Computer Communications 35 (2012) 628–636

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Computer Communications

journal homepage: www.elsevier .com/ locate/comcom

On optimal spectrum-efficient routing in TDMA and FDMA multihopwireless networks q

Mohamed Saad ⇑Department of Electrical and Computer Engineering, University of Sharjah, Sharjah, United Arab Emirates

a r t i c l e i n f o a b s t r a c t

Article history:Received 14 April 2011Received in revised form 11 September 2011Accepted 3 December 2011Available online 13 December 2011

Keywords:Multihop wireless networksSpectrum-efficient routingTDMAFDMAPolynomial-time algorithms

0140-3664/$ - see front matter � 2011 Elsevier B.V. Adoi:10.1016/j.comcom.2011.12.003

q A part of this paper was presented at IEEE SymCommunications (ISCC), June 2010 [13], and received⇑ Tel.: +971 6 505 0961; fax: +971 6 505 0872.

E-mail address: msaad@sharjah.ac.ae

This paper addresses the problem of finding the route with maximum end-to-end spectral efficiency,under the constraint of equal bandwidth sharing, in multihop wireless networks that use time divisionmultiple access (TDMA) or frequency division multiple access (FDMA). The conceptual difficulty of thisproblem arises from the fact that the associated routing metric is neither isotonic nor monotone, and,thus, it cannot be solved directly using shortest path algorithms. The author has recently presentedthe first polynomial-time algorithm that solves the problem to exact optimality for TDMA networks.The contribution of this paper is twofold. For TDMA networks, we present a new algorithm that achievesa significant improvement in the computational complexity as compared to the algorithms previouslyknown. For FDMA networks, we introduce the first polynomial-time algorithm that provides provablyoptimal routes. The proposed algorithms rely on the divide-and-conquer principle and a modifiedBellman–Ford algorithm for widest path computation. Our computational results further illustrate theefficiency of the proposed approach.

� 2011 Elsevier B.V. All rights reserved.

1. Introduction

Multihop wireless networks consist of a set of wireless devicesthat communicate with each other over multiple wireless hops,with intermediate nodes collaboratively routing ongoing traffic.The study of wireless multihop routing is a foundation for thedevelopment of emerging technologies such as [12]:

� infrastructure wireless mesh networks: wireless routers/accesspoints are interconnected to provide an infrastructure/back-bone for clients; or� WiMAX or IEEE 802.16: a number of fixed relays can be used to

extend the coverage area of a base station, and/or increase thecapacity of a wireless access system, as in IEEE 802.16j relay-based networks [7].

The recent study in [2] has introduced the problem of findingthe path with maximum spectral efficiency in multihop wireless net-works that use time division multiple access (TDMA), under theconstraint of equal bandwidth sharing. On the one hand, theauthors of [2] note that simple shortest path algorithms cannot

ll rights reserved.

posium on Computers anda Best Paper Award.

be used to solve the problem because the resulting routing metricis neither isotonic nor monotone [14]. On the other hand, exhaus-tive search has an exponential computational complexity becauseit involves pre-computing all paths joining a given node pair.Therefore, the study in [2] proposes two efficient, yet sub-optimalheuristics.

We have demonstrated in [12] that the spectrum-efficient rout-ing problem for TDMA networks (originally introduced in [2]) canbe, in fact, solved in polynomial-time. Our algorithm proposed in[12] is based on iteratively invoking (at most) M shortest path pro-cedures, where M is the number of network links. This leads to anoverall computational complexity of O(N2M) if the Dijkstra shortestpath algorithm is used in each iteration, where N is the number ofnodes. Since an N-node network has at most N(N � 1) directedlinks, the worst case complexity of the algorithm presented in[12] is O(N4).

Similarly to [2,12], this study combines tools from informationtheory and networking in an attempt to devise efficient spectrum-efficient routing algorithms that explicitly consider the impact ofthe physical layer. This is in contrast to studies from the informa-tion theory community (see, e.g., [10,17]) which focus on under-standing the fundamental performance limits of the network, butlead to protocols that are typically too complex to be implementedin practical systems [2]. Our work contrasts also with studies fromthe networking community (see, e.g., [3,6]) which are often builton link-level abstractions of the network without fully consideringthe impact of the physical layer [2].

M. Saad / Computer Communications 35 (2012) 628–636 629

The contribution of this paper is twofold.

� For TDMA networks, we present a new algorithm that achievesa significant improvement in the computational complexity ascompared to the algorithms previously known. In particular,we present an algorithm that provides provably optimal routesin O(N3) time.� For networks that use frequency division multiple access

(FDMA), we introduce the first polynomial-time algorithm thatprovides provably optimal solutions to the spectral-efficientrouting problem. The computational complexity of this algo-rithm is also O(N3).

Our proposed approach relies on the divide-and-conquer princi-ple, and the resulting algorithms (for FDMA and TDMA networks)are based on a single run of a modified Bellman–Ford algorithmfor widest path computation.

The remainder of this paper is organized as follows. Section 2provides a formal definition of the problem for TDMA networks.Section 3 provides the novel, improved polynomial-time algorithmfor TDMA networks. The problem formulation, and polynomial-time algorithm for FDMA networks are presented in Section 4.Numerical examples and results will be presented in Section 5.Section 6 concludes the paper.

2. TDMA problem definition

A multihop wireless network is modeled as a graph G = (V,E),where V represents the set of nodes (vertices) and E representsthe set of links (edges). We let l 2 E signify a link in the network.We also let N = jVj and M = jEj denote the number of nodes andlinks, respectively.

