SINR Diagram with Interference Cancellation

SINR Diagram with Interference Cancellation

Chen Avin Ben Gurion University, Beer-Sheva, Israel. {avin,coasaf,zvilo}@cse.bgu.ac.il, yoram.haddad@gmail.com    Asaf Cohen    Yoram Haddad Jerusalem College of Technology,Jerusalem, Israel    Erez Kantor The Technion, Haifa, Israel. erez.kantor@gmail.com    Zvi Lotker    Merav Parter    David Peleg The Weizmann Institute of Science, Rehovot, Israel. {merav.parter,david.peleg}@ weizmann.ac.il.Supported in part by a grant of the Israel Science Foundation and the I-CORE program of the Israel PBC and ISF (grant 4/11).Parts of this work appeared at the ACM-SIAM Symposium on Discrete Algorithms (SODA), Kyoto, Japan, 2012.
Abstract

In this paper we study the reception zones of a wireless network in the SINR model with receivers that employ interference cancellation (IC), a technique that allows a receiver to decode interfering signals, and cancel them from the received signal in order to decode its intended message. We first derive some important topological properties of the diagram describing the reception zones and their connections to high-order Voronoi diagrams and other related geometric objects. We then discuss the computational issues that arise when seeking an efficient description of the zones. Our main fundamental result states that although potentially there are exponentially many possible cancellation orderings (and consequently reception cells), in fact there are much fewer nonempty such cells. We prove a (tight) linear bound on the number of cells and provide a polynomial time algorithm to describe the diagram. Moreover, we introduce a novel measure, referred to as the Compactness Parameter, which influences the tightness of our bounds. We then utilize the properties established for reception diagrams to devise a logarithmic time algorithm for answering point-location queries for networks with IC.

Keywords: Interference cancellation, SINR, Voronoi diagram.

1 Introduction

1.1 Background and Motivation

Today, wireless communication is embedded in our daily lives, with an ever-growing use of cellular, satellite and sensor networks. The major advantage of wireless communication, namely, the broadcast nature of the medium, also creates its greatest obstacle, namely, interference. When a station has to decode a message (i.e., a signal) sent from a transmitter, it must cope with all other (legitimate) simultaneous neighboring transmissions.

Roughly speaking, two basic approaches to handling interference dominated the research community for many years [10]. One approache is orthogonalization. By using, e.g., time-division (TDMA) or frequency division (FDMA), the degrees of freedom in the channel can be divided among the participating transmitters. This generates an independent channel for each transmitter. The second approach is to treat the interference as noise. Taking this view, the interference, together with the ambient (or background) noise, disrupt the signal reception and decoding abilities. For the signal to be safely decoded, the Signal to Interference & Noise Ratio (SINR) must be large enough.

Due to the increasingly large number of users, the achievable rate or utilization of wireless networks has become the bottleneck of the communication. Consequently, the capacity of wireless networks, i.e., the maximum achievable rate by which stations can communicate reliably, has received increasing attention in recent years [2, 4, 11, 12, 13, 17, 21]. One of the main challenges for wireless network designers is to increase this rate and try to fully utilize the capacity of the network. In a sense, both of the aforementioned approaches treat interference in wireless communication as a foe, and try to either avoid it or overcome it. However, modern coding techniques suggest the ability to jointly decode several signals simultaneously, achieving a higher total capacity (see, e.g., Chapter 14 of [8]). This paper focuses on a relatively recent and promising method for such joint decoding called interference cancellation (IC) [1].

The basic idea of interference cancellation, and in particular successive interference cancellation (SIC), is quite simple. Consider a situation where a station receives a “combined” transmission composed of several interfering signal that were transmitted simultaneously. The station is interested in decoding one of those signals, the “intended” signal. However, it might not be the dominant signal in the combined transmission received by the station. The station can attempt the following technique for retrieving its intended signal. First, the strongest interfering signal is detected and decoded. Once decoded, this signal can then be subtracted (“canceled”) from the received (combined) signal. Subsequently, the next strongest interfering signal can be detected and decoded from the now “cleaner” combined signal, and so on. Optimally, this process continues until all stronger interferences are cancelled and we are left with the desired intended signal, which can now be decoded. This successive process seems prone to error propagation. Nevertheless, note that if the SINR while decoding a message at a given iteration is high enough (above a threshold ), then the process ensures correct decoding at that stage (at high probability), and using simple union bound, correct decoding at all stages. SIC is similar in spirit to several well known algorithms like the Gram-Schmidt process [25], solving triangular systems of linear equations, and fountain codes [6]. It should be noted that without using IC, every station can decode at most one transmitter (i.e., the strongest signal it receives). In contrast, with IC, stations can potentially decode more transmitters, or expressed dually, every transmitter can potentially reach more receivers. This clearly increases the utilization of the network.

The SINR model and interference cancellation are fairly well-studied from an information-theoretic point of view. The SINR model was used in the analysis of network capacity and throughput, e.g., [12] and the many papers which followed. Scheduling under the SINR model was discussed in [22], while random access techniques were given in [15]. In [27], the authors considered a stochastic SINR model, where the focus was on the outage probability - the probability that a receiver’s SINR is below a threshold . In [16], the sets of possible SINR values subject to linear power constraints were characterized. The zero-outage region, the region achievable regardless of the channel realization, was also considered. [16] also considered the problem of removing a subset of the users in order do achieve certain SINR demands. A suboptimal algorithm maximizing the number of active users was suggested.

Interference cancellation is the optimal strategy in several scenarios, such as strong interference [23, 7], corner points of a multiple access channel [8, Chapter 14], and spread spectrum communication (CDMA) [26], and it constitutes a key building block in the best known bounds for the capacity of the interference channel [10]. [28] studied the transmission capacity of wireless ad-hoc networks under successive interference cancellation. Although considering SIC under the SINR regime, [28] focused on the capacity and outage probability, rather than the geometric and algorithmic aspects of the reception zones. Thus, to the best of our knowledge, little is known about the structure and the properties of the reception zones under interference cancellation (namely, the areas in which transmitters can be decoded), as well as algorithmic issues for large wireless networks.

In this paper, we initiate the study of the topological properties of the reception zones in the context of the IC setting, discuss the computational issues arising when trying to compute these reception zones or answer queries regarding specific points, and devise polynomial-time algorithms to address these problems. This is done by extending the notion of SINR diagrams [3] to the setting of stations that can apply successive interference cancellation. The SINR diagram of a wireless network of transmitters partitions the plane into reception zones , one per station, and a complementary region of the plane where no station can be decoded, denoted . In [3], SINR diagrams have been studied for the specific case where all stations use the same transmission power, i.e., uniform power. It is shown therein that the reception zones have some “nice” properties, like being convex (hence connected) and “fat” (as defined later on). In [14] it was established that for a nonuniform power setting, the reception zones are not necessarily connected, but are (perhaps surprisingly) hyperbolically convex in a space of dimension higher by one than the network’s dimension. Turning to the stochastic setting, the relation between stochastic SINR diagram (formed by modeling the SINR as a marked point process) and classical stochastic geometry models such as Poisson–Voronoi tessellations, has been studied extensively. See [5] for a detailed analysis, results and applications of this approach.

(a) (b)

(c)

Figure 1: (a) Nonuniform power assignment where for , receiver is in the reception zone of transmitter . (b) In every uniform power assignment, is not in the reception zone of . (c) In a uniform power setting but with interference cancellation, station may be in the reception zone of transmitter , i.e., it could decode transmitter after canceling transmitter .

When adding SIC to SINR diagrams, the resulting structures, denoted SIC-SINR diagrams, become much more complex to present. However, they clearly reveal the benefits of the cancellation method. An example of this idea is illustrated in Figure 1. All three parts of the figure depict a network with two transmitters , and two receivers (or points in the plane) , with the requirement that and need to decode the signal transmitted by and respectively. The four nodes occur on a straight line in the order , similar to the known “nested links” example given in [19]. This example shows that in order to achieve the requirements, a nonuniform power assignment must be used by the two transmitters, thus demonstrating that the capacity (achievable rate) of nonuniform power assignments is higher than that of uniform power assignments. Figure 1(a) shows the reception zones and for and respectively in the nonuniform setting, which satisfy and . As mentioned, it can be proved that the two demands of the system cannot be satisfied when both and transmit with the same power. An SINR diagram with a uniform power assignment is shown in Figure 1(b). Note that here, , but . In contrast, when SIC can be employed at , it can first decode . Afterwards, it “cancels” from its received combined signal and then decodes . It follows that with SIC the two demands can be satisfied even with uniform powers! The SIC-SINR diagram presented in Figure 1(c) illustrates this by showing an additional zone, , the zone in which stations with SIC can decode after “canceling” . Note that, as explained later, is the intersection of two convex shapes, and , where the latter (shown as an empty circle) is the reception zone of if it had transmitted alone in the network. One clearly sees that the total reception area of with SIC is considerably larger than without SIC. In Subsection 2.5 we present an even more compelling and general motivating example, that shows the following.

Observation 1.1

There exists a wireless network for which any power assignment requires time slots to satisfy all the demands, while using SIC allows a satisfying schedule using a single time slot.

Despite the importance of IC, not much is known about its complexity. The goal of this paper is to take a first step towards understanding it, by studying reception maps under the setting of SIC. The starting point of our work is the observation that under the SIC setting, reception zones are no longer guaranteed to be convex, fat or even connected. This holds even for the “simplified” setting where stations transmit at the same power level and are aligned on a straight line (one dimensional map). The zones are also not hyperbolically convex as was shown for the nonuniform power setting without IC [14]. Moreover, while for SINR diagrams without IC there is a single polynomial that represents each of their reception zones, with IC the reception zone of each transmitter may depend on the cancellation order, which can lead to an exponential number of polynomials and cells. If this were the case, then even drawing the diagram might prove to be infeasible.

1.2 Main Contributions.

The study of SIC-SINR reception maps raises several immediate questions. The first is a simple “counting” question that has strong implications on algorithmic issues: What is the maximum number of reception cells that may occur in an SINR diagram of a wireless network with stations, where every point in the map is allowed to perform SIC? Is it indeed exponential? We address this question in two different ways. Initially we re-explore the intimate connections between wireless communication and computational geometry methods like higher-order Voronoi diagrams [24, 20]. In particular, we use a bound on the number of cells in ordered order- Voronoi diagram [20] to upper bound the number of reception zones by , where is the number of transmitters and is the dimension. In general, this bound is not tight, but interestingly we were able to tie the number of reception zones to a novel parameter of the network, termed the Compactness Parameter, , and achieve a much tighter bound when the compactness parameter is sufficiently high. The compactness parameter is a function of the two most important parameters of the wireless network model, namely, the reception threshold constant , which stands for the SINR threshold, and the path-loss parameter , and its value is . We then prove that when , the number of reception zones is linear for any dimension! This bound allows us to provide an efficient scheme for computing the cancellation order that gives the reception zones and therefore allows us to build and represent the diagram efficiently.

Our second question has a broader scope: Are there any “niceness” properties of reception zones that can be established in the SIC settings? Specifically, we aim toward finding forms of convexity satisfied by reception cells in SIC reception maps. Apart from their theoretical interest, these questions also have considerable practical significance, since having reception zones with some form of convexity might ease the development of protocols for various design and communication tasks [14]. We answer this question by using the key observation that zones are intersections of convex shapes, giving us some “nice” geometric guarantees.

Our third question is of algorithmic nature. We consider the point location task, where given a point and a station , one wants to know whether a receiver lacated at can receive ’s transmission using SIC. Applying the trivial computation in time, one can compute the set of stations that receives under the SIC setting. However, if the number of queries is large, an order of time per query might be too costly. To approach this problem we use the guarantees produced for the first two questions and present a scheme for answering point location queries approximately in logarithmic time.

We believe that the questions raised herein, as well as the results and techniques developed, can significantly contribute to the evolving topic of wireless topology and what we may refer to as computational wireless geometry.

The rest of the paper is organized as follows. In Section 3, we establish the basic properties of SIC-SINR diagrams and show its relation to higher-order Voronoi diagrams. We then derive a tight bound on the number of connected components in the reception map of a given station under SIC. Section 4 describes how one can construct SIC-SINR reception maps in polynomial time. Finally, Section 5 considers the point-location task and provides an efficient construction of a data structure that answers point-location queries (with predefined approximation guarantees) in logarithmic time.

2 Preliminaries

2.1 Geometric notions.

We consider the -dimensional Euclidean space (for ). The distance between points and is denoted by . A ball of radius centered at point is the set of all points at distance at most from , denoted by . Unless stated otherwise, we assume the 2-dimensional Euclidean plane, and omit . The maximal and minimal distances between a point and a set of points are defined, respectively, as and . The hyperplane , for , is defined by . Given a set of points , let the corresponding set of all hyperplanes be . A finite set of hyperplanes defines a dissection of into connected pieces of various dimensions, known as the arrangement of . The basic notions of open, closed, bounded, compact and connected sets of points are defined in the standard manner (see [3]).

We use the term zone to describe a point set with some “nice” properties. Unless stated otherwise, a zone refers to the union of an open connected set and some subset of its boundary. It may also refer to a single point or to the finite union of zones. A polynomial is the characteristic polynomial of a zone if for every .

Denote the area of a bounded zone (assuming it is well-defined) by . A nonempty bounded zone is fat if the ratio between the radii of the smallest circumscribed and largest inscribed circles with respect to is bounded by a constant.

2.2 Wireless Networks and SINR.

We consider a wireless network , where , is a set of transmitting radio stations embedded in -dimensional space, is a mapping assigning a positive real transmitting power to each station , is the background noise, is a constant serving as the reception threshold (to be explained soon), and is the path-loss parameter. The signal to interference & noise ratio (SINR) of at point is defined as

When the network is clear from the context, we may omit it and write simply . Throughout this paper we assume the uniform setting, where . Let be a network identical to except its dimension is . In our arguments, we sometimes refer to an ordered subset of stations, . Denote the last element in by . When the order is insignificant, we refer to this set as simply . The wireless network restricted to a subset of nodes is given by . The network is assumed to contain at least two stations, i.e., . The fundamental rule of the SINR model is that the transmission of station is received correctly at point if and only if its SINR at reaches or exceeds the reception threshold of the network, i.e.,

When this happens, we say that is heard at .

2.3 SINR diagrams (without SIC).

Let us now introduce the central notion of SINR maps. We refer to the set of points that hear station as the reception zone of , defined as

(Note that is undefined at points in and in particular at itself.) Analogously, the set of points that hear none of the stations (due to the background noise and interference) is defined as

An SINR diagram

is a “reception map” characterizing the reception zones of the stations. This map partitions the plane into zones; a zone for each station , , and the zone where none of the stations is received. It is important to note that a reception zone is not necessarily connected. A maximal connected component within a zone is referred to as a cell. Let denote the cell in . Hereafter, the set of points where the transmissions of a given station are successfully received is referred to as its reception zone. Hence the reception zone is a set of cells, given by

where is the number of cells in . Analogously, is composed of connected cells , for . Overall, the topology of a wireless network is arranged in three levels: The reception map is at the top of the hierarchy. It is composed of reception zones, , , and . Each zone is composed of reception cells. The following lemma is taken from [3].

Lemma 2.1 ([3])

Let be a uniform () power network where . Then is convex and fat for every .

In our arguments, we may sometimes refer to the wireless network induced on a subset of stations . The reception zone of in this induced network is denoted by . When is clear from context, we may omit it and write simply and . The following definition is useful in our later arguments. Let , , be the characteristic polynomial of , given by

(1)

Then if and only if .

2.4 Geometric diagrams in .

Throughout the paper we make use of the following types of diagrams.

Hyperplane Arrangements.

Given a set of of hyperplanes in , the arrangement of dissects into connected pieces of various dimensions. Let denote the number of connected components in . The following facts about are taken from [9].

Lemma 2.2 ([9])

(a) .
(b) can be constructed in time and maintained in space.

Given a set of points , we define to be the arrangement on , the set of all hyperplanes of pairs in . has an important role in constructing SIC-SINR maps, as will be described later on.

Corollary 2.3

.

Voronoi diagrams.

The ordinary Voronoi diagram on a given set of points is generated by assigning each point in the space to the closest point in , thus partitioning the space into cells, each consisting of the set of locations closest to one point in (referred to as the cell’s generator). Let denote the Voronoi cell of given a set of generators . Let , for , denote the Voronoi cell of in a system restricted to the points of .

Avin et al. [3] discuss the relationships between the SINR diagram on a set of stations with uniform powers and the corresponding Voronoi diagram on , and establishes the following lemma. Let .

Lemma 2.4 ([3])

for every .

Higher order Voronoi diagrams.

Higher order Voronoi diagrams are a natural extension of the ordinary Voronoi diagram, where cells are generated by more than one point. They provide tessellations where each region consists of the locations having the same (ordered or unordered) closest points in , for some given integer .

Order- Voronoi diagram.

The order- Voronoi diagram is the set of all non-empty order- Voronoi regions , where the order- Voronoi zone for an unordered subset , is defined as follows.

This can alternatively be written as

(2)

This alternate representation plays a role in this paper. Note that corresponds to the ordinary Voronoi diagram and that any , for , is a refinement of .

Ordered Order- Voronoi diagram.

Let be an ordered set of elements from . When the generators are ordered, the diagram becomes the ordered order- Voronoi diagram [20], defined as

where the ordered order- Voronoi region , , is defined as

Alternatively, as in [20],

(3)

Note that each is an intersection of convex shapes and hence it is convex as well.
The following claim is useful for our later arguments.

Lemma 2.5

For every such that there exist , such that the hyperplane separates and .

Proof: We focus on the case where and . Let denote the first index such that . First consider the case where . Then and , so separates the zones and the lemma holds. Otherwise, assume and let denote the longest common prefix of and . Let , and . First note that by Eq. (3), and . In addition, and are convex. Next, observe that correspond to distinct Voronoi regions in the system of points and therefore and are separated by . The lemma follows.    

2.5 Motivating Example: Interference Cancellation vs. Power Control.

The following motivating example illustrates the power of interference cancellation even in the uniform setting where all stations use the same transmission power. Consider a set of communication requests consisting of sender-receiver pairs embedded on a real line as follows. The receivers are located at the origin and the set of senders are positioned on an exponential node-chain, e.g., is positioned on (see Fig 2). Since all receivers share the same position, without SIC there exists no power assignment that can satisfy more than one request simultaneously, hence time slots are necessary for satisfying all the requests. We claim that by using SIC, all requests can be satisfied in a single time slot even with a uniform power assignment. We focus on a given receiver and show that it successfully decodes the signal from after successive cancellations of the signal transmitted by for every . Using the notation of Section 2.2, let denote the network imposed on the last stations, whose positions are to . Note that

for every . We therefore establish that there exists an instance such that any power assignment for scheduling requires slots, whereas using SIC allows a satisfying schedule using a single time slot.

So far, the literature on capacity and scheduling addressed mostly nonuniform powers, showing that nonuniform power assignments can outperform a uniform assignment [19, 18] and increase the capacity of a network. In contrast, examples such as Fig. 2 illustrate the power of interference cancellation even with uniform power assignments, and motivate the study of this technique from an algorithmic point of view. Understanding SINR diagrams with SIC may play a role in the development of suitable algorithms (e.g. capacity, scheduling and power control), filling the current gap between the electrical engineering and algorithmic communities with respect to SIC research.

Figure 2: The power of interference cancellation. Any power assignment requires time slots to satisfactorily schedule these requests in the SIC-free setting. Using SIC, all requests are satisfied in a single time slot (even when using uniform powers).

