Wireless Link Capacity under Shadowing and Fading

We consider the following basic link capacity (a.k.a., one-shot scheduling) problem in wireless networks: Given a set of communication links, find a maximum subset of links that can successfully transmit simultaneously. Good performance guarantees are known only for deterministic models, such as the physical model with geometric (log-distance) pathloss. We treat this problem under stochastic shadowing under general distributions, bound the effects of shadowing on optimal capacity, and derive constant approximation algorithms. We also consider temporal fading under Rayleigh distribution, and show that it affects non-fading solutions only by a constant-factor. These can be combined into a constant approximation link capacity algorithm under both time-invariant shadowing and temporal fading.

1 Introduction

Efficient use of networks requires attention to the scheduling of the communication. Successful reception of the intended signal requires attention to the interference from other simultaneous transmissions, and both are affected crucially by the vagaries in the propagation of signals through media. We aim to understand the fundamental capacity question of how much communication can coexist, and the related algorithmic aspect of how to select large sets of successfully coexisting links. The hope is to capture the reality of signal propagation, while maintaining the fullest generality: arbitrary instances, and minimal distributional assumptions.

The basic property of radio-wave signals is that they attenuate as they travel. In free space, the attenuation (or “pathloss”) grows with the square of the distance. In any other setting, there are obstacles, walls, ceilings and/or the ground, in which the waves can go through complex transforms: reflection, refraction (or shadowing), scattering, and diffraction. The signal received by a receiver is generally a combination of the multiple paths that it can travel, that are phase-shifted, resulting in patterns of constructive and destructive interference. The general term for variation of the received strength of the signal from the free-space expectation is fading.

The fading of signals can be a function of time, location, frequency and other parameters, of which we primarily focus on the first two. A distinction is often made between large-scale fading, the effects of larger objects like buildings and trees, and the small-scale fading at the wavelength scale caused by multiple signal-paths. We primarily distinguish between temporal fading, that varies randomly within the time frame of communication, and shadowing, that is viewed as invariant within the time horizon of consideration. Temporal fading is typically experienced at the small-scale as a combination of multi-path propagation and movement or other environment changes.

The most common way of modeling true fading is stochastic fading. To each point in space-time, we associate a random variable drawn from a distribution. Typically, this is given by a distribution in the logarithmic dBm scale, so on an absolute scale the distributions are exponential. There is a general understanding that log-normal shadowing (LNS), which is Gaussian on the dBm scale, is the most faithful approximation known of medium-large scale fading or even all atemporal fading [46, 8, 11]. Empirical models often add variations depending on the environment, the heights of the sender/receiver from the ground, and whether there is a line-of-sight (e.g., [30]). The most prominent among the many models proposed for small-scale and temporal fading are the ones of Rayleigh and Rice [39], with the former (latter) best suited when there is (is no) line-of-sight, respectively. The Rayleigh distribution mathematically captures the case when the signal is highly scattered and equally likely to arrive at any angle. Though probabilistic models are known to be far from perfect, they are generally understood to be highly useful for providing insight into wireless systems, and certainly more so than the free-space model alone.

Stochastic fading is the norm in generational models, such as for simulation purposes. For instance, LNS is built into the popular NS-3 simulator. It is also commonly featured in stochastic analysis, e.g. [29]. Worst-case analysis of algorithms has, however, nearly always involved deterministic models, either the geometric free-space model, or extensions to more general metric spaces, e.g., (cite various). One might expect such analysis to treat similarly arbitrary or “any-case” fading, but that quickly invokes the ugly specter of computational intractability [16].

Problems and setting. In the Link Capacity problem, we are given a set of links, each of which is a sender-receiver pair of nodes on the plane. We seek a maximum feasible subset of links in , where a set is feasible in the physical (or SINR) model if, for each link, the strength of the signal at the receiver is times larger than total strengths of the interferences from the other links. We consider arbitrary/any-case positions of links, aiming for algorithms with good performance guarantees, as well as characterizations of optimal solutions.

