Optimal error resilience of adaptive message exchange

We study the error resilience of the message exchange task: Two parties, each holding a private input, want to exchange their inputs. However, the channel connecting them is governed by an adversary that may corrupt a constant fraction of the transmissions. What is the maximum fraction of corruptions that still allows the parties to exchange their inputs? For the non-adaptive channel, where the parties must agree in advance on the order in which they communicate, the maximum error resilience was shown to be 1/4 (see Braverman and Rao, STOC 2011). The problem was also studied over the adaptive channel, where the order in which the parties communicate may not be predetermined (Ghaffari, Haeupler, and Sudan, STOC 2014; Efremenko, Kol, and Saxena, STOC 2020). These works show that the adaptive channel admits much richer set of protocols but leave open the question of finding its maximum error resilience. In this work, we show that the maximum error resilience of a protocol for message exchange over the adaptive channel is 5/16, thereby settling the above question. Our result requires improving both the known upper bounds and the known lower bounds for the problem.

[1]  Amiram H. Kaspi,et al.  Two-way source coding with a fidelity criterion , 1985, IEEE Trans. Inf. Theory.

[2]  Mark Braverman,et al.  Towards deterministic tree code constructions , 2012, ITCS '12.

[3]  Bernhard Haeupler,et al.  Interactive Channel Capacity Revisited , 2014, 2014 IEEE 55th Annual Symposium on Foundations of Computer Science.

[4]  Mark Braverman,et al.  Coding for Interactive Communication Correcting Insertions and Deletions , 2017, IEEE Transactions on Information Theory.

[5]  Amit Sahai,et al.  Efficient and Explicit Coding for Interactive Communication , 2011, 2011 IEEE 52nd Annual Symposium on Foundations of Computer Science.

[6]  Mark Braverman,et al.  Toward Coding for Maximum Errors in Interactive Communication , 2011, IEEE Transactions on Information Theory.

[7]  Prakash Ishwar,et al.  The Infinite-Message Limit of Two-Terminal Interactive Source Coding , 2009, IEEE Transactions on Information Theory.

[8]  Bernhard Haeupler,et al.  Optimal Error Rates for Interactive Coding II: Efficiency and List Decoding , 2013, 2014 IEEE 55th Annual Symposium on Foundations of Computer Science.

[9]  Klim Efremenko,et al.  Interactive error resilience beyond 2/7 , 2020, Electron. Colloquium Comput. Complex..

[10]  Klim Efremenko,et al.  Interactive coding over the noisy broadcast channel , 2018, Electron. Colloquium Comput. Complex..

[11]  Klim Efremenko,et al.  Maximal Noise in Interactive Communication Over Erasure Channels and Channels With Feedback , 2015, IEEE Transactions on Information Theory.

[12]  Ran Raz,et al.  Interactive channel capacity , 2013, STOC '13.

[13]  Amit Sahai,et al.  Adaptive protocols for interactive communication , 2013, 2016 IEEE International Symposium on Information Theory (ISIT).

[14]  Rafail Ostrovsky,et al.  Optimal Coding for Streaming Authentication and Interactive Communication , 2015, IEEE Transactions on Information Theory.

[15]  Ran Gelles,et al.  Coding for Interactive Communication: A Survey , 2017, Found. Trends Theor. Comput. Sci..

[16]  Alexander A. Sherstov,et al.  Optimal Interactive Coding for Insertions, Deletions, and Substitutions , 2017, 2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS).

[17]  Avi Wigderson,et al.  Explicit Capacity Approaching Coding for Interactive Communication , 2018, IEEE Transactions on Information Theory.

[18]  Gang Wang,et al.  Communication with partial noisy feedback , 2017, 2017 IEEE Symposium on Computers and Communications (ISCC).

[19]  Leonard J. Schulman.  Coding for interactive communication , 1996, IEEE Trans. Inf. Theory.

[20]  Ameya Velingker,et al.  Communication with Partial Noiseless Feedback , 2015, APPROX-RANDOM.

[21]  E. Berlekamp.  Block coding with noiseless feedback , 1964 .

[22]  Ran Gelles,et al.  Capacity of Interactive Communication over Erasure Channels and Channels with Feedback , 2015, SIAM J. Comput..

[23]  Leonard J. Schulman,et al.  Communication on noisy channels: a coding theorem for computation , 1992, Proceedings., 33rd Annual Symposium on Foundations of Computer Science.

[24]  Leonard J. Schulman,et al.  Deterministic coding for interactive communication , 1993, STOC.

[25]  Amit Sahai,et al.  Efficient Coding for Interactive Communication , 2014, IEEE Transactions on Information Theory.

[26]  Ameya Velingker,et al.  Bridging the Capacity Gap Between Interactive and One-Way Communication , 2017, SODA.

[27]  Bernhard Haeupler,et al.  Synchronization Strings: Channel Simulations and Interactive Coding for Insertions and Deletions , 2017, ICALP.

[28]  Madhu Sudan,et al.  Optimal error rates for interactive coding I: adaptivity and other settings , 2013, STOC.

[29]  Denis Pankratov,et al.  On the Power of Feedback in Interactive Channels , 2013 .

[30]  Yael Tauman Kalai,et al.  Fast Interactive Coding against Adversarial Noise , 2014, JACM.

[31]  Mark Braverman,et al.  List and Unique Coding for Interactive Communication in the Presence of Adversarial Noise , 2014, FOCS.

[32]  Imrich Chlamtac,et al.  On Broadcasting in Radio Networks - Problem Analysis and Protocol Design , 1985, IEEE Transactions on Communications.