Optimal Multicommodity Spectrum-Efficient Routing in Multihop Wireless Networks

Finding the route with maximum end-to-end spectral efficiency in multihop wireless networks has been subject to interest in the recent literature. All previous studies, however, focused on finding one route from a given source to a given destination under the constraint of equal bandwidth sharing. To the best of our knowledge, for the first time, this paper provides extensions to the multicommodity flow case, i.e., the case of multiple simultaneous source-destination (s-d) pairs. In particular, given an arbitrary number of s-d pairs, we address the problem of finding a route for every s-d pair such that the minimum spectral efficiency across all routes is maximized. We provide two alternative approaches, where one is based on fixed-sized time slots and the other is based on variable-sized time slots. For each approach, we derive the provably optimal routing algorithm. We also shed the light on the arising tradeoff between the complexity of network-layer route computation and the complexity of medium access control (MAC) layer scheduling of time slots, as well as the amenability to distributed implementation of our proposed algorithms. Our numerical results further illustrate the efficiency of the proposed approaches and their tradeoffs.


Introduction
Multihop wireless networks consist of a set of wireless devices that communicate with each other over multiple wireless hops, with participating nodes collaboratively relaying ongoing traffic.Wireless multihop relaying/routing is the foundation for the development and deployment of emerging technologies such as (i) client mesh networks: a set of client devices (tablets, phones, and/or laptops) form a multihop network with peer-to-peer relaying; (ii) infrastructure wireless mesh networks: wireless routers/access points are interconnected to provide an infrastructure/backbone for clients; (iii) millimeter-wave-based 5G networks: future 5G networks are envisioned to depend (among others) on ultradense small-cell base stations and the use of millimeter-(mm-) wave spectrum for transmission [1].The large bandwidth of mm-wave is also accompanied by a high path loss, which necessities the use of multihop relaying across the small-cell base stations [2].Intelligent routing methods will also be needed for the underlying applications of 5G, e.g., the Internet of Things (IoT) [3].
The end-to-end spectral efficiency (in bps/Hz) of a communication route is defined as the rate at which data can be transmitted over the route per unit bandwidth.Therefore, it is an indication of how efficient the channel bandwidth is utilized.Since the bandwidth is a scarce resource in wireless systems, this paper focuses on finding communication routes with maximum spectral efficiencies.In particular, given a multihop wireless network consisting of set of wireless devices and interconnecting wireless links and a set of sourcedestination (-) pairs of nodes, this paper addresses the problem of finding a path for each - pair such that the minimum spectral efficiency of all paths is maximized.

Related Work.
To the best of our knowledge, this is the first systematic, comprehensive study to address wireless spectrum-efficient routing in the case of multiple - pairs.Related work is presented in two categories: (1) wireless spectrum-efficient routing, (i) To the best of our knowledge, for the first time, we address the problem of spectrum-efficient routing in the case of multiple - pairs.In particular, given a multihop wireless network and an arbitrary set of - pairs, we address the problem of finding a route for every - pair such that the minimum spectral efficiency across all routes is maximized.
(ii) For the problem above, we provide two alternative approaches, where one is based on fixed-sized time slots and the other is based on variable-sized time slots.For each approach, we derive the provably optimal routing algorithm.En route, our study sheds the light on the arising tradeoff between the complexity of network-layer route computation versus the complexity of medium access control (MAC) layer scheduling of time slots.Our numerical results further illustrate the efficiency of the proposed approaches, and their tradeoffs.
The remainder of this paper is organized as follows.Section 2 discusses the basics and preliminaries.Section 3 presents the problem formulation and provably optimal algorithm for the fixed time slots approach.The variabletime-slots-based approach, its problem formulation, and provably optimal algorithm are presented in Section 4. Section 5 discusses the tradeoff between the two approaches.Numerical examples and results are presented in Section 6. Section 7 concludes the paper.