We treat Link Capacity in extensions of the standard physical model to stochastic fading. We separate the fading into temporal and atemporal (or time-invariant) aspects, which we refer to as temporal fading and shadowing, respectively. We generally assume independence across space in time-invariant distributions and across time in the temporal distributions. This is a simplification, aimed to tackle most pronounced aspects; where possible, we relax the independence assumptions, sometimes allowing for arbitrary (worst-case) values. Observe that the two forms can be arbitrarily correlated: the temporal results hold under arbitrary time-invariant fading.

Our results. We give a comprehensive treatment of link capacity under stochastic fading models. We give constant-factor approximation algorithms for both time-invariant and temporal stochastic models. These are complementary and can be multiplexed into algorithms for both types of fading.

For (time-invariant) shadowing, we allow for essentially any reasonable stochastic distribution. We show that shadowing never decreases the optimal link capacity (up to a constant factor), but can significantly increase it, where the prototypical case is that of co-located links. We give algorithms for general instances, that achieve a constant factor approximation, assuming length diversity is constant.

For temporal fading, we treat arbitrary instances that can have arbitrary pathloss/shadowing. We show that algorithms that ignore the temporal fading given by Rayleigh distribution achieve a constant factor approximation. The links can additionally involve weights and can be of arbitrary length distribution.

Besides the specific results obtained, our study leaves us with a few implications that may be of general utility for algorithm and protocol designers. One such lesson is that to achieve good performance for shadowing,

algorithms can concentrate on the signal strengths of the links,

and can largely ignore the strength of the interference between the links. Another useful lesson is that

algorithms can base decisions on time-invariant shadowing alone,

since the temporal fading will even out.

These appear to be the first any-case analysis of scheduling problems in general stochastic models. In particular, ours appears to be the first treatment of approximation algorithm for scheduling problems under shadowing or time-invariant fading.

Related Work. Gupta and Kumar [18] introduced the physical model, which corresponds to our setting with no fading. Their work spawned off a large number of studies on “scaling laws” regarding throughput capacity in instances with stochastic input distributions. First algorithms with performance guarantees in the physical model were given by Moscibroda and Wattenhofer [31]. Constant approximation for the Link Capacity problem were given for uniform power [16], linear power [14, 38], fixed power assignments [21], and arbitrary power control [24]. This was extended to a distributed setting [12, 2], admission control in cognitive radio [22], link rates [25], multiple channels [6, 41], spectrum auction [23], changing spectrum availability [9], and MIMO [43]. NP-hardness was established in [17]. Numerous works on heuristics are known, as well as exponential time exact algorithms (e.g., [35]).

The Link Capacity problem has been fundamental to various other scheduling problems, appearing as a key subroutine for shortest link schedule [42, 16, 20], maximum multiflow [40, 4], weighted link capacity [42, 25], and capacity region stability [3, 26].

Numerous experimental results have indicated that simplistic range-based models of wireless reception are insufficient, e.g., [15, 28, 46]. Significant experimental literature exists that lends support for stochastic models [33], especially log-normal shadowing, e.g., [46, 8, 11]. Analytic results on stochastic fading are generally coupled with stochastic assumptions on the inputs, such as point processes in stochastic geometry [29, 19, 44]. Most stochastic fading models though do not lend themselves to closed-form formulation; Rayleigh fading is a rare exception [7]. Log-normal shadowing has been shown to result in better connectivity [37, 32] and throughput capacity [36], but this may be artifact of the i.i.d. assumption [1].

The only work on Link Capacity with any-case instances in fading models is by Dams et al. [10], who showed that temporal Rayleigh fading does not significantly affect the performance of LinkCapacity algorithms, incurring only a -factor increase in performance for link capacity algorithms. We improve this here to a constant factor. Rayleigh fading has also been considered in distributed algorithms for local broadcast [45].

