Dynamic Programming for Scheduling a Single Route in Wireless Networks Gyouhwan Kim

Rohit Negi

[email protected] [email protected] Department of Electrical and Computer Engineering Carnegie Mellon University 5000 Forbes Ave., Pittsburgh, PA 15213

Abstract— Multi-slot resource scheduling in a general two dimensional wireless ad hoc network, is a hard problem with no known polynomial-time solution. Recent optimization theoretic analysis, structures this multi-slot scheduling problem into a polynomial number of particular single-slot sub-problems, thus isolating the hardness. We consider this hard sub-problem, under the useful topology restriction of a single route of many links. We introduced a novel interference model that allows for a dynamic programming (DP) algorithm to solve this sub-problem. The developed DP algorithm provides a solution with a provable accuracy in polynomial time. Incorporating this novel solution in a general resource scheduling framework, a multi-slot scheduling algorithm for single route networks is developed. The excellent accuracy of the algorithms is demonstrated through extensive simulations in various scenarios.

Keywords: Wireless networks, ad hoc networks, scheduling, MAC, dynamic programming, cross layer design. I. I NTRODUCTION In wireless ad hoc networks, distributed nodes communicate with each other through the shared wireless medium, causing interference to unintended nearby receivers, due to the broadcast nature of wireless transmission. Thus, the scheduling problem of assigning the transmission time to each node appropriately is crucial for the efficient use of limited wireless resource. In general, however, this scheduling problem is a hard non-convex problem since the link capacity in a wireless network, is a non-convex function of transmission powers of multiple nodes, due to interference [1], [2]. Even without power adaptation, it is an combinatorial problem with an exponential complexity [3]. Thus, an efficient polynomial-time algorithm with guaranteed accuracy is not available yet. Traditionally, the scheduling problem is therefore transformed into a simpler coloring problem using a disk graph model. In a disk graph, each link is represented by a disk with a certain radius, and two links are considered to interfere (or contend for the common resource) with each other, if the corresponding two disks intersect [4]. Thus, the interference is modeled in a pairwise manner instead of considering all interferers together. Given a disk graph with multiple disks representing links, the multi-slot scheduling problem is reduced into a graph coloring problem, which minimizes the This work was supported in part by the National Science Foundation under awards CNS-0347455 and CNS-0520153, and by Samsung Electronics.

chromatic number required to cover all the links while no two links with intersecting disks are assigned the same color. Here, each color would be interpreted as the equal-sized resource, such as a time slot of equal length. Thus, the disks prevent close-by links (i.e., the potential interfering links) from being assigned the same color (i.e., the same time slot). However, even with this geometric simplification of a wireless network, the graph coloring problem still remains NP-hard and hence, no polynomial-time algorithm with the chromatic number arbitrarily close the minimum, exists [4], [5]. Further, the disk graph either distorts or ignores many underlying physical layer aspects. In addition to the oversimplified pairwise interference model, the resources are assumed to be equal-sized and the transmitter and the receiver of a link are assumed to be colocated. The low efficiency of graph coloring in some physical layers has been shown in [6], demonstrating the importance of an accurate physical layer model. Towards solving the scheduling problem with an accurate modeling of the underlying physical layer, an optimization theoretic approach has been applied in one of our previous work [1]. In [1], instead of resorting to a geometric model, Shannon’s capacity formula [7] is used for link rate, allowing detailed physical layer aspects to come into play without distortion. The multi-slot max-min rate scheduling problem (MS MMRP) which maximizes the minimum link rate amongst all links, is formulated as an optimization problem and the dual problem is analyzed. Using the dual analysis, it has been shown that the MS MMRP can be viewed as a series of sub-problems requiring maximization of the sum rate of links in a single slot with a given weight vector, i.e., a singleslot weighted sum rate maximization problem (SS WSRP). Unfortunately, it turns out that the SS WSRP has the inherent hardness of handling interference and thus, does not allow any efficient algorithm with guaranteed accuracy. In some other optimization theoretic works such as [2], [3], it was shown that even with a different physical layer assumption (e.g., discrete rate or no power adaptation), and a different scope of the problem (e.g., joint routing and scheduling problem), the hardness of handling interference boils down to the SS WSRP. Thus, the lack of an efficient solution to the core bottleneck problem SS WSRP, makes it difficult to design an efficient algorithmic solution with guaranteed accuracy, for a general ad hoc networking problem which involves scheduling.

