Energy, Latency, and Reliability Tradeoffs in Coding Circuits

Using the Thompson circuit complexity model, it is shown that fully parallel encoding and decoding schemes with asymptotic block error probability that scales as <inline-formula> <tex-math notation="LaTeX">$O(f(n))$ </tex-math></inline-formula> have energy that scales as <inline-formula> <tex-math notation="LaTeX">$\Omega (n{-\ln f(n)}^{1/2})$ </tex-math></inline-formula>. In addition, it is shown that the number of clock cycles [<inline-formula> <tex-math notation="LaTeX">$T(n)$ </tex-math></inline-formula>] required for any encoding or decoding scheme that reaches this bound must scale as <inline-formula> <tex-math notation="LaTeX">$T(n)\ge {-\ln f(n)}^{1/2}$ </tex-math></inline-formula>. Similar scaling results are extended to serialized computation. A similar approach is extended to three dimensions by generalizing the Grover information-friction energy model. Within this model, it is shown that encoding and decoding schemes with probability of block error <inline-formula> <tex-math notation="LaTeX">$P_{\mathrm {e}}(n)$ </tex-math></inline-formula> consume at least <inline-formula> <tex-math notation="LaTeX">$\Omega (n(-\ln P_{\mathrm {e}}(n))^{({1}/{3})})$ </tex-math></inline-formula> energy.

[1]  Shachar Lovett,et al.  Bounded-Depth Circuits Cannot Sample Good Codes , 2011, 2011 IEEE 26th Annual Conference on Computational Complexity.

[2]  Robert G. Gallager,et al.  Low-density parity-check codes , 1962, IRE Trans. Inf. Theory.

[3]  Frank R. Kschischang,et al.  On Scaling Rules for Energy of VLSI Polar Encoders and Decoders , 2016, ArXiv.

[4]  Jason Cong,et al.  Three-Dimensional Integrated Circuit Design: EDA, Design and Microarchitectures , 2009 .

[5]  Mahtab Mirmohseni,et al.  Information Theory of Molecular Communication: Directions and Challenges , 2016, IEEE Transactions on Molecular, Biological and Multi-Scale Communications.

[6]  Frank R. Kschischang,et al.  Energy Consumption of VLSI Decoders , 2014, IEEE Transactions on Information Theory.

[7]  Rolf Landauer,et al.  Dissipation and noise immunity in computation and communication , 1988, Nature.

[8]  Frank Göring,et al.  Short proof of Menger's Theorem , 2000, Discret. Math..

[9]  Andrea J. Goldsmith,et al.  Fundamental limits on the power consumption of encoding and decoding , 2012, 2012 IEEE International Symposium on Information Theory Proceedings.

[11]  A. Turing.  On Computable Numbers, with an Application to the Entscheidungsproblem. , 1937 .

[12]  Shachar Lovett,et al.  Bounded-Depth Circuits Cannot Sample Good Codes , 2011, Computational Complexity Conference.

[13]  Sennur Ulukus,et al.  Optimal scheduling for energy harvesting transmitters under temperature constraints , 2015, 2015 IEEE International Symposium on Information Theory (ISIT).

[14]  Frank R. Kschischang,et al.  On the VLSI Energy Complexity of LDPC Decoder Circuits , 2015, IEEE Transactions on Information Theory.

[15]  Brendan J. Frey,et al.  Factor graphs and the sum-product algorithm , 2001, IEEE Trans. Inf. Theory.

[16]  D. Manjunath,et al.  Information friction limits on computation , 2014, 2014 52nd Annual Allerton Conference on Communication, Control, and Computing (Allerton).

[17]  Soummya Kar,et al.  Computing Linear Transformations With Unreliable Components , 2015, IEEE Transactions on Information Theory.

[18]  Frank R. Kschischang,et al.  Modeling and Energy Optimization of LDPC Decoder Circuits With Timing Violations , 2018, IEEE Trans. Commun..

[19]  Michael I. Jordan,et al.  Graphical Models, Exponential Families, and Variational Inference , 2008, Found. Trends Mach. Learn..

[20]  Ravi Jain,et al.  Towards a model of energy complexity for algorithms [mobile wireless applications] , 2005, IEEE Wireless Communications and Networking Conference, 2005.

[21]  Andrea J. Goldsmith,et al.  On the Total Power Capacity of Regular-LDPC Codes With Iterative Message-Passing Decoders , 2015, IEEE Journal on Selected Areas in Communications.

[22]  Christopher Graham Blake.  Energy Consumption of Error Control Coding Circuits , 2017 .

[23]  Sanjeev Arora,et al.  Computational Complexity: A Modern Approach , 2009 .

[24]  Erik D. Demaine,et al.  Energy-Efficient Algorithms , 2016, ITCS.

[25]  Pulkit Grover,et al.  Energy-adaptive polar codes: Trading off reliability and decoder circuit energy , 2017, 2017 IEEE International Symposium on Information Theory (ISIT).

[26]  S. Lloyd.  Ultimate physical limits to computation , 1999, Nature.

[27]  R. Landauer,et al.  Irreversibility and heat generation in the computing process , 1961, IBM J. Res. Dev..

[28]  K. Menger.  Zur allgemeinen Kurventheorie , 1927 .

[29]  Pulkit Grover.  Information Friction and Its Implications on Minimum Energy Required for Communication , 2015, IEEE Transactions on Information Theory.

[30]  Rüdiger L. Urbanke,et al.  Polar codes: Characterization of exponent, bounds, and constructions , 2009, 2009 IEEE International Symposium on Information Theory.

[31]  JOHN E. SAVAGE.  The complexity of decoders-II: Computational work and decoding time , 1971, IEEE Trans. Inf. Theory.

[32]  Mark R. Greenstreet,et al.  Modeling Energy-Time Trade-Offs in VLSI Computation , 2012, IEEE Transactions on Computers.

[33]  Mayank Bakshi,et al.  Energy-efficient Decoders for Compressive Sensing: Fundamental Limits and Implementations , 2015 .

[34]  Dmitri B. Chklovskii,et al.  Wiring Optimization in Cortical Circuits , 2002, Neuron.

[35]  John E. Savage,et al.  Complexity of decoders-I: Classes of decoding rules , 1969, IEEE Trans. Inf. Theory.

[36]  C. Mead,et al.  Fundamental limitations in microelectronics—I. MOS technology , 1972 .

[37]  Sennur Ulukus,et al.  Energy Harvesting Transmitters That Heat Up: Throughput Maximization Under Temperature Constraints , 2015, IEEE Transactions on Wireless Communications.

[38]  Charles H. Bennett,et al.  The thermodynamics of computation—a review , 1982 .

[39]  Frank R. Kschischang,et al.  On the Energy Complexity of LDPC Decoder Circuits , 2015 .