3 SIC-SINR Diagrams in Uniform Power Networks

In this section, we first formally define the reception zones under interference cancellation, forming the SIC-SINR Diagrams. We then take a first step towards studying the properties of these diagrams. We elaborate on the relation between the SIC-SINR diagram and the ordered order-k Voronoi diagram, and use it to prove convexity properties of the diagram and to bound the number of connected components in the SIC-SINR Diagrams. We then define the Compactness Parameter of the diagrams and use it to achieve tighter bounds on the number of connected components.

3.1 SINR diagrams with SIC.

Let . We now focus on the reception zone of a single station, say , under the setting of interference cancellation. In other words, we are interested in the area containing all points that can decode , possibly after some sequence of successive cancellations. To warm up, we start with the case of a single point and ask the following question: does successfully receive using SIC?

Let correspond to the set of stations ordered in nonincreasing order of received signal strength at point up to station , i.e., , where . Since all stations transmit with the same power, it also holds that

(4)

To receive correctly, must successively cancel the signals transmitted by station , for , from the strongest signal to the weakest. It therefore follows that successfully receives following SIC iff

(5)

for every . The reception zone of in a wireless network under the setting of SIC is denoted by , or simply when is clear from the context. It contains and the set of points obeying Equation (5), i.e.,

(6)

We now provide a more constructive formulation for , which becomes useful in our later arguments. Let be an ordering of stations . Let denote the reception area of all points that receive correctly after successive cancellation of . Formally, the zone is defined in an inductive manner with respect to the length of the ordering , i.e., number of cancellations minus one. For , . Otherwise, for ,

or

(7)

The following is a direct consequence of Eq. (7).

Corollary 3.1

Let , . Then

Finally, the reception zone of under SIC is given as follows. Let denote the collection of all cancellation orderings ending with , namely,

Then

(8)

The reception zone

is a set of cells. Although by definition it seems that might consist of an exponential number of regions for each , in what follows we show that this is not the case and that there are only polynomially many cancellation ordering that are relevant for . Note that the region of unsuccessful reception to any of the points, namely, , is unaffected by SIC. This follows by noting that SIC only affects the set of points . In other words, the successive signal cancellation allows points to “migrate” from the reception zone of station to that of station . However, points that hear nobody can cancel none of the signals. Overall, analogous to the SIC-free case, the topology of a wireless network under SIC is again arranged in three levels: The reception map, at the top of the hierarchy, is composed at the next level of reception zones, , and . Finally, at the lowest level, each zone is composed of reception cells.

Analogous to the SIC-free setting, we may refer to the wireless network induced on a subset of stations . The reception zone in this induced network is denoted by or .

Throughout the paper we consider a uniform power network of the form . By Lemma 2.1, reception zones of uniform SIC-free power maps are convex. However, once signal cancellation is allowed, the convexity (and connectivity) of the zones is lost, even for the simple case where stations are aligned on a line; see Figure 3 for an illustration of the SIC-SINR map of a 3-station system.

3.2 Higher-order Voronoi diagrams and SIC-SINR maps.

To understand the structure and the topological properties of SIC-SINR reception maps, we begin our study by describing the relation between SIC-SINR reception maps and ordered order- Voronoi diagram. Specifically, we prove that every SIC-SINR zone is composed of a collection of convex cells, each of which is related to a cell of the higher-order Voronoi diagram. To avoid complications, we assume our stations are embedded in general positions.

We begin by describing the relation between a nonempty reception region and an nonempty ordered order- polygon.

Lemma 3.2

For every , .

Proof: By Lemma 2.4, . Therefore by Eq. (7) it follows that

where the last inequality follows by Eq. (3).    

We now show that reception regions in that result from different cancellation orderings correspond to distinct connected cells.

Lemma 3.3

Every two regions correspond to two distinct cells.

Proof: By Eq. (8), is the union of regions for , i.e., where . By Lemma 3.2, and . Due to Claim 2.5, and hence also . The lemma follows.    