The non-geometric aspects of signal propagation have been modeled non-stochastically in various ways. One simple mechanism is to vary the pathloss constant [18]. A more general approach is to view the variation as deforming the plane into a general metric space [13, 21]. Also, the pairwise pathlosses can be obtained directly from measurements, inducing a quasi-metric space [6]. All of these, however, lead to very weak performance guarantees in the presence of the huge signal propagation variations that are seen in practice (although some of that can be ameliorated by identifying parameters with better behavior, like “inductive independence” [23]).

2 Models and Formulations

2.1 Communication Model

The main object of our consideration is a set of communication links, numbered from to . Each link represents a unit-demand communication request between a sender node and a receiver node , both point-size wireless nodes located on the plane.

We assume the links all work in the same channel, and all (sender) nodes use the same transmission power level (unless stated otherwise). We consider the following basic question, which is called the LinkCapacity problem: what is the maximum number of links in that can successfully communicate in a single time slot? We will refer to a set of links that can successfully communicate in a single time slot as feasible.

When a subset of links transmit at the same time, a given link will succeed if its signal (the power of the transmission of when measured at ) is larger than times the total (sum) interference from other transmissions, where is a threshold parameter, and the interference of link on link is the power of transmission of when measured at . We will denote by the received signal power of link and by the interference of link on link . In this notation, link transmits successfully if111In general, there should also be a Gaussian noise term in the success condition, which is omitted for simplicity of exposition. It may be noted that in expectation, the success of only a fraction of links will be affected by the noise.

(1)

2.2 Geometric Path-Loss

The Geometric Path-Loss model or GPL for short, defines the received signal strength between nodes and as , where is the power used by the sender , is the path-loss exponent and denotes the Euclidean distance. In particular, the signal strength/power of a link and the interference of a link on link are, respectively,

where denotes the length of link and is the distance from the sender node of link to the receiver node of link .

If the links in a set transmit simultaneously, the formula determining the success of the transmission on link is similar to (1), but we will use the slightly modified notation to indicate that GPL model is considered.

2.3 Shadowing

One of the effects that GPL ignores (or models only by appropriate change of the exponent ), is signal obstruction by objects, or shadowing. In generic networks shadowing is often modeled by a Stochastic Shadowing model, or SS for short, such as the Log-Normal Shadowing model, or LNS for short. In this case, there is a parametrized probability distribution , such that the signal strength of a link at is assumed to have been sampled from the distribution and , and similarly, for any two links , the interference is sampled from and . We assume that signals and interferences do not change in time (due to shadowing), at least during the time period when LinkCapacity needs to be solved. In this model too, signal reception is characterized by the signal to interference ratio, but we will use the notation to indicate that SS model is considered.

We shall be assuming independence among the random variables. This may lead to artifacts that are contrary to experience. It is nevertheless valuable to examine closely this case that might be considered the most extreme.

2.4 Temporal Fading

Another effect that is not described by the models above is the temporal variations in the signal, due to a combination of movement (of either transceivers or people/objects in the environment) and the scattered multipath components of the signal. We will concentrate on Rayleigh fading, where the signal power is distributed according to an exponential distribution with mean , i.e. , and similarly, the interference power is distributed according to an exponential distribution with mean , i.e., , where and are the signal and interference values, not necessarily from SS or GPL. Again, the success of transmission is described by the signal to interference ratio. Note, however, that in this case the success is probabilistic: the same set of links can be feasible in one time slot and non-feasible in another.

2.5 Computational Aspects

There is a striking difference between GPL and shadowing on one side, and temporal fading on the other side, from the computational point of view. This difference stems from the spatial nature of GPL and shadowing, and the time-variant nature of temporal fading. In the former case, an algorithm can be assumed to have access to, e.g., the signal strengths of links, which could be obtained by measurements. This, however, is impossible or impractical under temporal fading, which forces the algorithms to be probabilistic and base the actions solely on the expected values of signal strengths (w.r.t. fading distribution), and the performance ratio of algorithms is measured accordingly (see Sec. 5 for details).

