gives the sufficient conditions. To simplify notation we write
S(β) ={R:
e∈N(v)
f(e)
R(e) ≤β ∀v∈V}.
Given a β, the setS(β)is the feasible set of rate vectors that are feasible.
C. Overall Problem Formulation
We now give the objective function and the the constraints that have to be satisfied. We write it in the same format as problemP1outlined in the overall strategy section. Each time link e is active, the rate of energy consumption (power) for nodet(e)is: P(e) =N0W σ(e) 2RW(e) −1 .
Since the flow required on link e is f(e) and each time the link is used we get a rateR(e), linkeis only active R(e)f(e) time. Therefore, the total rate of energy consumption for link eis
e∈E
f(e) R(e)P(e).
Therefore the overall problem of minimizing the total power required to achieve the given link flows is given by
u∗=g(f, R) = min e∈E f(e) R(e) N0W σ(e) 2RW(e) −1 (f, R)∈ S(β). f ∈ T.
Since the problem is set up in the same format as P1, we are now ready to formulate and solve problem P2. For this purpose, we assume that we are given a fixedf˜∈ T and we want to solve the problem in theR variables.
VI. SOLVING THEFIXEDFLOWPROBLEM In this section we assume the following:
• We are given a fixedf˜∈ T and the problem is solved over the rate (R) variables.
• We also assume that the value of β is fixed. Therefore we suppressβ from the setS(β).
Consider the following problem (P2): θ( ˜f) = ming( ˜f , R)
( ˜f , R)∈ S
Letθ( ˜f)be the optimal solution value to this problem attained atR=γ( ˜f). We show how to find an3-approximate solution toP2. Specifically, we find a set of rate vectors for which the value of the objective function ofP2 is∆( ˜f)≤3θ( ˜f). From this it follows that
∆(f∗)≤3θ(f∗) = 3g(f∗, R∗) = 3u∗.
Therefore∆(f∗) is a3-approximate solution for P1. Thus if we knew f∗ then this would imply a 3-approximate solution for P1.
We compute the desired3-approximation step in two steps. In the first step we find an intermediate solution with total powerχ( ˜f) which is not necessarily feasible (the rate vector may not be schedulable without primary conflicts). We how-ever show that the solution is a2-approximate (infeasible) so-lution toP2and that this solution can be modified to a feasible solution with total power∆( ˜f)such that∆( ˜f)≤ 32χ( ˜f). Thus implying that the latter is a feasible3-approximate solution for P2.
A. Intermediate solution
In this section, we describe an approximation algorithm to solve the minimum power problem that gets a solution that uses at most twice the amount of power that is used by the optimal solution. However this solution may not have any link schedules that are free of primary conflicts.
We define an auxiliary set of minimum power problems, one for each node. The minimum power problem at node vis defined as follows. Let
φ(v,f˜) = min e∈N(v) ˜ f(e) R(e)P(e). e∈N(v) ˜ f(e) c(e) ≤β.
Here nodev’s minimum power problem is solved independent of all other nodes in the network resulting in the optimal solution φ(v,f˜). For now we assume that this problem can be solved optimally for each node. Later on in Section VI-B we show how to do this. LetRv(e)denote the rate allocated to linke∈N(v)when the minimum power problem is solved for nodev. Note thatRv(e)is only defined ife∈N(v). Note that each link gets two rate allocation, from the optimal solution of the minimum power problem for each of the two nodes that are incident on the link. The overall power allocation algorithm MIN POWER shown below is used to select one rate value for each of the link and is shown later to yield a good solution to problemP2:
• For each node v solve its min-power problem and determine the value of φ(v,f˜).
• ComputeRv(e)for alle∈N(v). • Set R(e) = max{Rt(v)(e), Rr(v) (e)}. • ComputeP(e) =N0W 2RW(e) −1 .
• Output χ( ˜f) = e∈E Rf(e)˜(e)P(e) as an
approxi-mate solution.
In the next theorem, we show that χ( ˜f) is at most twice the optimal solution value for the problem P2.
Theorem 3: Algorithm MIN POWER computes a
(infeasi-ble) solutionχ( ˜f)for P2 that is at most2θ( ˜f).
Proof: Let R∗(e)and P∗(e) represent the optimal rate
and power allocation, for problem P2, for the fixed f˜∈ T. Since φ(v,f˜) is the optimal solution to the optimization problem at node v, for problem P2, for the fixed f˜ ∈ T, we have e∈N(v) ˜ f(e) R∗(e)P ∗(e)≥φ(v,f˜) ∀v∈V.