This lemma establishes the following.

Lemma 3.4

For every two reception cells and , there are distinct orderings such that and .

For illustration of these relations, see Figures 3 and 4.

Figure 3: SIC-SINR reception map in for a 3-station network aligned on a line. The first line of colored segments corresponds to reception zones with no cancellations. The second line of segments in each figure represents the added reception cells by signal cancellation. , and are in middle, light and dark grey respectively.
Figure 4: Reception map of and ordered order- Voronoi diagram (for . (a) SINR map with no cancellation. Shown are , and . (b) Intermediate map of : Reception cells of following at most one cancellation. (c) Final map of : Reception cells of following two cancellations.

Next, this relation between the SIC-SINR reception maps and ordered order- Voronoi diagram is used to establish the convexity of cells and to bound the number of connected components in the zone. We first show that the reception cells of are convex.

Lemma 3.5

Every reception cell is convex.

Proof: Due to Lemma 3.3 it is enough to show that every nonempty is convex. Let and let . By Lemma 2.1, is convex for every and therefore by Eq. (7), is an intersection of convex and bounded shapes, hence it is convex (and bounded) as well.    

We now discuss the number of connected components in SIC-SINR diagrams. Without loss of generality we focus on station . By Lemma 3.3, every two distinct orderings correspond to distinct reception cells (though it might be empty). Since the number of distinct orderings of length is (i.e., the size of ) and each of those orderings might correspond to a distinct cell, it follows that the number of connected cells in might be exponential. Fortunately, the situation is much better due to Lemma 3.4. An ordering is defined as a nonempty cancellation ordering () if and only if is nonempty. Partition the collection of ’s into sets as follows. An ordering is in the set if and only if it is an and in addition .

We first consider the hyperplane arrangement and claim that any given cell intersects with at most one high-order Voronoi region , where .

Lemma 3.6

Let . Then there exists at most one such that .

Proof: Assume to the contrary that there are two distinct Voronoi regions corresponding to two orderings, and , where , that have a nonempty intersection with a common face , i.e., and . By Claim 2.5, there exists a hyperplane that separates and . As and are convex, it follows that this separating hyperplane must intersect as well. However, by the definition of arrangements, is not intersected by any separating hyperplane, contradiction.    

The following lemma shows that there are only polynomially many orderings in .

Lemma 3.7

(a) , and (b)  for .

Proof: Part (a) follows by combining Lemma 3.6 with Corollary 2.3. To prove Part (b) we provide a construction of an -station wireless network in that has cells corresponding to , asymptotically matching the upper bound for . Consider a set of points on the line where is positioned on . Select the points so that and (for every ). We now show that . Let be the mid point between and , i.e., . Consider an iterative network construction process in which we start with an empty set of stations , and at step we add a station at location . We claim that for each step the following holds:
(1) The cell exists, and
(2) point dissects into cells where and , for .
We prove these invariants by induction on . Consider . Then obviously exists and by adding to any we get that and . We now assume that (1) and (2) hold for step and consider step . By the inductive assumption, at step and according to (2), dissects into segments, one of which is , which establishes (1) for step . It is left to show that breaks into segments, i.e., . Note that , for , exists if and only if . To see that the latter fact indeed holds, observe that and that . Therefore it is sufficient to show that . Note that since , and that since , and therefore (2) holds as well.
So far, we showed that step of the iterative process, adds new cells, such that , due to , namely, . Observe now that at each step , there are newly added cells , where . This follows by noting that among the newly added cells correspond to , and therefore after steps we end with cells of . This establishes part (b) of the lemma.    

Exploiting the relation between SIC-SINR diagrams and high-order Voronoi diagrams, we establish the following.

Lemma 3.8

, for every .

Proof: By Lemma 3.4 and Eq. (8), a cancellation ordering might correspond to a distinct cell in only if it is in . Therefore the lemma follows immediately by Lemma 3.7.    

3.3 A Tighter Bound on the Number of Connected Components.

In this section we introduce a key parameter of a wireless network