Given the two facts mentioned above, 1) the SS WSRP is a bottleneck in various ad hoc networking problems, and 2) it is a hard problem for an ad hoc network with arbitrary topology in 2-dimensional space; this paper, first solves the SS WSRP after restricting the network topology to a single route. Solving the SS WSRP for a single route network still has significance, since no algorithm exists, which solves the scheduling problem even for networks completely in 1-dimensional space (i.e., a straight line). Also, it is an interesting problem by itself, since an algorithmic solution to the SS WSRP for a single route, can be used for solving the MS MMRP, which maximizes the end-to-end data rate of a multi-hop route, from the source to the destination. Since, we now consider only a single route, a dynamic programming (DP) technique may be applied to solve SS WSRP along the route. Intuitively, the sequence of links forming the route provides the concept of stages in a DP. Also, a new simplified interference model is introduced, to trade-off between the complexity of the DP and the accuracy of the solution. In the interference model, only the nearest K interferers on a route, on each side of a node, are considered instead of all the interferers. The 2K interferers are chosen as the state variables. Thus, a polynomial-time DP algorithm, similar to a Viterbi algorithm [8], is developed to find the optimal power sequence for links along the route, in stages. Using the relation between the considered interference model and the true interference, this paper shows that upper and lower bounds for the optimum weighted sum rate can be obtained from the solution of the DP algorithm. Further, the tightness of bounds is confirmed through intensive simulations on a large number of randomly generated single link networks. Next, the DP algorithm is applied to a general max-min resource sharing framework proposed by a recent optimization theoretic result [9], and an optimization algorithm for a multislot max-min rate problem (MS MMRP) for a single route, is obtained. Utilizing the DP algorithm as the core sub-problem (SS WSRP) solver, this optimization algorithm maximizes the minimum rate amongst all links constituting the route. The runtime of the optimization algorithm is polynomial in the number of links N , since it converges in a polynomial number of iterations and uses the polynomial-time DP algorithm in each iteration. The simulation result for a randomly generated single route network with 50 nodes, shows that the developed optimization algorithm provides tight performance bounds for the optimum. Therefore, this paper provides a performance verification and design tool for heuristic scheduling algorithms targeting practical use. Obtaining guidelines for practical protocol design is crucial, since significant change or complete redesign of current centralized multiple access control (MAC) protocols for ad hoc networks seems unavoidable, as shown in [10]. The rest of the paper is organized as follows. First, the interference model is introduced, and the appropriately modified link rate formula are described in Section II. Next, the DP algorithm to solve the SS WSRP is presented in Section III, with a proof on the accuracy of a posterior guarantee. In Section IV, optimization framework developed

in [9], is applied to the MS MMRP for single route networks, demonstrating the utility of DP in general optimization theoretic scheduling frameworks. The accuracy of the proposed algorithms is investigated through simulation results in Section V, and Section VI concludes the paper. II. I NTERFERENCE M ODEL As mentioned in the previous section, the intrinsic hardness of the scheduling problem is due to the interference between links. In a wireless network, the rate of a link is affected by all other transmitting links, through interference. Thus, technically, the optimal choice of transmitting links has to be made considering all the links together. However, in reality, a transmitted signal in wireless medium attenuates quickly with distance. Thus, the interferers in the vicinity of a receiver are expected to be dominant, especially in the case of no power adaptation with a common transmission power limit, as is the case of most current wireless local area network (WLAN) devices. In addition, the number of dominant interferers would be very small, since any efficient scheduling or MAC scheme probably does not allow simultaneous transmission of nearby nodes. Therefore, we consider a modified link capacity formula with only a few nearly placed interferers. The modified link rate ci of link i, after applying the interference model is written as, 
pi g(i, i + 1)  . (1) ci = W log 1 + (SN R−1 + j∈Ji pj g(j, i + 1))Γ W is the system bandwidth and each link is assumed to operate below Shannon capacity [7] by a gap Γ. pi is the normalized transmission power of node i, which takes 0 or 1 as its value, since we assume no power adaptation. g(i, j) is the gain from node i to j normalized by the closed-in distance gain g0 . No mobility or fading is considered. Thus, channel gains are constant and follow the simple path loss model g(i, j) = 1/d(i, j)α , with the path loss exponent α ≥ 2, where d(i, j) is the distance between nodes i and j normalized by the closed-in distance d0 . g(i, i + 1) is the normalized signal gain of link i, since the network topology considered in this paper is a single route, and we number the links and the nodes from the source to destination. SN R is the normalized signal to noise ratio defined as SN R = P0 g0 /W N0 , where P0 is the maximum transmission power and N0 is the background noise power spectral density (PSD). Ji is the set of the 2K interferers, K on each side of link i, on the route. Note that if Ji includes all the links instead of the 2K interferers, then the modified rate function becomes the same as Shannon capacity formula, with a gap Γ. Thus, the number of considered interferers is the only simplification we assume in this paper, for the interference. Therefore, though the interference model does not model the true interference exactly, it will be more accurate than the pairwise models in graph theoretic approaches, as long as the number of interferers considered is greater than 1. Also, the underlying physical layer aspects, ignored or distorted in disk graphs, are considered properly.

