Utilizing Multiple Radio Interfaces in Wireless Networks

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Utilizing Multiple Radio Interfaces in

Wireless Networks

Alex Kogan

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Utilizing Multiple Radio Interfaces in

Wireless Networks

Research Thesis

Submitted in Partial Fulfillment of the Requirements for the Degree of

Doctor of Philosophy

Alex Kogan

Submitted to the Senate of

the Technion – Israel Institute of Technology

Elul 5772 Haifa August 2012

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Acknowledgments

The research thesis was done under the supervision of Prof. Roy Friedman in the Computer Science Department.

The results presented in this thesis have been published in the following venues:

• R. Friedman and A. Kogan. Efficient power utilization in multi-radio wireless ad hoc networks. InProc. Int. Conf. on Principles of Distributed Systems (OPODIS), pages 159–173, 2009.

• R. Friedman and A. Kogan. Power aware management middleware for multiple radio interfaces. InProc. ACM/IFIP/USENIX Int. Middleware Conference, pages 288–307, 2009.

• R. Friedman and A. Kogan. Deterministic dominating set construction in networks with bounded degree. InProc. Int. Conf. on Distributed Computing and Networking (ICDCN), pages 65–76, 2011.

• R. Friedman, A. Kogan, and Y. Krivolapov. On power and throughput tradeoffs of WiFi and Bluetooth in smartphones. Accepted to appear in IEEE Transactions on Mobile Computing (TMC). Also inProc. IEEE Int. Conf. on Computer Communications (INFOCOM), pages 900– 908, 2011.

• R. Friedman and A. Kogan. Efficient and reliable multicast in multi-radio networks. Accepted to appear in Proc. IEEE Int. Symposium on Reliable Distributed Systems (SRDS), 2012.

First and foremost, I would like to thank Roy, who has been everything one can ask for from an advisor, and much beyond – supportive, encouraging, giving me the freedom and confidence to explore my own ideas and always being around to keep me on track with wise guidance. All these qualities made my years as a PhD student so enjoyable.

I would also like to thank Prof. Hagit Attiya, my advisor during my master studies, who shaped my love to research and was always available for fruitful conversations about my work.

I am thankful to Profs. Danny Raz and Reuven Cohen from the Technion and Prof. Danny Dolev from the Hebrew University of Jerusalem for many helpful comments on earlier versions of this thesis.

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I thank Prof. Erez Petrank for being a great collaborator, who taught me much and introduced me to new and fascinating areas of research.

I am grateful to my good friend Dr. Yevgeny Bar-Lev (Krivolapov), for numerous coffee breaks and lunches that led to excellent collaboration, which had a lot of impact on my work.

Last, but not least, I thank my parents, Maya and Roman, and my parents-in-law, Elena and Eduard, whose endless love, support and silent understanding made all this happen. But above all, I want to thank my wife Ella, for being such a devoted spouse and for patiently taking care of our beautiful sons, Eithan and Adam, throughout the long weeks and many weekends I spent studying.

The generous financial support of the Hasso Plattner Center, the Technion and the Israel Science Foun-dation (ISF) grant 1247/09 is gratefully acknowledged.

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List of Figures

1 Decreasing the transmission range may decrease the network capacity . . . 10

2 Example graph. . . 17

3 Example super-graph. . . 18

4 Approximation performance of Algorithm 1 vs. the na¨ıve1W CDSalgorithm . . . 26

5 Approximation performance of Algorithm 1 vs. Algorithm 2 . . . 27

6 The total number of messages sent by Algorithm 1 and Algorithm 2 . . . 27

7 The relative number of messages sent by Algorithm 1 and Algorithm 2 . . . 27

8 A (partial)2-ring graphR(n,2). . . 32

9 A2-ring graphR(n,2)where nodes are assigned values in{0,1}. . . 32

10 A subgraphG′ofR(n,2). . . 34

11 Architecture overview. . . 45

12 State transitions in OCM . . . 47

13 Performance of OCM in a static system . . . 53

14 Performance of OCM in a mobile system . . . 55

15 Performance of OCM in a mobile system with MAC notification mechanism . . . 56

16 Performance of OCM under two-phase mobility with hot-spots . . . 57

17 The circuit used to determine the power consumption. . . 61

18 A photo of the actual setup. . . 62

19 Instant power consumed by Omnia in non-communicating modes . . . 65

20 Instant power consumed by Spica in non-communicating modes . . . 66

21 Throughput-to-power ratio in UDP/TCP communication at Omnia . . . 70

22 Instant power consumed by Omnia in the access point network configuration. . . 75

23 Instant power consumed by Spica in the access point network configuration. . . 75

24 Power consumed by WiFi in UDP communication at Diamond . . . 76

25 Power consumed by streaming at Desire and Galaxy S II . . . 76

26 Same-radio vs. separate-radio ARQ-based multicasting . . . 88

27 Same-radio vs. separate-radio hybrid multicasting in a network with varying packet loss probability . . . 91

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28 Same-radio vs. separate-radio hybrid multicasting in a network with varying number of

receivers . . . 91

29 Same-radio vs. separate-radio ARQ-based multicasting in a network with heteroge-neous receivers . . . 94

30 Same-radio vs. separate-radio hybrid multicasting in a network with heterogeneous receivers . . . 94

31 Performance of the multi-NP protocol with varying retransmission buffer size . . . 98

32 Performance of the multi-NP protocol in a network with interfering nodes . . . 99

33 Screenshots of therunnerWMapplication . . . 118

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List of Tables

1 Density vs. number of nodes . . . 25 2 Comparison of results on distributed deterministicO(log ∆)-approximation of optimal

dominating sets. . . 31 3 The summary of simulation parameters for the OCM middleware . . . 51 4 Phones used in our experiments. . . 61 5 Power consumed by Bluetooth and WiFi radios in different non-communicating scenarios 64 6 Power consumed and throughput obtained by Bluetooth and WiFi radios of the

Win-dows Mobile-based phones . . . 67 7 Power consumed and throughput achieved by Bluetooth-based file transfer . . . 72 8 Power consumed and throughput achieved by FTP over WiFi radio . . . 73 9 Power consumed and throughput achieved by WiFi and Bluetooth radios in Desire and

Galaxy S II when transferring files with interference . . . 77 10 The summary of simulation parameters for the multi-NP protocol . . . 98 11 The capabilities of the original and the extendediperftool . . . 118

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Abstract

Contemporary mobile devices are equipped with multiple wireless interfaces, such as WiFi, Bluetooth, WiMax, ZigBee, NFC, etc. All these technologies differ dramatically one from another in their max-imum transmission range, bandwidth and power demands. Among all subsystems operating inside mobile devices, wireless communication is known as being particularly power-hungry, accounting for as much as50-70%of the total power consumption in small handheld devices, such as smartphones, and for10%in laptops. Given the varying characteristics of different wireless technologies, an obvious question arises: Can we utilize the presence of multiple interfaces on contemporary mobile devices in order to improve their wireless networking capabilities in general and power efficiency in particular?

