Online matrix-vector multiplication problem: Difference between revisions

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=== Algorithms for solving OMv ===

OMv can be solved in <math>O(n^3)</math> time by a naive algorithm that, in each of the <math>n</math> rounds, multiplies the matrix <math>M</math> and the new vector <math>v_i</math> in <math>O(n^2)</math> time. TheA fastest knownfaster algorithm for OMv is implied by a result of Williams and runs in time <math>O(n^3/\log^2 n)</math>.<ref>{{Cite journal |last=Williams |first=Ryan |date=2007-01-07 |title=Matrix-vector multiplication in sub-quadratic time: (some preprocessing required) |url=https://dl.acm.org/doi/10.5555/1283383.1283490 |journal=Proceedings of the ACM-SIAM Symposium on Discrete Algorithms |series=SODA '07 |___location=USA |pages=995–1001 |isbn=978-0-89871-624-5}}</ref> The fastest known algorithm for OMv runs in time <math>n^3/2^{\Omega{\sqrt{\log n}}}</math>, due to Larsen and Williams.<ref>{{Cite journal |last1=Larsen |first1=Kasper Green |last2=Williams |first2=Ryan |date=2017-01-16 |title=Faster online matrix-vector multiplication

|url=https://dl.acm.org/doi/10.5555/3039686.3039828 |journal=Proceedings of the ACM-SIAM Symposium on Discrete Algorithms |series=SODA '17 |___location=USA |pages=2182–2189 |isbn= 978-1-61197-478-2}}</ref>

=== Implications of conjectured hardness ===