資料傳輸排程之能耗與成本最佳化
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(2) Energy and Cost Optimization for Data Communication Scheduling by. PI-CHENG HSIU Supervised by. DR. TEI-WEI KUO Submitted to GRADUATE INSTITUTE OF COMPUTER SCIENCE AND INFORMATION ENGINEERING In Partial Fulfillment of the Requirements For the Degree of DOCTOR OF PHILOSOPHY At the. NATIONAL TAIWAN UNIVERSITY June 2009. c National Taiwan University 2009. All rights reserved. .
(3) This dissertation is dedicated to my parents, Fang-Yu Hsiu and Hsiu-Lan Chang.. ii.
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(6) N P-hardP = N P
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(8) . NS2.
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(11) N P-hardP = N P
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(14) AMBA :
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(16) Abstract Energy- and cost-efficiency designs in consumer electronics have been active research topics in the past decades. This dissertation is highly motivated by the rapid growth of data exchanges for both inter-device and intra-device communication, thus addressing energy conservation and cost reduction by targeting the communication components of portable devices. The study on communication subsystems aims at the design of a routing protocol for residual-energy maximization. A polynomial-time optimal algorithm is proposed for the multicast case. The aggregate case is proved to be N P-hard and, unless P = N P, its minimization version cannot be approximated within a ratio better than 2. A distributed algorithm and its realization, referred to as the Maximum-Residual Multicast Protocol (MRMP), are then developed. In MRMP, a transient multicast tree is derived based on the autonomous decisions of devices in the network, where no global information needs to be collected a priori. The derived tree is proved to be loop-free and theoretically optimal in the maximization of minimum residual energy. The capability of MRMP was evaluated over NS2, for which we have very encouraging results in essential performance metrics adopted for routing protocol evaluation. The study on communication architectures focuses on the proposing of a theoretical methodology for bus-layer minimization. Real-time tasks with chain-based precedence constraints are explored on multi-layer bus systems with an objective to minimize the communication cost. The target problem is proved to be N P-hard and, unless P = N P, it cannot be approximated within a ratio better than 1.5. A polynomial-time optimal algorithm is first proposed for a restricted case in which one multi-layer bus, and unit execution and communication time are considered. The result is then extended as a pseudo-polynomial-time optimal algorithm in the considerations of multiple multi-layer buses, arbitrary execution and communication time, and different timing constraints and objective functions. The capability of the proposed algorithm was evaluated over an AMBA-like system topology to provide more insights in system designs, compared to some popular heuristics. Keywords: Energy efficiency, cost efficiency, routing protocols, scheduling algorithms, data communication, networked embedded systems. v.
(17) Acknowledgment. “Research is a repeat search process to find out the unknown.” -Lui Sha. I would like to express my greatest gratitude to my advisor, Prof. Tei-Wei Kuo. On the way of research exist many different destinations. He always encourages me to pick a vital but tough one. To the destination might lead several roads. He always guides me to a blight one. Without his encouragement and guidance, I would not finish this dissertation. I would also like to express my sincere gratitude to Prof. Lui Sha, who introduced me to the real arena of research field. I would much appreciate the opportunity and experience of study in the University of Illinois at Urbana-Champaign. I would like to show my appreciation to Prof. Jane W.S. Liu, Prof. AiChun Pang, Prof. Chi-Sheng Shih, and Prof. Jian-Jang Huang for all the skills that I learnt from them during the cooperation in the industrial and research projects. I would also appreciate the fruitful achievements produced from the projects. I must offer the best thanks to my dissertation committee, Prof. Lui Sha, Prof. Ming-Syan Chen, Prof. Stephen J.H. Yang, Prof. Ai-Chun Pang, Prof. ChiSheng Shih, Prof. Shih-Hao Hung, and Dr. Ted Chang. Their valuable comments and suggestions make this dissertation more solid and complete.. vi.
(18) vii. I want to thank all the members in the Embedded Systems and Wireless Networking (NEWS) Laboratory. Special thanks go to Yuan-Hao Chang and YungFeng Lu for cheering one another on in times of need in the Ph.D program. Special thanks also go to Chin-Hsien Wu and Der-Nien Lee for their discussion and implementation on my research work. Many enjoyments were brought in cooperation with them. My exclusive blessings are given to Chuan-Yue Yang, Pei-Lun Suei, Po-Chun Huang, and Che-Wei Chang for their remaining Ph.D. lives. I also want to thank all the friends in the University of Illinois at Urbana-Champaign. Special thanks go to Ming-Young Nam, Mu Sun, and Chin-Fei Cheah for their particular help in both the research and daily life. I am very sorry for not having space to list all your names and deeply appreciative of all your help. I am grateful to National Science Council, Ministry of Education, and Yen Tjing Ling Industrial Research Institute for their travel grants to participate in the international conferences, and to my nation for the loans to pursue my Ph.D. degree. Particularly pay my most gratefulness to my patents, Fang-Yu Hsiu and Hsiu-Lan Chang, and my younger sister, Tzy-Yi Hsiu. Without their support and consideration, I might not sustain the pressure, tide over the difficulties, and finish the degree. This dissertation is dedicated to them. My special thanks to Pei-Shih Liu for her company in the past five years are beyond works.. Pi-Cheng Hsiu@NTU June, 2009.
(19) Contents. Abstract in Chinese. iv. Abstract. v. Acknowledgment. vi. Contents. viii. List of Figures. xii. List of Tables. xiii. 1 Introduction. 1. 1.1 Motivations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 2. 1.1.1. Energy-Efficient Routing . . . . . . . . . . . . . . . . . . . . .. 2. 1.1.2. Cost-Efficient Scheduling . . . . . . . . . . . . . . . . . . . . .. 3. 1.2 Objectives and Contributions . . . . . . . . . . . . . . . . . . . . . .. 4. 1.2.1. Residual-Energy Maximization. . . . . . . . . . . . . . . . . .. 4. 1.2.2. Bus-Layer Minimization . . . . . . . . . . . . . . . . . . . . .. 6. 1.3 Organization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 2 Related Work. 7 9. 2.1 Energy-Efficient Routing . . . . . . . . . . . . . . . . . . . . . . . . . 10. viii.
(20) ix. CONTENTS. 2.2 Cost-Efficient Scheduling . . . . . . . . . . . . . . . . . . . . . . . . . 14 3 Residual-Energy Maximization. 17. 3.1 Network Model and Problem Formulation . . . . . . . . . . . . . . . 18 3.2 An Optimal Algorithm for Maximum-Residual Multicasting . . . . . 21 3.3 A (2 − )-Inapproximability Result for Maximum-Residual Aggregating 24 3.4 Performance Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . 28 3.4.1. Algorithms for Comparison. . . . . . . . . . . . . . . . . . . . 28. 3.4.2. Experimental Setups and Performance Metrics . . . . . . . . . 29. 3.4.3. Experimental Results . . . . . . . . . . . . . . . . . . . . . . . 31. 3.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 4 Distributed Residual-Energy Maximization. 36. 4.1 Network Model and Problem Formulation . . . . . . . . . . . . . . . 37 4.2 A Distributed Optimal Algorithm for Maximum-Residual Multicasting 40 4.2.1. Algorithm Description . . . . . . . . . . . . . . . . . . . . . . 40. 4.2.2. Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44. 4.3 A Maximum-Residual Multicast Protocol . . . . . . . . . . . . . . . . 50 4.3.1. Routing Tables . . . . . . . . . . . . . . . . . . . . . . . . . . 51. 4.3.2. Route Discovery . . . . . . . . . . . . . . . . . . . . . . . . . . 53. 4.3.3. Route Establishment . . . . . . . . . . . . . . . . . . . . . . . 55. 4.3.4. Data Forwarding . . . . . . . . . . . . . . . . . . . . . . . . . 57. 4.4 Performance Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . 58 4.4.1. Experimental Setups and Performance Metrics . . . . . . . . . 58. 4.4.2. Experimental Results . . . . . . . . . . . . . . . . . . . . . . . 63. 4.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 5 Bus-Layer Minimization. 69.