2.6 Technical Preliminaries

Throughout this text, by “constants” we will mean fixed values, independent of the network size and topology (e.g. distances). Some examples are parameters , and the constants under big O notation.

Affectance.

In order to describe feasibility of a set of links with arbitrary signal and interference values, we will use the notion of affectance, which is more convenient than (but equivalent to) the signal to noise ratio. For two links we let and extend this definition to subsets: If is a set of links, then and . Then, a set of links is feasible if and only if holds for every link . When considering a particular SS distribution or GPL we will use superscripts and respectively, as before. We will use the following result of [5], which shows that feasibility is robust with respect to the threshold value .

Lemma 1.

If a set of links and number are such that for each link , then can be partitioned into at most feasible subsets.

Smooth Shadowing Distributions.

We will use quantiles of an SS distribution. Consider an SS distribution and assume links use uniform power assignment. For a probability and a link with , let denote the -quantile, i.e., a value such that . For a given number , the distribution is called -smooth, if there is a constant such that holds for each link . Note that all major distributions used to model stochastic shadowing satisfy such a smoothness condition.

3 Comparing Shadowing to GPL

We start by comparing the optimal solutions of LinkCapacity under SS and GPL for any given set of links. However, a particular instance drawn from an SS distribution is arguably not informative and can be hard to solve. Instead, we will be more interested in the gap between GPL optimum and a “typical” SS optimum, in the sense of expectation.

We denote by the size (number of links) of the optimal solution to LinkCapacity for a set of links under GPL. Similarly, we denote by the size of the optimal solution to LinkCapacity under SS model with distribution . We assume that for every link , the variables and (for all ) are independent, unless specified otherwise.

We will compare the expected value with , for a given set of links, where the expectation is over the distributions of random variables and for all links . In particular, we prove that the expected SS optimum is never worse than a constant factor of the GPL optimum. On the other hand, due to the presence of links with high signal strength that can appear as a result of shadowing, the capacity can considerably increase.

3.1 SS Does not Decrease Capacity

First, we show that , i.e., a typical optimum under SS is not worse than the optimum under GPL.

Theorem 1.

Let be a -smooth SS distribution with a constant , and let be any set of links. Then .

Proof.

Let be a maximum cardinality subset of that is feasible under GPL, and let us fix a link . Recall that and, by additivity of expectation, . Also, by smoothness assumption, there are constants and such that and . On the other hand, by Markov’s inequality,

Recall that we assumed that the random variable is independent from for each . Thus, we have that with probability at least , both and hold, implying that . By additivity of expectation, it follows that the expected size of a subset of with for each link is at least . On the other hand, such a set can be partitioned into at most feasible (under SS) subsets, by Lemma 1. This implies that . ∎

In the particular case of Log-Normal Shadowing, even independence is not necessary for the result above to hold, as shown in th next theorem. We use the standard notation and to denote log-normally and normally distributed random variables, respectively. In Log-Normal Shadowing model, we assume that for all links , and for appropriate positive values and . In particular, the second parameter is constant.

A log-normally distributed variable can be seen as , where is a normal random variable. We will use the fact that . We will also use the following basic fact.

Fact 1.

If and are normal random variables, then .

Proof.

By Cauchy-Schwartz,

Since and , using the formula for the expectation of a log-normal variable gives

Theorem 2.

For any set of links under Log-Normal shadowing , , even if the signal and interference distributions are arbitrarily correlated.

Proof.

Let be a feasible subset of under GPL. It is enough to show that the expected size of an optimal feasible subset of under LNS is . Consider an arbitrary link . The affectance of under LNS is:

where log-normal random variables and represent the signal of link and interference caused by link , respectively. Recall that and . Let us denote and and note that the variables have variance . Hence, using the expectation formula for log-normal variables, we can observe that and . Using this observation together with Fact 1, we obtain that for every pair ,

Using the fact that is feasible, and linearity of expectation, we have, for every , that . Thus, using Markov’s inequality, we obtain that

The latter implies that the expected number of links with is less than . It remains to note that by Lemma 1, a -th fraction of the remaining set of links will be feasible under LNS, i.e., we will have that for those links . ∎

3.2 SS Can Increase Capacity

Next we show that, perhaps surprisingly, there are instances for which : the typical optima under SS can be much better than the optimum under GPL. The intuition is that shadowing will create many links with higher signal strength than the expectation, which will be the main contributors to the increase in capacity.