In this thesis we investigate this question from several different perspectives: theoretical, more practical and fully experimental. The major body of this work considers the problem of energy con-sumption in multi-radio wireless networks. Unlike many previous attempts that consider each radio interface independently, we take an integrated approach specifically considering networks in which devices own at least two wireless interfaces, and formulate a novel optimization problem. A solution to this problem defines a network topology where some devices turn off their power-hungry interface while every device still remains connected to the rest of the network by at least one interface. To the best our knowledge, this is a first attempt to capture the problem of energy consumption in multi-radio wireless networks from a formal perspective. Through theoretical analysis and simulations, we show that the proposed approach may achieve significant energy savings that increase with the density of the network.

In the subsequent work, we develop a distributed middleware service that manages multiple wire-less interfaces and heuristically approximates the optimal power-efficient topology of mobile and unre-liable wireless networks. We describe the architecture of the solution, as well as a detailed simulations-based performance study. The study, which was performed with the complete Java implementation of the middleware and covered both static and mobile networks, validates the ability of the middleware to obtain considerable power savings while keeping the network connected and maintaining reasonable communication latency.

Our optimization problem generalizes another well-known graph theoretic problem of finding a minimum dominating set. As a part of this thesis, we make a contribution to general graph theory, showing how to improve state-of-the-art approximation solutions to the latter problem in graphs where nodes have a bounded degree. Such graphs arise in many infrastructure-less network settings, such

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as ad-hoc networks, wireless sensor networks or peer-to-peer networks, where dominating sets play a crucial role.

In our middleware evaluation study, we rely on the performance characteristics of WiFi and Blue-tooth reported in various papers. While examining these papers, we have discovered several important shortcomings, which motivated us to perform a combined power and throughput performance study of WiFi and Bluetooth in smartphones. In the process, we discover several interesting phenomena, some of which counter previous conventions, and draw some operative suggestions for researchers and smartphone developers. As a part of our study, we extendiperf, a widely-adopted open-source tool for measuring network performance, to support Windows Mobile operating system and Bluetooth communication.

Finally, we shift our focus from power saving topologies to another important problem in wireless computing, namely reliable multicast. This problem arises when a single sender needs to transmit a sequence of messages to a group of multiple receivers in a reliable fashion, e.g., when distributing software updates or delivering digital content from a base station to several mobile devices. In our proposed scheme, one radio interface is dedicated only for error recovery information transmissions, while other interfaces serve for transmitting the actual data. Using formal analysis and simulations-based evaluation, we demonstrate that with this approach each receiver needs to handle much fewer messages than in the common single radio approach, leading to better utilization of radio interfaces and energy conservation.

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Chapter 1

Introduction

Mobile devices equipped with multiple wireless interfaces are becoming increasingly common. Nowa-days, all contemporary laptops, smartphones, tablets, netbooks, PDAs, etc. are enabled with WiFi and Bluetooth (BT) radios. A recent market research report estimates that about70% across all mobile phones (including smartphones1) are equipped with Bluetooth, while80%of smartphones are enabled with WiFi [15]. Future devices are expected to include even more interfaces, supporting currently emerging standards, such as WiMax, ZigBee and Near Field Communication (NFC).

An important aspect of these technologies is their ability to work in infrastructure-less settings, allowing creation of so-called mobile ad-hoc networks (MANETs). MANETs enable direct wireless communication between devices, offering fast and easy deployment in situations where it is not possi-ble or not cost effective otherwise (e.g., military operations, temporary social and professional events, vehicular networks, etc.). Thus, it comes as no surprise that direct communication between nearby de-vices is of growing interest. Obvious examples include media streaming either between mobile dede-vices or between a mobile device and another nearby stationary wireless device (e.g., TV or computer) in a home or office environment. Another scenario includes ad-hoc social networking and communication, such as iPhone’s iGroups, Nokia’s Instant Community, Mobiluck, and WiPeer, to name a few.

Yet, mobile devices are typically operated by batteries. Thus, their operational lifetime before the depleted battery should be recharged or replaced is limited. Among all subsystems operating inside these devices, wireless communication is known as one of the major sources for power consump-tion. Researchers have found that wireless communication accounts for about10%of the total energy consumption in laptops [6, 80], growing up to50%and beyond in small handheld devices and smart-phones [6, 23, 39, 109]. For example, the study of Pering et al. [109] shows that WiFi and Bluetooth are responsible together for more than70%of the total power consumption of an idle connected handheld device. The availability of multiple wireless interfaces turns the problem of efficient power usage by the wireless communication subsystem even more acute.

All aforementioned technologies for wireless communication differ dramatically one from another in several performance parameters, such as maximum transmission range, bandwidth and power

de-1

Following the terminology of [15], we refer to a mobile phone that can run third-party software assmartphone.

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mands. For instance, the nominal bandwidth of WiFi is54Mb/s in the IEEE 802.11g standard (and 600Mb/s in the IEEE802.11n standard) with a transmission range of hundreds of meters, while BT has a much shorter transmission range and a nominal bandwidth of about2.1Mb/s (in version2.0). At the same time, previous studies suggest that BT is much more power efficient than WiFi [18, 22, 102, 109]. For example, Pering et al. [109] show that transmitting with a typical BT radio consumes13.3times less power than with a typical WiFi radio.

Given such varying characteristics of different wireless technologies, an obvious question arises: Can we utilize the presence of multiple interfaces on contemporary mobile devices in order to improve their wireless networking capabilities in general, and power efficiency in particular?

In this thesis we investigate this question from several different perspectives: theoretical, more prac-tical and fully experimental. The major body of this work considers the problem of energy consumption in multi-radio wireless networks with an emphasis on ad-hoc networks. Unlike many previous attempts that consider each radio interface independently, we take an integrated approach specifically consid-ering networks in which the devices own at least two wireless communication interfaces. We use the availability of WiFi and BT as the main motivation behind this work and the most natural scenario for the application of the methods that we introduce. Yet, the developed methods are mostly generic and can be applied for any two radio interfaces, while the achieved benefits will essentially depend on the actual tradeoffs between the performance characteristics of those interfaces. Moreover, most of our results can be easily extended to networks with devices having more than two interfaces; we elaborate on such extensions in relevant parts of the thesis.

In more detail, after reviewing the related work in Chapter 2, we introduce a formal approach for re-ducing the energy consumption of multi-radio wireless ad-hoc networks in Chapter 3 [45]. Specifically, we formulate a new optimization problem, calledk-Weighted Connected Dominating Set (kW CDS). A solution to this problem defines a network topology where some nodes turn off their (power-hungry) interface while every node still remains connected to the rest of the network by at least one interface. We show that thekW CDSproblem isN P-hard, and provide a centralized approximation algorithm along with two distributed ones. The performance of all algorithms is analyzed and compared in terms of their approximation ratio, running time and the number of produced messages. Furthermore, the per-formance of the distributed algorithms is simulated in ad-hoc network settings using typical parameters of WiFi and BT technologies. The simulation results show that the proposed approach may achieve significant energy savings, which increase with the size (and the density) of the network.