Following [2,12], we consider the setting in which all transmitdevices are constrained by the same symbol-wise average transmitpower P, and assume that all devices transmit with power P whentransmitting. A possible justification for this assumption is thatnodes in infrastructure wireless mesh networks are mostly immo-bile and connected with abundant power supplies. Therefore, fora link l 2 E, the signal-to-noise ratio (SNR) is given by:

SNRl ¼PGl

N0B; ð1Þ

where Gl is the path gain from the sender of link l to the receiver oflink l, N0 is the normalized one-sided power spectral density of theadditive white Gaussian noise (at any receiver in the network), andB is the finite bandwidth of the wireless channel.

To avoid the difficulty (NP-hardness) of joint optimal routing andmedium access control (MAC) layer scheduling, and following [2,12],it is assumed that a common channel is shared among all nodes usingtime division multiple access (TDMA) without spatial reuse. In otherwords, each node transmits in its own unique time slot, and uses theentire bandwidth (B) when transmitting. It is demonstrated in [2]that, even though a path is selected assuming no spatial reuse/inter-ference, applying a scheduling technique (separately) that allowssome spatial reuse to the selected path can further improve the spec-tral efficiency. Therefore, our framework can still benefit from spatialreuse. It is worth noticing that the MAC layer of the IEEE 802.16 meshprotocol, for example, is based on TDMA (see, e.g., [5]).

The spectral efficiency R(L) of an arbitrary path L in the networkis defined as the bandwidth-normalized end-to-end rate, i.e.,R(L) = CL/B (in bps/Hz), where CL is the end-to-end achievable datarate (in bps) and B is the channel bandwidth (in Hz) [2].1 Under the

1 In other words, the spectral efficiency is defined as the rate at which data can betransmitted over a path per unit bandwidth. Hence, it is an indication of how efficientthe channel bandwidth (which is a scarce resource) is utilized.

constraint of equal bandwidth sharing, the TDMA achievable end-to-end data rate for path L can be expressed using the well-knownShannon capacity formula as (see, e.g., [4,16]):

CL ¼minl2L

BjLj log 1þ PGl

N0B

� �; ð2Þ

where jLj is the number of links (hops) in path L. Note that the factor1/jLj results from the sharing of bandwidth equally among relaylinks, i.e., each link l along path L is allocated a time fraction 1/jLj.Consequently, the end-to-end spectral efficiency (R(L) = CL/B) ofpath L is given by [2]

RðLÞ ¼minl2L

1jLj log 1þ PGl

N0B

� �: ð3Þ

For a given path L, we define:

wðLÞ ¼minl2L

log 1þ PGl

N0B

� �: ð4Þ

Note that log 1þ PGlN0B

� �can be viewed as the width of link l 2 E.

Therefore, w(L) as defined by (4) is the width of the bottleneck linkalong path L, i.e., w(L) is the width of path L. In the sequel, we use

the terms link width and path width to refer to log 1þ PGlN0B

� �and

minl2L log 1þ PGlN0B

� �, respectively.

By combining (3) and (4), the end-to-end spectral efficiency ofany path L can be expressed as

RðLÞ ¼ wðLÞjLj : ð5Þ

Note that the spectral efficiency of any path L, as given by (5), can beviewed as the ratio of the width of path L to its hop-count.

Given a source–destination (s–d) pair of nodes (s,d) 2 V � V, theproblem of finding the route with maximum end-to-end spectralefficiency under the constraint of equal bandwidth sharing can,thus, be expressed as the following optimization problem:

maxL2Lsd

wðLÞjLj ; ð6Þ

where Lsd is the set of all paths L connecting the source–destinationpair of nodes (s,d) 2 V � V.

It was noted in [2] that Bellman–Ford or Dijkstra shortest pathalgorithms cannot be used to solve (6) directly because the routingmetric w(L)/jLj is neither isotonic nor monotone [14]. Problem (6),however, can be solved by an exhaustive search over all (possiblyexponentially many) paths. This exponential complexity makesexhaustive search prohibitive in networks with a moderate-to-large number of nodes. The study in [2] presents two efficient algo-rithms that provide suboptimal solutions to (6). In [12], however,we have shown that provably optimal solutions to the problemcan be obtained at an O(N4) worst-case complexity. In what fol-lows, we present an improved algorithm with a computationalcomplexity of O(N3), while still providing provably optimal solu-tions to the problem.

3. Optimal reduced-complexity algorithm

3.1. Main result

The main idea of the proposed algorithm is based on the follow-ing two facts:

630 M. Saad / Computer Communications 35 (2012) 628–636

� Fact 1. The hop-count jLj for any path L (in an N-node network)can only take the values 1,2, . . . ,N � 1. Therefore, (6) can besolved by searching over all possible values of jLj.� Fact 2. It is an implicit property of the Bellman–Ford algorithm

[1] that, at its hth iteration, it identifies the optimal path fromthe source to the destination, among paths of at most h hops.

In particular, for a given hop-count h, consider the followingproblem:

maxL2Lsd :jLj¼h

wðLÞjLj ¼ max

L2Lsd :jLj¼h

wðLÞh

: ð7Þ

Note that the equality in (7) results from substituting jLj = h in theobjective function.

Remark 1. Let paths L⁄ and Lh denote the optimal solutions to (6)and (7), respectively. Therefore, the spectral efficiencies R(L⁄) andR(Lh) denote the optimal objective function values of (6) and (7),respectively. Since the union

SN�1h¼1 fL 2 Lsd : jLj ¼ hg covers the set

of all paths Lsd, and by the divide-and-conquer principle [15], thefollowing is true:

RðL�Þ ¼ max16h6N�1

RðLhÞ: ð8Þ

Remark 1 indicates that the spectrum-efficient routing problem(6) can be solved by solving a sequence of subproblems (7). Inparticular, each subproblem considers only paths with a givenhop-count h. Subproblem (7), however, is still difficult to solve.Therefore, we consider its relaxation in which jLj = h is replacedwith jLj 6 h. In particular, consider the following relaxed problem:

maxL2Lsd :jLj6h

wðLÞh

: ð9Þ

Problem (9) is a relaxation of (7) in the sense that any feasiblesolution (path) for (7) is also feasible for (9).