In the remainder of this section we consider a set of links of the same length , and assume that all sender nodes are located at one point (for all) and all receivers are located at another point. We call such links co-located. Note that under GPL, any feasible subset of co-located links contains a single link. We show below that under SS, the capacity can significantly increase. Let us fix a -smooth shadowing distribution for a constant . Since is a set of co-located links of length , we have that for each , which we denote for short.

In a dense or co-located set of links, the only hope for an increase in capacity are links of signal strength higher than . Intuitively, if there is a feasible subset made of links of strength , then , since the total interference on each link is likely to be . The following definition essentially captures the maximum size of such a set of strong links there can be in .

For each link , denote the probability that link has signal strength at least times what is expected. For each integer , there is a maximal value such that , because is a non-increasing function of , and when . We will use the following slightly different definition: . For the case of log-normal distribution, , as shown in Cor. 1 below.

The following lemma (Lemma 2) gives an upper bound on the SS optimum, when the signal strengths of links are fixed and there are few links with “strong signal”. Essentially, it indicates that the main contribution to the capacity is by links with strong signal, and interestingly, using power control cannot change this.

We will use the following result from linear algebra.

Fact 2.

[34, 27] Let be an non-negative real matrix. Then , where is the largest eigenvalue of .

Lemma 2.

Let be a set of co-located links. Assume that the signal strengths are fixed, but interferences are drawn from a -smooth SS distribution , for a number . For a number , assume that there are at most links with , where for a large enough constant . Then holds w.h.p. with respect to interference distributions, even when links use power control.

Proof.

Let . Let denote the event that contains a feasible (under SS) subset of size . Let denote the event that a set is feasible. By the union bound, we have that . The sum is over subsets, so it is enough to prove that for each subset of size , .

Let us fix a subset of size for the rest of the proof. Note that contains a subset of links such that . Since is a -smooth distribution and all are identically distributed, there is a constant s.t. where is such that for every pair . For each pair of links , let denote the binary random variable that is iff . Note that are i.i.d. variables and . Consider the sum . We have that . By a standard Chernoff bound, we have that . Recall that is constant. We choose the constant so as to have , which gives .

It remains to prove that if , then the set is not feasible with any power assignment. Let denote the normalized gain matrix of the set , where for any pair of links we denote and . As shown in [47], the largest SIR ratio that can be achieved with power control is , where is the largest eigenvalue222It is important here that we defined the normalized gain matrix in terms of interferences and signal strengths with respect to uniform power assignment (i.e., the power level is “cancelled” in the ratio , leaving the gain ratio, since links and use the same power level). of . Thus, if we show that , then the set is not feasible even with power control. To that end, we will use the bound given in Fact 2: . We restrict our attention only to the terms with , as those will have sufficient contribution to the sum. Indeed, recall that for each pair with , we have . Since it also holds for each link , that , we have: . Since we also have , we obtain the bound:

Hence, for any fixed , we choose the constant so as to have , in which case and the set is infeasible. This completes the proof. ∎

Theorem 3.

Let be a set of co-located links under a -smooth SS distribution with associated sequence . There are constants , such that

where the expectation is taken w.r.t. interference and signal strength distributions. In particular, if , then . The upper bound holds even if power control is used.

Proof.

We begin by showing the first inequality. Note that the case trivially holds, so we assume .

Let us call a link strong if . By the definition of values and since , the probability that any fixed link is strong is at least and at most . Let be the subset of strong links. The observation above readily implies, by the linearity of expectation, that

Recall that for all links . Thus, the expected affectance on each link is

where the expectation is taken over the distribution of interferences . Thus, using Markov’s inequality, we get that the expected number of links in with is . This, together with Lemma 1 and the fact that , proves that .

