Abstract
We develop a new lower bound method for analysing the complexity of the Equality function (EQ) in the Simultaneous Message Passing (SMP) model of communication complexity. The new technique gives tight lower bounds of \(\varOmega {\left( \sqrt{n}\right) }\) for both EQ and its negation NE in the non-deterministic version of quantum-classical SMP, where Merlin is also quantum – this is the strongest known version of SMP where the complexity of both EQ and NE remain high (previously known techniques seem to be insufficient for this).
Besides, our analysis provides to a unified view of the communication complexity of EQ and NE, allowing to obtain tight characterisation in all previously studied and a few newly introduced versions of SMP, including all possible combination of either quantum or randomised Alice, Bob and Merlin in the non-deterministic case.
Some of our results highlight that NE is easier than EQ in the presence of classical proofs, whereas the problems have (roughly) the same complexity when a quantum proof is present.
D. Gavinsky—Partially funded by the grant P202/12/G061 of GA ČR and by RVO: 67985840.
H. Klauck—This work is funded by the Singapore Ministry of Education (partly through the Academic Research Fund Tier 3 MOE2012-T3-1-009) and by the Singapore National Research Foundation.
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Notes
- 1.
The communication complexity of EQ becomes n in the most of deterministic models, but those results are usually trivial and we do not consider the deterministic setup in this work (except for one special case where a “semi-deterministic” protocol has complexity \(O{\left( \sqrt{n}\right) }\) – that situation will be analysed in one of our lower bound proofs).
- 2.
Note that Merlin’s message is only seen by the referee, and not by Alice and Bob. Letting the players receive messages from Merlin prior to sending their own messages would contradict the “simultaneous flavour” of the SMP model. Practically, that would make NE trivial; while the case of EQ is less obvious, we believe that the techniques developed in this work would be useful there as well.
- 3.
The best lower bound that we were able to prove using combinations of known techniques is \(\tilde{\varOmega }(n^{1/3})\).
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Bottesch, R., Gavinsky, D., Klauck, H. (2015). Equality, Revisited. In: Italiano, G., Pighizzini, G., Sannella, D. (eds) Mathematical Foundations of Computer Science 2015. MFCS 2015. Lecture Notes in Computer Science(), vol 9235. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-48054-0_11
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