Remark 2. For a given hop-count h, let the paths Lh and bLh be theoptimal solutions to problems (7) and (9), respectively. Since Lh is

optimal for (7), it is also feasible for (9). Therefore, wðLhÞh 6

wðbLhÞh . This

implies that

wðLhÞ 6 wðbLhÞ: ð10ÞIn other words, the optimal path resulting from solving (9) is atleast as wide as that resulting from solving (7).

The main result follows.

Proposition 1. For a given hop-count h 2 {1,2, . . . ,N � 1}, let path bLh

denote the optimal solution to (9), and let RðbLhÞ ¼ wðbLhÞjbLh j

denote its

spectral efficiency. Also, let path L⁄ be the optimal solution to (6), and

let RðL�Þ ¼ wðL�ÞjL�j denote its spectral efficiency. Then

RðL�Þ ¼ max16h6N�1

RðbLhÞ: ð11Þ

Proof. For a given hop-count h, let path Lh denote the optimalsolution for problem (7). The following is true:

RðL�Þ ¼ max16h6N�1

wðLhÞh6 max

16h6N�1

wðbLhÞh6 max

16h6N�1

wðbLhÞjbLhj

: ð12Þ

Note that the first step follows from Remark 1, bearing in mind thatjLhj = h because Lh is optimal for (7). The second step follows fromRemark 2. The third step follows from the fact that jbLhj 6 h becausebLh is optimal for (9).

Now, (12) indicates that

RðL�Þ 6 max16h6N�1

RðbLhÞ: ð13Þ

Note that among all paths L 2 Lsd;RðL�Þ is the maximum possiblespectral efficiency. For any h 2 f1;2; . . . ;N � 1g; bLh 2 Lsd, i.e., bLh isa feasible path for problem (6). Therefore, (13) must hold with strictequality. This completes the proof. h

In light of Proposition 1, the spectrum-efficient routing problem(6) can be solved to optimality by solving a sequence of relaxedsub-problems as given by (9). Moreover, for any given hop-counth, maximizing wðLÞ

h (as in (9)) is equivalent to maximizing w(L).Therefore, problem (9) is a problem of finding the widest path froms to d, such that the path has at most h hops. Consequently, and inlight of Proposition 1, the spectrum-efficient routing problem (6)can be solved by the following procedure:

Procedure 1 (TDMA)

� For h = 1,2, . . . ,N � 1:– Find bLh, the widest path from s to d with at most h

hops, using log 1þ PGlN0B

� �as link labels.

– Let RðbLhÞ ¼ wðbLhÞ

jbLh j.

� Return the path with largest RðbLhÞ.

Now, it is well-known that the Bellman–Ford shortest pathalgorithm can be modified so as to compute the widest path in anetwork. See, e.g., [9]. Moreover, it is an implicit property of theBellman–Ford algorithm that, at its hth iteration, it identifies theoptimal (i.e., widest) path from the source to the destination,among paths of at most h hops. This implies that Procedure 1can be, in fact, implemented by invoking a widest path algorithmonly once (as opposed to N � 1 times). This will be discussed next.

3.2. Detailed implementation

Before we state the Bellman–Ford algorithm, we make thefollowing definitions:

wi,j width of link (i, j) 2 E, i.e., wi;j ¼ log 1þ PGi;j

N0B

� �:

Wih width of the widest path from the source node s 2 V to any

other node i 2 V such that the path has at most h hops.jbLi

hj hop-count of the widest path from the source node s 2 V toany other node i 2 V such that the path has at most h hops.

predih predecessor of node i 2 V on the widest path from the source

node s 2 V such that the path has at most h hops.

Note that we use l 2 E or (i, j) 2 E to signify a link in the network. Inparticular, (i, j) 2 E is the link connecting nodes i 2 V and j 2 V. There-fore, Gi,j is the path gain over the link (i, j). Note also that the width ofany path L, as defined by (4), is wðLÞ ¼minði;jÞ2L log 1þ PGi;j

N0B

� �.

The Bellman–Ford algorithm for widest path computation cannow be stated as follows [9]:

Algorithm Bellman–Ford

1. Initialization:

Let Wsh ¼ 1 for h = 0,1, . . .,N � 1, and Wi

0 ¼ 0 for

all i – s.2. Iteration:

for h = 1,2, . . . ,N � 1

for all i 2 V do Wih :¼Wi

h�1;for all (i, j) 2 E do:

if min Wih�1;wi;j

n o> Wj

h, then

Wjh :¼min Wi

h�1;wi;j

n o.

0 20 40 60 80 1000

10

20

30

40

50

60

70

80

90

4

1

3

2

5

Route with maximum end−to−endspectral efficiency

Fig. 1. Five-node network example. The network is fully-connected. For clarity, notall links are shown. Node 1 is the source, and node 5 is the destination. The routewith maximum end-to-end spectral efficiency (shown) is 1–3–2–5.

M. Saad / Computer Communications 35 (2012) 628–636 631

Note that the complexity of algorithm Bellman–Ford isdominated by the complexity of doing N � 1 passes over all M linksin the network. Therefore, its overall computational complexity isO(NM). Since an N-node network has at most N(N � 1)directed links, the worst case complexity of algorithm Bellman–Fordis O(N3).