Summing these equations over allv∈V, we get
v e∈N(v) ˜ f(e) R∗(e)P ∗(e)≥ v∈V φ(v,f˜). Note that v e∈N(v) ˜ f(e) R∗(e)P ∗(e) = 2 e∈E ˜ f(e) R∗(e)P ∗(e) = 2θ( ˜f).
Therefore 2θ( ˜f) ≥v∈V φ(v,f˜). However by construction
v∈V φ(v,f˜)> χ( ˜f). Therefore, 2θ( ˜f)≥
v∈V
φ(v,f˜)≥χ( ˜f) and the result follows.
Note that the above proof just uses the fact that each link gets rate allocation from exactly two node problems. It does not even require any convexity conditions. However, convexity is needed for solving the individual nodes optimal allocation problem. We now deal with this problem of determining the value ofφ(v,f˜)for all nodesv in the network.
B. Solving the Node Problem
The problem that each node has to solve in order to determine the value ofφ(v,f˜)is the following:
min e∈N(v) ˜ f(e) σ(e)R(e) 2RW(e) −1 N0W e∈N(v) ˜ f(e) R(e) ≤β.
This is a convex optimization problem and can be solved by writing the Karush-Kuhn-Tucker conditions for optimality and solving the problem. The derivative of the objective function with respect to R(e) is not necessarily invertible in closed form. However, it is easy to solve the problem numerically. Since our main objective is to solve the end-to-end flow problem, we approximate the objective function in order to solve the problem in closed form. This approximation is based on the following observation. Consider the expression 2RW(e). Note that R(e)W is the spectral efficiency of the system in bits/Hertz. In most practical systems the spectral efficiency is less than one. This is especially true in multi-hop systems where the nodes are not particularly sophisticated. Therefore we expand the expression in a Taylors series and take the first three terms in the series. We write
2RW(e) ≈1 + ln 2R(e) W + 1 2ln22 R2(e) W2 .
We substitute this approximation in the objective function of the optimization problem to get
N0ln 2 e∈N(v) ˜ f(e) σ(e)+N0 ln22 2W β e∈N(v) ˜ f(e)R(e) σ(e) .
Ignoring the first term which is independent of R(e)and the constant factors in the second term, we solve the following optimization problem: min e∈N(v) ˜ f(e)R(e) σ(e) e∈N(v) ˜ f(e) R(e)≤β.
Associating a dual multiplier λ > 0 with the constraint, the optimality conditions are
˜ f(e) σ(e)−λ ˜ f(e) R2(e) = 0 ∀e∈N(v). We can now solve for R(e).
R(e) = 1 β σ(e) e∈N(v) ˜ f(e) σ(e).
Substituting this value of R(e)in the objective function, we get that eachφ(v,f˜)equals
N0ln 2 e∈N(v) ˜ f(e) σ(e)+ N0ln22 2βW e∈N(v) ˜ f(e) σ(e) 2 . As mentioned earlier the solution of the individual nodes problems is used by algorithm MIN POWER to construct a solution to problem P2. In the next section, we outline how to schedule the given solution.
C. Power Control and Scheduling
In the previous sections we showed how to compute a (not necessarily feasible) solution for P2 with total power χ( ˜f) such that it is a 2-approximate solution. In this section we show how to modify this solution to yield a feasible solution toP2 with total power∆( ˜f)≤32χ( ˜f).
Assume that the current solution (with total power χ( ˜f)) has been determined for β = 1. There may be no feasible solution with β = 1 since this is only a necessary condition. An alternate approach is to be conservative and setβ =23 and solve MIN POWER. This solution can be scheduled without any primary conflict as shown in Lemma 2. The approach that we take is the following: We solve MIN POWER assuming β = 1to get the solution with total powerχ( ˜f). LetR(e)be the rate assigned to edge e by this solution. Next we choose the length of the time slot τ such that all values of Rf(e)˜(e)τ
are integral for all e∈E. Note that the value ofR(e)might be irrational since it involves σ(e). However, choosing τ sufficiently small will suffice in practice. We now introduce
˜ f(e)
R(e)τ edges between t(e) and r(e) and compute an edge
coloring of this graph. Let ∆ be the maximum degree of the graph andξ be the number of colors used to color the graph. Note that∆≤ξ≤32∆. (See Shannon [11] for a proof of this result). Thus 23 ≤ ∆ξ ≤ 1. We scale the values of R(e) in order to make the schedule feasible. Note that R(e)is linear
inβ, therefore we can setR(e) =R(e)∆ξ and compute the corresponding power. Taking this approach guarantees us that the maximum amount of scaling that we have to do is by a factor of 32. If we take the more conservative approach of setting β = 23 ensures that the scaling factor is 32. We now summarize the joint power control and scheduling algorithm POW SCHED as follows:
• Solve MIN POWER usingβ= 1.
