Error Correction for Message Streams

Authors Meghal Gupta , Rachel Yun Zhang



PDF
Thumbnail PDF

File

LIPIcs.ITCS.2025.59.pdf
  • Filesize: 0.77 MB
  • 18 pages

Document Identifiers

Author Details

Meghal Gupta
  • University of California Berkeley, CA, USA
Rachel Yun Zhang
  • MIT, Cambridge, MA, USA

Cite As Get BibTex

Meghal Gupta and Rachel Yun Zhang. Error Correction for Message Streams. In 16th Innovations in Theoretical Computer Science Conference (ITCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 325, pp. 59:1-59:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.ITCS.2025.59

Abstract

In the setting of error correcting codes, Alice wants to send a message x ∈ {0,1}ⁿ to Bob via an encoding enc(x) that is resilient to error. In this work, we investigate the scenario where Bob is a low space decoder. More precisely, he receives Alice’s encoding enc(x) bit-by-bit and desires to compute some function f(x) in low space. A generic error-correcting code does not accomplish this because decoding is a very global process and requires at least linear space. Locally decodable codes partially solve this problem as they allow Bob to learn a given bit of x in low space, but not compute a generic function f. 
Our main result is an encoding and decoding procedure where Bob is still able to compute any such function f in low space when a constant fraction of the stream is corrupted. More precisely, we describe an encoding function enc(x) of length poly(n) so that for any decoder (streaming algorithm) A that on input x computes f(x) in space s, there is an explicit decoder B that computes f(x) in space s ⋅ polylog(n) as long as there were not more than 1/4 - ε fraction of (adversarial) errors in the input stream enc(x).

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Coding theory
Keywords
  • error-correcting codes
  • streaming algorithms
  • space-efficient algorithms