(21) x. CONTENTS. 5.1 System Model and Problem Formulation . . . . . . . . . . . . . . . . 70 5.1.1. System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 70. 5.1.2. Problem Definition . . . . . . . . . . . . . . . . . . . . . . . . 72. 5.2 A (1.5 − )-Inapproximability Result . . . . . . . . . . . . . . . . . . 74 5.3 A Dynamic-Programming Approach . . . . . . . . . . . . . . . . . . . 77 5.3.1. A Basic Algorithm . . . . . . . . . . . . . . . . . . . . . . . . 77. 5.3.2. Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80. 5.4 Extension of the Basic Algorithm . . . . . . . . . . . . . . . . . . . . 82 5.4.1. Arbitrary Execution and Communication Time . . . . . . . . 82. 5.4.2. Multiple Multi-Layer Buses and Non-Preemptive Tasks/Transactions . . . . . . . . . . . . . . . . . . . . . . . . 86. 5.4.3. Different Timing Constraints and Objective Functions . . . . . 87. 5.5 Performance Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . 89 5.5.1. Experimental Setups and Performance Metrics . . . . . . . . . 89. 5.5.2. Experimental Results . . . . . . . . . . . . . . . . . . . . . . . 91. 5.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 6 Concluding Remarks. 97. 6.1 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 6.2 Future Research Directions . . . . . . . . . . . . . . . . . . . . . . . . 100 6.2.1. Residual-Energy Maximization. . . . . . . . . . . . . . . . . . 100. 6.2.2. Bus-Layer Minimization . . . . . . . . . . . . . . . . . . . . . 101. Bibliography. 102. Curriculum Vitae. 113. Publication List. 114.
(22) List of Figures 2.1 A simple ad hoc network . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.2 The multiple bus and multi-bus architectures . . . . . . . . . . . . . . 15 3.1 A network topology example . . . . . . . . . . . . . . . . . . . . . . . 19 3.2 An input instance example . . . . . . . . . . . . . . . . . . . . . . . . 25 3.3 Impacts of the network size (γ = 1% × 403 , Δ = 10 × 103 ) . . . . . . 32 3.4 Impacts of the energy consumption for reception (|V | = 50, Δ = 10 × 103 ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 3.5 Impacts of notification thresholds (|V | = 50, γ = 1% × 403 ) . . . . . . 34 4.1 The execution of MRMA on an example network . . . . . . . . . . . 42 4.2 The Maximum-Residual Multicast Protocol (MRMP) . . . . . . . . . 51 4.3 An F-shaped antenna . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 4.4 Network Lifetime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 4.5 Delivery Ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 4.6 Control Overhead . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 4.7 Propagation Delay . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 5.1 The multi-layer bus architecture . . . . . . . . . . . . . . . . . . . . . 71 5.2 A reduction example . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 5.3 An illustration of terminology . . . . . . . . . . . . . . . . . . . . . . 78. xi.
(23) LIST OF FIGURES. xii. 5.4 Splitting and grouping of a given precedence graph . . . . . . . . . . 83 5.5 Average number of layers needed by each bus . . . . . . . . . . . . . 91 5.6 Impacts of the relative weight of buses . . . . . . . . . . . . . . . . . 93 5.7 Preemptive task/transaction model vs. non-preemptive task/transaction model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94.
(24) List of Tables 4.1 A wireless adaptor specification . . . . . . . . . . . . . . . . . . . . . 60 5.1 Average running time (second) required per graph . . . . . . . . . . . 95. xiii.
(25) Chapter 1 Introduction. With the maturation of wireless communication and semiconductor technologies, human beings already unconsciously enjoy the convenience and interest brought by various consumer electronics in our daily life. Right because of the significant driving force from the social needs, low-power and low-cost design issues in embedded systems have been attractive and active research topics in the decade. Energy conservation and cost reduction are critical since portable devices are always operated with limited battery and usually optimized for specific functionality. Several technologies for energy conservation and cost reduction are being developed by targeting specific components of the portable devices, such as the MPU/DSP and transceiver [41]. The current technologies have gone a long way in this direction and the development appears to continue. Energy-efficient task scheduling and data communication are important topics in this area for limited resource management and arrangement. This dissertation is motivated by the rapid growth of data exchanges for both interdevice and intra-device communication, thus addressing optimization problems of energy consumption and cost from the perspective on data communication.. 1.
(26) 1.1. MOTIVATIONS. 1.1. 2. Motivations. Routing is the process in which a route from some nodes to others is derived and achieved for data communication in the networking system. Routing protocols designed for autonomous networking systems of portable devices must consider energy consumption of the devices in the network as a primary objective. Power-aware routing metrics serve a heuristic in deriving routes with the best energy efficiency. Identifying an appropriate routing metric, designing a routing algorithm for this metric, and implementing it with a routing protocol are essential problems in data communication. Communication architecture plays another important role in data communication among individual nodes in on-chip systems. Cost has been a major concern in the development of portable devices. Cost-efficient scheduling is essentially a critical problem in minimizing system cost so as to meet performance requirements at the system design stage.. 1.1.1. Energy-Efficient Routing. An ad hoc network is an autonomous system of wireless devices, known as nodes, connected by wireless links, where packets are transmitted via intermediate nodes, instead of an established infrastructure. The energy consumption required for a transmitter usually increases dramatically with the distance, and that for a receiver is considered as a constant [10, 39]. Because such a node is usually battery-powered, there is a strong demand in power-aware routing. With the popularity of mobile devices, routing becomes increasingly challenging because of the dynamic nature of network topologies and critical energy-efficiency considerations. The problem is further complicated by the existence of a huge population of devices and the needs.
(27) 1.1. MOTIVATIONS. 3. in good communication bandwidth utilization, such as the replacement of multiple unicasts with a multicast. Example applications are advertisement in shopping malls [58], tourist information distribution [16], taxi dispatching [34], and cooperative congestion monitoring [59, 78].. 1.1.2. Cost-Efficient Scheduling. As the number of data processing engines per system grows significantly in the next few years, how to resolve the communication problem among tasks has become a very challenging and important design issue. The overheads introduced by data exchanging among tasks could easily offset the abundant computing power introduced by many processing engines if related system design issues are not addressed carefully. As powerful system architectures are proposed to resolve the overhead problem, the design issues ironically become even more difficult because the complexity in real-time task scheduling grows significantly. Designs based on experience and/or empirical study could not provide rigid theoretic ground in the minimization of system design cost. Such an observation motivates this work on the proposing of optimal algorithms to minimize the bus cost of real-time embedded systems with multi-layer buses and precedence constraints. For the rest of this dissertation, we shall use terminologies cores, data processing engines, processing elements, and processors interchangeably when there is no ambiguity..
(28) 1.2. OBJECTIVES AND CONTRIBUTIONS. 1.2. 4. Objectives and Contributions. This dissertation targets energy-efficiency and low-cost issues in embedded system designs. We are interested in the components for data communication: (1) communication subsystems and (2) communication architectures. Our study on communication subsystems aims at not only the identification of the asymptotical hardness and performance of various power-aware routing metrics, but also the development of a distributed routing protocol adaptable to network topologies and resources that might change over time. For communication architectures, we focus on the proposing of a generic and theoretical methodology for performance/cost exploration on multi-layer bus systems.. 1.2.1. Residual-Energy Maximization. The concept of Maximum-Residual Routing was first raised by in [72, 73], where the minimum residual energy of nodes is maximized for each multicast. The objective is to prolong the first node failure time when network topologies and data traffic may change frequently in an unpredictable way. In this dissertation, Maximum-Residual Multicasting and Aggregating are explored in heterogeneous wireless ad hoc networks. We propose a Prim-like algorithm for Maximum-Residual Multicasting and prove its optimality when up-to-date topology and energy information are available. The proposed algorithm can be applied to many existing routing protocols, especially those based on the link-state approach, e.g., [28, 35]. In protocol designs, control messages for maintaining up-to-date topology and energy information need additional energy consumption. We show by experiments that the proposed algorithm for Maximum-Residual Multicasting intends to find the best route in a.