ThekW CDSproblem generalizes another well-known graph theoretic problem of finding a min-imum dominating set. Given a graph G, a dominating set of the graph is a set of nodes such that every node inG is either in the set or has a direct neighboring node in the set. This problem plays a significant role in many distributed applications, especially in those running over infrastructure-less networks, such as MANETs, wireless sensor networks and peer-to-peer networks [25, 60, 119, 124]. Interestingly, in many cases, the network graph is such that each node has a limited number of direct neighbors [37, 41, 69, 98]. In Chapter 4 [47] we investigate whether this fact can be used to improve ex-isting approximation algorithms (that have a running time complexity, which is linear inn, the number of nodes in the system [35, 92, 119]). In particular, we consider whether it is possible to get a local so-lution, i.e., one that has a running time that does not depend on the number of nodes in the system. We

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answer negatively for the latter, showing that any deterministic algorithm in networks with bounded degree that has a non-trivial approximation ratio requires Ω(log∗n)communication rounds. On the positive side, we show two deterministic algorithms that achievelog ∆and2 log ∆-approximation in

O(∆3+ log∗n) andO(∆2log ∆ + log∗n)time, respectively, where∆is the degree bound, i.e., the maximum number of neighbors each node can communicate directly with in the network. Further-more, we elaborate how these more efficient solutions to the dominating set problem help to improve our algorithms for thekW CDSproblem in networks with bounded degree.

The network settings in which solutions to thekW CDSand the dominating set problems are con-sidered in Chapters 3 and 4 are static and reliable. In Chapter 5 [46], we take a more practical approach and relax all these assumptions. In particular, we develop a distributed middleware service, called Over-lay Construction and Maintenance (OCM), that heuristically approximates the optimal power-efficient topology of a mobile, asynchronous and unreliable multi-radio wireless ad-hoc network. We describe the architecture of the solution, as well as a detailed simulations-based performance study. The study, which was performed with the complete Java implementation and covered both static and mobile net-works, validates the ability of the middleware to obtain considerable power savings while keeping the network connected and maintaining reasonable latency.

In our middleware evaluation study, we rely on the performance characteristics of WiFi and BT reported in various papers [24,39,109,126]. While examining these papers, we have discovered several shortcomings, with the two most important ones being as following. First, many of these studies were performed several years ago, and given the rapid change in technology, one cannot trust that they remain valid. Second, in most of these studies, each wireless interface was measured independently and in isolation in a lab setting. Hence, it is not clear that the results hold for a mobile phone, in which the two RF interfaces are packed in close proximity, sometimes on the same chip. These observations motivated us to perform a combined power and throughput performance study of WiFi and BT in smartphones. The results of this study are summarized in Chapter 6 [49]. In the process, we discover several interesting phenomena, some of which counter previous conventions, and draw some operative suggestions for researchers and smartphone developers. Furthermore, we use the obtained results to refine the design of the OCM middleware for smartphones.

In Chapter 7 [48], we shift our focus from power saving topologies and consider another impor-tant problem in wireless computing, namely thereliable multicast. This problem arises when a single sender needs to transmit a sequence of messages to a group of multiple receivers in a reliable fashion. This capability is useful in several domains, such as distribution of software updates or delivery of digital content (e.g., audio and video) from a base station to plurality of mobile devices [30, 100]. In our proposed scheme, one radio interface is dedicated only for error recovery information transmis-sions, while other interfaces serve for transmitting the actual data. We apply this concept to both ARQ and hybrid FEC+ARQ protocols, which are common techniques for enhancing reliability of multicast communication [90, 105, 112]. We compare the number of packets each receiver needs to process in both our approach and in the common single interface approach, where data and error recovery infor-mation are transmitted on the same interface. Using formal analysis and simulations-based evaluation, we demonstrate that with our approach each receiver needs to handle much fewer messages, leading to better utilization of radio interfaces and energy conservation.

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We conclude the thesis in Chapter 8, summarizing the obtained results and suggesting directions for future research. The technical information about various extensions implemented in standard tools used in the experimental parts of this thesis is given in the corresponding appendices. In particular, Ap-pendix A describes extensions implemented in the SWANS simulator [31] to support evaluation of the OCM middleware described in Chapter 5. Along with that, Appendix B [74] provides details of exten-sions implemented iniperf[104], a popular open-source tool for measuring network performance. These iperf extensions were heavily used in caring out experiments in smartphones, reported in Chapter 6. Finally, in Appendix C, we give a brief overview of additional results that we have achieved during the course of the work on this thesis, and which are not related directly to the main topic of the thesis.

To summarize, the rest of the thesis is organized as following. The related work is surveyed in Chapter 2. In Chapter 3 we introduce a novel optimization problem for multi-radio wireless ad-hoc networks and discuss several solutions for static settings. Our work on the approximation of mini-mum dominating sets in bounded degree graphs is summarized in Chapter 4. In Chapter 5 we present the design and the evaluation of a middleware service that constructs and maintains a power-efficient topology in multi-radio mobile ad-hoc networks. Our performance study of WiFi and Bluetooth in smartphones is given in Chapter 6. In Chapter 7 we analyze the benefits of utilizing multiple radios for reliable multicasting. Finally, we conclude the thesis with a discussion in Chapter 8.

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Chapter 2

Related Work

2.1

Multi-radio networks

Several papers discuss the differences between selected types of wireless technologies in terms of the properties we are interested in, i.e., transmission range, bandwidth and power usage. For example, Bahl et al. [12] compare between TR100 low-power radio and an IEEE 802.11b card and show tremendous differences in power consumption (e.g., the power consumed duringsendingstate has a ratio of 1:235). In another study by Ferro and Potorti [41], BT and IEEE 802.11b cards are compared, showing less dramatic, but still very significant differences in power consumption.

A few papers consider wireless networks consisting of devices equipped with multiple radio inter-faces. Bahl et al. [12, 116] outline the advantages behind systems that use two radios in an integrated manner. They investigate multi-radio solutions to several problems in wireless computing, such as energy management and capacity enhancement, and show significant benefits for such solutions over single-radio ones. Although their work does not rely on any particular radio technology, it requires that all radios of the sender should be able to communicate with all radios of the receiver, meaning effectively that devices should be placed close enough or all radios should have the same transmission range. In addition, the results are presented as a set of heuristic approaches, inappropriate for further generalization, e.g., for devices equipped with more than two radio interfaces.