After the algorithm terminates, Wdh will carry the width of the

widest path from the source s to the destination d such that the pathhas at most h hops (for any arbitrary 1 6 h 6 N � 1). In other words,

wðbLhÞ ¼Wdh: ð14Þ

The hop-count of the widest path from s to d such that the pathhas at most h hops can be explicitly computed by adding the fol-lowing update at the end of Step 2 of the algorithm (after the if-statement):

jbLjhj :¼ jbLi

h�1j þ 1:

Note that, initially, we set jbLshj ¼ 0 for h = 0,1, . . . ,N � 1 and jbLi

0j ¼ 1for all nodes i – s. After the algorithm terminates, jbLd

hj will carry thehop-count of the widest path from the source s to the destination dsuch that the path has at most h hops. In other words,

jbLhj ¼ jbLdhj: ð15Þ

The widest path (with at most h hops) from s to any other node jcan be explicitly constructed by adding the following update at theend of Step 2 of the algorithm (after the if-statement):

predjh :¼ i:

In particular, the predecessor of node j on the widest path (withat most h hops) from the source s is specified as node i. After thealgorithm terminates, the widest path (with at most h hops) from sto d can be constructed by starting at the destination d andfollowing the predecessors backwards until the source s is reached.In particular, the path can be constructed (in reverse order) as fol-lows: n1 ¼ d;n2 ¼ predn1

h ;n3 ¼ predn2h�1;n4 ¼ predn3

h�2; . . . ; s� �

.Our proposed Bellman–Ford-based spectrum-efficient routing

algorithm can, now, be stated as follows:

Algorithm TDMA Spectrum-Efficient Routing

1. Initialization:

Let Wsh ¼ 1 for h = 0,1, . . . ,N � 1, and Wi

0 ¼ 0 for all

i – s.

Let jbLshj ¼ 0 for h = 0,1, . . . ,N � 1 and jbLi

0j ¼ 1 for all

i – s2. Iteration:

for h = 1,2, . . . ,N � 1

for all i 2 V do Wih :¼Wi

h�1;for all (i, j) 2 E do:

if minfWih�1;wi;jg > Wj

h, then

Wjh :¼minfWi

h�1;wi;jg;jbLj

hj :¼ jbLih�1j þ 1;

predjh :¼ i.

Let wðbLhÞ ¼Wdh; jbLhj ¼ jbLd

hj and RðbLhÞ ¼ wðbLhÞjbLh j

.

3. Return the path with largest RðbLhÞ.

Note that, similarly to algorithm Bellman–Ford, the complexityof algorithm TDMA Spectrum-Efficient Routing is dominated by thecomplexity of doing N � 1 passes over all M links in the network.Therefore, its overall computational complexity is O(NM), i.e., atmost O(N3).

Finally, it is worth mentioning that [9] presented an improvedversion of algorithm Bellman–Ford with a worst-case complexityof O(N3/logN). This implies that the complexity of algorithm TDMASpectrum-Efficient Routing can be further reduced to O(N3/logN) ifthe improved Bellman–Ford implementation of [9] is used.

3.3. Example

To illustrate the successive iterations of algorithm TDMA Spec-trum-Efficient Routing, we consider the 5-node network shown inFig. 1, in which the nodes are located in a 100 m � 100 m two-dimensional area. In particular, the coordinates of the nodes are(13.51,24.12), (92.75,39.11), (51.13,9.29), (2.17,15.95), and(84.45,87.92). We let node 1 be the source node, and node 5 bethe destination node. Without loss of generality, we assume thatthe network is fully connected, i.e., there is a link between everypair of nodes. Note that from an information-theoretic perspective,any two nodes can communicate at a sufficiently low rate [2]. InSection 5 (numerical results), however, other network models willbe also considered. The link widths wi;j ¼ log 1þ PGi;j

N0B

� �for all links

(i, j) are given as follows:

½wi;j�ði;jÞ2E ¼

0 1:5818 3:5881 7:9984 1:10502:0658 0 3:3238 1:4260 3:96923:5660 3:5731 0 2:9655 1:45497:9853 0:4983 4:1506 0 0:88881:7220 3:2324 0:9626 0:2150 0

26666664

37777775:

Note that there is no link between a node and itself; all other linksexist and have positive widths. The iterations of algorithm TDMASpectrum-Efficient Routing are illustrated in Table 1. The last stepof the algorithm is to return the path 1–3–2–5, because it has thelargest spectral efficiency RðbLhÞ.

3.4. Comparison against the existing algorithm

The purpose of this subsection is to highlight the main differ-ences between the proposed algorithm and the existing optimalalgorithm presented in [12]. In [12], we proposed two provablyoptimal spectrum-efficient routing algorithms for TDMA networks:Algorithm A and Algorithm B. Both algorithms have a worst-casecomputational complexity of O(N4), but Algorithm B has a loweraverage computation time. For clarity of the presentation, we

Table 1Successive Iterations of algorithm TDMA Spectrum-Efficient Routing. Recall that bLh isthe widest path from source to destination with at most h hops. wðbLhÞ; jbLhj and RðbLhÞare the width, hop-count and spectral efficiency of path bLh , respectively.

Iteration (h) bLh wðbLhÞ ¼Wdh jbLhj RðbLhÞ ¼ wðbLhÞ

jbLh j

1 1–5 1.1050 1 1.10502 1–2–5 1.5818 2 0.79073 1–3–2–5 3.5731 3 1.19104 1–3–2–5 3.5731 3 1.1910

632 M. Saad / Computer Communications 35 (2012) 628–636

provide a comparison against Algorithm A. Since Algorithm B has asimilar structure (to Algorithm A) and the same worst-case com-plexity, the highlighted differences will not change if Algorithm Bis used for comparison. In Section 5 (numerical results), however,we use Algorithm B as a benchmark to assess the performance ofthe algorithm proposed in this paper. In what follows, we summa-rize Algorithm A from [12].