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads

References

  1. Kook Jin Ahn, Sudipto Guha, and Andrew McGregor. Analyzing Graph Structure via Linear Measurements, pages 459-467. SIAM, 2012. URL: https://doi.org/10.1137/1.9781611973099.40.
  2. Noga Alon, Yossi Matias, and Mario Szegedy. The space complexity of approximating the frequency moments. In Proceedings of the twenty-eighth annual ACM symposium on Theory of computing, pages 20-29, 1996. URL: https://doi.org/10.1145/237814.237823.
  3. Omri Ben-Eliezer, Rajesh Jayaram, David P Woodruff, and Eylon Yogev. A framework for adversarially robust streaming algorithms. ACM Journal of the ACM (JACM), 69(2):1-33, 2022. URL: https://doi.org/10.1145/3498334.
  4. Zvika Brakerski and Yael Tauman Kalai. Efficient Interactive Coding against Adversarial Noise. In 2012 IEEE 53rd Annual Symposium on Foundations of Computer Science, pages 160-166, 2012. URL: https://doi.org/10.1109/FOCS.2012.22.
  5. Zvika Brakerski and Moni Naor. Fast Algorithms for Interactive Coding. In Proceedings of the Twenty-Fourth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA '13, pages 443-456, USA, 2013. Society for Industrial and Applied Mathematics. URL: https://doi.org/10.1137/1.9781611973105.32.
  6. Mark Braverman. Towards Deterministic Tree Code Constructions. In Proceedings of the 3rd Innovations in Theoretical Computer Science Conference, ITCS '12, pages 161-167, New York, NY, USA, 2012. Association for Computing Machinery. URL: https://doi.org/10.1145/2090236.2090250.
  7. Mark Braverman and Klim Efremenko. List and Unique Coding for Interactive Communication in the Presence of Adversarial Noise. In 2014 IEEE 55th Annual Symposium on Foundations of Computer Science (FOCS), pages 236-245, Los Alamitos, CA, USA, October 2014. IEEE Computer Society. URL: https://doi.org/10.1109/FOCS.2014.33.
  8. Mark Braverman and Anup Rao. Towards Coding for Maximum Errors in Interactive Communication. In Proceedings of the Forty-Third Annual ACM Symposium on Theory of Computing, STOC '11, pages 159-166, New York, NY, USA, 2011. Association for Computing Machinery. URL: https://doi.org/10.1145/1993636.1993659.
  9. Moses Charikar, Kevin Chen, and Martin Farach-Colton. Finding frequent items in data streams. In International Colloquium on Automata, Languages, and Programming, pages 693-703. Springer, 2002. URL: https://doi.org/10.1007/3-540-45465-9_59.
  10. Moses Charikar, Kevin Chen, and Martin Farach-Colton. Finding frequent items in data streams. Theoretical Computer Science, 312(1):3-15, 2004. Automata, Languages and Programming. URL: https://doi.org/10.1016/S0304-3975(03)00400-6.
  11. Jiecao Chen and Qin Zhang. Bias-aware sketches. arXiv preprint, 2016. URL: https://arxiv.org/abs/1610.07718.
  12. Graham Cormode and S. Muthukrishnan. An improved data stream summary: the count-min sketch and its applications. Journal of Algorithms, 55(1):58-75, 2005. URL: https://doi.org/10.1016/j.jalgor.2003.12.001.
  13. Varsha Dani, Thomas P. Hayes, Mahnush Movahedi, Jared Saia, and Maxwell Young. Interactive Communication with Unknown Noise Rate, 2015. URL: https://arxiv.org/abs/1504.06316.
  14. Irit Dinur, Shai Evra, Ron Livne, Alexander Lubotzky, and Shahar Mozes. Locally testable codes with constant rate, distance, and locality. In Proceedings of the 54th Annual ACM SIGACT Symposium on Theory of Computing, pages 357-374, 2022. URL: https://doi.org/10.1145/3519935.3520024.
  15. Zeev Dvir, Parikshit Gopalan, and Sergey Yekhanin. Matching vector codes. SIAM Journal on Computing, 40(4):1154-1178, 2011. URL: https://doi.org/10.1137/100804322.
  16. Klim Efremenko. 3-query locally decodable codes of subexponential length. In Proceedings of the forty-first annual ACM symposium on Theory of computing, pages 39-44, 2009. URL: https://doi.org/10.1145/1536414.1536422.
  17. Klim Efremenko, Ran Gelles, and Bernhard Haeupler. Maximal Noise in Interactive Communication Over Erasure Channels and Channels With Feedback. IEEE Trans. Inf. Theory, 62(8):4575-4588, 2016. URL: https://doi.org/10.1109/TIT.2016.2582176.
  18. Klim Efremenko, Bernhard Haeupler, Yael Tauman Kalai, Gillat Kol, Nicolas Resch, and Raghuvansh R Saxena. Interactive coding with small memory. In Proceedings of the 2023 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 3587-3613. SIAM, 2023. URL: https://doi.org/10.1137/1.9781611977554.CH137.
  19. Klim Efremenko, Gillat Kol, and Raghuvansh R. Saxena. Binary Interactive Error Resilience Beyond ¹/₈ (or why (¹/₂)³ > ¹/₈). In 2020 IEEE 61st Annual Symposium on Foundations of Computer Science (FOCS), pages 470-481, 2020. URL: https://doi.org/10.1109/FOCS46700.2020.00051.
  20. Philippe Flajolet. Approximate counting: a detailed analysis. BIT Numerical Mathematics, 25(1):113-134, 1985. URL: https://doi.org/10.1007/BF01934993.
  21. G. Forney. Generalized minimum distance decoding. IEEE Transactions on Information Theory, 12(2):125-131, 1966. URL: https://doi.org/10.1109/TIT.1966.1053873.
  22. Sumegha Garg, Pravesh K Kothari, Pengda Liu, and Ran Raz. Memory-sample lower bounds for learning parity with noise. arXiv preprint, 2021. URL: https://arxiv.org/abs/2107.02320.
  23. Ran Gelles. Coding for Interactive Communication: A Survey. Foundations and Trends® in Theoretical Computer Science, 13:1-161, January 2017. URL: https://doi.org/10.1561/0400000079.
  24. Ran Gelles and Bernhard Haeupler. Capacity of Interactive Communication over Erasure Channels and Channels with Feedback. SIAM Journal on Computing, 46:1449-1472, January 2017. URL: https://doi.org/10.1137/15M1052202.
  25. Ran Gelles, Bernhard Haeupler, Gillat Kol, Noga Ron-Zewi, and Avi Wigderson. Towards Optimal Deterministic Coding for Interactive Communication, pages 1922-1936. SIAM, 2016. URL: https://doi.org/10.1137/1.9781611974331.ch135.