Preliminaries
To avoid the computational intractability of joint optimal routing and medium access control (MAC) layer scheduling, and following [4,[6][7][8][9], it is assumed that a common channel is shared among all nodes using TDMA without spatial reuse, i.e., each node transmits in its own unique time slot.It is demonstrated in [4] that, even though a path is selected assuming no spatial reuse/interference, applying a scheduling technique (separately) that allows some spatial reuse to the selected path can further improve the spectral efficiency.In other words, our framework can still benefit from spatial reuse.It is worth noticing that the MAC layer of the IEEE 802.16 mesh protocol, for example, is based on TDMA (see, e.g., [13]).
A multihop wireless network is modeled as a graph  = (, ), where  represents the set of nodes (vertices) and  represents the set of links (edges).We let  ∈  signify a link in the network.We also let  = || and  = || denote the number of nodes and links, respectively.
Following [4,6,8], we consider the setting in which all transmit devices are constrained by the same symbol-wise average transmit power  and assume that all devices transmit with power  when transmitting.A possible justification for this assumption is that nodes in infrastructure wireless mesh networks are mostly immobile and connected with abundant power supplies.Therefore, for a link  ∈ , the signal-to-noise ratio (SNR) is given by where   is the path gain from the sender of link  to the receiver of link ,  0 is the normalized one-sided power spectral density of the additive white Gaussian noise (at any receiver in the network), and  is the finite bandwidth of the wireless channel.Now, assume  simultaneous - pairs are using the network.Each - pair  = 1, 2, . . .,  has a source node   ∈  and a destination node   ∈ .We also let L  denote the set of all routes from   to   .Moreover, we let   ∈ L  signify a route from   to   .
Finally, the spectral efficiency of an arbitrary path  in the network is defined as the bandwidth-normalized end-to-end data rate [4].In other words, where () is the spectral efficiency (in bps/Hz) of path , () is the end-to-end achievable rate (in bps) for path , and  is the channel bandwidth (in Hz).

Fixed-Sized Time Slots
The studies in [4,[6][7][8][9] focused on a single - route and assumed the bandwidth is shared equally among its links via TDMA.In other words, each link transmits in its own unique time slot, where the time slots are of fixed size.One way to extend this equal bandwidth sharing to the multicommodity case is to maintain the assumption that the time frame is divided equally among all links of the different - routes.
Note that the factor 1/ ∑  =1 |  | comes from the sharing of the bandwidth equally among all links (of all routes), i.e., each link on any route is allocated a time fraction of 1/ ∑  =1 |  | for transmission.Note also that the minimum function in (3) results from the fact that the end-to-end data rate of any path   is equal to the data rate achieved by its bottleneck link.Using (2), the spectral efficiency of path   can, thus, be expressed as follows: Consequently, the minimum spectral efficiency,   , across all active routes ( Therefore, the problem of jointly finding routes for - pairs 1, 2, . . .,  such that the minimum spectral efficiency across all routes is maximized can be cast as the following optimization problem: max It is worth noting that problem (8) cannot be solved using standard shortest path methods as the resulting routing metric is not isotonic [5].In particular, even with one - pair, the routing metric of ( 8) is not isotonic.See, e.g., [4,6].In what follows, we develop a polynomial-time algorithm that provides provably optimal solutions to (8).
The main result follows.
Moreover, by substituting the equality constraint ( 1 ,  2 , . . .,   ) =  in its objective function, ( 12) is equivalent to max Now, it is readily seen that ( 10) is a relaxation of ( 14).Therefore, if (  1 ,   2 , . . .,    ) and ( L 1 , L 2 , . . ., L  ) are the optimal solutions to ( 10) and ( 14), respectively, then Now, the following is true: Note that the first equality comes directly from (13).The second equality comes from the fact that the route set ( L 1 , L 2 , . . ., L  ) is feasible for (12).The inequality comes from (15) and from the fact that the route set (  1 ,   2 , . . .,    ) is feasible for (10).Consequently, (16)  Recall that, in the above procedure,  is the set of link widths given by ( 9), and   is given by (7).The only remaining issue to show is how to solve (10).Note that, for a given  ∈ , maximizing Moreover, the latter is minimized if every - pair  minimizes |  |.Consequently, problem (10) is equivalent to finding the minimum-hop path for every - pair 1 ≤  ≤ , such that the minimum link width across all paths is not less than .Therefore, for a given value of , (10) can be solved as follows: (i) Remove all links  ∈  for which log(1 + P  / 0 ) < .In the remaining graph, obtain the minimum-hop path for every - pair 1 ≤  ≤ .
In light of the above discussion, problem (8) can be solved by the following algorithm.