Algorithm A from [12]:

� For every a 2 log 1þ PGlN0B

� �: l 2 E

n o, do:

– Remove all links l 2 E for which log 1þ PGlN0B

� �< a.

– In the remaining graph, find bLa, the minimum-hop

path from source s to destination d. Let RðbLaÞ ¼ wðbLaÞ

jbLa j.

� Return the path with largest RðbLaÞ.

It is worth noting that Algorithm A from [12] is based on itera-tively invoking a shortest path (minimum-hop path) procedure. Inevery iteration, however, some links are removed from the net-work graph. Therefore, the network graph changes from iterationto iteration. This implies that the algorithm can only be imple-mented by invoking the shortest path procedure multiple times.In particular, Algorithm A from [12] is based on invoking a shortestpath procedure at most jEj = M times, which is the same as the car-

dinality of the set log 1þ PGlN0B

� �: l 2 E

n o. This implies that the over-

all computational complexity is O(MN2), if the Dijkstra shortestpath algorithm is used in every iteration. Since an N-node networkhas at most N(N � 1) directed links, the worst case computationalcomplexity of Algorithm A from [12] is O(N4).

In light of the above discussion, the differences betweenAlgorithm A from [12] and our new proposed algorithm TDMASpectrum-Efficient Routing can be summarized as follows:

� Algorithm A from [12] is based on computing the shortest path.The algorithm presented in this paper, however, is based oncomputing the widest path.� Algorithm A from [12] is based on iteratively invoking a shortest

path procedure (at most) M times, where M is the number ofnetwork links. The algorithm presented in this paper, however,is implemented by invoking a modified Bellman–Ford algorithmfor widest path computation only once.� The worst-case computational complexity of Algorithm A from

[12] is O(N4), where N is the number of network nodes. Thealgorithm presented in this paper, however, has an improvedworst-case computational complexity of O(N3).

4. FDMA Spectrum-Efficient Routing

In this section, and following [8,11], we assume that the pathgain Gl of any network link l 2 E is constant over the availablebandwidth of the channel. This represents the case of flat fading,as in narrow band systems. In the FDMA case, relay links transmitsimultaneously, but are assigned non-overlapping frequency

bands. Under the constraint of equal bandwidth sharing, the FDMAachievable end-to-end data rate CL for an arbitrary path L can beexpressed using the well-known Shannon capacity formula as(see, e.g., [4,11]):

CL ¼minl2L

BjLj log 1þ PGl

N0B=jLj

� �: ð16Þ

Note that the factor 1/jLj (appearing inside and outside the argu-ment of the logarithm) results from the sharing of bandwidthequally among relay links. In other words, every link l along pathL is allocated a bandwidth of B/jLj. The spectral efficiency (RL = CL/B) of path L is, thus, given by:

RðLÞ ¼ 1jLj min

l2Llog 1þ PGljLj

N0B

� �: ð17Þ

Given a source–destination (s–d) pair of nodes (s,d) 2 V � V inan FDMA network, the problem of finding the route with maxi-mum end-to-end spectral efficiency under the constraint of equalbandwidth sharing can, thus, be expressed as the following optimi-zation problem:

maxL2Lsd

1jLj min

l2Llog 1þ PGljLj

N0B

� �: ð18Þ

Note that the hop-count jLj appears in two competing compo-nents of the objective function of (18). In particular, increasingthe hop-count jLjwould decrease the term 1

jLj, but would at the same

time increase the term minl2L log 1þ PGl jLjN0B

� �. This is the fundamental

difference between spectrum-efficient routing in TDMA networksand FDMA networks. To address this issue, the following lemma isneeded.

Lemma 1. Let x, y and a be non-negative real numbers with y P x.Then

1y

logð1þ ayÞ 6 1x

logð1þ axÞ: ð19Þ

Proof. We will proceed by proving that (1 + ay)x6 (1 + ax)y, for

x,y,a P 0 and x 6 y. Then, (19) will follow from the monotonicityof the logarithm function. It is obvious that (19) will hold withstrict equality if x = y. To exclude this trivial situation, we focuson the case x < y.

For any values x P 0 and y P 0 with x < y, define the followingfunction:

f ðaÞ ¼ ð1þ ayÞx

ð1þ axÞy: ð20Þ

It is not difficult to verify that

lima!0

f ðaÞ ¼ 1; ð21Þ

and

lima!1

f ðaÞ ¼ 0: ð22Þ

Eq. (21) is obvious, and (22) can be seen if f(a) is rewritten as

f ðaÞ ¼ ðax�yÞ ð1=aþyÞxð1=aþxÞy, which approaches zero as a ?1 (because x < y).

Now, to show that 0 6 f(a) 6 1, it suffices to prove that f(a) ismonotonically decreasing in the interval 0 6 a 61, for any x P 0and y P 0 with x < y. We prove this fact by showing that the firstderivative of f(a) is always zero or negative. In fact,

M. Saad / Computer Communications 35 (2012) 628–636 633

dfda¼ xyð1þ axÞyð1þ ayÞx�1 � xyð1þ ayÞxð1þ axÞy�1

ð1þ axÞ2y

¼ xyaðx� yÞð1þ axÞy�1ð1þ ayÞx�1

ð1þ axÞ2y : ð23Þ

Note that, in (23), (x � y) is a negative term, while all other termsare non-negative. Therefore, df

da 6 0. Consequently, f(a) is monotoni-cally decreasing in the interval 0 6 a 61, for any x P 0 and y P 0with x < y. Combining this fact with (21) and (22), we conclude that0 6 f(a) 6 1. By the definition of f(a) in (20), we conclude that

ð1þ ayÞx 6 ð1þ axÞy: ð24Þ

By taking the logarithm of both sides of (24), and bearing in mindthat the logarithm is a monotone function, (19) is proven. h

The main result for FDMA networks follows.

Proposition 2. For a given hop-count h 2 {1,2, . . . ,N � 1}, let path bLh

denote the optimal solution to

maxL2Lsd :jLj6h

1h

minl2L

log 1þ PGlhN0B

� �; ð25Þ

and let RðbLhÞ ¼ 1

jbLh jmin

l2bLhlog 1þ PGl jbLh j

N0B

� �denote its spectral effi-

ciency. Also, let path L⁄ be the optimal solution to (18), and let

RðL�Þ ¼ 1jL�jminl2L� log 1þ PGl jL�j

N0B

� �denote its spectral efficiency. Then

RðL�Þ ¼ max16h6N�1

RðbLhÞ: ð26Þ

Proof. The idea of the proof is similar to that of Proposition 1(going through FDMA-relevant versions of Remarks 1 and 2). How-ever, Lemma 1 is a crucial new component, which makes this resultpossible for FDMA networks. In what follows, we provide a sum-mary of the proof.

By the divide-and-conquer principle [15], and since the unionSN�1h¼1 fL 2 Lsd : jLj ¼ hg covers the set of all paths Lsd, the following

is true:

RðL�Þ ¼ max16h6N�1

RðLhÞ; ð27Þ

where, for any hop-count h, Lh is the optimal solution (path) result-ing from solving:

maxL2Lsd :jLj¼h

1jLj min

l2Llog 1þ PGljLj

N0B

� �: ð28Þ

R(Lh) is the spectral efficiency of path Lh, i.e., the optimal objectivefunction value of (28).

By incorporating the restriction jLj = h in the objective functionof (28), it can be easily seen that (28) is equivalent to:

maxL2Lsd :jLj¼h

1h

minl2L

log 1þ PGlhN0B

� �: ð29Þ

Problem (29) is still difficult to solve because it involves searchingover paths with a hop count of exactly h. Problem (25), however,is a relaxation of (29), in the sense that the optimal solution to(29) is also feasible for (25). Therefore, the optimal objective func-tion value of (25) is at least at large as that of (29). Now, recall thatLh and bLh denote the optimal solutions (paths) of (29) and (25),respectively. Therefore,

minl2bL log 1þ PGlh

N0B

� �P min

l2Lh

log 1þ PGlhN0B

� �: ð30Þ

h

Moreover, the following is true:

RðbLhÞ ¼1

jbLhjminl2bLh

log 1þ PGljbLhjN0B

!P

1h

minl2bLh

log 1þ PGlhN0B

� �

P1h

minl2Lh

log 1þ PGlhN0B

� �¼ RðLhÞ; ð31Þ

where the first inequality follows from Lemma 1 (withx ¼ jbLhj; y ¼ h and a ¼ PGl

N0B), the second inequality follows from(30), and the last equality follows from the fact that jLhj = h.

By combining (27) and (31), the following is true:

RðL�Þ 6 max16h6N�1

RðbLhÞ: ð32Þ

Note that among all paths L 2 Lsd;RðL�Þ is the maximum possiblespectral efficiency. For any h 2 f1;2; . . . ;N � 1g; bLh 2 Lsd, i.e., bLh isa feasible path for problem (18). Therefore, (32) must hold withstrict equality. This completes the proof. h

Note that, for any given hop-count h, the objective function of

(25) is equivalent to maximizing minl2L log 1þ PGlhN0B

� �. Therefore,

formulation (25) is a problem of finding the widest path from s

to d with at most h hops, and using log 1þ PGlhN0B

� �as link labels.

Consequently, and in light of Proposition 2, the FDMA spectrum-efficient routing problem (18) can be solved as follows:

Procedure 2 (FDMA)

� For h = 1,2, . . . ,N � 1:– Find bLh, the widest path from s to d with at most h

hops, using log 1þ PGlhN0B

� �as link labels.

– Let RðbLhÞ ¼ 1

jbLh jmin

l2bLhlog 1þ PGl jbLh j

N0B

� �.

� Return the path with largest RðbLhÞ.

Recall that it is an implicit property of the modified Bellman–Ford algorithm that, at its hth iteration, it identifies the widest pathfrom the source to the destination, among paths of at most h hops.It is worth noting, however, that the set of link labels (widths) usedby Procedure 2 depend on h, and, thus, changes from iteration toiteration. This implies that Procedure 2 can be implemented byinvoking the modified Bellman–Ford algorithm N � 1 times (as op-posed to only one time in the TDMA case). This leads to an overallcomplexity of O(N4). The following simple remark, however, leadsto the reduction in complexity to O(N3).

Remark 3. By the monotonicity of the logarithm function, the

widest path from s to d using log 1þ PGlhN0B

� �as link labels is the

same as the widest path from s to d using PGlhN0B as link labels. For any

given hop-count h, the latter is also the same as the widest path

using PGlN0B as link labels. Mathematically,

argmaxL2Lsd :jLj6h

minl2L

log 1þ PGlhN0B

� �¼ argmax

L2Lsd :jLj6hmin

l2L

PGl

N0B: ð33Þ

Therefore, Procedure 2 is equivalent to:

Procedure 3 (FDMA-Modified)

� For h = 1,2, . . . ,N � 1:– Find bLh, the widest path from s to d with at most h

hops, usingPGlN0B as link labels.

– Let RðbLhÞ ¼ 1

jbLh jmin

l2bLhlog 1þ PGl jbLh j

N0B

� �.

� Return the path with largest RðbLhÞ.