Now, let us demonstrate the second inequality of the claim. Let us assume that power control is allowed, and, as before, let denote the signal strength of link and denote the interference from link to link when all links use uniform power . Denote , where is a large enough constant. We will show that . Let be the event that there are at least strong links. Recall that the expected number of strong links is at most . Since , applying standard Chernoff bound gives , i.e., with high probability, holds: There are at most strong links.

Given that holds, Lemma 2, applied with and (assuming the constant is suitably large), tells us that gives that with high probability, . This completes the proof. ∎

Corollary 1.

For a set of co-located equal length links with Log-Normal Shadowing distribution ,

Proof.

First, let us estimate . Let and be the parameters associated with the LNS distribution of each link . Since the links have equal lengths, those parameters are the same across all links in . Recall that for each link . We will use the fact that the tail probability of a log-normal variable with parameters is as follows: where is the tail probability of the standard normal distribution . There is no closed form expression for , but it can be approximated as follows for all :

Using these formulas, we obtain: Denote for some . Then, by the definition of , we must have that . A simplification gives . It remains to show that LNS is a smooth distribution, i.e., for some constant , . To this end, note that the mean of a log normal variable with parameters is and the median is , so assuming is fixed, we can take . ∎

Remark.

Theorem 3 can be extended in two ways. First, we can assume that the sender nodes and receiver nodes are not in exactly the same location, but are within a region of diameter smaller compared to the link length. Second, we can assume that the links do not have exactly equal lengths, but the lengths differ by small constant factors. These modifications incur only changes in constant factors. This follows from Lemma 1. That is, the theorem predicts increased capacity due to shadowing not only for co-located sets of links, but also sets that contain dense parts, i.e., have subsets that are “almost co-located”.

The results above assert that under SS, the main contributor to the capacity increase in a co-located set of links are the “strong” links. The following result demonstrates that significant capacity increase (though not as dramatic as above) can happen even when all links have signal strength fixed to the GPL value. This phenomenon is due to interference distributions.

Theorem 4.

Let be a set of co-located links, all of length . Assume that the signal strengths are fixed and equal to but the interferences are drawn from Log-Normal shadowing model . Then,

Proof.

Assume, for simplicity, that . Consider any fixed subset of size for a fixed to be specified later, and let . From the definition of LNS we have,

for a constant , where we used the fact that . Denote . Thus, the probability that holds for all pairs , is . Note that if the latter event happens then is a feasible set. Let us partition into subsets of size . From the discussion above we have that each of those subsets is feasible with probability at least . Since those subsets are disjoint, the probability that none of them is feasible is at most . In order to have the latter probability smaller than , it is sufficient to set , which holds when

The following result demonstrates the limitations of Thm. 3. Namely, it shows that the slack in the upper bound cannot be replaced with constant in general: We show that there is an SS distribution with , such that the gap between SS and GPL capacities is at least a factor of for co-located equal length links. It also shows that is not necessary for increase in capacity compared to GPL.

Theorem 5.

There is an SS distribution with for each , such that holds for a set of co-located links.

Proof.

Let denote the expected signal strength of each link, as before. Consider the following discrete SS distribution , where for each link and , , where . It is easy to show that and for each . Consider an arbitrary subset of size and fix a link . It follows from the definition of , that . For each link , and . Thus, by independence of interferences, and

Thus, set is feasible with probability at least . Split into disjoint subsets of size and let denote the event that is feasible. Note that the events are independent, and . By the union bound and independence, . Thus, in order to have , it suffices to take . ∎

4 Computing Capacity under Shadowing

In this section we study the algorithmic aspect of LinkCapacity under SS. Namely, our aim is to design algorithms that perform well in expectation under SS, compared with the expected optimum under SS. In the first part, we handle (nearly) co-located links, while the second part discusses the more general case when the links are arbitrarily placed on the plane. In both cases, we obtain constant factor approximations for links of bounded length diversity, under general SS distributions satisfying weak technical assumptions. This holds against an optimum that can use arbitrary power control.