Variable-Sized Time Slots
An alternative approach to accommodating multiple - pairs under the condition of equal bandwidth sharing is to divide the time frame equally among - pairs (as opposed to dividing the time equally among the links).In other words, every - pair/path will transmit for a fraction of 1/ of the time (assuming  - pairs).Consequently, if - pair  uses path   , then every link along this path will transmit for a fraction of 1/|  | of the time.Since different - pairs may use paths with different hop-counts, their respective links may use time slots of different sizes.
In this case, the end-to-end spectral efficiency of route   serving - pair  can be expressed as Note that the factor 1/|  | comes from the fact that every link  ∈   transmits for a fraction of 1/|  | of the time.Note also that the minimum function in (18) results from the fact that the end-to-end data rate of any path   is equal to the data rate achieved by its bottleneck link.It is worth noting that the spectral efficiency for - pair  depends on the hop-count |  | of its own path   only.This is in contrast to the case of equal time slots, where the spectral efficiency for any - pair  depends on the hop-counts of all - paths  1 ,  2 , . . ., Ł  .
The problem of maximizing the minimum spectral efficiency across all - pairs can, thus, be expressed as max where (  ) is given by (18).It is not hard to see that the minimum spectrum efficiency will be maximized if every - pair  maximizes its individual spectral efficiency (  ).In other words, every - pair  solves the following optimization problem: max Moreover, since, for any number of - pairs,  is a constant, solving (20) is equivalent to solving the single - pair problem.The best known algorithm for solving (20) has been introduced in [8], and can be summarized as follows.
(2) Return the path with largest ( ℎ  ).The following observations are in order regarding algorithm Variable-Time-Slots.
(i) The algorithm can be implemented by each - pair in a completely independent manner.In other words, - pairs are completely isolated and there is no need for coordination and/or a centralized component.(ii) The number of - pairs  does not affect the computation of the optimal paths.In particular, for any number of - pairs,  is a constant, and thus argmax In other words, every - pair computes its optimal path regardless of how many other - pairs exist, and the optimal paths do not change with the change of the number of - pairs.(iii) The number of - pairs , however, is needed for MAC layer scheduling.In particular, every - pair transmits for 1/ of the time, and every link  ∈   used by - pair  transmits for a fraction of 1/|  | of the time.(iv) Since the paths used by different - pairs may have different hop-counts, the resulting time slots may be of different durations.
(v) The algorithm can be implemented by the different - pairs in parallel.Note also that shortest path algorithms can be modified to compute the widest path.See, e.g., [15].Moreover, it is worth noticing that the Bellman-Ford shortest path algorithm, in its ℎ ℎ iteration, computes the shortest (or widest) path with at most ℎ hops.Consequently, algorithm Variable-Time-Slots can be implemented by invoking the Bellman-Ford procedure only once.The overall complexity of algorithm Variable-Time-Slots is, thus, ( 3 ).
In contrast, however, assume that algorithm Equal-Time-Slots results in paths  1 ,  2 , and  3 for the 3 - pairs, where and | 3 | = 4, respectively.In this case, the time frame will be divided into 3 + 2 + 4 = 9 slots, where the normalized size of each slot is precisely 1/9.

Tradeoffs
In this section we summarize the main differences and tradeoffs between algorithms Equal-Time-Slots and Variable-Time-Slots.
(i) Distributed implementation: algorithm Equal-Time-Slots has a component that requires centralized knowledge of the paths computed by all - pairs (in every iteration of the algorithm).Alternatively, coordination and exchange of information is required among - pairs.Algorithm Variable-Time-Slots, however, can be implemented in a fully distributed fashion among - pairs.No coordination is required among - pairs in their path computation.
(ii) Synchronization: as described above, algorithm Equal-Time-Slots is based on the centralized (or coordinated) knowledge of the paths    for all - pairs  = 1, 2, . . ., .This implicitly implies that all existing - pairs must make their path computation decisions (i.e., invoke their algorithms) in a synchronized way; any - pair cannot obtain its optimal path without the results of the other - pairs.Algorithm Variable-Time-Slots, however, does not require this synchronization.Any - pair can compute its optimal path regardless of the other - pairs.In fact, optimal path computation does not even require the knowledge of the number of existing - pairs .The optimal path in case of only one - pair would remain optimal in the presence of any number of - pairs .Knowing  is necessary in the transmission (MAC) scheduling phase only.
(iii) MAC scheduling: algorithm Equal-Time-Slots results in TDMA transmission frames with equal time slots, while algorithm Variable-Time-Slots results in TDMA transmission frames with variable time slots.
(iv) Complexity: the complexity of Equal-Time-Slots is ( 4 ), while the complexity of Variable-Time-Slots is ( 3 ).
It is worth noting that both algorithms (Equal-Time-Slots and Variable-Time-Slots) require the value of the link SNRs (  / 0 ) to be known at the source nodes computing their respective paths.In practice, the link SNR can be directly measured by received signal strength indicators available on most devices [4], and fed back to the transmitters.Nodes can then exchange their knowledge about the values of   / 0  for their outgoing links using a distributed link-state protocol.Please refer to [9] for further elaboration.
In the following section we provide a numerical study on the performance of both proposed approaches and their tradeoffs.