• Form the scheduling graph and determine the num-ber of colors ξ.
• Compute the ratio ∆ξ.
• Scale all the rates by this ratio to obtain the rates for the new solution.
• Output∆( ˜f)as the total power used by this solution.
Theorem 4: The algorithm POW SCHED obtains a solution
toP2 that is not more than a factor of three from the optimal solution.
Proof: If there areξcolors in the solution to the problem,
we assume that system operates in a cycle with periodξ. In each time slot, all links having the same color are scheduled for transmission. Since the coloring ensures that there are no primary conflict, all these transmissions will be successful. For β= 1, let the (not necessarily feasible)2-approximate solution for P2 (obtained in the previous section) have total power χ( ˜f). Here we scale down all the rate values for this solution by at most a factor of 32 to yield a new (feasible) solution with total power∆( ˜f). As shown earlier in the previous section that the total power is proportional to
v e∈N(v) ˜ f(e)R(e) σ(e) .
Thus the new solution may use at most 32 more power. Hence
∆( ˜f) ≤ 32χ( ˜f). Together with the proof of Theorem 3 this yields ∆( ˜f)≤3 2χ( ˜f)≤ 3 2 v∈V φ(v,f˜)≤3θ( ˜f). Thus establishing the desired result.
VII. ROUTINGEND-TO-ENDFLOWS
In this section we develop an efficient 3-approximation algorithm for the generalized routing, scheduling and power control problem.
Theorem 5: There exists an efficient 3-approximation
al-gorithm for the generalized routing, scheduling and power control problem.
Proof: Recall that in Section VI we fix a f = ˆf ∈ T,
and solve the following optimization problem for the fixedf: θ( ˆf) = ming( ˆf , R)
( ˆf , R)∈ S(β).
This is just the minimum power problem for the given set of link flows fˆ. From Theorem 4, we know that we can find a solution to this problem of total power∆( ˆf)where
∆( ˆf)≤32
v∈V
φ(v,fˆ)≤3θ( ˆf).
Note thatθ(f∗) =g(f∗, R∗).Therefore whenf =f∗ is used to setup the same optimization problem.
∆(f∗)≤3
2
v∈V
φ(v, f∗)≤3g(f∗, R∗).
Of course, we cannot evaluate this problem since we do not know the value of f∗. However, since we know φ(v, f) in closed form, we now formulate the following optimization problem.
γ= min
v∈V
φ(v, f) f ∈ T.
Let f˜be an optimal solution to this problem. We show later that we can compute f˜efficiently. Note that by the definition
of f˜ v∈V φ(v,f˜)≤ v∈V φ(v, f∗). Thus ∆( ˜f)≤ 32 v∈V φ(v,f˜)≤ 32 v∈V φ(v, f∗)≤3g(f∗, R∗). implying that by using f˜as the flow vector for routing thes totflow, and by using the results developed in Section VI we get the desired 3-approximation algorithm.
In order to simplify some notation we let A=N0ln 2andB= N0ln
22 2W β .
We now show how to solve f˜ in Theorem 5 efficiently. Therefore we solve the following optimization problem (see Section VI): min v∈V A e∈N(v) f(e) σ(e)+B e∈N(v) f(e) σ(e) 2 e∈Nin(v) f(e) = e∈Nout(v) f(e) ∀v=s, t. e∈Nout(s) f(e) =T.
This is a convex (non-separable) optimization problem over linear constraints. Also, note that this is a Quadratically Constrained Quadratic Programmming problem [9], [14], since the objective function can be written as(Xf)T·(Xf) +y·f, for a matrix X, a vector y and the flow vector f (whose components aref(e)). This also shows that the above is also
a Semidefinite Programming Problem [14]. Thus the above problem can be solved efficiently in polynomial time [9], [14]. This establishes that our overall algorithm runs in polynomial time.