(29) 1.2. OBJECTIVES AND CONTRIBUTIONS. 5. dynamic fashion to prolong network lifetime. The experimental results show that it is excellent in the improvements of network lifetime and load balance, in comparison with other routing metrics, e.g., [50, 72, 77, 85]. On the other hand, we show that Maximum-Residual Aggregating is N P-hard and approximation algorithms for this problem are unlike to exist. We then prove that, unless P = N P, its minimization version cannot be approximated in polynomial time within a ratio of (2 − ) for any > 0, where the ratio bound is with respect to the maximum remaining energy before routing. In protocol designs, the maintenance of global routing information is highly challenging because of the dynamic nature of network topologies and energy resources. The problem is exaggerated when applications with a huge population of mobile devices are under consideration. We first propose a distributed algorithm for Maximum-Residual Multicast and prove its optimality without the considerations of node movements and control overheads. When mobility and control message collisions are taken into consideration, it is shown that every derived route remains loop-free and converges toward an optimal solution in the maximization of the minimum residual energy. Based on the proposed algorithm, we then develop a source-initiated on-demand routing protocol, referred to as Maximum-Residual Multicast Protocol (MRMP), which is adaptable to network topologies and resources that may change over time. In MRMP, no periodic control message is employed to collect routing information a priori or repair link breakages. Neither group membership nor neighbor relationship is maintained at a node by explicit control messages. When desiring a route, a source invokes a route-discovery procedure over the network, and the individual decisions of intermediate nodes form a loop-free multicast tree naturally. For the performance evaluation, the protocol was implemented over NS2 [26], and simulations were conducted extensively with parameters set based.
(30) 1.2. OBJECTIVES AND CONTRIBUTIONS. 6. on a realistic commercial wireless device [2]. We have very encouraging results in essential performance metrics adopted generally for routing protocol evaluation [22].. 1.2.2. Bus-Layer Minimization. The multi-layer bus architecture provided by ARM [1] further improves the communication concurrency and flexibility [54]. The increasing number of bus layers implies the overheads of the cost per area, power consumption and design complexity. In this dissertation, the problem of scheduling real-time tasks with chain-type precedence constraints is explored over multi-layer bus systems with an objective to minimize the bus cost, referred to as the tardiness-bounded layer minimization problem. The objective is to minimize the total cost contributed by the needed bus layers without any violation of timing constraints. The contributions of this work start with fundamental but negative results on the N P-hardness of the target problem and its inapproximability ratio. To be more specific, we show that, unless P = N P, it is not possible to have any approximation algorithm with an approximation ratio better than 1.5. A polynomial-time optimal algorithm, based on dynamic programming, is then proposed for a restricted case, when tasks only have unit execution times, and any communication delay is of one time unit. The algorithm is later extended as a pseudo-polynomial-time algorithm for general cases when tasks have arbitrary execution times, communication delay can be of any time units, any selected task and/or communication can be preemptive/non-preemptive, and other timing cosntaints/objective functions are considered. The proposed algorithm was evaluated against a list-scheduling algorithm [75], over an AMBA-based system topology [53] with task generated by TGFF [80], to provide better insights to this work..
(31) 1.3. ORGANIZATION. 1.3. 7. Organization. The rest of this dissertation is organized as follows: Chapter 2 provides background information on energy-efficiency issues in data communication and reviewed literature related to this dissertation. Chapter 3 targets the asymptotical hardness of various power-aware routing metrics and the comparison of their performance. In this chapter, a polynomial-time optimal algorithm is proposed for Maximum-Residual Multicasting. It is then shown that Maximum-Residual Aggregating is N P-hard and that, unless P = N P, its minimization version cannot be approximated within a ratio of (2 − ) for any > 0. The performance of the routing metric was evaluated by a series of experiments, for which it demonstrated itself being effective and efficient in network lifetime and load balance, in comparison with other routing metrics. Chapter 4 focus on the development of a routing protocol adaptable to dynamic network topologies and resources. In this chapter, a distributed algorithm and its realization, referred to as MRMP, are proposed for residual-energy maximization. In MRMP, a transient multicast tree is established on demand and derived based on the autonomous decisions of intermediate nodes. The derived tree is proved to be loop-free and theoretically optimal in the maximization of minimum residual energy. The performance of the MRMP was evaluated over NS2 with a series of simulations, for which we have very encouraging results in essential performance metrics adopted generally for routing protocol evaluation. Chapter 5 aims at the proposing of a theoretical methodology for tackling communication cost optimization for tasks with performance requirements and.
(32) 1.3. ORGANIZATION. 8. precedence constraints. The tardiness-bounded layer minimization problem is explored in embedded systems with multi-layer buses. In this chapter, we show the N P-hardness of the problem and the best possible approximation ratio of approximation algorithms. A polynomial-time optimal algorithm is first proposed for a restricted case in which one multi-layer bus, and unit execution and communication time are considered. The result is then extended as a pseudo-polynomial-time optimal algorithm in the considerations of multiple multi-layer buses, arbitrary execution and communication time, and different timing constraints and objective functions. The capability of the proposed algorithm was evaluated to provide more insights in system designs, compared to some popular heuristics. Chapter 6 concludes this dissertation and provides further research directions in energy and cost optimization in data communication..
(33) Chapter 2 Related Work. This dissertation is inspired by exciting recent results of two research topics for energy-efficiency in data communication: (1) energy-efficient routing and (2) costefficient scheduling. A large number of researchers have been dedicated to poweraware design of network protocols for the ad hoc networking environment. They share a common objective to reduce energy consumption for communication subsystems and, consequently, prolong network lifetime. The main focus of the first part in this chapter is on the survey of various power-aware routing metrics and protocols for wireless ad hoc networks with special attention to Maximum-Residual Routing. The second part reviews literature related to the problems and technologies on cost-efficient scheduling. With the appearance of emerging communication architectures, this research topic has been attractive and active in the decades and many research results over various communication architectures have been proposed in the literature.. 9.
(34) 2.1. ENERGY-EFFICIENT ROUTING. 2.1. 10. Energy-Efficient Routing. The study on energy-efficient routing can be classified into static or dynamic routing. In static routing, a static network topology is considered, and the data traffic is assumed to be known a priori, e.g., [13, 14, 39, 62, 69]. A well-known example problem is Maximum-Lifetime Routing, where packets are routed according to a pre-determined routing plan until the energy of some node drains away. In dynamic routing, both network topologies and the data traffic may change dynamically in an unpredictable way, and nodes may play the role as a source in some dynamic way. In such a routing problem, we have no knowledge on future arrivals. An optimal route is determined on demand based on the network status (e.g., network topology and battery information) at the time being, when a source has packets to route. In the past decade, various dynamic power-aware routing metrics have been proposed for the prolongation of network lifetime, and a class of fundamental optimization problems were defined, e.g., [50, 72, 73, 77, 85]. Among those routing metrics, Minimum-Energy Routing is proposed to minimize the total energy consumption in packet routing. Minimum-Energy Routing can be explored in terms of unicasting or multicasting. Minimum-Energy Unicasting is polynomial-time solvable [73]. However, the minimization of the total energy consumption may result in the rapid energy exhaustion of some specific nodes. In order to avoid such a problem, researchers started proposing algorithms to explore Minimum-Energy Unicasting on a sub-network that excludes nodes with remaining energy lower than a designated threshold (e.g., Conditional Max-Min battery Capacity Routing [77]). Another example approach is Max-Min ZPmin Routing [50], which derives a routing path by avoiding to route packets via low-energy nodes, where the total energy consumption is at most Z times that of Minimum-Energy Unicasting..