Numerical Results
We consider multihop wireless networks, in which the nodes are located at random positions in a 100 × 100 twodimensional area.Without loss of generality, it is assumed that any two nodes can directly communicate; i.e., the network is fully connected.Note that, from an information theoretic point of view, two nodes can always communicate at a sufficiently low rate [4,17].The path gain   of each link  is assumed to be given by where   is the length of link ,  0 is the reference distance for the far-field,   is a log-normally distributed random variable (with 0-dB mean and 8-dB log-variance) that reflects shadowing, and  is a constant.Without loss of generality, we set  0 = 0.1 and  = 0.01.We test our proposed algorithms on random and independent network realizations, where in each realization the horizontal and vertical coordinates of each node are chosen randomly (and independently) according to a uniform distribution between 0 and 100, and the path gains are generated randomly (and independently) according to (22).Among the randomly generated nodes, a set of - pairs is chosen at random.Furthermore, for each tested scenario, we average our results over 10 4 random network realizations.
In other words, every point in each of the following result figures is averaged over 10 4 random network realizations.
The simulation parameters are summarized in Table 1 (note that (⋅) and (⋅) denote the expected value and variance of a random variable, respectively).

Effect of the Network
Size.First, we vary the number of nodes in the network () from 5 to 30.For every value of ,  we let the number of - pairs () be 5, and we set the network SNR (/ 0 ) to 80 dB. Figure 1 depicts the minimum spectral efficiency among all - routes obtained using algorithms Equal-Time-Slots and Variable-Time-Slots, respectively.It is clearly seen that algorithm Equal-Time-Slots results in higher worst-case spectral efficiencies.In particular, the minimum source-destination (-) spectral efficiency resulting from algorithm Equal-Time-Slots is from 29.23% to 39.2% higher than that of algorithm Variable-Time-Slots. Averaged over all experiments for different values of , the minimum - spectral efficiency resulting from algorithm Equal-Time-Slots is 36% higher than that of algorithm Variable-Time-Slots.
Moreover, Figure 2 depicts the average spectral efficiency across all - routes obtained using algorithms Equal-Time-Slots and Variable-Time-Slots, respectively.It is straightforward to see that algorithm Variable-Time-Slots results in higher average spectral efficiencies.In particular, the average - spectral efficiency resulting from algorithm Variable-Time-Slots is from 29.88% to 64.2% higher than that of algorithm Equal-Time-Slots.Averaged over all experiments for different values of , the average - spectral efficiency resulting from algorithm Variable-Time-Slots is 57.27% higher than that of algorithm Equal-Time-Slots.In short, although algorithm Equal-Time-Slots has a better worst-case performance (as seen in Figure 1), algorithm Variable-Time-Slots has a significantly better average performance (as seen in Figure 2).
Finally, we provide a comparison between the running times of algorithms Equal-Time-Slots and Variable-Time-Slots per - pair.For fairness of comparison, we compare the overall running time of algorithm Variable-Time-Slots with the running time of the distributed component of algorithm Equal-Time-Slots (i.e., Step (1a), which can be implemented by each - pair in isolation).In other words, the running time of the centralized component of algorithm Equal-Time-Slots is excluded from the comparison.The running times of both algorithms are depicted in Figure 3.In fact, the (per  - pair) average running time of algorithm Variable-Time-Slots is 0.44 milliseconds, while that of algorithm Equal-Time-Slots is 0.13 seconds.In other words, although the centralized component of algorithm Equal-Time-Slots was not considered in this comparison, the running time of algorithm Variable-Time-Slots is on average 99.23% lower than that of algorithm Equal-Time-Slots.