With this interference model, a DP to solve the SS WSRP for single route networks is devised in the next section. III. DP FOR W EIGHTED S UM R ATE M AXIMIZATION DP is an optimization technique for solving a problem as a series of sub-problems [8]. An example of a DP is the Viterbi algorithm, where each feasible solution is a sequence of states (“path through a trellis”). The Viterbi algorithm finds the path with the smallest path metric, where the path metric is defined as the sum of branch metrics. To apply a DP algorithm to the SS WSRP, we first need to define the state properly, so that the SS WSRP can be framed as a DP. We can define the sequence of stages in the trellis as the sequence of nodes along the route, from source to the destination. At each stage i, we define the current state as . (1) (2) (K) (1) (2) (K) (k) si = (si , si , ..., si , di , di , ..., di , pi ), where si ’s for k = 1, 2, ..., K are the node indices of the nearest K interferers to the left of link i (i.e., between the source node and (k) transmitter of link i). di ’s are the node indices of the nearest K interferers to the right of link i (i.e., between the receiver of link i and the destination node). pi ∈ {0, 1} is the normalized transmission power of link i, which is binary because there is no power adaptation. Notice that the assumption of K nearest (also dominant) interferers on both sides of a receiver in (1) implies that state si uniquely determines the rate of link i. However, this itself is not sufficient for si to be considered a state variable of a DP. But, with the following additional condition on the route, si becomes a state variable for our problem. d(i, j) ≤ d(i − i0 , j + j0 ), ∀i < j & i0 , j0 ≥ 0,

(2)

The above condition guarantees that as one moves from left to right in the trellis, interfering nodes that are discarded because they are not dominant, do not have to be reintroduced at a subsequent stage. Assuming that the route satisfies this condition, si is a state variable, because si is completely determined by si−1 and si+1 . With the trellis defined by the state si , the problem is to find the path (determined by the sequence of powers pi ) throughthe trellis, that maximizes N the weighted sum rate λT c = i=1 λi ci . This is a classic problem that can be solved by a DP. The DP algorithm effectively examines all possible sequences of pi ’s leading to each state, keeping track of only the best sequence up to the state. Let λ be the weighting vector of the SS WSRP, Si be the set of feasible states in stage i, and Si−1 (si ) be the set of states in stage i − 1, from which a transition to the state si in stage i is possible. The DP algorithm for the SS WSRP can be written as below.

Fig. 1.

Trellis diagram with N = 5, K = 1 and example path

Ri (s) is the maximum weighted sum rate up to stage i resulting from the power sequence {p1 , p2 , . . . , pi }, such that the . path terminates in state s of stage i. Thus, R∗ = max RN (s) is s∈SN

the maximum weighted sum rate. The corresponding optimum power sequence p∗ obtained by DPsolver (3), solves the SS WSRP. The runtime of DPsolver depends on the number of  stages N  and number of states per stage |Si | =  Kthe N K −i )( ) and thus, is O(N 2K+1 ). 2( k=0 i−1 k=0 k k Fig. 1 shows a part of an example trellis with the path corresponding to a power sequence {1, 0, 0, 1, 0}. In particular, note that the set of dominant interferers of the receiver of link i is completely determined by those corresponding to links i − 1 and i + 1, due to assumption (2). Thus, si is a state variable, as required for DP. DPsolver solves the SS WSRP not for the true rate function (with the total interference from all transmitters), but for the modified rate formula (1), which only includes the 2K dominant interferers. Thus, in general, p∗ is not optimal for SS WSRP with the true rate function. The true weighted sum rate, say C(p∗ ), evaluated by the solution p∗ of DPsolver, is less than the optimal weighted sum rate C ∗ achieved by the true optimal power vector q∗ . Interestingly, however, the following theorem holds for the true weighted sum rate C(p∗ ) and the optimum C ∗ . Theorem 1: If

C(p∗ ) R∗

= ρ, then C(p∗ ) ≥ ρC ∗ .

