Abstract
We generalize the deterministic simulation theorem of Raz & McKenzie (Combinatorica 19(3):403–435, 1999), to any gadget which satisfies a certain hitting property. We prove that inner product and gap-Hamming satisfy this property, and as a corollary, we obtain a deterministic simulation theorem for these gadgets, where the gadget’s input size is logarithmic in the input size of the outer function. This yields the first deterministic simulation theorem with a logarithmic gadget size, answering an open question posed by Göös, Pitassi & Watson (in: Proceedings of the 56th FOCS, 2015).
Our result also implies the previous results for the indexing gadget, with better parameters than was previously known. Moreover, a simulation theorem with logarithmic-sized gadget implies a quadratic separation in the deterministic communication complexity and the logarithm of the 1-partition number, no matter how high the 1-partition number is with respect to the input size—something which is not achievable by previous results of Göös, Pitassi & Watson (2015).
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Acknowledgements
Part of the research for this work was done at the Institut Henri Poincaré, as part of the workshop Nexus of Information and Computation Theories.
The research leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP/2007-2013)/ERC Grant Agreement n. 616787. The first author was partially supported by a Ramanujan Fellowship of the DST, India, and the last author is partially supported by a TCS fellowship and by European Research Council (ERC) under the European Unions Horizon 2020 research and innovation programme under grant agreement No. 715672.
The research leading to these results has also received funding from the Foundation for Science and Technology (FCT), Portugal, grant number SFRH/BPD/116010/2016. This work is partially funded by the ERDF through the COMPETE 2020 programme within project POCI-01-0145-FEDER-006961, and by National Funds through the FCT as part of project UID/EEA/50014/2013.
We are thankful to the anonymous referees for their thorough reading and numerous suggestions for improving the paper.
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Chattopadhyay, A., Koucký, M., Loff, B. et al. Simulation Theorems via Pseudo-random Properties. comput. complex. 28, 617–659 (2019). https://doi.org/10.1007/s00037-019-00190-7
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DOI: https://doi.org/10.1007/s00037-019-00190-7
Keywords
- Communication complexity
- lifting theorem
- simulation theorem
- Inner-product
- gap-Hamming
Subject classification
- Theory of computation
- Communication Complexity