Pering et al. [109] introduce the CoolSpot system, which enables traditional WiFi hot-spots with Bluetooth interfaces and with a policy for multi-radio management. The paper considers single-hop communication between a wireless mobile device and such a hot-spot, where a closely located device is able to switch automatically between two interfaces in order to reduce its power consumption. The authors also provide an empirical evaluation of several policies that effectively manage radio switching. The combination of WiFi and Bluetooth was recently considered in several other papers. Anantha-narayan and Stoica [4] utilize the presence of Bluetooth devices to optimize detection of WiFi access points. Yoo and Park [128] explore a technique to save energy by clustering network and allowing nearby devices in the same cluster to communicate on Bluetooth, while one of the nodes in the cluster

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remains connected to an access-point on WiFi. Interestingly, this idea is very similar to the one we have also considered [45, 46] (prior to [128]), but the setting (and methods) are different. While our work in [45, 46] considers ad-hoc and multi-hop communication, Yoo and Park assume infrastructure-based one-hop setting. Since the power consumed by idle WiFi connected to access-point is quite comparable to the power consumed by idle Bluetooth [49], the rationale for a protocol considered by Yoo and Park is doubtful.

Yet another recent research direction is to combine WiFi with an alternative low-power wireless technology, namely ZigBee. For instance, Zhou et al. [130] try to save power wasted by WiFi on network discovery. Their idea is to use ZigBee to monitor unique interference signatures generated by beacons produced by WiFi access-points. In another related paper, Jin et al. [66] use ZigBee-enabled smartphones and access-points to reduce power consumed by an idle WiFi radio and by low bit-rate communication. Similarly to Pering et al. [109], their idea is to switch WiFi off and use ZigBee whenever possible.

It is worth to mention several recent papers that consider the issue of utilizing identical interfaces (e.g., IEEE 802.11) on multiple channels at the MAC layer [86, 117]. These works propose protocols for efficient assignment of available radio channels to these interfaces in order to improve network capacity, and do not address the energy efficiency of the network.

2.2

Energy preservation in wireless networks

Due to the importance of efficient energy management in wireless networking, the preservation of energy is a widely studied topic. A recent extensive work by Anastasi et al. [5] surveys the subject in a related, but different environment of wireless sensor networks. In this section, we summarize the most relevant techniques and particular solutions proposed for single-radio wireless ad hoc networks.

2.2.1 Overlay-based methods

This set of solutions builds upon the observation that when the density of nodes is sufficient, only a small portion of nodes, forming the overlay, need to have their radios turned on to forward traffic of active connections, while other nodes may turn their radios off. The underlying assumption of this approach, supported by several works [12, 24, 39, 109], is that power consumed by radios inidlestate is non-negligible compared to active communication (i.e.,sendingandreceivingstates).

Xu et al. [126] propose the GAF algorithm, which partitions a wireless network into small virtual grids. Based on location information provided by a GPS or similar systems, GAF selects one node in each grid to remain active, while the rest of the nodes are put into sleep. Friedman and Korland [50] use the idea of partitioning a network into a logical grid along with using synchronized clocks in order to construct several energy-effective scheduling protocols. These protocols assign for each cell of the grid time slots in which the cell should be active, i.e., send or receive messages. All nodes in inactive cells are put into sleep, saving energy. The scheduling is devised to reduce message collisions as well.

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Chen et al. [24] present SPAN, a probabilistic power-saving technique where a node decides to join an overlay if it discovers two neighboring nodes that cannot communicate directly or through another node in the overlay. Similar ideas are used by Wu et al. [124] in order to build an overlay consisting of nodes with high remaining energy levels.

Awerbuch et al. construct the Pulse protocol for multi-hop wireless infrastructure access [10] and for ad-hoc networks [11]. The design is based on periodical flooding of special messages, calledpulses, originated by designated nodes, orpulse sources, and propagated throughout the wireless network. If a node needs to communicate, it responds with a reservation message propagated back to the pulse source on the same path the node received the pulse. Nodes that have forwarded a reservation message stay active in order to forward traffic, while all other nodes may turn off their radios until the next pulse. Thus, the overlay is in fact a tree reconstructed every pulse interval and spans only nodes on active routes. While in an infrastructure-based network access points serve as a pulse source, in an ad-hoc network some distributed selection of such source is required.

The main drawback of the discussed overlay-based methods is lost connectivity: since these works consider a single wireless interface, when a node turns its radio off, it is unable to receive incoming messages. Consequently, some infrastructure support is required, or else the number of lost messages is eventually increased or overlay nodes are required to store messages for relatively long durations, in-creasing the latency and the space required for message buffering. Thus, this approach is inappropriate in ad-hoc networks with heavy communication patterns.

2.2.2 Variable transmission range

Another extensively studied power-saving technique is to control topology by varying the transmission range of the radio (see, for example, [56, 73, 91] and a survey in [114]). The main tradeoff in the topol-ogy control comes from the fact that the power of the received signal,P, is in an inverted proportion to the distance from the sender,d, exponentially raised to adistance-power gradient[106]. The value of the gradient in ideal conditions is two, that isP ∼d−2, while in realistic settings it may increase to six and even more [73, 114]. As a result, decreasing the transmission range decreases the power consump-tion at least quadruply and can potentially reduce the interference due to a sparseness argument (which is shown below to be untrue in some cases). On the other hand, it may also hurt network connectivity and increase the diameter of the network (in terms of the number of hops), which in turn may increase the communication latency and decrease the network capacity [59]. Figure 1 demonstrates such a situation. Moreover, power consumption in the idle state still remains a problem [12, 24, 39, 109].

In addition to the assumption made by this technique that the transmission power of radio interfaces can be set to an arbitrary level between0and some maximal power, it should be noted that even when long range, power-expensive links are removed from the network, the remaining links may produce a significant interference. Burkhart et al. [20], for instance, note that low degree networks do not imply low interference by showing graphs with a constant degree where a transmission on one edge interferes with all other edges. This observation implies that reducing power consumption and interference might be conflicting goals, leading to a series of works trying to employ topology control in order to reduce the interference instead of the power consumption (e.g., [17, 20, 101]).

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6

1

2

3

5

4

Figure 1: When the transmission range of node1 is decreased s.t. the link to node4 is broken, the network is still connected, but packets between nodes1and6now contend with packets between nodes 2and5, reducing the total capacity.

2.2.3 IEEE 802.11 power-saving mode

The IEEE 802.11 standard defines a power-saving mode (PSM) [64], which allows to switch the wire-less card into sleeping mode in case of no activity and wake it periodically to probe an access point for pending messages. More specifically, when a node enters PSM, it notifies the access point and turns its radio off. Every predefined period, the access point transmits abeacon frame, which contains a map indicating nodes in PSM that have at least one pending message buffered for them. Consequently, the nodes in PSM should stay synchronized with their access points, turn their radios on every beacon frame and stay active to receive pending messages, if those exist. It should be noted that although the PSM is fully standardized for infrastructure networks, in ad hoc setting nodes may still use this mode, but their active neighbors should store the traffic intended for them in some unspecified way.

While PSM can reduce significantly the power consumption during idle periods, it was found that in certain common interactive applications, it may cause substantial increase in latency and even in power [3, 79]. This is because PSM does not adapt well to varying idle time periods and requires nodes to stay listening at constant intervals. Thus, when an application generates traffic in bursts, PSM causes increased delay inside the bursts, while much power is wasted between the bursts on listening to beacons. In order to cope with these limitations, several enhancements were proposed, tailored mainly to application specific domains, such as web-browsing [6, 79] and distributed file systems [3].