Proof: Suppose R(q∗ ) is the weighted sum of the modified rate functions ci ’s, obtained by the true optimal power sequence q∗. Then, R∗ ≥ R(q∗ ), because DPsolver solves the SS WSRP with respect to the modified rate function optimally. Also, the modified link rate ci in (1) cannot be smaller than the true rate for any power vector p, since the interference model considering the nearest 2K interferers underestimates the true interference. This implies R(q∗ ) ≥ C ∗ . Therefore, R∗ ≥ C ∗ and hence, C(p∗ ) ≥ ρC ∗ if C(p∗ ) = ρR∗ .

DPsolver: R1 (s) = λ1 c1 (s) Ri (s) =

s ∈ S1 ,

max Ri−1 (ζ) + λi ci (s),

ζ∈Si−1 (s)

s ∈ Si , i = 2, 3, ..., N.

(3)

Theorem 1 shows that the solution p∗ of DPsolver provides an upper bound R∗ , as well as a lower bound C(p∗ ) to the optimum C ∗ . If these two bounds are close to each other, then the solution to the SS WSRP is accurate. Also, the run-time of DPsolver is O(N 2K+1 ). Therefore, DPsolver is polynomial-

time algorithm with a provable guarantee on the accuracy. While no accuracy can be guaranteed apriori (because of the underlying NP hard problem), a provable accuracy can be obtained after running the DP. In the next section, we apply DPsolver to a multi-slot scheduling problem, to complete the design of the scheduler for single route networks. IV. E XTENSION TO M ULTI - SLOT MAX - MIN RATE PROBLEM In general, transmitters of links located close together cannot send their signals at the same time, since they cause interference to each other. Thus, the shared wireless resource, i.e., time and/or frequency, needs to be divided into multiple slots and assigned to the contending links. As shown in [1], [2], [3], this multi-slot scheduling problem can be viewed as a series of single-slot problems which are, in fact, SS WSRPs with different weighting vectors. Therefore, we can apply DPsolver (3), obtained in Section III, to various multi-slot problems presented in [1], [2], [3], for the network topology satisfying (2). Among the various scheduling problems, we choose to solve the multi-slot scheduling problem, which maximizes the minimum rate amongst all links (i.e., the MS MMRP), since this is the same as maximizing the throughput between the source and the destination nodes in single route networks considered in this paper. Towards solving the MS MMRP efficiently, DPsolver is applied to an optimization framework for a general max-min resource sharing problem from [9]. For this purpose, we first review the optimization framework in [9] briefly, and check its applicability to the MS MMRP. The max-min resource sharing problem considered in [9] is, γ ∗ = max{γ : f (t) ≥ γe, t ∈ T },

(4)

where f is a vector of N functions fn ’s, and e is a vector of 1’s with appropriate length. Thus, this problem maximizes the minimum among all fn ’s. In [9], assuming certain conditions, it is shown that the interior point method based optimization algorithm in [9], converges to a solution with an approximation ratio of (1 − ), in at most O(N (ln N + −2 )) iterations, for any 0 <  < 1. i.e., the objective value is at least (1 − ) times of the optimum value. The conditions assumed in [9] are: 1) fn ’s are non-negative, continuous concave functions of t, 2) T is a non-empty, convex, compact set, and 3) there exists an exact solver for the following sub-problem Λ(λ) that needs to be solved in each iteration of the optimization algorithm. Λ(λ) = max{λT f (t) : t ∈ T } ,

(5)

for λ ∈ U = {λ : eT λ = 1, λ ≥ 0}. One can easily derive Λ(λ) by taking the dual of the convex problem (4) since Λ(λ) is the Lagrange dual function of (4) [11], [9]. Now, we check the applicability by writing the MS MMRP in the same form as this general optimization framework and showing that the required three conditions are satisfied for

MS MMRP. MS MMRP written in the form of (4) is γ ∗ = max{γ : Ct ≥ γe, t ∈ T }

(6)

T = {t : eT t = 1, t ≥ 0} ,

(7)

γ, t

where t is the normalized time slot size in the unit simplex T . C is a N -by-2N matrix containing link rate vectors c’s as its columns. Each c is a constant vector corresponding to a binary power vector p of length N . Thus, Ct is a convex combination of all possible rate vectors, representing the rate of links obtained by a general continuous size multi-slot scheduling. The objective γ is the minimum rate amongst all links, thus, (6) is the MS MMRP. Next, we compare (6), (7) with (4), to show that the conditions for the applicability of the optimization algorithm in [9] to the MS MMRP are satisfied. Condition 1) holds, since Ct corresponding to the function f of (4) is linear in t and non-negative. Condition 2) on the set T is clearly satisfied, since T is a unit simplex in finite (N ) dimension (7). Also, for the MS MMRP, Λ(λ) = max{λT c}, since maximizing λT Ct over the unit simplex T is the same as finding c, and the corresponding p maximizing λT c, i.e., exactly SS WSRP. Thus, condition 3) is satisfied, since DPsolver (3) solves the SS WSRP. Consequently, one can get a solver for the MS MMRP, (let’s call it MS MMRPsolver) using the optimization algorithm in [9], and utilizing DPsolver as the sub-problem solver. Furthermore, due to the properties of the optimization framework in [9] and DPsolver, the following theorem holds. Theorem 2: MS MMRPsolver provides both upper and lower bounds to the optimum of the MS MMRP in polynomial-time. Proof-Part 1: MS MMRPsolver solves the MS MMRP with respect to the modified rate function ci in (1), with the approximation ratio (1 − ). Thus, the solution of MS MMRPsolver, say t, results in the corresponding γ(t) ≥ (1 − )γ ∗ . As in the proof of Theorem 1, due to the interference model underestimating the true interference, the true optimum maxmin rate, say θ∗ , is less than or equal to γ ∗ and hence, γ(t) ≥ (1 − )θ∗ . Next, once the solution t is obtained by MS MMRPsolver; using the solution, one can easily calculate the true minimum rate amongst all links, by considering all interferers. We call it θ(t). Now, if θ(t)/γ(t) = ρ, then this and γ(t) ≥ (1 − )θ∗ give us θ(t) ≥ ρ(1 − )θ∗ . Therefore, θ(t) and lower MS MMRPsolver provides an upper bound ρ(1−) bound θ(t) to the true max-min rate θ∗ , with a approximation ratio ρ(1 − ). Proof-Part 2: The runtime of MS MMRPsolver is O(N (2K+2) (ln N + −2 )), i.e., polynomial in N . This is because the optimization algorithm in [9] guarantees convergence to a solution with approximation ratio (1 − ), in O(N (ln N + −2 )) and the runtime of DPsolver is O(N (2K+1) ). Next, the accuracy of the developed algorithms is investigated through simulations in various cases.

SNR/SNRUWB = −20 dB

1

0

50

100 150 SNR/SNRUWB = 0 dB

200

37

0

208 SNR/SNRUWB = 40 dB

37

Minimum Approximation Ratio

37

0.8

0.6

0.4

0 −20 0

Fig. 2.

Sample route and transmitting nodes: K = 2, α = 3

0 10 20 SNR/SNRUWB (dB)

30

40

Accuracy of DP solution: minimum approximation ratio

25

0.8

0.6

0.4 α=2, K=1 α=2, K=2 α=3, K=1 α=3, K=2

0.2

0 −20

Fig. 4.

Mean Number of Transmitting Links

Mean Approximation Ratio

−10

208

1

Fig. 3.