(35) 2.1. ENERGY-EFFICIENT ROUTING. 11. Multicasting and aggregating are techniques suggested to solve the implosion and overlap problems in packet routing. They attempt to minimize the number and the size of duplicate packets and thus to reduce energy consumption. Multicasting transmits packets from a source to multiple destinations by using a group address for the destinations. Aggregating fuses packets coming from multiple destinations enroute to a source. Minimum-Energy Multicasting and Minimum-Energy Aggregating proved to be N P-hard [20, 42]. Some theoretical analysis was presented in [7, 12, 17, 20, 42, 84], and heuristic algorithms were proposed in different variations. Example results include the minimization of the maximum energy consumption of nodes for routing packets [72], the minimization of the maximum energy consumption of a path for forwarding packets to/from one of the destinations [85], and the maximization of remaining energy before packets are routed [50, 77]. They will be introduced in more detail and were adopted for performance comparison in Section 3.4. In order to prolong network lifetime, the data packets should be routed so that the energy consumption is balanced among the nodes in proportion to their remaining energy, instead of routing to minimize the total energy consumption [15]. Therefore, keeping the minimum remaining energy of nodes as high as possible appears to be a more applicable routing metric. In this dissertation, we explore the maximization of the minimum remaining energy of all nodes after packets are routed. The closest related work is that on the maximization of the minimum remaining energy (or a function relative to remaining energy) before packets are routed [50, 77]. We use Figure 2.1 to illustrate the difference between the maximization of the minimum remaining energy of nodes before and after routing. The gray areas in the figure represent the antenna patterns of four heterogeneous nodes with their ˆ respectively communication ranges shown. Let the budget functions β() and β().
(36) 12. 2.1. ENERGY-EFFICIENT ROUTING. Figure 2.1: A simple ad hoc network represent the remaining energy of each node before and after routing. Assume that the weight on each edge represents the energy consumed for the corresponding transmission and that the energy consumption for each reception is ε, where ε B. Consider the two routing paths P1 = s w v and P2 = s u v for node s to send a packet to v. The minimum remaining energy of nodes before routing on P1 and P2 is β(s) = β(w) = β(v) = B + ε and β(u) = B respectively. When the maximization of the remaining energy before routing is considered, P1 is chosen ˆ for routing. As a result, β(w) = 0. However, if the maximization of the remaining energy after routing is considered, then P2 should be picked because the minimum ˆ ˆ remaining energy (among all nodes) after routing is β(w) = 0 and β(u) = B−2ε via P1 and P2 , respectively. In this example, the number of packets from s to v (under the metric on the maximization of the remaining energy after routing) is. B 2ε. times. that when the maximization of the remaining energy before routing is considered. The maximization problem of the minimum remaining energy among all nodes after routing is referred to as Maximum-Residual Routing.. The idea of. Maximum-Residual Routing was first proposed in [73]; however, designing a routing algorithm for this metric and implementing it in a routing protocol were not addressed in the paper. This routing metric is intuitively believed to be effective for prolonging network lifetime [63, 72, 73, 91], but has not been widely studied yet, in.
(37) 2.1. ENERGY-EFFICIENT ROUTING. 13. contrast with Minimum-Energy Routing and Maximum-Lifetime Routing. In [88], a Dijkstra-like heuristic algorithm for Maximum-Residual Unicasting was proposed to demonstrate its effectiveness in prolonging network lifetime. In [56], a Dijkstra-like algorithm for Maximum-Residual Broadcasting was proposed and, in particular, its optimality was also proved when energy consumption for receivers of nodes is the same, i.e., homogeneous wireless ad hoc networks. However, an ad hoc network is usually composed of various mobile devices and not necessarily all nodes desire to be destinations of a session. In addition, they are essentially algorithms relying on the knowledge of the entire topology and the remaining energy information of all nodes, which is highly challenging in routing protocol design. In the past decades, many excellent routing protocols have been proposed for mobile ad hoc networks, e.g., [5, 23, 48]. Each of them tried to optimize some routing and performance metrics for different application scenarios. Popular metrics include the propagation delay and the delivery ratio, where many studies target applications similar to multiplayer online gaming and teleconferencing [23, 83]. In order to save the network bandwidth, multicast protocols were also widely explored, e.g., [18, 27, 30, 36, 37, 47, 66, 86, 87]. Among those excellent solutions, MAODV [66], ODMRP [47], and DDM [37] are examples of the best ones and were submitted to the IETF MANET Working Group as candidates for standardization. MAODV discovers tree-based routes on demand using a broadcast route-discovery mechanism. MAODV is sensitive to node mobility because it actively tracks and reacts to changes in routes so as to repair link breakages [66]. ODMRP is a mesh-based protocol that provides alternative paths to adapt to topology changes. Control messages are flooded periodically to refresh group membership and update routes, and redundant routes are exploited for data delivery, which makes ODRMP scale not well with network sizes [47]. In DDM, each source is responsible for the maintenance of each.
(38) 2.2. COST-EFFICIENT SCHEDULING. 14. multicast group. The list of destinations is placed in packet headers for self-routing over an underlying unicast protocol. DDM is meant for small multicast groups operating in dynamic networks of any size [37]. Routing over mobile ad hoc networks is complicated by the considerations of energy efficiency [15, 50, 72, 73, 77], while shortest paths are not favored in routing. Many approaches were presented in the literature. However, most of the existing results rely on the knowledge of certain global information, such as the remaining energy of all nodes and/or the minimum transmission power between every pair of nodes. The maintenance problem of similar global information is highly challenging in protocol designs because of the difficulty and cost in the maintenance of up-todate information. As a result, various assumptions, such as static network topologies and/or fixed traffic patterns, are made to reduce the problem complexity in poweraware routing.. 2.2. Cost-Efficient Scheduling. In the last decades, researchers have been exploring architecture designs and task scheduling methodologies for multi-core systems [19, 93]. One major and classical model is the fully connected architecture, where the communication between every two processors goes through a dedicated bus, and bus contention is ignored [25, 81, 82, 89]. Due to the challenges on the scalability and cost, the multiple bus architecture was proposed, where a set of processors is connected by a collection of buses [60], as shown in Figure 2.2(a). Such an architecture introduces bus contention in task scheduling problems[71]. In this direction, researchers have proposed excellent theoretical results, e.g., [43, 57, 70]. Popular objectives in the optimization.
(39) 15. 2.2. COST-EFFICIENT SCHEDULING. are such as the pin/wiring minimization (for the cost consideration) [43, 70] and the makespan minimization (for the performance consideration) [57]. A popular and practical communication architecture in recent years is the multi-bus architecture, which have buses of different bandwidths connected by bridges/switches to provide different quality-of-service degrees to processing elements of different performance levels or purposes, as shown in Figure 2.2(b). With the popularity of such an architecture for embedded-system designs, how to schedule task with performance guarantees and bus-cost minimization becomes a very important design issue. Because of the difficulty in task scheduling, existing research results are mainly based on heuristics or search-based solutions, e.g., [52, 61].. p2. p3. p1. p2. p3. RAM. bridge. p1. p4. p5. RAM. (a) Multiple Bus. p4. (b) Multi-Bus. Figure 2.2: The multiple bus and multi-bus architectures The multi-layer bus architecture provided by ARM [1] further improves the communication concurrency and flexibility [54], as shown in Figure 5.1. The multi-layer bus architecture is effectively utilized by many practitioners to fit the characteristics and needs of various embedded systems [93]. Example cases include MPEG decoding systems that adopt a high-speed bus to connect a digital signal processor and a memory controller and another peripheral bus to connect an analogto-digital converter and serial ports, where traffics of different characteristics are separated [79]. In many of such embedded systems, tasks are often compiled to run on some specific (types of) processors and expected to execute with deadline constraints. Although excellent design methodologies and EDA tools have been.
(40) 2.2. COST-EFFICIENT SCHEDULING. 16. developed to address (hardware) architecture synthesis issues in processor allocation, e.g., [9, 65], and bus partitioning, e.g., [46, 68], little work has been done in the exploration of the trade-off between the cost and performance in system designs. The trade-off issue has become even more critical with the increasing number of bus layers because it implies the overheads of the cost per area, power consumption and design complexity..
