Scheduling with interference decoding: Complexity and algorithms

Ad Hoc Networks 11 (2013) 1732–1745

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Ad Hoc Networks journal homepage: www.elsevier.com/locate/adhoc

Scheduling with interference decoding: Complexity and algorithms q O. Goussevskaia a,⇑, R. Wattenhofer b a b

Department of Computer Science, Federal University of Minas Gerais, Brazil Computer Engineering and Networks Laboratory, ETH Zurich, Switzerland

a r t i c l e

i n f o

Article history: Received 25 June 2012 Received in revised form 28 February 2013 Accepted 7 March 2013 Available online 28 March 2013 Keywords: Wireless networks Scheduling Analog network coding Successive interference cancelation Complexity Physical interference model Algorithms

a b s t r a c t In this article we study the problem of scheduling wireless links in the physical interference model with interference decoding capability. We analyze two models with different decoding strategies that explore the fact that interfering signals should not be treated as random noise, but as well-structured signals. The first model makes use of successive interference cancelation, which allows the strongest signal to be iteratively decoded and subtracted from a collision, thus enabling the decoding of weaker simultaneous signals. The second model explores the fact that routers are able to forward the interfered signal of a pair of nodes that wish to exchange a message and these nodes are able to decode the collided messages by subtracting their own contribution from the interfered signal. We prove that the scheduling problem remains NP-complete in both models. Moreover, we propose a polynomial-time scheduling algorithm that uses successive interference cancelation to compute short schedules for network topologies formed by nodes arbitrarily distributed in the Euclidean plane. We prove that the proposed algorithm is correct in the physical interference model and provide simulation results demonstrating the performance of the algorithm in different network topologies. We compare the results to solutions without successive interference cancelation and observe that considerable throughput gains are obtained in certain scenarios.  2013 Elsevier B.V. All rights reserved.

1. Introduction The problem of scheduling wireless links is fundamental in determining the communication capacity of a wireless network. Given a set of communication requests between senders and receivers, arbitrarily distributed in space, the problem consists in determining which subsets of links (requests) can be scheduled concurrently, such that no collisions occur among them, and how these sub-

q This work is based on the preliminary conference papers [1] and [2]; the project was supported in part by CNPq and Fapemig. ⇑ Corresponding author. Address: Departamento de Ciencia da Computacao, UFMG, Predio do ICEx, sala 3010, Av. Antonio Carlos, 6627, 31270-010 Belo Horizonte, Brazil. Tel.: +55 31 3409 5860; fax: +55 31 3409 5858. E-mail addresses: [email protected] (O. Goussevskaia), wattenhofer@ tik.ee.ethz.ch (R. Wattenhofer).

1570-8705/$ - see front matter  2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.adhoc.2013.03.010

sets must be chosen, such that all links are able to transmit successfully in minimum time. The link scheduling problem has been studied in a variety of interference models, such as graph-based models (e.g. the protocol interference model) and fading channel models (e.g. the physical interference model). The physical model, to which we also refer as the SINR (Signal to Interference plus Noise Ratio) model, provides a more realistic representation of wireless interference than graph-based models, however, it might be more challenging when it comes to complexity analysis and algorithm design. In [3] it was shown that the link scheduling problem with uniform power assignment is NP-complete in the physical interference model by a reduction from the Integer Partition problem. One of the key concepts on which wireless interference models rely is the definition of a successful transmission.

O. Goussevskaia, R. Wattenhofer / Ad Hoc Networks 11 (2013) 1732–1745

In the SINR model, a transmission is considered successful if the signal power of the sender at the intended receiver sufficiently exceeds the sum of signal powers of all concurrent transmissions in the network (considered as interference) at the receiver, i.e., it is assumed that a receiver successfully decodes one, and only one, message at a time. This ‘‘traditional’’ approach of handling wireless interference by avoiding or reducing it is implemented in most of the existing MAC protocols for wireless LANs, which are based on carrier sensing, e.g. 802.11. In carrier sense, a transmission is deferred if the sender device senses another transmission in progress. The idea is to eliminate interference and allow higher SINR at the receiver. The problem with this approach is that it discourages spatial reuse, thus wasting available bandwidth. Since radio devices can typically detect an ongoing transmission over very large areas, a single ongoing transmission can block a great number of potential concurrent transmissions spread over a large area. This basically results in linear scheduling, i.e., one node transmits at a time, whereas many concurrent transmissions could be scheduled simultaneously. More recently, the fact that wireless interference is harmful and that a receiver can only decode the strongest signal at a time has been revised. Techniques such as cochannel separation, analog network coding, MIMO, and successive interference cancelation have radically changed the definition of a successful transmission. Cochannel separation techniques allow the receiver to decode several signals simultaneously under the assumption that these signals differ significantly in their strength [4]. Analog network coding makes it possible to simultaneously decode two signals of similar strength, under the assumption that the receiver knows one of the interfered signals by having overheard or forwarded it earlier [5]. MIMO systems explore space–time coding techniques by means of antenna arrays and coherent combining at a receiver [6,7]. In successive interference cancelation, one sender’s packet with the strongest signal is first decoded and ‘‘subtracted’’ from the received collision, such that weaker simultaneous signals can be then decoded in an iterative fashion [8]. Given these alternative ways of defining a successful transmission, an interesting question that rises is whether the computational complexity of the scheduling problem remains the same. This work presents some initial steps into the study of this issue. We analyze two models with different decoding strategies that use interference decoding. In one model, we assume that a receiver is able to perform successive interference cancelation, i.e., to decode several signals simultaneously, provided that these signals differ in strength significantly. In the second model, we assume that routers are able to forward the superposition of two interfering signals of nodes that wish to exchange a message, and nodes are able to decode the ‘‘collided’’ message by subtracting their own contribution from the interfered signal. For each network coding definition, we construct an instance of the scheduling problem in the geometric physical interference model, in which nodes are distributed in the Euclidean plane, and present NPcompleteness proofs for both scenarios.

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Moreover, we propose a polynomial-time scheduling algorithm that uses successive interference cancelation to compute short schedules obeying the SINR constraints for network topologies formed by nodes arbitrarily distributed in the Euclidean plane. We provide simulation results demonstrating the performance of the algorithm in different network topologies. We compare the obtained schedule lengths to solutions that do not employ successive interference cancelation and observe that throughput gains of up to 20% are obtained in certain scenarios. In Section 2 we discuss some related work. In Section 3 we define two different models of SIDSMerence decoding. To the first model we refer as successive interference cancelation and to the second model we refer as interference decoding by signal mixing. To the scheduling problems defined in each of these models we refer as Scheduling with Successive Interference Cancelation (SSIC) and Scheduling with Interference Decoding by Signal Mixing (SIDSM). In Sections 4 and 5 we present NP-completeness proofs for SIDSM and SSIC, respectively. In Section 6 we present the polynomial algorithm for the SSIC problem and the corresponding proof of correctness. In Section 7 we discuss the simulation results. Finally, in Section 8 we present our conclusions.

2. Related work The fundamental problem of scheduling wireless requests has been extensively studied in many models. Most of the hardness results are derived in graph-based models [9,10] or in non-geometric physical interference (SINR) models, where the values in the gain matrix are chosen arbitrarily, i.e., are not constrained by triangular inequality [11]. The problem of scheduling wireless requests with uniform power assignment in the SINR model was shown to be NP-complete in [3]. After this first hardness result, the analysis of the problem in the physical model has received a lot of attention, generating an interesting body of work. In [12] a constant approximation algorithm was devised for the on-slot scheduling problem under uniform power assignment. More recently, a constant approximation algorithm for the joint problem of scheduling and power control was obtained in [13]. A detailed survey of these and related recent results can be found in [14]. The possibility of decoding collided simultaneous signals has been explored in a variety of contexts. Some techniques, such as cochannel signal separation, explore differences in the characteristics of interfered signals, such as signal’s strength, to decode several signals simultaneously [4,15]. Other techniques, such as MIMO, allow multiple concurrent transmissions using antenna arrays and space–time coding to decode multiple concurrent transmissions simultaneously [7]. Interference cancelation is a type of multi-user detection technique in wireless communication networks [16]. Many of such techniques were originally proposed for CDMA systems [17,18] and exploit several resources of the cellular infra-structure, such as separate areas of reception and centralized synchronization and power control. In [19] a proof of concept of interference cancelation applied

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to wireless LANs was presented, where a collision between signals with similar power could be decoded. In [8] a simpler technique called successive interference cancelation was implemented for the 802.15.4 (ZigBee) physical layer. In this approach, collided signals have to differ significantly in power in order to be decoded, and the strongest signal is successively subtracted from a collision, allowing the decoding of weaker simultaneous signals. Successive interference cancelation has been also applied in MIMO LANs: in [20] it was used in combination with interference alignment to improve throughput, and in [6] it was used for channel training and transmission control to avoid coordination and increase throughput. In [21] Gelal et al. analyzed how topology control can be used to improve the efficiency of successive interference cancelation in MIMO networks. In a more recent work, Avin et al. [22] analyzed the effect of interference cancelation on the topology of ‘‘reception zones’’, i.e., regions where a signal can be successfully decoded in the SINR model. Another group of related work deals with network coding in the physical layer, or analog network coding (ANC). ANC exploits the fact that, in a wireless network, often a receiver has prior knowledge about some packets destined to other nodes, by having overheard or forwarded them earlier. It similar in spirit to digital network coding [23,24], however, it operates on the raw analog signal, instead of first decoding and then mixing packets in a bitwise manner. In [25] the impact of such knowledge in combination with nested coding on the capacity region was analyzed. In [5,26] decoding algorithms are proposed for the scenario, where pairs of nodes that wish to exchange packets through a relay node are encouraged to transmit simultaneously. The relay node, without decoding the collided signal, amplifies and forwards it. The destination nodes then extract the packet destined to them by filtering out their own contribution from the mixed signal. In [27] the concept of ANC was generalized by showing how collided packets can be recovered in a 802.11 network using ZigZag decoding, provided that the network operates at sufficiently high SNR, not too many packets are involved in a collision and there are enough retransmissions containing the same packets. A collision is treated as a linear equation over the involved packets, which can be recovered if the resulting system of linear equations has a unique solution. The asymptotic optimality of ANC in high SNR regimes was recently proved in [28], and its asymptotic optimality in terms of degrees of freedom was shown in [29]. Finally, very recently, the problem of data rate maximization using ANC for unicast communication over a so-called ‘‘layered’’ wireless relay network with directed links was analyzed in [30].

3. Model In this work we study the problem of scheduling wireless requests in the physical interference model. In order to capture interference decoding capability, we work with two different decoding strategies, or two definitions of a successful transmission: Scheduling with Successive Interference Cancelation (SSIC) and Scheduling with Interference Decoding by Signal Mixing (SIDSM).

The input to the link scheduling problem is a set of links L = l1, . . ., ln, where each link lx represents a communication request from a sender sx to a receiver rx. We assume that nodes live in Euclidean plane. The distance between two nodes sx, ry is denoted by dxy = d(sx, ry). The length of a link lx is denoted by dxx. The received power Prx ðsx Þ of a signal transmitted by sender sx at receiver rx is defined as

Prx ðsx Þ ¼

P a ; dxx a

where P is the transmission power, and dxx is the propagation attenuation (link gain). The path-loss exponent a > 2 is a constant, whose exact value depends on external conditions of the medium, such as humidity, obstacles, etc. In the physical interference model, a receiver rx successfully decodes a transmission from a sender sx iff

SINRrx ðS t Þ ¼ P

Prx ðsx Þ

sy 2S t ;sy –sx P r x ðsy Þ

þN

P b;

ð1Þ

where S t is the set of nodes concurrently transmitting in time-slot t, and b is the minimum signal-to-interferenceplus-noise-ratio (SINR) required for a successful message decoding. Typically, it is assumed that b > 1. The objective of the Scheduling problem is to compute a minimum-length schedule S ¼ fS 1 . . . S T g of size T, such that all links in every time slot S t 2 S can be scheduled successfully according to inequality (1). We consider the case where all nodes transmit with the same power level P, i.e., we assume a uniform power assignment scheme. Moreover, we assume that nodes are distributed in an arbitrary, possibly worst-case, way in the Euclidean space. We consider one-hop communication requests in the first model (SSIC) and two-hop communication requests with a relay in the second model (SIDSM). The communication requests can have multiple receivers and multiple sources. In order to capture the capability of decoding several signals from a collision, we work with alternative definitions of a successful transmission, which we introduce below. 3.1. Successive interference cancelation Successive interference cancelation is based on the fact that interfering signals, unlike noise, obey a certain structure, determined by the data being transmitted. The idea is to exploit this structured nature in order to decode several collided signals. Firstly, the basic SINR model is used to decode the strongest signal received in a collision, and then the bits decoded from the signal are used to generate an approximated model used to subtract its contribution from the rest of the received signal plus interference and noise. As described in [8], a typical receiver would perform the following steps: 1. Detect a collision, by scanning for strong amplitude variations in the incoming signal. 2. Decode the strongest signal using inequality 1. 3. Use the decoded digital data from (2) and the properties of the physical layer standard, e.g. 802.15.4, to develop an approximated model for the received signal.

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4. Use the model developed in (3) to cancel out the strongest signal from the total interference by iterating through the data and aligning the phase of the model and the received samples. 5. Iterate to decode the remaining packets. In order to model successive interference cancelation, we assume that a receiver r is able to decode several signals simultaneously, provided that these signals differ in strength significantly. Consider a set of concurrently scheduled links S t , and a subset of k signals sorted in decreasing order of power received at a node r: ! = {Pr(s1), Pr(s2), . . ., Pr(sk)}. We assume that the receiver r is able to decode all k signals in ! if and only if the following condition holds "x 2 {1, . . ., k}:

P

Pr ðsx Þ P P b; P ðs Þþ r y sy 2 S t ; sz 2 S t Pr ðsz Þ þ N Pr ðsy Þ 2 !; Pr ðsz Þ R ! Pr ðsy Þ < Pr ðsx Þ

ð2Þ

where the first component of the denominator is the accumulated interference caused by transmissions in !, which have weaker power level than Pr(sx); the second component of the denominator is the accumulated interference of all other concurrent transmissions in the network, which are not in !; N is the ambient noise; and b is the minimum SINR threshold. The idea is that, one by one, each signal Pr(sx) 2 ! can be filtered out from the accumulated interference, provided that the SINR between this signal and the remaining interference is above the threshold b. The key point here is that a receiver r is able to decode not only the strongest signal, as in the traditional physical interference model, but also a relatively weak signal, provided that each of the stronger signals has been filtered out. Therefore, a signal Pr(sx) can be correctly decoded if and only if all concurrently scheduled stronger signals (Pr(sy)) obey the following constraints:

P r ðsx Þ þ

P

Pr ðsy Þ Pr ðsz Þ þ N sz 2 S t sz –sx Pr ðsz Þ < Pr ðsy Þ

8s y 2 S t ;

where Pr ðsy Þ > Pr ðsx Þ; P

Pr ðsx Þ Pr ðsz Þ þ N sz 2 S t Pr ðsz Þ < Pr ðsx Þ

P b;

and

ð3Þ

P b:

ð4Þ

The objective of the Scheduling with Successive Interference Cancelation problem (SSIC) is to compute a minimum-length schedule S ¼ fS 1 . . . S T g of size T, such that all links in every time slot S t 2 S can be scheduled successfully according to inequality (2), or equivalently, inequalities (3) and (4).

two packets collide, nodes often know one of the colliding packets due to having forwarded it earlier or having overheard it. Consider a situation where two nodes A and B wish to send a message to each other (see Fig. 1). Due to the interference of concurrent transmissions or due to the ambient noise, A and B cannot communicate directly, but only through a relay node R. Instead of scheduling four sequential transmissions A ? R, R ? B, B ? R, R ? A, as in the traditional approach, by using analog network coding, A and B can transmit simultaneously, allowing their transmissions to interfere at R. The router, not being able to decode the collided packets, can simply amplify and forward the interfered signal. It has been shown in [5] that A (as well as B) is able to decode B’s packet by subtracting the contribution of its own packet from the interfered signal, even if the two transmissions are not fully synchronized and the wireless channel distorts the signals. As a result, only two time slots are sufficient to schedule these requests. In order for such a signal mixing to result in two successful transmissions, the following SINR conditions must hold in two time slots ti, tj, j > i:

PR ðAÞ P b; sy 2 S ti PR ðsy Þ þ N sy R fA; Bg P R ðBÞ P P b; s 2 S PR ðsy Þ þ N P

y

ð6Þ

ti

sy R fA; Bg PA ðRÞ P P b; sy 2 S t PA ðsy Þ þ N

ð7Þ

j

sy –R PB ðRÞ P P b: s 2 S PB ðsy Þ þ N y

ð8Þ

tj

sy –R This means that in order for A (and B) to be able to decode the mixed signal (PR(A) + PR(B)) amplified and forwarded by R in time-slot tj, the signals received by R in time-slot ti from both B and A must, individually, obey SINRlAR ðSti n fBgÞ P b and SINRlBR ðSti n fAgÞ P b. Note that the relative signal strength of A and B does not have to exceed any threshold. In fact, it has been shown in [5] that even when PR(A) = PR(B), the signals can still be correctly decoded by their receivers. However, note that the mixed signal sent by R must still have SINRlRA ;lRB ðS tj Þ P b at both receivers A and B in time-slot tj. For those transmissions that occur without employing signal mixing, we define a successful transmission as in traditional physical interference model (see Eq. (1)).

t1

A

3.2. Interference decoding by signal mixing Our second definition of interference decoding is similar to analog network coding, introduced in [26,5]. This model explores the fact that in a wireless network, when

ð5Þ

t4

t2

R

t1

A

t2

t3

B

t1

R

t2

B

Fig. 1. Interference decoding by signal mixing.

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triple1 \ triple2 \    \ tripleq ¼ £;

R

ð9Þ

triple1 [ triple2 [    [ tripleq ¼ A [ B [ C;

P = min(Pa q,Pbq)-

ði þ jÞ ¼ k; 8ði; j; kÞ 2 triplel ; 8l 2 f1; . . . ; qg: Note that, for a solution to exist, we need:

bq

aq

A

B a1 P = ai

b1 a

b

P = bj

R2

c1

P = *ck

P = *(min(Paq,Pbq)- ) e1..q cq

Fig. 2. Reduction from NMTS: all 4q links can be scheduled successfully in 2q time slots if and only if senders a1, . . ., aq, b1, . . ., bq, c1, . . ., cq, are partitioned into q triples (ai, bj, ck) such that (i + j) = k.

For the sake of simplicity, in Sections 4 and 5 we set N = 0 and ignore the influence of (constant) ambient noise in the calculation of SINR, given that this has no significant effect on the results. In Section 6, where we present an algorithm to solve the SSIC problem, however, we do consider the case of non-zero ambient noise N.

4. Complexity of SIDSM In this section we prove that scheduling with interference decoding by signal mixing is NP-complete in the physical interference model, where nodes live in a Euclidean space (geometric SINR model). To see that the decision version of the problem is in NP is straightforward. To decide whether a schedule of a given size T is feasible, we have to verify, for every transmission, whether it employed signal mixing at a relay node or not. If yes, conditions (5)–(8) must be satisfied for each triple of participating nodes A, B, R. If a transmission was scheduled without network coding, then only the condition (1) has to be verified. Since computing the SINR level for each transmission in its time slot can be done in O(n2) time, a schedule is an efficiently verifiable witness for this problem. The hardness proof is by reduction from the well known NP-complete numerical matching with target sums problem (NMTS) [31], which can be formulated as follows: Given 3 sets A; B; C of positive integers, is it possible to match each element i 2 A to a distinct element j 2 B, such that their sum (i + j) equals to each of the elements k 2 C? The triples (i, j, k) must form a partition in the sense that they are disjoint and cover A [ B [ C. NMTS problem: Find q triples triplel ¼ ði; j; kÞ; i 2 A ¼ fi1 ; . . . ; iq g; j 2 B ¼ fj1 ; . . . ; jq g; k 2 C ¼ fk1 ; . . . ; kq g, such that:

X X X iþ j¼ k: i2A

j2B

ð10Þ

k2C

Theorem 4.1. NMTS is reducible to SIDSM in polynomial time. Proof. The proof proceeds as follows. First, we define a many-to-one reduction from any instance of NMTS to an instance of SIDSM. Then, we argue that the instance of SIDSM cannot be scheduled in T 6 2q time slots, but can be scheduled in T = 2q time slots if and only if there is a solution to the NMTS problem instance. Consider an instance of NMTS defined by A ¼ fi1 ; . . . ; iq g; B ¼ fj1 ; . . . ; jq g; C ¼ fk1 ; . . . ; kq g. The instance of SIDSM is constructed by placing (4q + 2) nodes in the plane in the following way (see Fig. 2). First, two nodes R and R2 are placed at positions (0,r(R)) and (0,0), respectively. Thereafter, q nodes, corresponding to integers j 2 B are placed on a straight line originating at R2 at angle (p/2  hb); q nodes, corresponding to integers i 2 A are placed on a line originating at R2 at angle (p/2 + ha); and q nodes, corresponding to integers k 2 C are placed on a line originating at R2 at angle p/2. The polar coordinates of each of these 3q + 2 nodes are:

rðRÞ maxðamax ; bmax Þ þ ;

hðRÞ ¼ p=2;

ð11Þ

rðR2 Þ ¼ 0; hðR2 Þ ¼ 0;  1=a 1 ; hðai Þ ¼ p=2 þ ha ; 8i 2 A; rðai Þ ¼ i  1=a 1 ; hðbj Þ ¼ p=2  hb ; 8j 2 B; rðbj Þ ¼ j  1=a 1 rðck Þ ¼ ; hðck Þ ¼ p=2; 8k 2 C; bk where  is a small positive constant, and angles are defined as follows:

    ha ¼ min h1a ; h2a ; hb ¼ min h1b ; h2b ; where   0 2 1 ba a2min þ rðRÞ2  a2min  rðel Þ2 1 @ A ha ¼ arccos 2 2ba amin rðRÞ þ 2amin rðel Þ 0  2 1 2 cmin þrðRÞ 2 a þ rðRÞ  1=a min b B C h2a ¼ arccos @ A; 2amin rðRÞ

ð12Þ

 0 2 2 1 2 ba bmin þ rðRÞ2  bmin  rðel Þ2 1 @ A hb ¼ arccos 2 2ba bmin rðRÞ þ 2bmin rðel Þ 0  2 1 2 2 cmin þrðRÞ b þ rðRÞ  1=a min b B C h2b ¼ arccos @ A; 2bmin rðRÞ where amin ¼ ð1=imax Þ1=a ; imax ¼ maxi2A ðiÞ; bmin ¼ ð1=jmax Þ1=a ; jmax ¼ maxj2B ðjÞ; cmin ¼ ð1=kmax Þ1=a ; kmax ¼ maxk2C ðkÞ; amax ¼ ð1=imin Þ1=a ; imin ¼ mini2A ðiÞ; bmax ¼ ð1=jmin Þ1=a ; jmin ¼ minj2B ðjÞ.