The interior point algorithms that are used to solve con-vex Quadratic Programmming or Semidefinite Programming problems although have a worst case polynomial running time tend to ignore the underlying (network) structure of the problem. We therefore, present an alternative practical approach to solving the above problem which may have worst case exponential complexity, but is efficient in practice. This approach is based on the Frank-Wolfe method [7]. This is an iterative method which solves the non-linear problem via a sequence of linear programs. Let the link flow vector, fk, where fk(e) is the flow on link e, represent the solution at the end of iteration kof the algorithm. We now linearize the objective function by expanding it in a Taylor’s Series about the current solution and using only the linear term. Let
H = v∈V A e∈N(v) f(e) σ(e)+B e∈N(v) f(e) σ(e) 2 and wk(v) =4N0ln 22 2W β e∈N(v) fk(e) σ(e) .
Therefore, the objective function in iteration kis
e∈E ∂H ∂f(e) f=fk−1 f(e).
We now solve this linear programming problem over the constraint setT. Letgk+1denote the optimal solution vector to this problem. We now determinefk+1 such that it minimizes the objective function in the interval [fk, gk+1]. This is an optimization problem in a single variable. This method is particularly well suited for our problem because:
• With the linear objective function, the flow problem between the source and the destination in iteration k reduces to solving a shortest path problem between the source and the destination with cost of
∂H ∂f(e) =
2A
σ(e)+wk(t(e)) +wk(r(e)),
on link e.
• Since the objective function is quadratic, determining fk+1 givenfk andgk+1 can be done in closed form. Therefore, the method just boils down to solving a sequence of shortest path problems. We outline this algorithm more formally below:
Set k= 1 andf1(e) = 0 ∀e. For k= 1,2, . . . Computew(v) =2N02W βln22 e∈N(v)√fkσ(e)(e) ∀v Computel(e) =2N0σ(e)ln 2 +wk(t(v)) +wk(r(v)). Compute the shortest pathP∗ betweensandt
using link length l(e). Setgk(e) =T−fk(e)e∈P∗ andgk(e) =−fk(e)otherwise.
Solve for λ∗= minλ∈[0,1]H(fk+λgk). Setfk+1(e) =fk(e) +λ∗gk(e) ∀e∈E end For
The for loop is repeated until the value of fk and fk+1
are close to each other. In most practical cases the algorithm converges after very few iterations.
A. Deriving a Feasible Solution
Once we have this flow from the approximation algorithm, it is easy to construct a feasible solution to the original optimization problem as follows:
• Using the flows given by the Frank-Wolfe method as the link flows use the result in Section VI to derive the optimum rates for all the links.
• Given these link rates, we can formulate the link coloring problem to get the schedule for these links in order to derive the feasible rates for all the links.
• Once the rate and the flow on the links are known, it is easy to get the power that is needed at the links. We will elaborate more on this procedure in the full version of the paper.
VIII. MULTIPLESOURCEDESTINATIONPAIRS In the case where there are multiple source destination pairs, then all the results in the paper carry through directly. The only difference is that when we use the Frank-Wolfe method, we have to solve one shortest path problem for each source-destination pair. The rest of the algorithm as well as the approximation results remain the same as before. Therefore, we can now compute the routing, scheduling and power control for the case where different sources have to send traffic to different destinations.
IX. ILLUSTRATIVEEXAMPLE
We now show the execution of the algorithm on an illustra-tive example. The example is shown in Figure 2. The number next to the links represents the channel gain on the link scaled by a factor of1013. For example, the channel gain on link1−2 is 8.79×10−13. We assume that N0 = 4×10−18mW/Hz and the total bandwidth W = 1M Hz. The source is node
1 2 3 4 5 8.79 7.96 4.81 5.26 4.59 9.54 6.46
Fig. 2. Illustrative Example.
0 2 4 6 8 10 12 0 100 200 300 400 500 600 700 800 900 Total Power (mW)
Desired Rate between 1 and 5 (Kb/s)
Feasible Solution Lower Bound
Fig. 3. Rate versus Power
1 and the destination is node 5. Figure 3 shows the rate-power curve for the example in Figure 2. The x-axis of the graph is the rate from node 1 to 4 and the y-axis is the minimum power required to achieve this rate. In the graph we show both the lower bound on the power requirement and the power requirement of a feasible solution. The solution shows interesting characteristics. If the desired rate is below about 260 kbps, then the optimal path is1−3−5and if the desired rate is above 260 Kbps then part of the rate is achieved on the path 1−2−5 and part of the rate is achieved along
1−3−5. This example illustrates two points. First it shows the performance of the algorithm and second it illustrates the complexity of the joint routing-scheduling-power control problem.