It is worth noting that besides being developed solely for the WiFi radio, PSM is orthogonal to the approaches considered in our research work, and thus can be used in addition to them.

2.3

Connected dominating sets

The dominating set problem plays an important role in many distributed algorithms as well as in this thesis. The generalization of this problem, the kW CDS problem [45], serves as a theoretical foundation for our protocols for improving power efficiency in wireless networks (cf. Chapters 3 [45] and 5 [46]). Moreover, while working on power-related issues, we have discovered that there is a place for improvement of state-of-the-art algorithms and lower bounds for dominating set approximation in networks with bounded degree (cf. Chapter 4 [47]). In the rest of this section, we survey the extensive body of previous research on the dominating set and related problems in various networking models.

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The problem of finding a dominating set that has a minimum number of nodes is known to beN P -complete [52], and, in fact, it is also hard for approximation [40]. The approximation ratio ofO(log ∆) (where∆is the node degree bound) was found to be the best possible (to within a lower order additive factor, unless NP has annO(log logn)-time deterministic algorithm [40]). For general graphs, the best distributed deterministicO(log ∆)-approximation algorithms have linear running time [35, 92, 119]. In fact, these algorithms perform no better than a trivial approach where each node collects a global view of the network by exchanging messages with its neighbors and then calculates locally a domi-nating set approximation by running, e.g., the centralized algorithm of Guha and Khuller [58]. The only lower bound known for general graphs is due to Kuhn et al. [82], which states that at least Ω(√logn/log logn)communication rounds are needed to find a constant or polylogarithmic approx-imation2_{. Recently, this bound was improved by the same authors, who showed that}_Ω(√_log_n₎

com-munication rounds are needed to find a constant approximation [85]. Nevertheless, their proofs rely on a construction of a special family of graphs in which the maximal node degree depends on the size of the graph. Thus, this construction cannot be realized in the bounded degree model.

Another extensive body of work considers restricted types of graphs, such as unit-disk graphs (UDG), planar graphs, graphs with bounded growth, etc. Although the dominating set problem re-mains NP-hard in these models, some approximation algorithms with a constant ratio are known (e.g., [33, 34, 83, 87]). In planar graphs, for example, Lenzen et al. [87] have shown that O(1) ap-proximation is possible in O(1) rounds (with the exact apap-proximation ratio of74). At the same time, Czygrinow et al. [33] prove that no deterministic algorithm that runso(log∗n)rounds in a planar graph and finds (5−δ)-approximation for the minimum dominating set (for anyδ >0) may ever exist.

In the UDG model, Lenzen and Wattenhofer [88] showed that anyf-approximation algorithm for the dominating set problem runs ing(n)time, wheref(n)g(n)∈Ω(log∗n). In contrast to this result, in Chapter 4 [47] we consider a different model of graphs with bounded degree nodes, in which ∆ is not a constant number, but rather an independent parameter of the problem. This enables us to obtain a more refined lower bound. Specifically, we show that while obtainingO(∆)-approximation for the optimal dominating set in our model is possible even without any communication, anyo (∆)-approximation algorithm requiresΩ(log∗n) time. Yet, since the proof of the lower bound by Lenzen and Wattenhofer [88] relies on the ring graph (which has a bounded degree), it carries to our model as well. In this thesis, however, we show an alternative, shorter and more straight-forward way to obtain our lower bound, but in a slightly relaxed model (for details, see Chapter 4). It is also worth noting that the same ring structure is used in the proof of the lower bound for planar graphs by Czygrinow et al. [33], but their arguments are completely different.

The dominating set problem in bounded degree networks was considered by Chlebik and Chlebikova [28], who derive explicit lower bounds on the approximation ratios of centralized solutions. The only previous work on distributed approximation of dominating sets in bounded degree networks that we are aware of is by Astrand et al. [8]. They consider models of colored graphs and derive upper and lower bounds on the approximation ratios of distributed local algorithms, i.e., algorithms with the running time that may depend on∆, but not onn. In addition to the dominating sets, several related problems were considered in the setting of bounded degree graphs. Recently, Astrand and Suomela

2

Kuhn et al. [82] assume unbounded local computations.

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et al. provided distributed deterministic approximation algorithms to a series of such problems, e.g., vertex cover [7, 9] and set cover [9]. Panconesi and Rizzi considered maximal matchings and various colorings [107].

It is worth mentioning several randomized approaches that have been proposed for the general graph model and which can also be applied in the setting of networks with bounded degree. For instance, Jia et al. [65] propose an algorithm withO(lognlog ∆)running time, while Kuhn et al. [84] achieve even better O(log2∆) running time. These solutions, however, provide only probabilistic guarantees on the running time and/or approximation ratio (for example, the former achieves the approximation ratio ofO(log ∆)in expectation andO(logn) with high probability), while our approach deterministically achieves the approximation ratio oflog ∆[47].

2.4

WiFi and BT performance measurements

In the experimental part of this thesis, we evaluate the energy and throughput characteristics of WiFi and BT in smartphones (cf. Chapter 6 [49]). Energy efficiency is recognized as a paramount property of any mobile device, in particular for smartphones. Consequently, several papers try to address the question of where and how the energy is consumed in smartphones. Carroll and Haiser [23], for example, design a set of micro-benchmarks to independently associate the power costs with a particular part of the smartphone system, such as CPU, RAM, WiFi, Audio, etc. in the Openmoko Neo Freerunner phone. While their work considers multiple wireless interfaces available on the devices under test, the interfaces are not profiled beyond enabled and disabled mode, and only their power cost in various general benchmarks is considered.

Few recent works examine energy consumption of the WiFi interface in mobile phones. Balasub-ramanian et al. [13] compare between WiFi and cellular interfaces, such as 3G and GSM. They utilize OS-specific methods to collect information on power usage, such as Nokia Energy Profiler. In compar-ison, the framework we present in Chapter 6 [49] is generic and can be used virtually with any mobile phone. Additionally, the (IEEE 802.11b) WiFi interface considered in [13] is outdated and is profiled only in the access point mode of operation. In another work, Xiao et al. [125] model and measure the power consumption of IEEE 802.11g WiFi interfaces in three mobile phones. Their experiments also consider only the access point operation mode and are conducted with TCP traffic only.

Power measurements of a WiFi card in various modes of operation are reported in [39, 80] (more than a decade ago). In [39], for instance, a thorough analysis of the behavior of the WiFi interface in an ad-hoc setting is given. Unlike these works, we have chosen to measure the performance of wireless interfaces integrated on-board. We believe that for an end-user, the performance of the integrated radio in different network settings (including various hidden costs, e.g., running kernel code and copying buffers) is even more relevant than the performance of a particular wireless card.

