Network Coding and Wireless Physical-layer ... - Jacobs University

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Chapter 4: Unequal Erasure Protection (UEP) in Network Coding 47 4.8 Subtree Decomposition Technique for Complexity Reduction of the GEK Assignment Problem S A B C D E F G R 2 R 3 R 1 R 4 Figure 4.4: Network example with scalable data multicast T 1 T 2 T 3 SA SB SC AR 1 AR 2 AR 3 AD BD BR 2 BE CE CR 1 CR 3 CR 4 T 4 T 5 DF EG FR 1 FR 3 FR 4 GR 2 GR 4 Figure 4.5: Line graph derived from Fig. 4.4 Consider the network in Fig. 4.4. The source S would like to multicast an ordered scalable message M = [m 1 , m 2 , m 3 ], of which elements m 1 , m 2 , and m 3 have dependency levels of 1, 2, and 3, respectively, to four sink nodes R 1 , R 2 , R 3 , and R 4 . Every edge in the graph is capable of transmitting one symbol per time unit. Before assigning a GEK to each edge, we can simplify the graph in Fig. 4.4, using
48 Chapter 4: Unequal Erasure Protection (UEP) in Network Coding Fragouli, Soljanin, and Shokrollahi’s approach [14]. Figure 4.4 is first transformed into a line graph shown in Fig. 4.5, where each node represents an edge from Fig. 4.4. Any two nodes in Fig. 4.5 are connected if the corresponding edges in Fig. 4.4 are adjacent. The nodes in Fig. 4.5 are grouped into five subtrees, each of which is bounded by dashed lines, such that the members in each subtree are forced by the topology to have the same GEK. For example, in the subtree T 1 , SA and AD must have the same GEK, since, according to Fig. 4.4, the node A has only one incoming edge SA and thus can do nothing but copy the received symbols and forward the copies to all outgoing edges AR 1 , AR 2 , AR 3 , and AD, hence the same GEK among them. Accordingly, our problem of assigning GEKs to twenty-one edges is reduced to that of assigning GEKs to five subtrees. The minimum subtree graph is shown in Fig. 4.6. T 1 T 4 T 2 T 3 T 5 Figure 4.6: The minimum subtree graph derived from Fig. 4.5 In this case, the subtrees T 1 = {SA, AD, AR 1 , AR 2 , AR 3 }, T 2 = {SB, BD, BE, BR 2 }, T 3 = {SC, CE, CR 1 , CR 3 , CR 4 }, T 4 = {DF, F R 1 , F R 3 , F R 4 }, T 5 = {EG, GR 2 , GR 4 }. The edges connecting T 1 and T 2 to T 4 , as well as T 2 and T 3 to T 5 in Fig. 4.6 imply that the GEK of T 4 must be derived from those of T 1 and T 2 whereas that of T 5 must be derived from those of T 2 and T 3 , i.e., the sets of vectors {f T1 , f T2 , f T4 } and {f T2 , f T3 , f T5 }, where f Tx denotes the GEK of the subtree T x , must be linearly dependent. In a similar manner, linear independence constraints can be represented by the sets {f T1 , f T2 , f T3 }, {f T1 , f T3 , f T4 }, {f T1 , f T2 , f T5 }, and {f T3 , f T4 , f T5 }. The three GEKs in each set must be linearly independent in order to ensure a full-rank system of linear equations at each of the source and the sinks, when there is no erasure. For example, the source S is connected to the subtrees T 1 , T 2 , and T 3 , hence the linearly independent constraint {f T1 , f T2 , f T3 }. The next two sections show how to solve the problem under these constraints with different optimization objectives. In Section 4.9, the objective is equity of received data
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Network Coding and Wireless Physica
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Abstract The general abstraction of
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Contents Abstract i Abstract ii Dec
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vi CONTENTS 5.4 Degree Distribution
Page 10 and 11: List of Figures 2.1 A basic digital
Page 12 and 13: x LIST OF FIGURES 5.11 Comparison o
Page 14 and 15: Part I Motivation, Overview, and In
Page 16 and 17: Chapter 1: Motivation and Overview
Page 18 and 19: Chapter 2 Introduction to Digital C
Page 20 and 21: Chapter 2: Introduction to Digital
Page 30 and 31: Chapter 3: Introduction to Graphs a
Page 46 and 47: 33 The emergence of scalable video
Page 48 and 49: Chapter 4: Unequal Erasure Protecti
Page 72 and 73: Chapter 5: Network Coding with LT C
Page 88 and 89: Part III Wireless Physical-layer Se
Page 90 and 91: Chapter 6 Wireless Physical-layer S
Page 92 and 93: Chapter 6: Wireless Physical-layer
Page 104 and 105: Chapter 7 Physical-layer Key Encodi
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Chapter 7: Physical-layer Key Encod
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Chapter 8: Summary, Conclusion, and
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Bibliography [1] 3GPP, “3GPP; Tec
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BIBLIOGRAPHY 121 [21] F.A. Hayek,
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BIBLIOGRAPHY 123 [43] M. Friedman a
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BIBLIOGRAPHY 125 [63] S. Jaggi, P.
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BIBLIOGRAPHY 127 [81] X. Sun, X. Wu
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