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Next we position the last q nodes {e1, . . ., eq} at the following location:

rðel Þ ¼

rðRÞ 1

ba

;

hðel Þ ¼ p=2; l 2 f1; . . . ; qg:

ð13Þ

The communication requests are defined as follows: nodes {c1, . . ., cq}, and {e1, . . ., eq} all demand to transmit to the same receiver R2; nodes {a1, . . ., aq} and {b1, . . ., bq} are grouped into two groups A and B, respectively, and wish to transmit q messages {m(a1), . . ., m(aq)} from group A to group B and q messages {m(b1), . . ., m(bq)} from group B to group A. The exact recipient of a message m(ai) is not set, being enough to transmit successfully to any node bj 2 B. The same holds for a message m(bj), originated at node bj 2 B, which has to be transmitted to any node ai 2 A. Having defined the geometric instance of SIDSM for any instance of NMTS, we show that it cannot be scheduled in T < 2q time slots using signal mixing interference decoding. It is enough to look at the 2q transmissions from nodes {c1, . . ., cq, e1, . . ., eq} to receiver R2. Given that signal mixing interference decoding allows simultaneous decoding of two signals only when one of the signals is already known by the receiver, and at time t = 0 receiver R2 does not know any of the considered 2q signals, it needs at least 2q time slots to receive and successfully decode each of them. We proceed by showing that the problem instance defined in equations (11)–(13) can be scheduled in T = 2q time slots using signal mixing interference decoding if and only if there is a solution to the NMTS problem. ()) For the first part of the claim, assume we know q triples ði; j; kÞ; i 2 A; j 2 B; k 2 C, such that conditions (9) and (10) are satisfied. To construct a 2q-slot schedule, we assign transmissions ai ? R, bj ? R, ck ? R2, "(i + j) = k to every odd slot {t1, t3, . . ., t2q1} (we refer to the odd-timeslot schedules as S 2tþ1 ). Note that the relay node R receives a collided signal (PR(ai) + PR(bj)). To every even slot {t2, t4, . . ., t2q} we assign the transmissions R ? {ai, bj} and el ? R2 (we refer to the even-time-slot schedules as S 2t ). In this way we schedule all 4q requests in 2q time slots. Now we prove that the obtained schedule is valid, i.e., all messages are decoded successfully. First we look at the odd time slots. The SINR at receiver R2 is equal to: P

SINRR2 ðS 2tþ1 Þ ¼

PR2 ðck Þ Pbk rðc Þa ¼ P k P ¼ PR2 ðai Þ þ PR2 ðbj Þ rða Þa þ rðb Þa Pði þ jÞ i

j

¼ b: Now we check the conditions (5) and (6):

P

P

SINRR2 ðS 2t Þ ¼

PR2 ðel Þ rðel Þa ¼ P ¼ b: PR2 ðRÞ rðRÞ a

And finally we check the conditions (7) and (8):

Pai ðRÞ P a ðRÞ dðel ; ai Þa ¼ i ¼ Pai ðel Þ dðR; ai Þa sj –R Iai ðsj Þ

P

a

ðrðai Þ2 þ rðel Þ2  2rðai Þrðel Þ cosðp  ha ÞÞ2  a2 rðai Þ2 þ rðRÞ2  2rðai ÞrðRÞ cos ha  a2 a2min þ rðel Þ2 þ 2amin rðel Þ cos ha P  a2 ¼ b: a2min þ rðRÞ2  2amin rðRÞ cos ha ¼

Condition (8) is proved in the same way, only using bi instead of ai and hb instead of ha. To sum up, we showed that in every odd time slot, conditions (5) and (6) hold for every relay node R participating in signal mixing; in every even time slot, conditions (7) and (8) hold for every sender ai and bj participating in signal network coding; every mixed packet forwarded by the relay node R can be decoded by at least one node in each group A and B, since exactly one node in every group is the sender of one of the mixed packets; and condition (1) holds for every transmission {c1, . . ., cq, e1, . . ., eq} ? R not employing network coding. This proves that our schedule guarantees successful decoding for all transmissions scheduled in each time slot t 2 {t1, . . ., t2q}. () For the second part of the claim, we need to show that if no solution to the NMTS problem exists, we cannot find a 2q-slot schedule for the SIDSM instance. No solution to NMTS implies that for at least one triple ði; j; kÞ; i 2 A; j 2 B; k 2 C, it holds that (i + j) > k. Assume we could still find a valid schedule with only 2q slots. As we have already pointed out, transmissions from nodes {c1, . . ., cq, e1, . . ., eq} to receiver R2 have to be scheduled sequentially. So let’s assume we have q time slots, in which senders {c1, . . ., cq} are scheduled, and another q time slots, in which senders {e1, . . ., eq} are scheduled. We will show that there is no way to schedule the remaining senders {a1, . . ., aq, b1, . . ., bq} in parallel. First we look at time slots t with an assigned sender el. Assume that at least one sender ai or bj transmits simultaneously. The SINR at R2 would be:



PR ðai Þ PR ðai Þ dðck ; RÞa ¼ ¼ a sj –ai IR ðsj Þ PR ðck Þ dðai ; RÞ

SINRR2 ðS t Þ ¼

sj –bj ¼

The last inequality holds by plugging in the value of ha, defined in (12) (here we assume that ha is acute enough, s.t. d(R, amin) > d(R, amax).). Condition (6) is proved as in (14), using bi instead of ai and hb instead of ha. Now we look at the even time slots. The SINR at receiver R2 is equal to:

ðrðRÞ þ rðck ÞÞa a

ðrðai Þ2 þ rðRÞ2  2rðai ÞrðRÞ cos ha Þ2

ðrðRÞ þ cmin Þa P a2 ¼ b: ð14Þ a2min þ rðRÞ2  2amin rðRÞ cos ha

PR2 ðel Þ ¼ PR2 ðai =bi Þ

P rðel Þa P rðai =bi Þa

6

P

maxðamax ;bmax Þþ 1 ba

P a=bamax

a < b;

where ai/bi and a/b is an abuse of notation, meaning ‘‘ai or bi’’ and ‘‘amax or bmax’’, respectively. This means that all 2q senders {a1, . . ., aq, b1, . . ., bq} have to be scheduled in the remaining q time slots, together with senders {c1, . . ., cq}. Since signal mixing interference

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O. Goussevskaia, R. Wattenhofer / Ad Hoc Networks 11 (2013) 1732–1745

CK dmin r1 P = *( * + ) C K-1

sk-1,1 sk-1,2

C1

2..kP(ring)+

s1

sk,1 P = *( * + )

C 2 P = *( s1,1

P = i1

P= * P= *

s2

sk,2

r2 dmin

P = i2

) sk,3 s3

P = i3

R

r3

s1,m P = *(

2..kP(ring)+

) P = *( * + ) sk-1,m

P = in P= *

sn dmin rn

sk,m

Fig. 3. Reduction from 3-partition: all (K  m + n) links can be scheduled successfully in m time slots if and only if the senders s1, . . ., sn, corresponding to the integers i1, . . ., in, are partitioned into m subsets, each summing up to exactly r.

decoding only applies to 2 simultaneous transmissions, one from set A and another from set B, exactly 2 senders {ai, bj} have to be scheduled in each of these q time-slots. Now consider the time slot t, correspondent to triple (ai, bj, ck)j(i + j > k). The SINR at receiver R2 is: P

SINRR2 ðS t Þ ¼

PR2 ðck Þ Pbk rðc Þa < b; ¼ P k P ¼ P R2 ðai Þ þ P R2 ðbj Þ rða Þa þ rðb Þa Pði þ jÞ i

j

i.e., at least one transmission ck ? R2 cannot be decoded correctly within 2q time slots if there is no solution to the NMTS problem. This completes the proof. h 5. Complexity of SSIC In this section we prove that scheduling with successive interference cancelation (SSIC) is NP-complete in the geometric SINR model. To see that the decision version of the problem is in NP is straightforward. To decide whether a schedule of a given size T is feasible, we have to verify, for every transmission, whether there is a time slot assigned to it and if the condition (2), or equivalently, conditions (3) and (4), are satisfied, considering the concurrently scheduled transmissions in each time-slot. Since computing the SINR level for each transmission in its time slot can be done in O(n2) time, a schedule is an efficiently verifiable witness for this problem. We proceed by presenting a polynomial-time reduction from 3-partition, a problem closely related to the subset

sum problem. 3-partition was proved to be NP-complete by Garey and Johnson in [32] and can be formulated as follows: Given a set I of integers, is it possible to partition this set into m subsets I 1 ; . . . ; I m , such that the sum of the numbers in each subset is equal? The subsets I 1 ; . . . ; I m must form a partition in the sense that they are disjoint and they cover I . Let r denote the (desired) sum of each subset I i , or equivalently, let the total sum of the numbers in I be mr. The 3-partition problem remains NP-complete when every integer in I is strictly between r/4 and r/2, in which case, each subset I i is forced to consist of exactly three elements [32]. 3-partition problem: Find I 1 ; . . . ; I m  I ¼ fi1 ; . . . ; in g s.t.:

I 1 \ I 2 \    \ I m ¼ £; I 1 [ I 2 [    [ I m ¼ I ; and X X X 1X ij ¼ ij ¼    ¼ ij ¼ ij : m i 2I i 2I i 2I i 2I j

1

j

2

j

m

j

Theorem 5.1. 3-partition is reducible to SSIC in polynomial time. Proof. The proof proceeds as follows. First, we define a many-to-one reduction from any instance of 3-partition to a geometric (Euclidean) instance of SSIC. Then, we argue that the instance of SSIC cannot be scheduled in T 6 m time slots, but can be scheduled in T = m time slots if and only if the instance of 3-partition is solved.

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Consider a set I ¼ fi1 ; . . . ; in g of positive integers, where

ij <

j¼1

r 2

;

8ij 2 I :

The instance of SSIC is constructed by placing (K  m + n) senders and (n + 1) receivers in the plane in the following way (see Fig. 3). First, one receiver R is placed at position (0, 0). Thereafter, K circles are drawn around R, and m senders are placed on each circle’s circumference. The outermost circle CK has radius (P/b  r)1/a. Each inner circle’s radius is recursively determined as

rðC i Þ ¼ ba 

k X

1 aþr rðC jÞ j¼iþ1

rðsi;j Þ ¼ rðC i Þ;

1.1 1 0.8

;

8i 2 fK  1; . . . ; 1g:

8i 2 f1    Kg;

j 2 f1    mg;

ð16Þ

All the positioned m  K senders have as intended receiver the receiver R. Now we place the remaining n senders s1, . . ., sn and n receivers r1, . . ., rn. For each integer ij in I , we set the radial coordinate of sj to (P/ij)1/a and leave its angular coordinate free.

 1=a 1 ; ij

hðsi Þ is free: Next we position the receivers ri, 1 6 i 6 n at distance dmin to their corresponding senders si:

1 ðimax 1Þ1=a



where 1 1=a imax

ð17Þ

1

1 þ ððn þ K  1ÞbÞa

hðr i Þ is free; and imax is the maximal value of the integers in set I . Having defined the geometric instance of SSIC for any instance of 3-partition, we proceed by showing that it cannot be scheduled in T < m time slots using successive interference cancelation. For that, consider any pair of senders si,x, si,y positioned at the same circumference Ci. Since they are equidistant from their intended receiver R, the power perceived at R is the same:

P R ðsi;x Þ ¼ 1 < b; PR ðsi;y Þ

6

8

10

ApproxLogN GreedyPhysical ApproxDiversity 2.5

2

1.5

1.1 1 500

1000

1500

2000

2500

Number of Links Fig. 4. Random receiver placement. Gain obtained with SSIC. (Number of nodes n = 1000 when variable number of link length classes numLengthClasses, numLengthClasses = 10 when variable number of nodes n, a = 5, b = 1.2.)

8ij 2 I ;

rðr i Þ ¼ rðsi Þ þ dmin ;

4

Number of Length Classes

!1a

hðsi;j Þ is free:

dmin ¼

1.5

ð15Þ

The polar coordinates of each of m senders si,1, . . ., si,m placed on circumference Ci are:

rðsi Þ ¼

2

2

1 rðC K Þ ¼ br 1

ApproxLogN GreedyPhysical ApproxDiversity

1a

Gain = T(algo) / T(coding)



Gain = T(algo) / T(coding)

n X ij ¼ m  r;

2.5

8si;x ; si;y 2 C i ;

i 2 f1    Kg:

Given that the power levels of any pair of such transmissions do not differ, SINR conditions (3) and (4) cannot be fulfilled, and R cannot decode them simultaneously. Since this argument applies to any pair of senders belonging to the same circumference, and that there are m senders in each circumference, at least m time slots are needed to schedule any m-tuple of such requests.

To proceed with the proof, we first need Lemma 5.2, in which we show that each receiver ri 2 {r1, . . ., rn}, corresponding to an integer i 2 I , is close enough to its respective sender to guarantee successful transmission, regardless of other links scheduled simultaneously. Since no two senders si,x, si,y positioned at the same circumference Ci can be scheduled simultaneously, we assume that at most (K + n) senders are scheduled in the same time slot as ri, i.e., one sender in each of K circumferences, plus n senders si, corresponding to the n integers in I . Lemma 5.2. Consider a time slot t, in which the schedule S t contains (n + K) senders (one sender in each of K circumferences, plus n senders si, corresponding to the n integers in I ). It holds that for every receiver ri 2 {r1, . . ., rn}, ri decodes its message successfully, i.e., constraints (3) and (4) are satisfied. Proof. We start by establishing a minimal distance between a receiver ri 2 {r1, . . ., rn} and any interfering server sj, j – i or sx,y, x 2 {1, . . ., K}, y 2 {1, . . ., m}. Since the positions of senders s1, . . ., sn depend on the integers i1, . . ., in, we can determine the minimum distance between two sender nodes si, sj.

 1  1     1 a 1 a    dðsi ; sj Þ ¼ dðsi ; RÞ  dðsj ; RÞ ¼     ii ij  1 1 1 P  1=a ¼ dmin ð1 þ ððn þ K  1ÞbÞa Þ: 1=a ðimax  1Þ imax

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O. Goussevskaia, R. Wattenhofer / Ad Hoc Networks 11 (2013) 1732–1745

We proceed by showing that any sender sx,y positioned on a circumference Cx, x 2 {1, . . ., K}, is even farther away:

dðsi ; sx;y Þ ¼ jdðsi ; RÞ  dðsx;y ; RÞj 1 1 P 1=a  1=a imax ðb  rÞ 1 1  a P ðimax  1Þ1=a i1= max

ð18Þ

where (18) and (19) hold because r > 2  imax, b > 1, and imax P 1. (i.e., ((r  b)  imax) P (imax  (imax  1))) By triangular inequality, we have: 1

8i; j

dðsj ; r i Þ P dðsi ; sj Þ  dmin ¼ dmin  ððn þ K  1ÞbÞa ; i – j:

This suffices to show that constraints (3) and (4) are satisfied for any receiver ri, i 2 {1, . . ., n}. Since d(sj, ri) > dmin = d(si, ri), the power received at ri from si is stronger than from any other concurrent transmissions. Therefore, constraint (3) does not apply, and we only need to show that constraint (4) is satisfied: P

P ri ðsi Þ

P ri ðsj Þ sj 2 S t ; P ri ðsj Þ < P ri ðsi Þ

P

P damin

ðn þ K  1Þ  dðs P;r Þa j

i

P

1 damin

¼ b: 

ðnþK1Þ 1

ðdmin ððnþK1ÞbÞa Þa

Having proved that successful transmission is guaranteed for receivers r1, . . ., rn under concurrent transmission of K senders sx,y positioned at different circumferences Cx, x 2 {1, . . ., K} and any number of senders sj, j 2 {1, . . ., n} corresponding to the integers in the 3-partition instance, we now return to the proof of Theorem 5.1. We claim that there exists a solution to the 3-partition problem if and only if there exists an m-slot schedule for the problem instance defined in Eqs. (15)–(17). ()) For the first part of the claim, assume we know m subsets I 1 ; . . . ; I m  I , whose elements sum up to r. To construct an m-slot schedule, 8ij 2 I 1 , we assign the corresponding sender sj to time slot 1, along with K senders s1,1, s2,1, . . ., sK,1. For every ij 2 I 2 , we assign the corresponding sender sj to time slot 2, along with K senders s1,2, . . ., sK,2. And so on until senders sj corresponding to ij 2 I m are assigned to time slot m, along with K senders s1, m, . . ., sK,m. In this way we scheduled all mK + n requests in m time slots. Now we prove that the obtained schedule is valid, i. e., all messages are decoded successfully. Due to Lemma 5.2, we can assume that all senders si, i 2 {1, . . ., n} transmit successfully and focus our analysis on the senders s1,t, . . ., sK,t, t 2 {1, . . ., m}. Since in each time slot t only K senders positioned on distinct circumferences are scheduled together, the situation is the same in each t. Therefore, we only look at one time slot and show that all K transmissions are decoded successfully at receiver R. The signal power R receives from each sender si,t, i 2 {1, . . ., K} is equal to

P PR ðsi;t Þ ¼ ¼ rðsi Þa

P b

K X j¼iþ1

1 rðC j Þa

IR ðsi;t Þ ¼

!: þr

K X

X

PR ðsj;t Þ þ

PR ðsj Þ ¼

sj 2I t

j¼iþ1

! 1 ¼P þr : rðC j Þa j¼iþ1

K X

X P P  ij aþ rðsj;t Þ sj 2I t j¼iþ1

K X

ð19Þ

¼ minðdðsi ; sj ÞÞ;

2 I;

The interference R experiences from concurrently scheduled senders is

Therefore, using the notation introduced in Section 3, we show that condition (2) holds "sx,t 2 ! = {s1,t, . . ., sK,t} and, therefore, all K senders in ! transmit successfully to receiver R in time slot t:

PR ðsx;t Þ PR ðsi;t Þ P ¼ IR ðsi;t Þ ðs Þ þ P ðs Þ P R y;t R z P ðs ÞR ! PR ðsy;t Þ 2 !; R z PR ðsy;t Þ < PR ðsx;t Þ P P 

P

K

1 þ j¼iþ1 rðC j Þa

b

¼

P

P K

1 j¼iþ1 rðC j Þa

r

þr

 ¼ b;

which, in combination with Lemma 5.2, proves that our schedule guarantees successful decoding for all transmissions scheduled in each time slot t 2 {1, . . ., m}. () For the second part of the claim, we need to show that if no solution to the 3-partition problem exists, we cannot find an m-slot schedule for our scheduling instance. No solution to 3-partition implies that for every partition of I into m subsets, the sum of one set I t is greater than r. Assume we could still find a schedule with only m slots. As we have already pointed out, senders positioned on the same circumference Ci, i 2 {1, . . ., K} have to be scheduled separately. Therefore, in each time slot t 2 {1, . . ., m}, exactly one sender positioned on each circumference Ci has to be scheduled. We argue that it is not possible to schedule n senders sj correspondent to the integers ij 2 {1, . . ., n} concurrently. Consider a time slot t, a sender sK,t, positioned on the outermost circumference CK, and a subset P I t of integers such that ij 2I t ij > r. To prove that sK,t’s transmission cannot be decoded correctly at receiver R, we can ignore the (K  1) senders positioned on inner circumferences and only analyze the senders sj correspondent to the integers ij 2 I t . We show that neither condition (3) nor (4) are satisfied at receiver R. To show that (3) does not hold, we observe that the ratio of the power levels of sK,t and sj is always below b and, therefore, sj’s signal cannot be filtered out at receiver R, 8ij 2 I t .

 PR ðsj Þ 6 PR ðsK;t Þ 

P

a

1 1=a i max

P

1 ðbrÞ1=a

¼

a

imax < b; br

where the last inequality holds since imax < r/2. Now we show that (4) also does not hold, since the sum of set I t is greater than r.



P

1 ðbrÞ1=a

P ðs Þ XR K;t ¼ X  PR ðsj Þ P

sj 2I t

ij 2I t

a 1

1=a 1=ij

a

<

br

r

¼ b;

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O. Goussevskaia, R. Wattenhofer / Ad Hoc Networks 11 (2013) 1732–1745 3.3

Gain = T(algo) / T(coding)

Gain = T(algo) / T(coding)

5

4

3

2 1.1 1

ApproxLogN GreedyPhysical ApproxDiversity 0 1

3

6.25

12.5

ApproxLogN GreedyPhysical ApproxDiversity

3

2.5

2

1.5 1.2 1 0.8 2.5

25

3.5

4.5

Gain = T(algo) / T(coding)

Gain = T(algo) / T(coding)

5

4

3

2 1.1 1

ApproxLogN GreedyPhysical ApproxDiversity 2

3

4

5

6.5

6

7

ApproxLogN GreedyPhysical ApproxDiversity 2.5

2

1.5

1.1 1 0.9 1.2

Number of Length Classes

1.7

2.2

2.7

3.2

SINR threshold Fig. 6. Influence of SINR parameters. (Random receiver placement, number of nodes n = 1000, number of link length classes numLengthClasses = 10.)

5

Gain = T(algo) / T(coding)

5.5

Path-loss alpha

Cluster radius

4

3

2 1.1 1

ApproxLogN GreedyPhysical ApproxDiversity 2

3

4

5

6

7

Number of Length Classes Fig. 5. Clustered receiver placement. Gain obtained with SSIC. (a) number of nodes n = 1000, number of link length classes numLengthClasses = 7; (b) n = 500, cluster radius clusterRadius = 1.0; (c) n = 1000, clusterRadius = 1.0; in all figures: a = 5,b = 1.2, 3 6 k 6 3 + numLengthClasses (we consider k P 3 to avoid scenarios with very long links, but preserve the same link length diversity numLengthClasses).