Although several papers explore traoffs between WiFi and Bluetooth interfaces in mobile de-vices (e.g., [4, 46, 102, 109]), with the exception of [109], none of them actually measures the perfor-mance of the latter. In fact, we are not aware of any such study for smartphones. Several other works consider the power cost of Bluetooth in the context of sensor networks [18, 22]. Interestingly, the ratio

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of power cost for transmission to the idle state, as specified by these papers, is much lower than actually measured in our experiments. As elaborated in Chapter 6, this result has an important implication on the design of multi-radio protocols and applications for smartphones, such as energy-aware file transfer. As was mentioned above, recently there has been an effort to utilize the emerging ZigBee radio standard to improve the operation of WiFi. For instance, Zhou et al. [130] show that ZigBee radios can be used to discover the existence of WiFi access points by identifying periodic beacons produced by the access points. In another paper, Jin et al. [66] develop a system where an external ZigBee radio is used to establish and maintain a low bit-rate connection between ZigBee-enabled mobile phones and access points. These papers consider only certain modes of WiFi communication, e.g., searching mode in [130] and access point-based communication in [66]. Moreover, their experiments are carried out on mobile devices extended by external ZigBee radio cards. Thus, it is not clear whether conclusions from performance results of these systems will be valid if and when the experiments will be conducted on devices that have ZigBee radios integrated on-board.

2.5

Reliable multicast

Typical techniques for enhancing the reliability of message transmission over an unreliable link include retransmitting lost packets and utilizing resilient encodings. The first approach involves some feedback mechanism that notifies the sender whether its messages have been received, known aspositive acks

(ACKs), or when some of its messages have been lost, known as negative acks(NACKs) [89]. A self-explanatory generic name for many such mechanisms isAutomatic Repeat reQuest (ARQ). The second approach, known as Forward Error Correction (FEC), incorporates error recovery data into certain packets such that data of a lost packet can be recovered from other received packets, provided that enough of them have been received successfully. ARQ works very well when the message loss probability is low, while FEC is considered better when the message loss is high. A known drawback of FEC is that the required redundancy is not known a-priory and it may even change dynamically. Thus, it is hard to calibrate FEC-based protocols to achieve reliability and efficiency together. Conse-quently, hybrid approaches that combine ARQ and FEC have been proposed (e.g., [1,53,105,112,113]). Such hybrid approaches commonly outperform pure ARQ and FEC protocols under various network conditions [78, 105].

In Chapter 7, we propose a new reliable multicast protocol, which utilizes the availability of mul-tiple channels by dedicating one of them exclusively for recovery transmissions (i.e., ARQ retransmis-sions, FEC packets or both). The most relevant previously studied approach is that of Kasera et al. [72], which was proposed for wired networks. There, retransmission of packets is done on separate multi-cast channels that can be dynamically joined or left by receivers. Specifically, when a sender detects a loss of packeti, it retransmits it over channel Ai. The multicast channels are implemented either

by using multiple IP multicast groups or by modifying routers’ functionality. Both methods, however, are unsuitable for wireless networks. Moreover, a na¨ıve idea of using a separate radio for every multi-cast channel allows the approach to be applied only in networks with a very large number of available radios.

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In general, reliable multicast protocols have been extensively studied in the context of the (wired) Internet; a good survey can be found in [110]. Some of these protocols are claimed to work well in wireless networks as well, e.g., [112]. In a recent study, Koutsonikolas and Hu compare and evaluate several protocols for reliable multicast in wireless mesh networks [78]. They consider only single-radio settings, and the comparison is performed only empirically (by simulations).

Several recent papers propose ways to utilize multiple available interfaces in order to improve multicast performance [57, 93, 129]. Most of these papers, however, deal with the question of how to assign interfaces to available wireless channels in an efficient way that will reduce interference on the one hand and improve throughput on the other hand. Such approaches do not solve the problem of energy waste, since all interfaces remain active all the time. Moreover, these works are focused on latency and throughput issues, and are mostly unconcerned with multicast reliability.

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Chapter 3

Efficient Power Utilization in Multi-Radio

Ad Hoc Networks

This chapter considers the problem of power utilization in multi-radio ad-hoc networks and makes several contributions. Our first contribution is the introduction of a formal approach for reducing the energy consumption of wireless networks consisting of nodes owning two interfaces, one of which has a smaller transmission range and a lower power consumption than the other. The problem of energy consumption in multi-radio networks was considered by a few papers [4, 12, 109]; all of them take heuristic approaches. We are not aware of any previous work that tries to capture the problem from a formal theoretic perspective. Thus, in addition to the novelty of the solutions proposed hereafter, the approach taken to model the problem is also new.

Specifically, we formulate a new optimization problem, which we callk-Weighted Connected Dom-inating Set(kW CDS). It is a generalization of the well-known graph theoretic problem of finding a minimumConnected Dominating Set(CDS). In the definition ofkW CDS, we distinguish between

shortandlongcommunication edges, corresponding to the interfaces with shorter and longer transmis-sion ranges, respectively. A solution to thekW CDSproblem is a set of nodes, so that every node in the system is close enough (up tokshort edges) to some node in the set, while all nodes in the set form a sub-network connected by long edges. An arbitrary parameterkcontrols the latency that applications running on devices may experience (e.g., instead of passing through one long edge, a message may pass through up tokshort edges). Each node in the system is assigned a weight, which captures its remaining battery power, and we seek a solution having minimum total weight of nodes in the selected set. Consequently, an optimal solution to thekW CDS problem provides a power-efficient topology where nodes in the selected set have high remaining power and stay with both interfaces turned on, while all other nodes turn off their power-hungry long range interface.

Second, we provide a centralizedkW CDS algorithm with a proven approximation factor. This protocol is based in part on ideas presented by Guha and Khuller for CDS [58] and includes two phases: building ak-Weighted Dominating Set (kW DS) and then extending it to akW CDS. For the second phase, we provide a deterministic construction through the calculation of a spanning tree. Yet,

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we also prove that whenever nodes are uniformly distributed, everykW DS is w.h.p. alsokW CDS. This is regardless of how thekW DSwas obtained.3 The significance of this third contribution of our work is that in many practical settings, the second phase of the protocol can be skipped, and akW CDS

is obtained very efficiently.

Our fourth contribution includes presenting two distributed asynchronous protocols for the

kW CDSproblem. The first of these is a distributed version of the centralized algorithm with a proven approximation factor, which is directly derived from the centralized algorithm. The second protocol is heuristic. It does not have a proven approximation factor, but in practice behaves similarly in most settings, yet is much more message efficient. A formal time and message communication complexity analysis is provided for both.

Finally, we simulate the performance of our algorithms with typical parameters of WiFi and BT technologies and show that as the number of nodes in the system increases, a large portion of nodes may turn off their WiFi radios while remaining connected to the rest of the network at the BT level. In the last section of this chapter we explain how thekW CDSproblem can be extended to networks with an arbitrary number of interfaces and outline a solution for the extended problem.

The rest of this chapter is organized as following: The detailed system model and formal definition of thekW CDS problem are given in Section 3.1. Section 3.2 presents a centralized approximation algorithm and proves its approximation factor. Section 3.3 introduces the distributed algorithms for approximating a solution to thekW CDSproblem. Simulation results of these distributed algorithms are provided in Section 3.4. We summarize the chapter in Section 3.5.