  26. Ran Gelles and Siddharth Iyer. Interactive coding resilient to an unknown number of erasures. arXiv preprint, 2018. URL: https://arxiv.org/abs/1811.02527.
  27. Peter Gemmell and Madhu Sudan. Highly resilient correctors for polynomials. Information Processing Letters, 43(4):169-174, 1992. URL: https://doi.org/10.1016/0020-0190(92)90195-2.
  28. Mohsen Ghaffari and Bernhard Haeupler. Optimal Error Rates for Interactive Coding II: Efficiency and List Decoding. Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS, December 2013. URL: https://doi.org/10.1109/FOCS.2014.49.
  29. Meghal Gupta, Venkatesan Guruswami, and Mihir Singhal. Tight bounds for stream decodable error-correcting codes, 2024. URL: https://doi.org/10.48550/arXiv.2407.06446.
  30. Meghal Gupta and Rachel Yun Zhang. Efficient interactive coding achieving optimal error resilience over the binary channel. arXiv preprint, 2022. URL: https://doi.org/10.48550/arXiv.2207.01144.
  31. Meghal Gupta and Rachel Yun Zhang. The Optimal Error Resilience of Interactive Communication Over Binary Channels. In Symposium on Theory of Computing, STOC 2012, New York, NY, USA, June 20 - June 24, 2022, STOC '22. ACM, 2022. URL: https://doi.org/10.1145/3519935.3519985.
  32. Venkat Guruswami. Error-correcting codes: Constructions and algorithms, lecture no. 11, 2006. URL: http://www.cs.washington.edu/education/courses/533/06au/.
  33. Venkatesan Guruswami and Madhu Sudan. List decoding algorithms for certain concatenated codes. In Proceedings of the Thirty-Second Annual ACM Symposium on Theory of Computing, STOC '00, pages 181-190, New York, NY, USA, 2000. Association for Computing Machinery. URL: https://doi.org/10.1145/335305.335327.
  34. Bernhard Haeupler. Interactive Channel Capacity Revisited. In 55th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2014, Philadelphia, PA, USA, October 18-21, 2014, pages 226-235, 2014. URL: https://doi.org/10.1109/FOCS.2014.32.
  35. Richard W Hamming. Error detecting and error correcting codes. The Bell system technical journal, 29(2):147-160, 1950. Google Scholar
  36. Piotr Indyk and David Woodruff. Optimal approximations of the frequency moments of data streams. In Proceedings of the thirty-seventh annual ACM symposium on Theory of computing, pages 202-208, 2005. URL: https://doi.org/10.1145/1060590.1060621.
  37. William Johnson and Joram Lindenstrauss. Extensions of lipschitz maps into a hilbert space. Contemporary Mathematics, 26:189-206, January 1984. URL: https://doi.org/10.1090/conm/026/737400.
  38. Swastik Kopparty, Shubhangi Saraf, and Sergey Yekhanin. High-rate codes with sublinear-time decoding. J. ACM, 61(5), September 2014. URL: https://doi.org/10.1145/2629416.
  39. Chaoyi Ma, Haibo Wang, Olufemi Odegbile, and Shigang Chen. Noise measurement and removal for data streaming algorithms with network applications. In 2021 IFIP Networking Conference (IFIP Networking), pages 1-9. IEEE, 2021. URL: https://doi.org/10.23919/IFIPNETWORKING52078.2021.9472845.
  40. Andrew McGregor, Atri Rudra, and Steve Uurtamo. Polynomial fitting of data streams with applications to codeword testing. Symposium on Theoretical Aspects of Computer Science (STACS2011), 9, March 2011. URL: https://doi.org/10.4230/LIPIcs.STACS.2011.428.
  41. Morteza Monemizadeh and David P Woodruff. 1-pass relative-error lp-sampling with applications. In Proceedings of the twenty-first annual ACM-SIAM symposium on Discrete Algorithms, pages 1143-1160. SIAM, 2010. URL: https://doi.org/10.1137/1.9781611973075.92.
  42. Robert Morris. Counting large numbers of events in small registers. Communications of the ACM, 21(10):840-842, 1978. URL: https://doi.org/10.1145/359619.359627.
  43. David E Muller. Application of boolean algebra to switching circuit design and to error detection. Transactions of the IRE professional group on electronic computers, 3:6-12, 1954. URL: https://doi.org/10.1109/IREPGELC.1954.6499441.
  44. N. Nisan. Pseudorandom generators for space-bounded computations. In Proceedings of the Twenty-Second Annual ACM Symposium on Theory of Computing, STOC '90, pages 204-212, New York, NY, USA, 1990. Association for Computing Machinery. URL: https://doi.org/10.1145/100216.100242.
  45. I. S. Reed and G. Solomon. Polynomial codes over certain finite fields. Journal of the Society for Industrial and Applied Mathematics, 8(2):300-304, 1960. URL: https://doi.org/10.1137/0108018.
  46. Irving S Reed. A class of multiple-error-correcting codes and the decoding scheme. IEEE Transactions on Information Theory, 4(4):38-49, 1954. URL: https://doi.org/10.1109/TIT.1954.1057465.
  47. Atri Rudra and Steve Uurtamo. Data stream algorithms for codeword testing. In International Colloquium on Automata, Languages, and Programming, pages 629-640. Springer, 2010. URL: https://doi.org/10.1007/978-3-642-14165-2_53.
  48. Leonard J. Schulman. Communication on noisy channels: a coding theorem for computation. In Proceedings., 33rd Annual Symposium on Foundations of Computer Science, pages 724-733, 1992. URL: https://doi.org/10.1109/SFCS.1992.267778.
  49. Leonard J. Schulman. Deterministic Coding for Interactive Communication. In Proceedings of the Twenty-Fifth Annual ACM Symposium on Theory of Computing, STOC '93, pages 747-756, New York, NY, USA, 1993. Association for Computing Machinery. URL: https://doi.org/10.1145/167088.167279.
  50. Leonard J. Schulman. Coding for interactive communication. IEEE Transactions on Information Theory, 42(6):1745-1756, 1996. URL: https://doi.org/10.1109/18.556671.
  51. Claude E. Shannon. A mathematical theory of communication. The Bell System Technical Journal, 27(3):379-423, 1948. URL: https://doi.org/10.1002/j.1538-7305.1948.tb01338.x.
  52. Sergey Yekhanin et al. Locally decodable codes. Foundations and Trendsregistered in Theoretical Computer Science, 6(3):139-255, 2012. URL: https://doi.org/10.1561/0400000030.
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail