Abstract
Henzinger et al. posed the so called Online Boolean Matrix-vector Multiplication (OMv) conjecture and showed that it implies tight hardness results for several basic partially dynamic or dynamic problems [STOC’15].
We show that the OMv conjecture is implied by a simple off-line conjecture. If a not uniform (i.e., it might be different for different matrices) polynomial-time preprocessing of the matrix in the OMv conjecture is allowed then we can show such a variant of the OMv conjecture to be equivalent to our off-line conjecture. On the other hand, we show that the OMV conjecture does not hold in the restricted cases when the rows of the matrix or the input vectors are clustered.
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References
Andoni, A., Indyk, P.: Nearest neighbours in high-dimensional spaces. In: Goodman, J.E., O’Orourke, J., Toth, C.D. (eds.) Handbook of Discrete and Computational Geometry, 3rd edn. CRC Press, Boca Raton (2017)
Bansal, N., Williams, R.: Regularity lemmas and combinatorial algorithms. Theory Comput. 8(1), 69–94 (2012)
Björklund, A., Lingas, A.: Fast Boolean matrix multiplication for highly clustered data. In: Dehne, F., Sack, J.-R., Tamassia, R. (eds.) WADS 2001. LNCS, vol. 2125, pp. 258–263. Springer, Heidelberg (2001). https://doi.org/10.1007/3-540-44634-6_24
Chakraborty, D., Kamma, L., Larsen, K.G.: Tight cell probe bounds for succinct Boolean matrix-vector multiplication. In: Proceedings of STOC 2018, pp. 1297–1306 (2018)
Floderus, P., Jansson, J., Levcopoulos, C., Lingas, A., Sledneu, D.: 3D rectangulations and geometric matrix multiplication. Algorithmica 80(1), 136–154 (2018)
Gąsieniec, L., Lingas, A.: An improved bound on Boolean matrix multiplication for highly clustered data. In: Dehne, F., Sack, J.-R., Smid, M. (eds.) WADS 2003. LNCS, vol. 2748, pp. 329–339. Springer, Heidelberg (2003). https://doi.org/10.1007/978-3-540-45078-8_29
Larsen, K.G., Williams, R.R.: Faster online matrix-vector multiplication. In: Proceedings of SODA 2017, pp. 2182–2189 (2017)
Henzinger, M., Krinninger, S., Nanongkai, D., Saranurak, T.: Unifying and strengthening hardness for dynamic problems via the online matrix-vector multiplication conjecture. In: Proceedings of STOC 2015 (also presented at HALG 2016)
Imase, M., Waxman, B.M.: Dynamic Steiner tree problem. SIAM J. Discrete Math. 4(3), 369–384 (1991)
Williams, R.: Matrix-vector multiplication in sub-quadratic time (some preprocessing required). In: Proceedings of SODA 2007, pp. 995–2001 (2007)
Acknowledgements
CL, JJ and MP were supported in part by Swedish Research Council grant 621-2017-03750.
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Gąsieniec, L., Jansson, J., Levcopoulos, C., Lingas, A., Persson, M. (2019). Pushing the Online Matrix-Vector Conjecture Off-Line and Identifying Its Easy Cases. In: Chen, Y., Deng, X., Lu, M. (eds) Frontiers in Algorithmics. FAW 2019. Lecture Notes in Computer Science(), vol 11458. Springer, Cham. https://doi.org/10.1007/978-3-030-18126-0_14
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DOI: https://doi.org/10.1007/978-3-030-18126-0_14
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