α=2, K=1 α=2, K=2 α=3, K=1 α=3, K=2

0.2

−10

0(UWB) 10 20 SNR/SNRUWB (dB)

30

40

Accuracy of DP solution: mean approximation ratio

V. S IMULATION R ESULTS We first verify the accuracy of DPsolver by solving SS WSRP with equal weight for all links and then investigate the performance of MS MMRPsolver. As mentioned in Theorem 1, DPsolver provides a solution to SS WSRP with an approximation ratio (i.e., the ratio between the upper and lower bounds) which is unknown before the simulation. Thus, verifying the accuracy of DPsolver requires an intensive simulation for various cases. Since the interference model considers only a few dominant interferers, the accuracy of DPsolver will be determined by the ratio between the considered and neglected interference. This ratio is also dependent on the number of transmitting links on a route and the path loss exponent. Therefore, we chose SN R and α as the simulation parameters. Intuitively, a very large noise value (equivalently, very low SN R) renders the interference negligible, allowing simultaneous transmission on a large number of links. The number of considered interferers on each side of a link K, was also chosen to be a simulation parameter, to determine how many interferers need to be considered to model the interference accurately. To verify the accuracy of DPsolver for most systems of practical interest, we performed the simulation over the SN R range −20dB ≤ SN R/SN RU W B ≤ 40dB, where SN RU W B is the the SN R of a Ultra wide band (UWB) system using 7.5GHz under −41.3dBm/MHz PSD limit [12]. SN R/SN RU W B for a system using Unlicensed National Information Infrastructure (UNII) 100MHz at 5.2GHz under

α=2, K=1 α=2, K=2 α=3, K=1 α=3, K=2

20

15

10

5

0 −20

−10

Fig. 5.

0 10 20 SNR/SNRUWB (dB)

30

40

Mean number of transmitting links

the transmission power limit of 40mW is 37.3dB. Thus, the SN R range covers both systems that operate in a very low SNR regime (e.g., UWB) and those that operate in a high SNR regime (e.g., 802.11a). K was set to 1 or 2 and α was set to 2 or 3. Other parameters were kept fixed throughout the simulation. Γ was set to 16.3dB based on the sensitivity requirement in one of the UWB standard proposals [13]. Current 802.11a/g WLAN devices also operate about 16dB below the Shannon capacity, though the standards have set requirements further below the capacity [14]. g0 , the gain at closed-in distance d0 , was set to −46.9dB considering d0 = 1m and the center frequency of 5.3GHz, i.e., the center of UWB frequency band from 3.1GHz to 7.5GHz. We randomly generated 1000 single route networks with 50 links satisfying the condition (2). The length of each link normalized by d0 , was chosen randomly with a uniform distribution between 1 to 10. Fig. 3 and 4 show the mean and the minimum approximation ratios obtained from the solutions of DPsolver, for 1000 sample networks, respectively. For all cases, the mean is greater than 0.9 and even the minimum is greater than 0.8 demonstrating the robustness and accuracy of DPsolver. In particular, when K = 2, the mean approximation ratio is greater than 0.975. This shows that in single route networks, the true interference from all interferers can be modeled very accurately, with only 2K = 4 interferers. The effect of SN R and path loss exponent on the accuracy

1

Approximation Ratio

0.8

0.6

0.4 α=2, K=1 α=2, K=2 α=3, K=1 α=3, K=2

0.2

0 −20

Fig. 6.

−10

0(UWB) 10 20 SNR/SNRUWB (dB)

30

40

Accuracy of multi-slot scheduling solution

ifying the approximation ratio of the optimization algorithm in [9] was set to 0.1. Thus, the maximum possible approximation ratio that MS MMRPsolver can achieve is 0.9 in this case, though we can increase it by reducing  further. Fig. 6 shows that the approximation ratio is close to 0.9 in most of the cases, and especially when K = 2. Thus, DPsolver performed well in conjunction with the general optimization framework in [9], demonstrating its utility as a core sub-problem solver for a general ad hoc networking problems which involve scheduling for a single route. In summary, the performance of both DPsolver and MS MMRPsolver were verified. VI. C ONCLUSION