(41) Chapter 3 Residual-Energy Maximization. In this chapter, Maximum-Residual Multicasting and Aggregating are explored in heterogeneous wireless ad hoc networks. We propose a Prim-like algorithm for Maximum-Residual Multicasting and prove its optimality when up-to-date topology and energy information are available. The proposed algorithm can be applied to many existing routing protocols, especially those based on the link-state approach, e.g., [28, 35]. In practical implementation, control messages for maintaining up-todate topology and energy information need additional energy consumption. We show by experiments that the proposed algorithm for Maximum-Residual Multicasting intends to find the best route in a dynamic fashion to prolong network lifetime. The experimental results show that it is excellent in the improvements of network lifetime and load balance, in comparison with previous related work, e.g., [50, 72, 77, 85]. On the other hand, we show that Maximum-Residual Aggregating is N P-hard and approximation algorithms for this problem are unlike to exist. We then prove that, unless P = N P, its minimization version cannot be approximated in polynomial time within a ratio of (2 − ) for any > 0, where the ratio bound is with respect. 17.
(42) 3.1. NETWORK MODEL AND PROBLEM FORMULATION. 18. to the maximum remaining energy before routing. The rest of this chapter is organized as follows: In Section 3.1, the network model under consideration is described. We then formulate the Maximum-Residual Multicasting and Aggregating problems in a more formal way. Sections 3.2 and 3.3 provide a positive and negative result for the two problems, respectively. Section 3.4 summarizes the experimental results and provides observations. Section 3.5 gives a brief summary.. 3.1. Network Model and Problem Formulation. Maximum-Residual Routing is explored in a heterogeneous wireless ad hoc network with nodes deployed in a 3-dimensional area. Each node is equipped with a wireless transceiver. Transmission by a node can be received by all nodes that lie within its communication range (depending on its antenna pattern and transmission power levels). Reception of a node from different nodes cannot proceed together in one reception. The energy consumption model is similar to that in [33]. The energy consumed by a transmitter u is proportional to τ (u) + α × δ(u, v)κ, where τ (u) is a constant based on the transmit amplifier, δ(u, v) is the Euclidean distance between nodes u and v, α ≥ 1 is the transmission-quality parameter depending on the antenna designs, and κ ∈ [2, 4) is the distance-power gradient depending on the environment conditions. The energy consumed by the receiver v, on the other hand, is a constant γ(v), related to the time and the energy that its transceiver spends and consumes in receive mode. Nodes may have different values of γ because wireless devices may be equipped with different transceivers produced by different vendors. Each node in the network could be either a source or a router. When having packets.
(43) 3.1. NETWORK MODEL AND PROBLEM FORMULATION. 19. to multicast to or aggregate from a set of destinations, the source needs to establish a multicast or an aggregate tree with the destinations for routing the packets. A multicast tree is a directed tree having one source without any incoming edge and all other nodes with exactly one. An aggregate tree is a directed tree having one source without any outgoing edge and all other nodes with exactly one. Figures 3.1(a) and 3.1(b) show an example network topology on which a multicast tree and an aggregate tree are determined, respectively.. (a) A multicast tree. (b) An aggregate tree. Figure 3.1: A network topology example The problems under discussion can be defined as follows: A network topology G = (V, E) is a directed graph in which V is the set of nodes, and E is the set of communication links between pairs of nodes. For each node u ∈ V , a budget β(u) denotes the remaining energy of u before routing. When a source s has packets to multicast or aggregate, for each directed edge (u, v) ∈ E, let a weight ω(u, v) denote the energy consumption of u for transmitting the packets to v and a constant γ(v) denote the energy consumption of v for receiving the packets. Because of the wireless medium, transmitting packets to multiple nodes lying in the communication range needs only one transmission, while receiving packets from multiple nodes needs to consume energy for each reception. Therefore, the remaining budget of u.
(44) 20. 3.1. NETWORK MODEL AND PROBLEM FORMULATION. (after routing on T ) can be defined as βˆT (u) = β(u) − max ω(u, v) − (u,v)∈T. . γ(u) ,. (w,u)∈T. where (u, v) ∈ T and (w, u) ∈ T are the outgoing and incoming edges of u on T , respectively. Note that each node, except for s, in a multicast tree T , has exactly one incoming edge and may have several outgoing edges. The case is exactly the reverse in an aggregate tree (refer to Figure 3.1). This is the reason why the asymptotic hardness of the two problems is so different. For the convenience of discussions, let β(V ) and βˆT (V ) denote min{β(u) | ∀ u ∈ V } and min{βˆT (u) | ∀ u ∈ V }, respectively. In the following, we call a tree T rooted at a source a maximum-residual multicast (or aggregate) tree if T spans all of the vertices in a given destination set and βˆT (V ) ≥ βˆT (V ), where T is any multicast (or aggregate) tree rooted at the source that also spans all of the vertices in the destination set.. Maximum-Residual Multicasting (Aggregating) Problem Input instance: A network topology G = (V, E), where each vertex u ∈ V has a budget β(u) and a constant γ(u), and each edge (u, v) ∈ E has a weight ω(u, v). A source s ∈ V and a destination set R ⊆ V . Objective: A multicast (or an aggregate) tree T rooted at s that spans all vertices in R such that βˆT (V ) is maximized..
(45) 3.2. AN OPTIMAL ALGORITHM FOR MAXIMUM-RESIDUAL MULTICASTING. 3.2. 21. An Optimal Algorithm for Maximum-Residual Multicasting. This section is to present our algorithm for Maximum-Residual Multicasting. Given a network topology G = (V, E) with a budget function β, a weight function ω, a reception function γ, a source s ∈ V , and a destination set R ⊆ V , an optimal algorithm shown in Algorithm 1, referred to as MRMT, is proposed to produce a maximum-residual multicast tree T from s to R. For convenience, we call a partition of V a cut (U, V − U) and an edge (u, v) a stingy edge crossing the cut (U, V − U) if u ∈ U and v ∈ V −U such that min{β(u)−ω(u, v)−γ(u), β(v)−γ(v)} is maximized. An observation is that every spanned vertex, except for s, has to consume energy for its reception. For the simplification of discussions, we add γ(s) to β(s) in the beginning (Line 1) and subtract additional γ(s) from β(s) at the end (Line 9). It can be imaged that an auxiliary source s with β(s ) = ∞ and an edge (s , s) with ω(s, s) = 0 are introduced and have been spanned. Then, MRMT starts from adding s into VT , a set of vertices spanned by T (Line 3), and goes on spanning other vertices in V − VT until all the vertices in R are spanned (Line 4). At each step, a stingy edge crossing the cut (VT , V −VT ) is added to T (Lines 5-7). When all vertices in R are spanned, there are |VT | vertices spanned and exactly |VT | − 1 edges added to T . Eventually, T forms a multicast tree. According to T , each vertex u in VT is actually assigned the power level such that the packets can be multicasted from s to R, and has a remaining budget βT (u) = β(u) − max{ω(u, v) | ∀ (u, v) ∈ ET } − γ(u) (Lines 8-9). The time complexity of MRMT depends on the time required to find a stingy edge to grow T . A straightforward method finds such an edge by searching.
(46) 3.2. AN OPTIMAL ALGORITHM FOR MAXIMUM-RESIDUAL MULTICASTING. 22. Algorithm 1 MRMT: a polynomial-time optimal algorithm for Maximum-Residual Multicasting Input: A network topology G = (V, E) with a budget function β, a weight function ω, and a reception function γ, a source s ∈ V , and a destination set R ⊆ V Output: A maximum-residual multicast tree T = (VT , ET ) from s to R 1: β(s) ← β(s) + γ(s) 2: ET ← φ 3: VT ← {s} 4: while R VT do 5: Find an edge (u, v) crossing the cut (VT , V −VT ) such that min{β(u)−ω(u, v)− γ(u), β(v) − γ(v)} is maximized. 6: ET ← ET ∪ {(u, v)} 7: VT ← VT ∪ {v} 8: for all u ∈ VT do βˆT (u) ← β(u) − max{ω(u, v) | ∀ (u, v) ∈ ET } − γ(u) 9: the adjacency lists of the vertices in V . Each step costs O(E) time, and we conclude a total running time of O(|E||V |). It is not hard to see that MRMT can be improved to run in O(|E| + |V | log |V |) time by using Fibonacci heaps, since MRMT bears similarity to Prim’s algorithm for computing minimum-spanning trees [21], i.e., a tree that spans all of the vertices in V with the minimum total weight of edges. The optimality of MRMT, however, is not so obvious as the time complexity that can directly be derived from Prim’s algorithm. A variant of Minimum-Spanning Tree is referred to as Minimum-Steiner Tree, which is also required to span only a subset of vertices and is proved to be N P-complete [29]. In the variant, a challenge is to determine which vertices should serve as the intermediates so as to form an optimal solution. It results in the N P-completeness of Minimum-Steiner Tree and would be a challenge for Maximum-Residual Multicasting as well. The following theorem proves that MRMT is an optimal algorithm for Maximum-Residual Multicasting.. Theorem 3.1 Algorithm MRMT always derives a maximum-residual multicast tree from s to R..