X. CONCLUSIONS
In this paper we studied the joint routing, scheduling and power control problem for multi-hop wireless networks. To the best of our knowledge this is the first attempt at determining the most energy efficient way of communicating data at a given rate, over point to point links in a multi-hop wireless network. The problem is a complex interplay of its three individual components. We formulated the overall problem as an
opti-mization problem with linear objective function and non-linear constraints. We designed an efficient 3-approximation algorithm for the optimization problem, which yields not only a lower bound on the total energy consumption, but also computes a feasible solution to the joint routing, scheduling and power control problem.
REFERENCES
[1] Baker, D.J., Wieselthier, J.E., and Ephremides, A., “A Distributed Algorithm for Scheduling the Activation of Links in a Self-Organizing Mobile Radio Networks”,IEEE Int. Conference Communications, 1982, pp. 2F6.1-2F6.5.
[2] Bambos, N., “Toward Power-sensitive Network Architectures in Wire-less Communications: Concepts, Issues and Design Aspects”, IEEE Personal Communications, 5(3), pp. 50-59, 1998. 1990. Elements of Information Theory, John-Wiley and Sons, Inc., 1991.
[3] Cover, T., and Thomas, J.A., Elements of Information Theory, John-Wiley and Sons, Inc., 1991.
[4] Cruz, R.L. and Santhanam, A.V., “Optimal Routing, Link Scheduling and Power Control in Multi-hop Wireless Networks”,IEEE INFOCOM 2003.
[5] ElBatt, T., and Ephremides, A., “Joint Scheduling and Power Control for Wireless Ad-hoc Networks”,IEEE INFOCOM 2002.
[6] Hajek, B., and Sasaki, G., “Link Scheduling in Polynomial Time”,IEEE Transactions on Information Theory, 34(5), pp. 910-917, 1988. [7] Minoux, M.,Mathematical Programming: Theory and Algorithms,
John-Wiley and Sons, Inc., 1986.
[8] Neely, M.J., Modiano, E. and Rohrs, C.E., “Dynamic Power Allocation and Routing for Time Varying Wireless Networks”,IEEE INFOCOM 2003.
[9] Nesterov, Y. and Nemirovskii, A., “Interior-Point Polynomial Algorithms in Convex Programming”,SIAM Studies in Applied Mathematics, SIAM, 1994.
[10] Post,M.J, Kershenbaum, A.S. and Sarachik, P.E., “ Scheduling Multihop CDMA Networks in the Presence of Secondary Conflicts”,Algorithmica, 1989, pp. 365-393.
[11] Shannon, C.E., “A Theorem on Coloring the Lines of a Network”,J. of Math. Physics, 28, pp. 148-151, 1949.
[12] Wieselthier, J.E., Barnhart, C.M., and Ephremides, A., “A Neural Net-work Approach to Routing Without Interference in Multihop NetNet-works”,
IEEE Transactions on Communications, 1994.
[13] Xiao, M., Johansson, M., and Boyd, S.P., “Simultaneous Rout-ing and Resource Allocation via Dual Decomposition”, 2002. http://www.stanford.edu/ boyd/srra.html
[14] Vandenberghe L. and Boyd. S. “Semidefnite programming”, SIAM Review, Vol 38, pp 49-95, 1996.
XI. APPENDIX
Lemma 6: Let P(e, j) > 0 represent the transmission
power for linkein some time slotj when linkeis active. In any minimum power schedule P(e, j)is independent ofj.
Proof: Consider two time slotsj andkwhen edgee is
active. LetP(e, j)andP(e, k)=P(e, j)be the transmission power in these two time slots. We want to show that by setting the powers to some P(e) = P(e, j) = P(e, k) we can satisfy all the constraints while reducing the power consumption. This follows directly from the fact that the rate is a concave function of the power. More formally LetR(e, j) andR(e, k) represent the rates on linke in time slots j and k with transmission powersP(e, j)and P(e, k)respectively. Then R(e, j) =W τlog2 1 + σ(e)P(e, j) N0W and R(e, k) =W τlog2 1 + σ(e)P(e, k) N0W .
The power consumed over the two time slots is
0.5 (P(e, j) +P(e, k)) and the average rate over the two time slots is 0.5(R(e, j) + R(e, k)). By Jensen’s inequality (since the rate is a concave function of the power), it is easy to show that transmitting at the average power
0.5 (P(e, j) +P(e, k)) in both time slots j and k results in a mean rate greater than0.5(R(e, j) +R(e, k)).This implies that it is possible to achieve the same mean rate using less power.