Effect of Number of 𝑠-𝑑
Pairs.Now, we vary the number of - pairs () from 5 to 20, while the number of nodes is fixed at  = 20 and the network SNR is fixed at 80 dB.The results for minimum and average spectral efficiencies across all - routes are depicted in Figures 4 and 5, respectively.Again, it can be easily seen that algorithm Equal-Time-Slots has a better worst-case performance (as seen in Figure 4), while algorithm Variable-Time-Slots has a significantly better average performance (as seen in Figure 5).In particular, the minimum spectral efficiency across all - routes obtained by algorithm Equal-Time-Slots is from 38.17% to 59.66% higher than that obtained by algorithm Variable-Time-Slots. Averaged over all experiments, the minimum spectral efficiency resulting from algorithm Equal-Time-Slots is 50.62% higher than that resulting from algorithm Variable-Time-Slots.On the other hand, the average spectral efficiency across all - routes resulting from algorithm Variable-Time-Slots is from 62.29% to 78.19% higher than that resulting from algorithm Variable-Time-Slots. Averaged over all experiments, algorithm Variable-Time-Slots results in 73.50% higher average spectral efficiencies than algorithm Equal-Time-Slots.6.3.Effect of the Network SNR.Now, we vary the network SNR from -20 dB to 80 dB, while the number of nodes is fixed at  = 20 and the number of - pairs is fixed at  = 5.The results for minimum and average spectral efficiencies across all - routes are depicted in Figures 6 and 7, respectively.In consistency with all other results, algorithm Equal-Time-Slots consistently shows a better worst-case performance (as seen in Figure 6), while algorithm Variable-Time-Slots shows a consistently and significantly better average performance (as seen in Figure 7).In particular, the improvement in worstcase spectral efficiencies due to algorithm Equal-Time-Slots is between 37.96% and 38.24% (with an average improvement of 37.93% across all experiments).However, the improvement in average spectral efficiencies due to algorithm Variable-Time-Slots is between 65.18% and 136.37% (with an average improvement of 129.20% across all experiments).

Comparison against
Benchmarks.Finally, we assess the performance of our proposed algorithms in comparison to existing techniques.To this end, we use the following two benchmarks.as compared to DSER (with an average improvement of 45% across all experiments).Moreover, our algorithm results in 6.5% to 39% higher average spectral efficiencies as compared to direct link routing (with an average improvement of 24% across all experiments) and results in 2.9% to 27% higher  average spectral efficiencies as compared to DSER (with an average improvement of 15% across all experiments).
In summary, all our experiments (of varying the network size, number of - pairs, and network SNR) indicate a similar trend of a better worst-case performance for algorithm Equal-Time-Slots versus a better average/typical performance for algorithm Variable-Time-Slots.It is worth noting, however, that the improvement in average results due to algorithm Variable-Time-Slots is always more significant.Moreover, algorithm Variable-Time-Slots enjoys a significantly (more than 99%) lower running time.Moreover, our proposed algorithm Variable-Time-Slots has shown a significantly superior worst-case and average performance as compared to DSER from [4], and as compared to direct link routing.

Conclusion
To the best of our knowledge, previous work on finding the path with maximum end-to-end spectrum efficiency was restricted to a single - pair.This paper proposed two alternative approaches for the spectrum-efficient routing problem in the multicommodity flow regime, i.e., in the case of multiple active - pairs.The routing objective was to maximize the minimum spectrum efficiency achieved across all active - pairs.The first approach was based on dividing the time frame into equal-sized slots, while the second approach allows dividing the time frame into variablesized slots.For each approach, we derived the provably optimal routing algorithm.We also shed the light on the arising tradeoff between the resulting routing algorithms.In summary, the routing algorithm induced by equal time slots enjoys a better worst-case performance (i.e., higher worst-case spectrum efficiencies), at the price of a higher computational complexity and the existence of a centralized component requiring coordination and synchronization among all - pairs in the route computation phase.However, the routing algorithm induced by variable time slots has the advantages of (1) a significantly lower computational complexity (more than 99% reduction in running time), (2) a significantly better average/typical performance (i.e., higher average achieved spectral efficiencies), (3) a significantly better worst-case and average spectral efficiency performance as compared to existing methods, and (4) being entirely distributed with no need for coordination or synchronization among - pairs.In fact the routing algorithm induced by variable time slots does not even require the knowledge of the number of active - pairs; the optimal path in case of the existence of a single - pair remains optimal in case of any number of - pairs.Knowledge of the number - pairs is required only in the phase of MAC layer scheduling of time slots.This concludes that algorithm Variable-Time-Slots might be preferred for practical implementation.