of DPsolver was also observed in Fig. 3 and 4. A high approximation ratio was maintained over a wide range of SN R. In particular, regardless of the value of K, when SN R was very low or α = 3, the approximation ratio was very close to 1, indicating that in these cases the solution of DPsolver is almost as good as the optimum. This is because when the SN R is very low, noise is much larger than the interference, and when α = 3, interference from the transmitters farther than the K nearest interferers on each side of a link, is negligible due to the large path loss. While Fig. 3 and 4 demonstrate the accuracy of DPsolver in terms of the resulting approximation ratio, Fig. 2 and 5 show the behavioral correctness of DPsolver. First, Fig. 2 shows the transmitting nodes (the circled nodes) in a sample network. As expected, the transmitting nodes are fairly regularly distanced. This supports the highly accurate results for α = 3 shown in Fig. 3, 4. If we assume roughly equaldistanced transmitting links, as observed in Fig.2, the portions of interference caused by nodes farther than the first and the second interferer are 0.20205 and 0.0775, respectively. These values computed using the Riemann’s zeta function, as ∞ were −n k = 1.20205, for n = 3. In Fig. 2, we also noticed k=0 that no consecutive links were selected for transmission. This is because simultaneous transmission on consecutive links requires simultaneous transmission and reception on a node, causing infinite amount of interference to itself. Thus, no more than half of the total number of links, 25 in this case, were selected by DPsolver. Fig. 5 shows the trend that more links transmit in low SN R cases, which matches well with the intuition about the effect of SN R on the number of transmitting links. Further, it was observed that DPsolver was more conservative when K = 2 than when K = 1, in allowing links to transmit simultaneously (the solid lines are below the dashed lines). This result also conforms with our intuition because more interferers are considered when K = 2. Consequently, DPsolver was verified not only in terms of its accuracy, but also in terms the properties of the obtained solutions. Next, MS MMRPsolver was applied to solve the multi-slot max-min rate scheduling problem (i.e., MS MMRP) for the one sample single route network shown in Fig. 2. The  spec-

For single route ad-hoc networks, we considered the singleslot sub-problem, the key step in solving the general multi-slot scheduling problem. A dynamic programming (DP) algorithm for this single-slot sub-problem was developed on the basis of a novel interference model. The developed DP algorithm provides a solution with provable accuracy in polynomial time. Further, the DP algorithm applied to a general optimization framework as a core sub-problem solver, resulting in a polynomial-time algorithm which solves the max-min rate scheduling problem for single route networks. The accuracy of the algorithms was verified by simulations in a wide range of parameters and sample networks. The simulation results also demonstrated the accuracy of the assumed low-complexity interference model. Thus, this algorithmic tool may be used to design practical scheduling and MAC protocols for wireless ad hoc networks. R EFERENCES [1] A. Rajeswaran, G. Kim, and R. Negi, “A scheduling framework for UWB & cellular networks,” Broadnet ’04, Oct. 2004. [2] M. Johansson and L. Xiao, “Cross-layer optimization of wireless networks using nonlinear column generation,” Tech. Report, RTI, Stockholm, Nov. 2003. [3] B. Radunovic and J. L. Boudec, “Optimal power control,scheduling and routing in UWB networks,” Proc. IEEE JSAC, Dec 2004. [4] M. V. Marathe, H. Breu, H. B. Hunt III, S. S. Ravi, and D. J. Rosenkrantz, “Simple heuristics for disk graphs,” Networks, 25:5968, 1995. [5] T. Erlebach, K. Jansen, and E. Seidel, “Polynomial-time approximation schemes for geometric graphs,” Symp. on Discrete Algorithms (SODA), pp. 671-679, 2001. [6] R. Negi and A. Rajeswaran, “Physical layer effect on MAC performance in wireless ad hoc networks,” Proc. IASTED Conf. CIIT ’03, July 2003. [7] T. Cover and J. Thomas, Elements of Inform. Theory, John Wiley, 1991. [8] D. P. Bertsekas, Dynamic programming and optimal control, Athena Scientific, 2000. [9] K. Jansen and L. Porkolab, “On preemptive resource constrained scheduling: polynomial-time approximation schemes,” Proc. Integer Prog. Comb. Optim., 9th IPCO Conference, pp. 329-349, May 2002. [10] S. Xu, and T. Saadawi, “Does the IEEE 802.11 MAC protocol work well in multihop wireless ad hoc networks?,” IEEE Communication Mag., June, 2001. [11] S. Boyd and L. Vandenberghe, Convex Optimization, Cambridge, 2004. [12] Federal Communications Commission, “Revision of Part 15 of the Commission’s rules regarding Ultra-Wideband transmission systems,” FCC 02-08, Apr. 2002. [13] IEEE 802.15 WPAN High Rate Alternative PHY Task Group 3a (TG3a), accessible at http://grouper.ieee.org/groups/802/15/pub/TG3a.html. [14] Maxim Integrated Products, Inc, MAX2828/2829 Single/Dual-Band 802.11a/b/g Transceiver ICs, accessible at http://www.maxim-ic.com