3.1

System model and preliminaries

The system consists of a set of nodes, communicating by exchanging messages over a wireless network. Each node is equipped with two wireless network interfacesAandB, with transmission rangesRand

r, respectively, so that R ≥ r. We assume that, under same conditions, the power consumed by B

is lower than the power consumed byA. In other words, it is preferred to use interface B overAfor communication whenever possible. Note that we do not make any assumptions on the technology type ofAandB. In particular, our protocols do not require the values ofRandrto be known.

The communication network is modeled as an undirected graphG= (V, El∪Es), whereV

repre-sents the set of nodes andEl(Es) represents the set of edges between two nodes that can communicate

directly using interfaceA(B). The communication network is connected at the level of the interface

A, i.e.,(V, El)is a connected graph. Each node is assigned a positive weight w, which is set to the

reciprocal of its remaining battery power. Communication links of both types are bidirectional, reliable and FIFO. Each node has a unique identifier,ID, and executes asynchronously.

In the k-Weighted Connected Dominating Set (kW CDS) problem, the input is a graph G = (V, El∪Es)and a positive weight functionwdefined on the nodes where:

3_{Note that this is in contrast to the weighted}_CDS_{problem, where most weighted dominating sets are not connected.}

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long edge

1

2

3

4

5

8

9

7

6

10

short edge

Figure 2: Example graph.

• V is a set of nodes. Denoten=|V|.

• El(Es) is a set oflong(short) edges.

• Each short edge also appears in the set of long edges, i.e.,(v, u)∈Es⇒(v, u)∈El. • The graph(V, El)is connected.

We refer the graphGl = (V, El)(Gs = (V, Es)) as along (short) instanceofG. For each node

v∈V, we introduce the notation ofk-short-neighborhood,Nk(v), referring the set of nodes at distance

kor less (in the number of edges) fromvinGs, except forvitself. Also, we denote byδsthe maximal

degree of any node inGs.

The objective in thekW CDSproblem is to find a setSof nodes such that:

(1) Every nodev ∈ V is either inS, or has a path of at mostkshort edges to some node inS, i.e.,

∃u∈Ss.t. v∈Nk(u). We say thatu k-coversv,visk-coveredbyuandS k-coversall nodes

inG.

(2) The subgraph induced byS onGlis connected.

(3) The total weight of nodes inSis minimal out of all sets standing in (1) and (2).

When S satisfies requirements (1) and (2), it is called a k-Weighted Connected Dominating Set

(kW CDS) ofG. If only requirement (1) is satisfied,Sis called ak-Weighted Dominating Set(kW DS) ofG.

For example, consider the graph depicted in Figure 2, where the weight of a node is equal to its ID. Both sets of nodes{1,6,8,10}and {1,3,4,6,8} are2W CDS, but the latter has the minimum weight. Additionally, the set{1,6,8}is2W DS, but not2W CDS, since node8is not connected by a long edge to the rest of the nodes in the set, although the nodes in{1,6,8}2-cover all other nodes in the graph. Note that for the case ofk = 1andEl = Es, thekW CDSproblem reduces to theCDS

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7

1

6

4

Figure 3: Example super-graph.

problem in vertex weighted graphs. Thus, finding the optimal solution for thekW CDS problem is

N P-hard.

Note that fork = 1, a na¨ıve approach for calculating the1W CDSset exists. We could run any weightedCDS algorithm (e.g., [58]) on the input graph Gand then define1W CDS as a set of all nodes in the system, except for any nodev that is not in the calculatedCDS set, but has a neighbor (fromN1(v)) in theCDSset. The following example, however, shows that such an approach may end

up with a1W CDSset that isΩ(n)from optimal: Consider a graphGconsisting ofn−1nodes with a short (and thus, also a long) edge between any pair of nodes, and an additional nodevthat only has a long edge to every other node inG. Assume the weight of every node inGexcept forvis2, and the weight ofvis1. The weightedCDS algorithm may choose nodev as the only node in the weighted

CDS (in fact, this is what the algorithm in [58] will do) and thus, the1W CDS set will contain all nodes in the system, having the total weight of2n−1. At the same time, the weight of the optimal 1W CDSset is3, which is the weight ofvand some other node inG.

For our approximation algorithm, we define an auxiliarysuper-graphstructureSG(k)in the fol-lowing way. Given the graphGand a subset of its nodesA, SG(k) = (A, E)is a graph, where an edge exists between two nodesuandv if and only if there is a long edge betweenuor any node in itsk-short-neighborhood and betweenvor any node in its k-short-neighborhood, accordingly. More precisely,(u, v) ∈E ⇔ ∃u′ ∈Nk(u)∪ {u} ∧ ∃v′ ∈ Nk(v)∪ {v}s.t. (u′, v′) ∈El. Note that each

edge of the super-graph corresponds to at most2k+ 2nodes and 2k+ 1 long edges of the original graph. The super-graphSG(2)for the graph in Figure 2 and a set{1,4,6,7}is shown in Figure 3. For example, there exists an edge between nodes4and6, since10∈N2(4)and(6,10)∈El.

3.2

Approximation algorithm

We show first that thekW DSset can be approximated by a factor ofklnδsby a reduction to a weighted

set-cover problem4 _{out of}_G

s. Next, we argue that for practical systems, the calculatedkW DSset is 4

In the set-cover problem, one is given several sets of elements. The objective is to select the minimum number of the sets so that their union contains (covers) all elements that are contained in any of the input sets. The set-cover and dominating set problems are known to be equivalent (with respect to L-reductions) [71].

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already connected by long edges, thus, w.h.p., no further calculation is required. Finally, we discuss how to deterministically connect nodes in thekW DS set by running an instance of a spanning tree algorithm.

3.2.1 Construction ofkW DS

The following simple procedure constructs aklnδs-approximation for thekW DSset.

1. Create an instance of a weighted set-cover problem by making each node an element, creating a set for each nodevwith all its neighbors inNk(v)and assigning the weight of the set to the

weight ofv.

2. Run a greedy algorithm that constructs a set-cover by choosing sets based on the ratio of their weight to the number of new elements they cover. The chosen sets correspond to thekW DSset inG.

Since every set that is created for each node in Step1contains at mostδk

s elements, the weight of

kW DS found in Step2 is at most lnδ_sk·w(OP T), wherew(OP T) is the weight of the minimum

kW DSinG(see [29] for the detailed proof).

Lemma 1. The algorithm presented above is aklnδs-approximation for thekW DSproblem. 3.2.2 FromkW DS tokW CDS

Probabilistic guarantees: We present a formal analysis of the probability for a kW DS set to be unconnected. For simplicity, we assume that nodes are spread in two-dimensional space; our analysis, however, can be easily extended to three dimensions. Interestingly, the analysis does not depend on the algorithm used to construct thekW DS set, but only on the system parameters, i.e., k,r, R and

n. Showing that for practical values for these parameters every kW DS set is connected w.h.p., we claim that the simple algorithm presented above can also be used in practice to approximate w.h.p. the

kW CDSset by a factor ofklnδs.