(47) 3.2. AN OPTIMAL ALGORITHM FOR MAXIMUM-RESIDUAL MULTICASTING. 23. Proof. The theorem is proved by showing an invariant that each step of MRMT always derives a subtree of some optimal solution. To be more specific, given a tree that is a subtree of any maximum-residual multicast tree from s to R and has not yet spanned all vertices in R, MRMT always includes a new edge that will not violate the invariant. When all vertices in R are spanned, a maximum-residual multicast tree is thus derived. Let T be a subtree of some maximum-residual multicast tree T ∗ from s to R, VT be the set of vertices that T spans, and (u, v) be a stingy edge crossing the cut (VT , V − VT ). If T has not yet spanned all of the vertices in R, we show that the inclusion of the stingy edge (u, v) to T will not result in the violation of the invariant. Suppose that T ∗ does not contain (u, v) (if it does, we are done). We show another maximum-residual multicast tree T that includes T ∪ {(u, v)} can be constructed from T ∗ . We delineate two cases, depending on whether v ∈ T ∗ or not: Case 1: Suppose v ∈ T ∗ . We remove the incoming edge of v from T ∗ and then add the stingy edge (u, v) to T ∗ . It is clear that T is a legal multicast tree that contains T ∪ {(u, v)} and spans all vertices in R. We now show that βˆT (V ) ≥ βˆT ∗ (V ). The removing of an edge will not let any vertex decrease its budget, and the inclusion of the edge (u, v) only has a budget impact on u. Note that the budget of v does not change in this case. Because T ∗ spans all vertices in R, T ∗ must contain an edge (x, y) that crosses the cut (VT , V − VT ). Moreover, min{β(x) − ω(x, y) − γ(x), β(y) − γ(y)} ≤ min{β(u) − ω(u, v) − γ(u), β(v) − γ(v)}, since (u, v) is a stingy edge that crosses the cut (VT , V − VT ). It means that β(u) − ω(u, v) − γ(u) ≥ min{β(x) − ω(x, y) − γ(x), β(y) − γ(y)} ≥ βˆT ∗ (V ). Therefore, we conclude that βˆT (V ) ≥ βˆT ∗ (V ). Case 2: Suppose that v ∈ / T ∗ . We just add the stingy edge (u, v) to T ∗ . The.
(48) 3.3. A (2 − )-INAPPROXIMABILITY RESULT FOR MAXIMUM-RESIDUAL AGGREGATING. 24. inclusion of the new edge (u, v) only affects the budgets of u and v. The correctness of β(u) − ω(u, v) − γ(u) ≥ βˆT ∗ (V ) follows directly from the analysis in Case 1. On the other hand, since v ∈ / T ∗ , the remaining budget of v on T , i.e., βˆT (v), is β(v) − γ(v). There must be an edge (x, y) ∈ T ∗ crossing the cut (VT , V − VT ) such that min{β(u)−ω(u, v)−γ(u), β(v)−γ(v)} ≥ min{β(x)−ω(x, y)−γ(x), β(y)−γ(y)}. In other words, β(v) − γ(v) ≥ βˆT ∗ (V ). Because T ∗ is a maximum-residual multicast tree, T must be another maximum-residual multicast tree that contains T ∪{(u, v)}. Throughout, MRMT always adds stingy edges to T . Since the initial tree T = s must be a subtree of some maximum-residual multicast tree, and MRMT terminates when the derived subtree spans all vertices in the destination set, MRMT derives a maximum-residual multicast tree from s to R.. We must point out that the multicast tree derived by MRMT may have some redundant edges to prune without losing the spanning of all vertices in R. Redundant edges can be pruned by simply traversing the multicast tree to prune edges from the vertices if neither themselves nor their descendants are in R. It can be done in O(|V |) time. Note that edge pruning does not result in the budget decreasing of any vertex, and the derived multicast tree after the above edge pruning remains a maximum-residual multicast tree.. 3.3. A (2−)-Inapproximability Result for MaximumResidual Aggregating. Unlike Maximum-Residual Multicasting, Maximum-Residual Aggregating is unfortunately N P-hard. In this section, we present the hardness of Maximum-Residual.
(49) 3.3. A (2 − )-INAPPROXIMABILITY RESULT FOR MAXIMUM-RESIDUAL AGGREGATING. 25. Aggregating.. Lemma 3.1 Maximum-Residual Aggregating is N P-hard. Proof. We prove this lemma by a reduction from the decision version of Hamiltonian Path, which is known to be N P-complete [29]. The input to the Hamiltonian Path problem is an undirected graph G = (V, E). The output is YES if and only if G has a simple path that contains every vertex in V. In order to distinguish directed edges from undirected edges, let us denote the undirected edge incident on u and v in G by [u, v].. (a) Hamiltonian Path. (b) Maximum-Residual Aggregating. Figure 3.2: An input instance example For any given instance G = (V, E) of Hamiltonian Path, an example shown as Figure 3.2(a), we show how to construct, in polynomial time, an instance
(50) G = (V, E), β, ω, s, R, γ of Maximum-Residual Aggregating such that G has a Hamiltonian path if and only if G contains an aggregate tree T with βˆT (V ) = k, where k is a non-negative constant. The construction is as follows: First, we add a new vertex a∈ / V. Let V = V ∪ {a} and E = {(u, v), (v, u) | ∀ [u, v] ∈ E} ∪ {(u, a) | ∀ u ∈ V}, as shown in Figure 3.2(b). Then, we assign the corresponding values to each vertex and edge. For each (u, v) ∈ E, let ω(u, v) = w, where w ≥ 0. For each u ∈ V , let γ(u) = r, where r > 0. Let β(a) = B − w and β(u) = B for each u ∈ V − {a}, where.
(51) 3.3. A (2 − )-INAPPROXIMABILITY RESULT FOR MAXIMUM-RESIDUAL AGGREGATING. 26. B = w + |V| × r. Let s = a and R = V − {s}. The instance can be constructed in a polynomial time of |V| and |E|. To complete the proof, we now show that G has a Hamiltonian path if and only if G contains an aggregate tree T with βˆT (V ) = B − w − r. Suppose that G has a Hamiltonian path. Since (u, v) and (v, u) ∈ G if [u, v] ∈ G, G must contain an aggregate tree T in which every vertex has at most one outgoing edge and at most one incoming edge. Thus, βˆT (V ) = B − w − r. On the other hand, suppose that G has no Hamiltonian path. Any aggregate tree T of G must contain a vertex with at least two incoming edges, and thus βˆT (V ) ≤ B − w − 2r. Since this construction can be done in polynomial time, the existence of a polynomial-time algorithm for Maximum-Residual Aggregating implies the same for Hamiltonian Path. We conclude that Maximum-Residual Aggregating is N P-hard. Since Maximum-Residual Aggregating is N P-hard, there is no polynomialtime optimal algorithm, unless P = N P. Moreover, approximation algorithms for this problem are equally unlike to exist. Indeed, since the optimum can be arbitrary close to zero, the approximation ratio of any polynomial-time algorithm becomes arbitrarily large. A simple observation allows us to sidestep this problem and provide a meaningful baseline for approximation algorithms: The optimal solution to maximize the minimum remaining budget, i.e., βˆT (V ), is equivalent to the optimal solution to minimize the maximum budget utilization, i.e., B − βˆT (V ), for some sufficiently large constant B. We are interested in approximation algorithms for the minimization of the maximum budget utilization with respect to the maximum remaining budget before routing, i.e., B = max{β(u) | u ∈ V }, and refer to the problem as the minimization version of Maximum-Residual Aggregating. Note that an α-approximation algorithm for the minimization version does not translate into.
(52) 3.3. A (2 − )-INAPPROXIMABILITY RESULT FOR MAXIMUM-RESIDUAL AGGREGATING. 27. an α-approximation algorithm for Maximum-Residual Aggregating (and vice versa).. Theorem 3.2 The minimization version of Maximum-Residual Aggregating cannot be approximated in polynomial time within a ratio of (2 − ) for any > 0, with respect to the maximum remaining budget before routing B, unless P = N P. Proof. The proof follows from Lemma 3.1. By multiplying βˆT (V ) with −1 and the adding of B, the reduction in Lemma 3.1 can be rephrased as follows: G has a Hamiltonian path if and only if G contains an aggregate tree T with B − βˆT (V ) = w + r. This theorem can be proved by contradiction. Suppose that there were a polynomial-time ρ-approximation algorithm A for the problem, for some approximation ratio ρ < 2. We show how to use the hypothetical algorithm A to decide whether G has a Hamiltonian path. Because A is an approximation algorithm with a ratio ρ, A will output an aggregate tree T with B − βˆT (V ) ≤ ρ(w + r) if G has a Hamiltonian path; otherwise, A will output T with B − βˆT (V ) ≥ w + 2r. It implies that A can be used to decide whether G has a Hamiltonian path if ρ <. w+2r . w+r. Hence, unless P = N P, no polynomial-time algorithm can be guaranteed to derive − ) × (B − βˆT ∗ (V )), for any > 0 and an aggregate tree T with B − βˆT (V ) ≤ ( w+2r w+r any optimal solution T ∗ . The ratio bound approaches (2 − ), as. w r. approaches 0.. Note that the negative results hold for general network topologies. In some specific applications, nodes are applied in suburban areas with omnidirectional antenna patterns. In that case, a network topology is obtained by considering circular communication ranges assigned to nodes in the 3-dimensional Euclidean space. It remains open whether this geometric special case can be solved in polynomial time, despite the NP-hardness of its general graph version..
(53) 3.4. PERFORMANCE EVALUATION. 3.4. 28. Performance Evaluation. In Section 3.2, an optimal algorithm is proposed for Maximum-Residual Multicasting. The problem explored in this work needs the information of the network topology and the remaining energy of nodes up to a certain precision. However, the network topology and the remaining energy information of nodes may change with time. In practice, the algorithm can be applied to use in existing link-state routing protocols with similar implementations/designs, e.g., STAR [28] and OLSR [35], where network topology information is collected and maintained in an up-to-date way by broadcasting the link-state costs of adjacent nodes to other nodes [5]. The broadcasting of energy information can be done in a similar way, except that we have to define a notification threshold on the remaining energy changing of nodes. The smaller the threshold, the more precise the energy information but the heavier the control traffic. We explore the impacts of the notification threshold values on the performance of the algorithm in the experiments.. 3.4.1. Algorithms for Comparison. We evaluated the capability of the proposed algorithm MRMT in the prolongation of network lifetime, in comparison with four multicast algorithms [50, 72, 77, 85] presented as follows:. • The maximization of the minimum remaining energy of nodes before routing [50, 77], i.e., the maximization of minu∈T β(u), is to avoid routing packets over nodes that have low remaining energy, where T denoted the derived routing tree. The Bellman-Ford algorithm was revised for the derivation of routing.
(54) 3.4. PERFORMANCE EVALUATION. 29. trees for the performance evaluation (referred to as MMEMT ). • The minimization of the maximum energy consumption of nodes for routing packets [72], i.e., the minimization of maxu∈T {β(u) − βˆT (u)}, is to avoid the consumption of too much energy at a single node for routing the packets. Prim’s algorithm was revised in performance evaluation (referred to as MSMT ). • While Minimum-Energy Multicasting has proved to be N P-hard [7, 12, 20], excellent heuristics were proposed to minimize the total energy consumption in the network for multicasting, i.e., the minimization of u∈T (β(u) − βˆT (u)). The heuristic algorithm BIP, proposed in [85], was adopted in performance evaluation (referred to as BIPMT ). • Another algorithm for comparison is on the minimization of the maximum en ergy consumption of any path [85], i.e., the minimization of maxv∈R { u∈sv (β(u)− βˆT (u))}, where s v is the path from s to v in T . It is to minimize the total energy consumed in forwarding packets to one of the destinations. Dijkstra’s algorithm was adopted in the derivation of a routing tree (referred to as SPMT ).. 3.4.2. Experimental Setups and Performance Metrics. The experimental environment consisted of a specified number of nodes (|V | = 10, 20, ..., 100). Nodes were randomly placed in a 3-dimensional rectangular area (100 × 100 × 20). Each node was equipped with an omnidirectional antenna (transmission range 0-40) and a power supplier (battery capacity 1 × 108 ). When having packets to multicast to a set of destinations, a source had to establish a session with.
(55) 3.4. PERFORMANCE EVALUATION. 30. the destinations. In the experiments, we assume that the number of packets routed in a session was fixed and equal to 100 (but the algorithms can be used without this restriction). We adopted an energy consumption model similar to the model in [33]. The energy consumed for transmitting all the packets of a session within distance δ was set as 1% × 403 + 1 × δ 3 , where 1% × 403 , i.e., 1% of the energy consumption for the transmitter at the maximum transmission power, was consumed by the transmit amplifier. The impacts of different energy values consumed in receiving all the packets of a session was also reported in the experiments (γ = (1%, 2%, ..., 10%)×403). For a network, one node was randomly chosen as the source every time, and the source had packets that need to establish 1-100 sessions to multicast to a destination set (randomly chosen from the rest). The uniform distribution was adopted for the random choices in the experiments. This process was repeat until any node had its energy exhausted. In addition to data packets, extra control messages were needed to advertise energy information when MRMT and MMEMT were adopted as routing algorithms, since MRMT and MMEMT considered the remaining energy of nodes to derive routing trees (while the other three did not). The control messages allowed all nodes in the network to have an identical view so that all nodes can determine consistent routes in a distributed way. A threshold Δ was set for the notification of the change in the remaining energy of a node. Whenever the remaining energy of a node decreased by an amount equal to Δ, the node broadcasted a control message (with size equal to that of a data packet) to all reachable nodes so as to update the energy information. The impacts of different notification thresholds were also explored (Δ = (5, 10, ..., 50) × 103 ). For the fairness in performance comparison, every routing algorithm was.
(56) 3.4. PERFORMANCE EVALUATION. 31. evaluated over the same data sets. In the study, two performance metrics were adopted: (1) Network Lifetime (measured in terms of the average number of sessions a node can establish) and (2) Load Balance (measured in terms of the standard deviation of the remaining energy of nodes). Other performance metrics adopted generally for routing protocol evaluation, e.g., propagation delay and delivery ratio, highly depend on the routing protocol which the routing algorithms are incorporated with, and are not studied here. For each routing algorithm, the experimental results were derived as an average value of those of 100 independent experiments.. 3.4.3. Experimental Results. Figures 3.3(a) and 3.3(b) show the network lifetime and the load balance impacted by the network size (|V | = 10, 20, ..., 100) under γ = 1% × 403 and Δ = 10 × 103. As shown in Figure 3.3(a), the network lifetime increases, in general, as the number of nodes increases. The main reason behind the observation is that a smaller network size in the same 3-dimensional space may result in fewer alternatives in routing packets and earlier draining away of energy of some nodes. As the network size increases in the same 3-dimensional space, a higher node density of nodes may result in more available paths between nodes, and more energy-efficient routes could be determined. When |V | ≥ 50, MRMT could improve the network lifetime by 48% to 220%, compared with different other algorithms. Because of a similar reason, the standard deviation of the remaining energy of nodes decreases as the number of nodes increases, as shown in Figure 3.3(b). When |V | ≥ 50, MRMT could reduce the variance by 39% to 98%, compared with different other algorithms. These observations imply that the attempt in keeping the minimum remaining energy as high as possible has a more balance degree in the remaining energy of nodes and a.
(57) 32. 3.4. PERFORMANCE EVALUATION. 20000. 3.5e+007. Standard Deviation of Remaining Energy. MRMT. 18000. BIPMT. Average Number of Sessions. MSMT SPMT. 16000. MMEMT. 14000 12000 10000 8000 6000 4000 2000. 10. 20. 30. 40. 50. 60. 70. 80. Number of Nodes (|V| = 10, 20, ..., 100). (a) Network Lifetime. 90. 100. MRMT BIPMT. 3e+007. MSMT SPMT MMEMT. 2.5e+007 2e+007 1.5e+007 1e+007 5e+006 0. 10. 20. 30. 40. 50. 60. 70. 80. 90. 100. Number of Nodes (|V| = 10, 20, ..., 100). (b) Load Balance. Figure 3.3: Impacts of the network size (γ = 1% × 403 , Δ = 10 × 103 ) longer network lifetime. The attempt also tends to do an even better job when a network becomes more dense. Figures 3.4(a) and 3.4(b) show the network lifetime and the load balance of different algorithms when different amounts of energy consumption for receiving packets were considered, i.e., γ = (1%, 2%, ..., 10%)×403) with |V | = 50 and Δ = 10 × 103 . As shown in Figure 3.4(a), the network lifetime decreases when the energy consumption for reception increases. Because more energy was consumed for each reception, we observed quicker exhaustion of node energy. Nevertheless, it is worth mentioning that MRMT achieves different degrees of improvement with different γ values. When γ = 1% × 403 and 10% × 403 , MRMT could improve the network lifetime by 70% to 161% and 29% to 76%, respectively, compared with different other algorithms. We have to point out that the setup value of the notification threshold (i.e., Δ = 10 × 103 ) may play a big role in these experiments. An improper setup of the notification threshold might result in heavy energy consumption for control traffic, and the exhaustion of node energy would be accelerated (see the following experiments on the impacts of notification thresholds). As shown in Figure 3.4(b), the amount of energy consumption for reception has an insignificant impact on the load balance issue. MRMT and MMEMT can significantly reduce the variance in.
(58) 33. 3.4. PERFORMANCE EVALUATION. 13000. 2.4e+007. Standard Deviation of Remaining Energy. MRMT BIPMT. 12000. Average Number of Sessions. MSMT. 11000. SPMT MMEMT. 10000 9000 8000 7000 6000 5000 4000 3000. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10 3. Energy Consumption for Reception (γ = (1%, 2%, ... 10%)∗40 ). (a) Network Lifetime. 2.2e+007 2e+007 1.8e+007 1.6e+007 MRMT. 1.4e+007. BIPMT MSMT. 1.2e+007. SPMT MMEMT. 1e+007 8e+006 6e+006 4e+006 2e+006. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10 3. Energy Consumption for Reception (γ = (1%, 2%, ... 10%)∗40 ). (b) Load Balance. Figure 3.4: Impacts of the energy consumption for reception (|V | = 50, Δ = 10×103 ) the remaining energy of nodes (with better load balancing), compared with other algorithms. It is because MRMT and MMEMT consider the remaining energy of nodes when determining routes, and they attempt to route packets in a loadbalancing way. Moreover, MRMT further outperforms MMEMT in the load balance since MRMT considers the energy consumption for transmissions (i.e., the weights of edges in a network topology), but MMEMT does not. Figures 3.5(a) and 3.5(b) show the network lifetime and the balance of remaining energy of nodes with respect to the notification thresholds, i.e., Δ = (5, 10, ..., 50) × 103 with |V | = 50 and γ = 1% × 403 . As shown in Figure 3.5(a), MRMT has the best performance improvement when Δ = 10 × 103 . The reason for the improvement hiking from Δ = 5 × 103 to Δ = 10 × 103 is that the notification threshold was unduly small under the network setting when Δ = 5 × 103 , and it may result in frequent dissemination of control messages. We must point out that the larger value the notification threshold is, the more out-of-date remaining energy information is used in determining routes. That is why the network lifetime decreases as the notification threshold increases when Δ ≥ 10 × 103 . With the same reason, out-of-date information may result in improper routing of packets over nodes with lower remaining energy. That also results in the increasing of the variance of.
(59) 34. 3.4. PERFORMANCE EVALUATION. 13000. 2.4e+007. Average Number of Sessions. Standard Deviation of Remaining Energy. MRMT BIPMT. 12000. MSMT SPMT. 11000. MMEMT. 10000 9000 8000 7000 6000 5000 4000. 5. 10. 15. 20. 25. 30. 35. 40. 3. Notification Threshold (Δ = (5, 10, ..., 50)*10 ). (a) Network Lifetime. 45. 50. 2.2e+007 2e+007 1.8e+007. MRMT BIPMT. 1.6e+007. MSMT SPMT. 1.4e+007. MMEMT. 1.2e+007 1e+007 8e+006 6e+006 4e+006. 5. 10. 15. 20. 25. 30. 35. 40. 3. 45. 50. Notification Threshold (Δ=(5, 10, ..., 50)*10 ). (b) Load Balance. Figure 3.5: Impacts of notification thresholds (|V | = 50, γ = 1% × 403 ) the remaining energy of nodes. As shown in the experiments, the setup of the notification threshold has no impacts on BIPMT, MSMT, and SPMT because they do not consider the remaining energy of nodes and have no control messages for such information collection. The experimental results show the importance in the design of protocols in the information maintenance of the remaining energy of nodes. The network lifetime and the balance in the remaining energy of nodes can be thus significantly improved. Another interesting observation is: The consideration of energy consumption in packets transmissions is more influential in the prolongation of network lifetime, while the consideration of remaining energy of nodes is more influential in the improvement of load balance. Based on the experimental results, we surmise that Maximum-Residual Routing would also have excellent performance when the delay in the partitioning of the network is considered (instead of the delay in the first node failure time). It is because the partitioning of the network results from a series of node failures, and this routing metric tends to prolong the node failure time..
(60) 3.5. SUMMARY. 3.5. 35. Summary. This chapter targets power-aware routing in heterogeneous wireless ad-hoc networks. The routing metric to maximize the minimum remaining energy of nodes after packets of a session are routed is referred to as Maximum-Residual Routing. The objective is to keep the minimum remaining energy of all nodes as high as possible so as to delay the first failure time of nodes in the network. In this paper, we propose an algorithm for Maximum-Residual Multicasting and prove its optimality when up-todate topology and energy information are available. The proposed algorithm is easy to implement and can be applied to use in existing link-state routing protocols for wireless ad-hoc networks, especially for those considering shortest-path routing by using Dijkstra’s algorithm. We prove that Maximum-Residual Aggregating is N Phard and that, unless P = N P, its minimization version cannot be approximated in polynomial time within a ratio of (2−) for any > 0, with respect to the maximum remaining energy before routing. We have conducted extensive simulations to better understand the properties of the routing metric. Based on the experiments, we provide some interesting observations and conclude that using our routing metric to find routes is very beneficial because (1) network lifetime is significantly improved and (2) variance in remaining energy is significantly reduced, compared with other work [50, 72, 77, 85]..
(61) Chapter 4 Distributed Residual-Energy Maximization. In this chapter, we are interested in Maximum-Residual Routing, where the minimum residual energy of nodes is maximized for each multicast. The objective is to prolong the first node failure time when network topologies and data traffic may change frequently in an unpredictable way. This concept is first raised by Singh, et al. [72, 73], where no algorithm design and protocol implementation are presented in the work. Other closely related results are a heuristic algorithm for unicasting [88] and an optimal algorithm for broadcasting [56]. They are essentially algorithms relying on the knowledge of the entire topology and the remaining energy information of all nodes. Unlike the past work, we consider applications with a huge population of mobile devices such that no global information can be efficiently maintained at any node. We first propose a distributed algorithm for Maximum-Residual Multicast and prove its optimality without the considerations of node movements and control overheads. When mobility and control message collisions are taken into considera-. 36.
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