Let us start with several definitions. For any set of links, let denote the max/min link length ratio in . A set is called equilength if .

An equilength set of links is called cluster if there is a square of side length that contains the sender nodes of all links in , where .

4.1 Clusters

Consider a cluster of links with minimum length . The algorithm is based on Thm. 3, which suggests that choosing only the strong links is sufficient for a constant factor approximation to the expected optimum. We prove that a similar result holds in a more general setting, where the signal strengths of links are fixed and arbitrary, while interferences are drawn from a -smooth SS distribution for a constant . The algorithm is as follows.

Algorithm.

Let set be constructed by iterating over the set in a decreasing order of link strength, and adding each link to if its strength satisfies . Output the set of successful links in .

The theorem below shows that this strategy results in only additive expected error, and yields a fully constant factor approximation when signal strengths are drawn from an SS distribution with , such as LNS.

Theorem 6.

Let be a cluster of links. Assume that the signal strengths of links are fixed and arbitrary, but interferences are drawn from a -smooth SS distribution for a constant . Then,

where the expectation is taken only w.r.t. interference distributions. When the signal strengths are also drawn from and we further have , then

where expectation is taken w.r.t. signal and interference distributions.

Proof.

First, let us note that . Indeed, take a link . We have that and . Thus, with probability at least , , i.e., . By additivity of expectation, this implies that . It remains to show that . But this simply follows from Lemma 2 with and an appropriate value . The second part of the theorem follows from Thm. 3, which asserts that . ∎

4.2 General Equilength Sets

The algorithm presented in the previous section can be extended to general sets of equilength links, which are not necessarily clusters. The essential idea is to partition such a set into clusters, solve each cluster separately, then combine the solutions. This, however, requires some technical elaboration.

Let be an arbitrary equilength set. First, we partition into a constant number of well-separated subsets, where an equilength set is called well-separated if is a disjoint union of subsets such that for each , is a cluster and for each and links and , , where .

Proposition 1.

Any equilength set on the plane can be split into a constant number of well-separated subsets.

Proof.

Let . Partition the plane into squares of side with horizontal and vertical lines. Consider only the squares intersecting the bounding rectangle of links . Assign the squares integer coordinates in the following way: if two squares with coordinates and share a vertical (horizontal) edge then and ( and , resp.). Then split into subsets , , where . Let us fix for some arbitrary indices and let be any links. It remains to note that if the sender nodes of and are in different squares, then , which via the triangle inequality (and using the assumption that is equilength) implies that . ∎

Algorithm.

Partition into well-separated subsets, solve the capacity problem for each subset separately (as described below) and output the best solution obtained.

To process a well-separated subset , observe first that is a disjoint union of clusters , by definition. Run the algorithm from Sec. 4.1 on each cluster separately, obtaining a subset . Denote . Let be a constant, as indicated in the proof of Thm. 7, and let be the set of all links in such that . Partition into at most feasible subsets (under SS) using Lemma 1 and let be the largest of those.

Theorem 7.

Let be a set of equilength links (arbitrarily placed on the plane) under a -smooth SS distribution , for a constant , with . Then,

Proof.

First, note that performing the step of partitioning into well-separated subsets and selecting the best one, we lose at most a constant factor against the optimum. So we concentrate on a well-separated set , consisting of clusters . Recall that be the subsets obtained by the algorithms on the clusters and .

We first show that the expected number of links the algorithm chooses from is , where the expectation is only w.r.t. interference distributions. To that end we show that the out-affectance from any link in to all the other links is constant under GPL. That means that the expected affectance on a link in , even under , is constant, which yields the claim after some sparsification of .

For any two indices , , define the representative affectance by set on set as the largest GPL affectance by a link in on a link in :