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Network encoding complexity: Exact values, bounds, and inequalities
1. | Department of Electrical and Computer Engineering, Texas A & M University, College Station, TX 77843, USA |
2. | Department of Mathematics, Zhengzhou University, Zhengzhou, Henan 450001, China |
3. | Department of Mathematics, University of Hong Kong, Pokfulam Road, Hong Kong, China |
For an acyclic directed network with multiple pairs of sources and sinks and a set of Menger's paths connecting each pair of source and sink, it is known that the number of mergings among these Menger's paths is closely related to network encoding complexity. In this paper, we focus on networks with two pairs of sources and sinks and we derive bounds on and exact values of two functions relevant to encoding complexity for such networks.
References:
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M. Bicknell,
A primer on the Pell sequence and related sequences, Fibonacci Quart., 13 (1975), 345-349.
|
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K. Cai, K. B. Letaief, P. Fan and R. Feng,
On the solvability of 2-pair networks – a cut-based characterization, Phys. Commun., 6 (2013), 124-133.
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T. M. Cover and J. A. Thomas,
Elements of Information Theory 2nd edition, Wiley-Interscience, New York, 2006. |
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J. Edmonds and R. M. Karp,
Theoretical improvements in algorithmic efficiency for network flow problems, J. ACM, 19 (1972), 248-264.
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Ramsey Theory 2nd edition, John Wiley Sons, New York, 1990. |
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S. Jaggi, P. Sanders, P. A. Chou, M. Effros, S. Egner, K. Jain and L. Tolhuizen,
Polynomial time algorithms for multicast network code construction, IEEE Trans. Inf. Theory, 51 (2005), 1973-1982.
doi: 10.1109/TIT.2005.847712. |
[8] |
M. Langberg, A. Sprintson and J. Bruck,
The encoding complexity of network coding, IEEE Trans. Inf. Theory, 52 (2006), 2386-2397.
doi: 10.1109/TIT.2006.874434. |
[9] |
S.-Y. R. Li, R. W. Yeung and N. Cai,
Linear network coding, IEEE Trans. Inf. Theory, 49 (2003), 371-381.
doi: 10.1109/TIT.2002.807285. |
[10] |
K. Menger,
Zur allgemeinen Kurventheorie, Fundam. Math., 10 (1927), 96-115.
|
[11] |
W. Song, K. Cai, R. Feng and R. Wang,
The complexity of network coding with two unit-rate multicast sessions, IEEE Trans. Inf. Theory, 59 (2003), 5692-5707.
doi: 10.1109/TIT.2013.2262492. |
[12] |
W. Song, K. Cai, C. Yuen and R. Feng, On the solvability of 3s/nt sum-network – a region
decomposition and weak decentralized code method, preprint, arXiv: 1502.00762 |
[13] |
A. Tavory, M. Feder and D. Ron,
Bounds on linear codes for network multicast, Electr. Colloq. Comput. Complexity, 10 (2003), 1-28.
|
[14] |
E. L. Xu, W. Shang and G. Han, A graph theoretical approach to network encoding
complexity, in Proc. 2012 Int. Symp. Inf. Theory Appl. , 2012,396–400.
doi: 10.1007/978-1-4614-1800-9_176. |
[15] |
L. Xu and G. Han, Bounds and exact values in network encoding complexity with two sinks,
in Proc. 49th Ann. Allerton Conf. Commun. Control Comput. , 2011,1462–1469. |
[16] |
R. W. Yeung,
Information Theory and Network Coding Springer-Verlag, New York, 2008. |
[17] |
R. W. Yeung, S. -Y. R. Li, N. Cai and Z. Zhang,
Network Coding Theory Now Publishers, Delft, Netherlands, 2006. |
show all references
References:
[1] |
M. Bicknell,
A primer on the Pell sequence and related sequences, Fibonacci Quart., 13 (1975), 345-349.
|
[2] |
K. Cai, K. B. Letaief, P. Fan and R. Feng,
On the solvability of 2-pair networks – a cut-based characterization, Phys. Commun., 6 (2013), 124-133.
|
[3] |
T. M. Cover and J. A. Thomas,
Elements of Information Theory 2nd edition, Wiley-Interscience, New York, 2006. |
[4] |
J. Edmonds and R. M. Karp,
Theoretical improvements in algorithmic efficiency for network flow problems, J. ACM, 19 (1972), 248-264.
|
[5] |
R. Graham, B. Rothschild and J. Spencer,
Ramsey Theory 2nd edition, John Wiley Sons, New York, 1990. |
[6] |
G. Han, Menger's paths with minimum mergings, in Proc. 2009 IEEE Inf. Theory Workshop
Netw. Inf. Theory, 2009. 271–275. |
[7] |
S. Jaggi, P. Sanders, P. A. Chou, M. Effros, S. Egner, K. Jain and L. Tolhuizen,
Polynomial time algorithms for multicast network code construction, IEEE Trans. Inf. Theory, 51 (2005), 1973-1982.
doi: 10.1109/TIT.2005.847712. |
[8] |
M. Langberg, A. Sprintson and J. Bruck,
The encoding complexity of network coding, IEEE Trans. Inf. Theory, 52 (2006), 2386-2397.
doi: 10.1109/TIT.2006.874434. |
[9] |
S.-Y. R. Li, R. W. Yeung and N. Cai,
Linear network coding, IEEE Trans. Inf. Theory, 49 (2003), 371-381.
doi: 10.1109/TIT.2002.807285. |
[10] |
K. Menger,
Zur allgemeinen Kurventheorie, Fundam. Math., 10 (1927), 96-115.
|
[11] |
W. Song, K. Cai, R. Feng and R. Wang,
The complexity of network coding with two unit-rate multicast sessions, IEEE Trans. Inf. Theory, 59 (2003), 5692-5707.
doi: 10.1109/TIT.2013.2262492. |
[12] |
W. Song, K. Cai, C. Yuen and R. Feng, On the solvability of 3s/nt sum-network – a region
decomposition and weak decentralized code method, preprint, arXiv: 1502.00762 |
[13] |
A. Tavory, M. Feder and D. Ron,
Bounds on linear codes for network multicast, Electr. Colloq. Comput. Complexity, 10 (2003), 1-28.
|
[14] |
E. L. Xu, W. Shang and G. Han, A graph theoretical approach to network encoding
complexity, in Proc. 2012 Int. Symp. Inf. Theory Appl. , 2012,396–400.
doi: 10.1007/978-1-4614-1800-9_176. |
[15] |
L. Xu and G. Han, Bounds and exact values in network encoding complexity with two sinks,
in Proc. 49th Ann. Allerton Conf. Commun. Control Comput. , 2011,1462–1469. |
[16] |
R. W. Yeung,
Information Theory and Network Coding Springer-Verlag, New York, 2008. |
[17] |
R. W. Yeung, S. -Y. R. Li, N. Cai and Z. Zhang,
Network Coding Theory Now Publishers, Delft, Netherlands, 2006. |












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