Since neither condition (3) nor (4) are satisfied for link sK,t ? R when the sum of subset I t is greater that r, the transmission cannot be decoded successfully and the schedule needs more than m time slots. This completes the proof of Theorem 5.1. h 6. Algorithm for SSIC In this section we present a scheduling algorithm that exploits successive interference cancelation in the physical interference model. The algorithm greedily schedules links, checking for successive interference cancelation opportu-

nities at each step. The result is a schedule of length T, where in each time slot all transmissions can be decoded successfully according to Eq. (2). Note that we do not provide approximation guarantees for this algorithm, i.e., we do not know how well it performs in comparison to an optimal solution to the SSIC problem. However, in Section 7, we compare its performance to scheduling algorithms that do not employ interference cancelation techniques through simulations, and observe a significant improvement in throughput. 0 We start by defining a function SSIC(r, !, I, b ), which returns true iff a receiver r is able to decode all signals in a 0 given set ! with SINR threshold b . More precisely, given a set of k signals (sorted in decreasing order of power received by r) ! = {Pr(s1), Pr(s2), . . ., Pr(sk)}, a set of all other 0 concurrent signals I, and an SINR threshold b , SSIC(r, !, I, 0 b ) = true iff the following condition holds "x 2 {1, . . ., k}:

P

sy 2 S t ; Pr ðsy Þ 2 !; P r ðsy Þ < Pr ðsx Þ

P r ðsx Þ P P b0 : Pr ðsy Þ þ Pr ðsz Þ 2 IPr ðsz Þ þ N

0

Algorithm 1 starts by setting two constants: b = 3b/2, a slightly higher SINR threshold than the original b; and c, a constant defined in (21). The algorithm schedules links in non-decreasing order of their length.

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O. Goussevskaia, R. Wattenhofer / Ad Hoc Networks 11 (2013) 1732–1745

Once a link lx is selected to be scheduled in time slot t (line 10), some of the remaining links ly (those that have not been scheduled yet) are eliminated from the current time slot (and put into set D) in two steps. To do that, the signals which have already been scheduled in this time-slot (li 2 S t ) are divided into two subsets: !, containing signals from senders located within distance dyy of receiver ry (line 12), and I, containing signals from the remaining senders in S t (line 13). Note that the set ! contains all senders in S t that are ‘‘stronger’’ than sy, i.e., incur more interference on ry than sy, and therefore can potentially benefit from SSIC. In the first elimination step (line 14), all links ly that do 0 not meet the decoding condition SSIC(ry, !, I, b ), i.e., cannot benefit from SSIC, and have an SINRry ðS t Þ (ratio of signal to the interference from senders in S t , plus noise) lower than 0 b , i.e., cannot toletare any additional interference from links potentially selected later into S t , are deleted (put into set D). In the second elimination step (line 15), all links whose senders are within distance c  dxx from receiver rx, i.e., are too close and therefore cause too much interference on the selected link lx, are removed. This link selection and deletion procedure is repeated until all links have been either scheduled in time-slot t or deleted. The whole process (one-slot schedule maximization) is repeated using the deleted links as input, until all links in L have been scheduled in exactly one time-slot of schedule S. Algorithm 1. SSIC Algorithm

1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21.

input: Set of links L = {l1, . . ., ln}; output: Schedule S ¼ fS 1 ; . . . ; S T g of length T,meeting feasibility conditions SSIC (2); Set c according to (21); 0 b :¼3b/2; t:¼0; repeat D:¼£; S t :¼ £; repeat S t :¼ S t [ flx g, where lx ¼ argminli 2LnfSt [Dg dii ; for ly 2 L n fS t [ Dg do ! :¼ fP ry ðsi Þ; si 2 S t jdðry ; si Þ 6 dyy g; I :¼ fPry ðsi Þ; si 2 S t n !g; 0 if!SSIC(ry, !, I, b ) and SINRry ðS t Þ < b0 then D:¼D [ {ly}; else if d(rx, sy) 6 c  dxx then D:¼D [ {ly}; end for until L n fS t [ Dg ¼ £ L :¼ L n S t ; t:¼t + 1; until L = £ return S;

In the following theorem we prove that the schedule S obtained by Algorithm 1 is correct, i.e., all selected links can be scheduled concurrently without collisions using successive interference cancelation.

Theorem 6.1. Algorithm 1 produces a valid schedule according to SSIC feasibility conditions, defined in (2).

Proof. Consider a time-slot t and an arbitrary link lx scheduled in S t . Let S x be the set of links shorter than lx, i.e., those added to S t before lx, and Sþ x be the set of links longer than lx, i.e., those added after lx. When a link lx is added to the solution, two conditions hold: (1) the signal from the intended sender sx can be decoded with SINR threshold 0 0 b = 3b/2, since lx either satisfies SSICðr x ; !; S x n !; b Þ or 0  SINRrx ðSx Þ P b , where ! ¼ fPrx ðsi Þjsi 2 S t jdðsi ; rx Þ 6 dxx g; and (2) senders in Sþ x are located outside the disk of radius c  dxx. It remains to show that the additional interference from Sþ x is small enough to allow the signal from sx to be decoded with SINR threshold b. We need to show that either þ  þ SSICðrx ; !; fS x n !g [ Sx ; bÞ=true or SINRr x ðSx [ Sx Þ P b, i.e.:  SSICðrx ; !; S t n !; bÞ or  SINRrx ðS t n sx Þ P b. In order to bound the interference from Sþ x we use the fact that, by the second elimination criterion of the algorithm, disks of radius c  djj around each receiver r j 2 Sþ x do not contain any sender sz – sj. Using this fact and the triangular inequality, we can lower bound the distance between any two senders ðsj ; sz Þ 2 Sþ x as d(sj, sz) P d(rj, sz)  djj P c  djj  djj = djj(c  1) P dxx(c  1). Therefore, disks Dj of radius dxx(c  1)/2 around senders in Sþ x do not intersect. We partition the space into concentric rings Ringk of width c  dxx around the receiver rx. Each ring Ringk contains all senders sj 2 Sþ for which k(c  dxx) 6 d(sj, rx) x, 6 (k + 1)(c  dxx). We know that the first ring Ring0 does not contain any sender. Consider all senders sy 2 Ringk for some integer k > 0. All disks of radius dxx(c  1)/2 around each sj must be located entirely in an extended ring Ringk of area

AðRing k Þ ¼ ½ðdxx ðk þ 1Þc þ dxx ðc  1Þ=2Þ2 2

 ðdxx kc  dxx ðc  1Þ=2Þ2 p < ð2k þ 1Þdxx 2c2 p: Since disks of area A(Dy) P (dxx(c  1)/2)2p around senders in Sþ x do not intersect, and the minimum distance between rx and sy 2 Ringk, k > 0 is k(c  dxx), we can use an area argument to bound the number of senders inside each ring. The total interference coming from ring Ringk, k P 1 is then bounded by

Irx ðRing k Þ 6

X

Irx ðsy Þ 6

sy 2Ring k

6

AðRing k Þ P AðDy Þ ðkcdxx Þa

ð2k þ 1ÞP23 c2 a a k d ca ðc xx

 1Þ

2

1

6 k

ða1Þ

P25 3 ; a dxx cðaÞ

where the last inequality holds since k P 1 ) 2k + 1 6 3k and c P 2 ) (c  1) P c/2. Summing up the interferences over all rings yields

Irx ðSþx Þ < <

1 1 X X 1 P25 3 Irx ðRing k Þ 6 a1 a ðaÞ dxx c k¼1 k¼1 k

a  1 P25 3 Prx ðsx Þ ; 6 3b a  2 daxx cðaÞ

ð20Þ

O. Goussevskaia, R. Wattenhofer / Ad Hoc Networks 11 (2013) 1732–1745

where the last two inequalities hold since a > 2 and c is defined as follows:

 a1 ! 5 2 a1 c ¼ max 2; 2 3 b : a2

ð21Þ

If we define !þ i to be the set of signals in ! coming from senders located closer to rx than si, we know that, since  0 0 SSICðr x ; !; S x ; b Þ=true or SINRr x ðSx Þ P b , the following bounds on interference hold:

Irx ðSx n !þi Þ þ N 6

Prx ðsi Þ 2Prx ðsi Þ ; 6 3b b0

8Prx ðsi Þ 2 !;

in case successive interference cancelation is used, and

Irx ðSx Þ þ N 6

Prx ðsx Þ 2Prx ðsx Þ ; ¼ 3b b0

in case no interference cancelation is performed. In both cases, by using the bound (20) on Irx ðSþ x Þ (and the fact that P rx ðsi Þ P P rx ðsx Þ; 8Prx ðsi Þ 2 !), we obtain

2Prx ðsi Þ Prx ðsx Þ þ 3b 3b Prx ðsi Þ 6 ; 8Prx ðsi Þ 2 !: b

Irx ðfSx n !þi g [ Sþx Þ þ N 6

) SSICðr x ; !; S t n !; bÞ for encoded transmissions, and

Irx ðSx [ Sþx Þ þ N 6

2Prx ðsx Þ P rx ðsx Þ Prx ðsx Þ þ 6 : 3b 3b b

) SINRrx ðS t n sx Þ P b for non-encoded transmissions. This completes the proof. h 7. Simulation results In this section we present some simulation results to illustrate the gain in throughput obtained by using successive interference cancelation. We generated two kinds of topologies. In both scenarios, nodes are distributed on a square field of size W = 1000, and links have different levels of variance in length. More precisely, numLengthClasses length classes were defined, such that the link length lk in each class ck,1 6 k 6 numLengthClasses is uniformly distributed between lmax = W/2k and lmin = W/2k+1 + W/2k+2. In the first scenario, to which we refer as Random receiver placement, receiver nodes were distributed uniformly at random in the deployment field, and the respective senders were positioned uniformly at random at distance lk from their intended receivers. In the second scenario, to which we refer as Clustered receiver placement, numClusters cluster center positions were selected uniformly at random on the plane, and n/numClusters receivers were positioned uniformly at random inside disks of radius clusterRadius around each of them. Thereafter, the respective senders were positioned uniformly at random at distance lk from their intended receivers. The clustered topology aims to simulate a scenario of heterogeneous density distribution. With high-link-length-diversity and highly-clustered topologies we tried to simulate scenarios, where more interference cancelation opportunities would arise.

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We compare the performance of Algorithm 1 to the performance of three scheduling algorithms without coding: GreedyPhysical (proposed in [33]), ApproxDiversity (proposed in [3]), and ApproxLogN (proposed in [12]). As the SSIC Algorithm, all these algorithms are polynomial in time and are specifically designed for the SINR model. In all experiments, the number of simulations was chosen large enough to obtain at least 95% confidence intervals. Firstly, we look at the Random receiver placement topology. We analyze the size of the obtained schedule as a function of the number of length classes (see Fig. 4a). It can be seen that the more diverse the link lengths, the more interference cancelation opportunities exist in the network, and the higher the gain of the successive interference cancelation approach relative to non-interference cancelation scheduling algorithms. In Fig. 4b we analyze the influence of the total number of links on the relative performance of the algorithms. Since the number of length classes is maintained constant, the number of successive interference cancelation opportunities does not significantly change. Therefore, the gain in throughput due to network coding does not vary much with varying network density. In Fig. 5 we look at the Clustered receiver placement topology. First, we analyze the size of the obtained schedule as a function of the cluster radius (see Fig. 5a). It can be seen that the smaller the size of the clusters, the more interference cancelation opportunities exist in the network, and the higher the gain of the SSIC approach, especially relative to ApproxDiversity and GreedyPhysical. In Fig. 5b and c we analyze the influence of the number of length classes on the relative performance of the three scheduling algorithms, for n = 500 and n = 1000 nodes, respectively. We can see that, once again, the more diverse the link lengths, the higher the gain of the SSIC approach. Moreover, compared to the Random receiver placement scenario, the gains relative to ApproxDiversity and GreedyPhysical are much higher. In Fig. 6a we analyze the impact of the path-loss exponent a in the Random receiver placement topology. It can be observed that when a < 3, the GreedyPhysical algorithm achieves slightly better performance than the SSIC algorithm. This can be explained by the fact that the second elimination step of Algorithm 1 depends on the constant c, defined in (21), which increases sharply when a approaches 2. For higher values of a, however, the successive interference cancelation approach becomes increasingly more efficient. In Fig. 6b we analyze the impact of the SINR threshold b. It can be seen that the value of b does not influence the performances of the algorithms, which is expected, given that b is just a ratio. In Figs. 4a Fig. 5, Fig. 6b, it can be observed that the throughput gain of coding is the smallest relative to algorithm ApproxLogN. This is due to the fact that ApproxLogN outperforms the other algorithms. Nevertheless, the coding approach achieves gains of up to 20% relative to ApproxLogN (when the number of length classes is 10 and pathloss exponent a is 6), 450% relative to GreedyPhysical (in the Clustered receiver placement scenario with 1000 links, when the cluster radius is 63 or the number of link length

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classes is P7), and around 500% relative to ApproxDiversity (in the Clustered receiver placement scenario with 1000 links, when the cluster radius is 66 or the number of link length classes is P6.). Overall, the simulation results showed that the gain of successive interference cancelation depends both on the topology of the network and on the SINR parameters. The more interference cancelation opportunities a network topology generates, the more explicit the gain of the SSIC algorithm over traditional scheduling algorithms is. In particular, since network topologies with high link length diversity and heterogeneous node densities exhibit stronger power variation between simultaneous packets at the receivers, they support higher throughput gains due to successive interference cancelation.

8. Conclusion In this work we wanted to obtain some understanding of the complexity of scheduling wireless links with interference decoding capability. Given that network coding changes the definition of a successful transmission, allowing a receiver to decode several messages simultaneously, it is interesting to analyze whether the complexity of the scheduling problem is altered. By showing that the problem remains NP-complete, we can conclude that the basic difficulties of scheduling wireless requests in a global interference model, such as geometric SINR, remain challenging even with coding capability. Our second goal was to obtain an efficient algorithm for scheduling wireless links in the physical interference model with successive interference cancelation capability and to analyze whether significantly shorter schedules can be obtained for nodes arbitrarily distributed in the Euclidean space. We proposed a scheduling algorithm that explores interference cancelation opportunities in the network and showed through simulations that better throughput can be obtained in certain network topologies. In particular, the stronger the signal power variation between simultaneous packets at the receivers in a network topology, the higher are the induced throughput gains due to successive interference cancelation.

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O. Goussevskaia, R. Wattenhofer / Ad Hoc Networks 11 (2013) 1732–1745 [32] M.R. Garey, D.S. Johnson, Complexity results for multiprocessor scheduling under resource constraints, SIAM Journal on Computing (4) (1975) 397–411. [33] G. Brar, D. Blough, P. Santi, Computationally efficient scheduling with the physical interference model for throughput improvement in wireless mesh networks, in: Proc. of the 12th International Conference on Mobile Computing and Networking (MOBICOM), 2006.

Olga Goussevskaia has received her PhD in Computer Science from ETH Zurich and now works as an Assistant Professor at the Federal University of Minas Gerais, Brazil. Her main area of research is modeling and algorithmic aspects of wireless networks.

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Roger Wattenhofer has received his PhD in Computer Science from ETH Zurich and now works as a Full Professor at the Computer Engineering and Networks Laboratory at ETH Zurich, Switzerland. His areas of research include distributed computing and theoretical computer science.