Denote the sub-graph induced by thekW DSset onGlasG′. We note that ifG′ is unconnected,

then there must be at least one edge(u, v) ∈ Gls.t. (u, v) ∈/ G′ andthere is no path inG′ between

any nodek-coveringuand any nodek-coveringv. Denote a nodek-coveringuasu′; similarly, denote a nodek-coveringv asv′. It is possible thatu′ = u and/orv′ = v. The maximal physical distance betweenu′ andv′ isR+ 2kr. If there is no path inG′ betweenu′ andv′, then, in particular, there is no node in thekW DS set that is located within the long transmission range of bothu′ andv′. In other words, there is no node in the kW DS set that is located within distance R from bothu′ and

v′. Consequently, there is no node inGl that is located within distanceR−kr from both u′ andv′

(otherwise, it should be covered by some node inkW DSwhich is within distanceRfromu′ andv′). Denote this event asE.

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Assuming uniformly distributed random and independent positioning of nodes on the field, we get thatPr(E) ≤(1−Pr(e))n−4, whereeis the event of a node positioned within distanceR−krfrom bothu′andv′andn−4is the total number of nodes in the system, except foru,v,u′ andv′.Pr(e)is given by the ratio between the area of the intersection of two circles with centers at distanceR+ 2kr

and radiusR−kr, and between the area of the field. The area of the intersection of two circles is given by

A(R′, d′) =R′2cos−1(d

′

R′)−d

′√_R′2₋_d_′2

whereR′ is the radius of the circles andd′ is a half of the distance between their centers (taken from mathematical books). SubstitutingR′ =R−krandd′= 1₂R+kr, and denotingm= kr_R, we get that

Pr(e) =A(R−kr, 1 2R+kr) A(f ield) = R2₍₁₋_m)2_cos−1₍12+m 1−m)− √ 3 2 ( 1 2+m)R 2√₁₋_4m A(f ield)

As long asR >4kr, the area of the intersection of two circles is properly defined andPr(e)>0. Given suchR,randkand given the area of the field,Pr(E)is bounded bycn−4, wherec∈(0,1)is a constant. Recall thatEldenotes the set of edges inGl. Using union bound, we get

Pr(kW DSis not connected)≤ |El| ·P r(E)≤n2·cn−4 (1)

which approaches0 whenngrows. For example, takingR = 150m, r = 10m, field area of500× 500 = 250,000m2 and2000nodes, we get that the probability for akW DSset to be unconnected is less than2.88·10−24fork= 1and less than4.76·10−9 fork= 2. Not surprisingly, this theoretical upper bound is echoed by the results of the simulations reported in Section 3.4.

Deterministic guarantees: Here we provide a deterministic extension of the algorithm in Sec-tion 3.2.1 for constructing akW CDS.

3. Create a super-graphSG(k)fromGandkW DSand find a spanning tree (ST). The edges of the

STcorrespond to nodes (and long edges) in the original graphG, which along with the nodes in thekW DSset form akW CDSset.

Referring back to the example graph in Figure 2, when calculating2W CDS, the greedy algorithm in Step2will choose first node1, then node4,6and, finally, node7. The resultingSG(2)super-graph is the one depicted in Figure 3. The spanning tree algorithm in Step3 will choose the edge (4,7) (adding node 5 or a pair of nodes 8 and 9 to 2W CDS, depending on which nodes were chosen to represent the edge(4,7)in SG(2)) and two of the three remaining edges inSG(2). For example, edges (1,6) and(1,4)will add node 3 or a pair of nodes2 and3 to2W CDS. Thus, the possible resulting2W CDSset is{1,2,3,4,5,6,7}.

This example suggests that it is possible to slightly optimize Step 3 of the algorithm. We may assign a weight to each edge(v, u)in the super-graphSG(k)as the weight of nodes in the lightest path betweenvanduinGl (if the nodesu andvare neighbors inGl, the edge is assigned weight0), and

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run a minimum spanning tree algorithm (MST) in Step3. Such an approach, however, will not change the asymptotic approximation factor, and thus is not discussed further.

Denote bywmax andwmin the maximal and minimal weights of any node in the system,

accord-ingly. The proof of the following theorem uses Lemma 1 and the fact that each edge in the calculated spanning tree corresponds to at most2k+ 2nodes in the graphG,2kof which are not inkW DS. Theorem 2. The extended algorithm is aklnδs(1 + 2kw_wmax_min)-approximation for thekW CDS prob-lem.

Proof: LetOP T be the minimumk-weighted connected dominating set inG; denote its weight as

w(OP T). By Lemma 1, the weight ofkW DSfound in Step2is at mostklnδs·w(OP T).

The super-graphSG(k)created in Step3is connected. This follows from the definition of the graph

Gthat requires the long instance ofG(i.e.,Gl) to be connected. The number of nodes inSG(k)is at

most(klnδs·w(OP T))/wmin. Thus, the number of edges inSTis at most(klnδs·w(OP T))/wmin−

1. Each edge corresponds to at most2k+ 2nodes in the graphG,2kof which are not inkW DS. Thus, the total weight of nodes chosen by the algorithm in Step3is at most klnδs·w(OP T)·2k·w_wmax_min.

Adding the weight ofkW DSgives the required bound.

Whenk= 1and the ratio wmax

wmin is close to1, our result matches asymptotically the approximation factor of3 lnnachieved by Guha and Khuller [58] for the related problem ofCDSin vertex weighted graphs.

3.3

Distributed construction of the

kW CDS

set

3.3.1 Algorithm1

The lack of a centralized administration coupled with the potentially large number of nodes in ad hoc networks requires thekW CDSset to be constructed in an effective distributed way. We present such an algorithm based on the centralized approximation algorithm given in the previous section.

The distributed algorithm runs in two phases. The first phase corresponds to steps 1 and 2 of the centralized algorithm, while the second phase corresponds to step 3. Thus, if the probabilistic guarantees of the first phase suffice, the algorithm may terminate right after it (without the second phase).

In the first phase, the dominating setkW DS in the graphGsis iteratively constructed. Initially,

kW DS is empty. For every node v, let δ∗(v) denote the number of nodes inNk(v) that are not k

-covered by any node inkW DS(initially,δ∗(v) = |Nk(v)|). In each iteration, each nodevcalculates

parameterα(v) = w(v)/δ∗(v). Then,vsends itsα(v)and gathersα(u)from allu at distance2kor less fromv inGs. Nodev joinskW DS if itsα(v)is smaller than any collectedα(u) (using IDs to

break ties).

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In the second phase, all nodes in thekW DSset are connected by running the distributed minimum spanning tree algorithm [51] on the super-graphSG(k)created fromGandkW DS. More specifically, only nodes in thekW DSset run the distributed spanning tree algorithm. Every edge(v, u)inSG(k) is emulated by nodes on the shortest path betweenv andu inGl,

Utilizing Multiple Radio Interfaces in Wireless Networks