Figure 1 :
Figure 1: Minimum spectral efficiency among all - routes obtained using algorithms Equal-Time-Slots and Variable-Time-Slots, respectively, versus the number of network nodes.

AverageFigure 2 :
Figure 2: Average spectral efficiency among all - routes obtained using algorithms Equal-Time-Slots and Variable-Time-Slots, respectively, versus the number of network nodes.

Figure 3 :Minimum
Figure 3: Running times of algorithms Equal-Time-Slots and Variable-Time-Slots.

Figure 4 :
Figure 4: Minimum spectral efficiency among all - routes obtained using algorithms Equal-Time-Slots and Variable-Time-Slots, respectively, versus the number of - pairs.

Figure 5 :
Figure 5: Average spectral efficiency among all - routes obtained using algorithms Equal-Time-Slots and Variable-Time-Slots, respectively, versus the number of - pairs.
Number of s-d Pairs, KAlgorithm Variable-Time-Slots Algorithm Equal-Time-Slots Direct Link Routing

Figure 9 :Figure 10 :
Figure 9: Average spectral efficiency among all - routes obtained using algorithms Equal-Time-Slots and Variable-Time-Slots compared against direct link routing.

Average
Spectral Efficiency across all s-d Pairs [bps/Hz]Algorithm Variable-Time-Slots DSER from[5] Direct Link Routing

Figure 11 :
Figure11: Average spectral efficiency among all - routes obtained using algorithm Variable-Time-Slots compared against DSER from[4] and direct link routing.
In particular, if - pairs  = 1, 2, . . .,  are served by routes  1 ,  2 , . . .,   , respectively, then any link on any of the  routes will transmit for a fraction of 1/ ∑  =1 |  | of the time, where |  | is the number of hops/links in route   .
1 ,  2 , . . .,   ) can be expressed as   ( 1 ,  2 , . . .,   ) +   / 0 ) can be viewed as the width of any link .Consequently, min ∈{  :=1,2,...,} log(1 +   / 0 ) is the smallest link width used by the set of routes  1 ,  2 , . . .,   .In other words, the latter represents the width of the narrowest route among  1 ,  2 , . . .,   .To simplify our notation and algorithm development, we use the following substitution:  ( 1 ,  2 , . . .,   ) = min ( 1 ,  2 , . . .,   ) is the smallest link width used by the set of routes  1 ,  2 , . . .,   ; i.e., it represents the width of the narrowest route among  1 ,  2 , . . .,   using log(1 +   / 0 ) as link widths.Consequently, the minimum spectral efficiency,   , across all active routes ( 1 ,  2 , . . .,   ) can be rewritten as Remove all links  ∈  for which log(1 +   / 0 ) < .(ii)In the remaining graph, find    , the minimum-hop path from source   to destination   .The following observations are in order regarding algorithm Equal-Time-Slots.(i)Step(1a) of algorithm Equal-Time-Slots can be implemented by each  −  pair independently, and without any coordination with the other  −  pairs.In particular,  −  pair  (or more precisely source node   ) obtains its path    independently.(ii)Step (1b), however, requires knowledge about all  −  pairs.Therefore, it can be implemented by a centralized entity which knows the hop-count |   | of every path    .Alternatively, it requires that all  −  pairs (or source nodes) exchange their information about |   | with all other nodes using flooding, or any other means of all-to-all communication.(iii)At the MAC layer, the resulting set of paths will require dividing the time frame into ∑  =1 | *  | equalsized time slots.This simplicity of MAC layer scheduling comes at the expense of the necessity of coordination between  −  pairs during network-layer path computation.
Minimum Spectral Efficiency across all s-d Pairs [bps/Hz]