634 M. Saad / Computer Communications 35 (2012) 628–636

Now, since the link labels do not change throughout its itera-tions, Procedure 3 can be implemented by invoking the modifiedBellman–Ford algorithm only once (similarly to the TDMA case).Before stating the full algorithm, consider the followingdefinitions:

wi,j width/label of link (i, j) 2 E, i.e., wi;j ¼PGi;j

N0B (in the FDMA case).Wi

h width of the widest path (using the above link labels) fromthe source node s 2 V to any other node i 2 V such that thepath has at most h hops.

jbLihj hop-count of the widest path from the source node s 2 V to

any other node i 2 V such that the path has at most h hops.predi

h predecessor of node i 2 V on the widest path from the sourcenode s 2 V such that the path has at most h hops.

Our proposed Bellman–Ford-based spectrum-efficient routingalgorithm for FDMA networks can, now, be stated as follows:

Algorithm FDMA Spectrum-Efficient Routing

1. Initialization:

Let Wsh ¼ 1 for h = 0,1, . . . ,N � 1, and Wi

0 ¼ 0 for all

i – s.

Let jbLshj ¼ 0 for h = 0,1, . . . ,N � 1 and jbLi

0j ¼ 1 for all

i – s2. Iteration:

for h = 1,2, . . . ,N � 1

for all i 2 V do Wih :¼Wi

h�1;for all (i, j) 2 E do:

if min Wih�1;wi;j

n o> Wj

h, then

Wjh :¼ min Wi

h�1;wi;j

n o;

jbLjhj :¼ jbLi

h�1j þ 1;

predjh :¼ i.

Let jbLhj ¼ jbLdhj and RðbLhÞ ¼ 1

jbLh jlog 1þWd

hjbLhj� �

.

3. Return the path with largest RðbLhÞ.

Note that, after the algorithm terminates, jbLdhj will be the hop-

count of the widest path (using PGlN0B as link labels) from node s to

node d such that the path has at most h hops. Therefore,jbLhj ¼ jbLd

hj, which is used at the end of Step 2 of algorithm FDMASpectrum-Efficient Routing to compute the hop-count of the pathobtained at each iteration.

Moreover, after the algorithm terminates, Wdh will carry the

width of the widest path (using PGlN0B as link labels) from node s to

node d such that the path has at most h hops. Therefore,

Wdh ¼ min

l2bLdh

PGl

N0B: ð34Þ

Consequently, the following is true:

RðbLhÞ ¼1

jbLhjminl2bLh

log 1þ PGljbLhjN0B

!

¼ 1

jbLhjlog 1þmin

l2bLh

PGljbLhjN0B

!

¼ 1

jbLhjlog 1þmin

l2bLh

WdhjbLhj

!; ð35Þ

where the second equality follows from the monotonicity of thelogarithm function, and the third equality follows from (34). Note

that (35) is used at the end of Step 2 of algorithm FDMA Spec-trum-Efficient Routing to compute the spectral efficiency of the pathobtained at each iteration.

Note that the complexity of algorithm FDMA Spectrum-EfficientRouting is dominated by the complexity of doing N � 1 passes overall M links in the network. Therefore, its overall computationalcomplexity is O(NM), i.e., at most O(N3).

Finally, to implement algorithm FDMA Spectrum-Efficient Rout-ing (respectively, algorithm TDMA Spectrum-Efficient Routing), how-ever, the value of PGl

N0B has to be known for each link l. In practice, the

link SNR PGlN0B

� �can be directly measured by received signal strength

indicators available on most devices [2], and fed back to the trans-mitters. Nodes can then exchange their knowledge about the

values of PGlN0B, or log 1þ PGl

N0B

� �, for their outgoing links using a

link-state protocol.

5. Numerical results

5.1. Network model

We consider multihop wireless networks, in which the nodesare located at random positions in a 100 m � 100 m two-dimen-sional area. For each link l in the network, it is assumed that thepath gain is given by

Gl ¼ c � Al � ðdl=d0Þ�4; ð36Þ

where dl is the length of link l, d0 is the reference distance, Al is alog-normally distributed random variable (with 0-dB mean and 8-dB log-variance) that reflects shadowing, and c is a constant. With-out loss of generality, we set d0 = minl{dl} and c = 1/maxl{Al}. Theresulting path gains have, thus, non-negative values smaller thanone.

We test our proposed algorithms on random and independentnetwork realizations, where each network realization is character-ized as follows:

� the horizontal and vertical coordinates of each node are chosenrandomly (and independently) according to a uniform distribu-tion between 0 and 100 m;� there exists a link between two nodes if the distance between

the nodes is at most 25 m;� the path gains are generated randomly (and independently)

according to (36); and� two arbitrary nodes are chosen as the s–d pair.

All results were obtained using Matlab implementationsperformed on a 1.86 GHz Intel� Core2 processor and 1 GB ofmemory.

5.2. Performance of the proposed TDMA algorithm

First, we consider random networks with 20 nodes. Withoutloss of generality, the network SNR (P/N0B) is assumed to be40 dB. We generate 100 random network realizations. For eachnetwork realization, we obtain the optimal spectrum-efficientroute using:

� Algorithm TDMA Spectrum-Efficient Routing; and� Algorithm B from [12], i.e., the fastest optimal algorithm known

so far.

As theoretically justified by Proposition 1 and [12], thespectral efficiencies attained by our proposed algorithm TDMASpectrum-Efficient Routing and Algorithm B (from [12]) were iden-

0 20 40 60 80 1000

0.005

0.01

0.015

0.02

0.025

0.03

Network Realization

Alg

orith

m R

unni

ng T

ime

(in s

econ

ds)

TDMA Spectrum−Efficient RoutingAlgorithm B from [12]

Fig. 2. Reduced running time of algorithm TDMA Spectrum-Efficient Routing.

10 20 30 40 50 6010−4

10−3

10−2

10−1

100

Number of Nodes

Ave

rage

Alg

orith

m R

unni

ng T

ime

(in s

econ

ds) TDMA Spectrum−Efficient Routing

Algorithm B from [12]

Fig. 3. Reduced average running time of algorithm TDMA Spectrum-Efficient Routingfor different numbers of nodes. Every point in the figure is obtained by averagingover 104 random (and independent) network realizations.

−20 −10 0 10 20 30 40160

170

180

190

200

210

220

230

240

Network SNR (dB)

Rel

ativ

e Im

prov

emen

t in

Spec

tral

Effi

cien

cy (i

n %

)

Fig. 4. Percentage improvement in spectral efficiency caused by FDMA relative toTDMA. Every point in the figure is obtained by averaging over 104 random (andindependent) network realizations.

M. Saad / Computer Communications 35 (2012) 628–636 635

tical in every experiment. In particular, both algorithms result inthe provably optimal spectral efficiency. The running time resultsare depicted in Fig. 2. Obviously, the running time of our proposedimproved algorithm is consistently lower than that of Algorithm B.In particular, the average running time was 5.1253� 10�4 s foralgorithm TDMA Spectrum-Efficient Routing, and 0.0149 s for Algo-rithm B. In other words, our proposed algorithm reduced the aver-age running time by 96.56% as compared to Algorithm B.

Note also that, averaged over 104 additional random networkrealizations, the running time was 4.5452 � 10�4 s for algorithmTDMA Spectrum-Efficient Routing, and 0.0134 s for Algorithm B.In other words, our proposed algorithm reduced the averagerunning time by 96.61% as compared to Algorithm B.

Fig. 3 depicts the running times of the proposed algorithmTDMA Spectrum-Efficient Routing and that of Algorithm B from[12], when the number of nodes N is varied from 10 to 60. Foreach value for N, the running time of each algorithm is obtainedby averaging over 104 random network realizations. Without lossof generality, we assume that the network SNR is 20 dB in all

experiments. Again, the running time of algorithm TDMA Spec-trum-Efficient Routing is considerably lower than that of AlgorithmB. Moreover, the reduction in running time becomes more signif-icant as the number of nodes increases. In particular, algorithmTDMA Spectrum-Efficient Routing reduced the average time to ob-tain an exact optimal path by 94.46–98.73% relative to AlgorithmB.

5.3. TDMA vs. FDMA

Finally, we compare the maximum spectral efficienciesachieved in TDMA and FDMA networks, respectively. In this exper-iment, and without loss of generality, we assume that the networkis fully connected, i.e., any two nodes can directly communicate.We also let the network SNR (P/N0B) vary from �20 dB to 40 dB.For each network SNR value, we obtain:

� the maximum spectral efficiency achieved in TDMA networksusing algorithm TDMA Spectrum-Efficient Routing; and� the maximum spectral efficiency achieved in FDMA networks

using algorithm FDMA Spectrum-Efficient Routing.

In particular, the (TDMA and FDMA) spectral efficiencies are ob-tained, for each network SNR value, by averaging over 104 random(and independent) 20-node network realizations. Note that eachnetwork realization is characterized as before, except that thereis a link between every pair of nodes. We noticed that the maxi-mum spectral efficiencies achieved in FDMA networks exceededthose achieved in TDMA networks. Fig. 4 depicts the percentage in-crease in spectral efficiencies in FDMA networks relative to TDMAnetworks. For example, when the network SNR is equal to 20 dB,we observed that the spectral efficiency for FDMA networks ex-ceeded that for TDMA networks by 227.28%. The explanation forthis is simple. Let L be a path in the network. Recall that, underthe constraint of equal bandwidth sharing, the data rate achieved

on any link l 2 L is BjLj log 1þ PGl

N0B

� �in the case of TDMA, and

BjLj log 1þ PGl jLj

N0B

� �in the case of FDMA. The reason is that the noise

power at the receiver of link l 2 L is N0B in the case of TDMA andN0BjLj in the case FDMA. Therefore, it is expected that FDMA will lead

to higher data rates and, thus, higher spectral efficiencies as com-pared to TDMA.

636 M. Saad / Computer Communications 35 (2012) 628–636

Moreover, we noticed that the relative increase in spectral effi-ciency (of FDMA relative to TDMA) becomes less significant at highnetwork SNRs. For example, when the network SNR was set to100 dB, the relative increase in spectral efficiency was only10.48%. In fact, the relative increase in spectral efficiencies be-comes less significant for high network SNRs because the logarith-mic data rate (Shannon capacity) reflects an effect of diminishingreturns.

6. Conclusion

This paper addressed the problem of finding the path with max-imum end-to-end spectral efficiency in multihop TDMA and FDMAwireless networks with equal bandwidth sharing. The author hasrecently presented in [12] the first polynomial-time algorithmsthat solve the problem to exact optimality for TDMA networks. Inparticular, the algorithms presented in [12] have a computationalcomplexity of O(N4). The contribution of this paper is twofold. First,we demonstrated that the spectrum-efficient routing problem canbe, in fact, solved to exact optimality in only O(N3) time for TDMAnetworks. Our computational results illustrated that the proposedalgorithm for TDMA reduces the average computation time bymore than 94% as compared to the algorithms previously known.Second, we introduced the first polynomial-time algorithm thatprovides provably optimal routes for FDMA networks in O(N3) time.Our proposed approach relies on the divide-and-conquer principleand a modified Bellman–Ford algorithm for widest path computa-tion. Finally, it is worth mentioning that the complexity of our pro-posed algorithms for TDMA and FDMA can be further reduced toO(N3/logN) if the improved Bellman–Ford implementation of [9]is used.

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