{"@context":"https:\/\/schema.org\/","@type":"ScholarlyArticle","@id":"#article11975","name":"The Orthogonal Vectors Conjecture for Branching Programs and Formulas","abstract":"In the Orthogonal Vectors (OV) problem, we wish to determine if there is an orthogonal pair of vectors among n Boolean vectors in d dimensions. The OV Conjecture (OVC) posits that OV requires n^{2-o(1)} time to solve, for all d=omega(log n). Assuming the OVC, optimal time lower bounds have been proved for many prominent problems in P, such as Edit Distance, Frechet Distance, Longest Common Subsequence, and approximating the diameter of a graph.\nWe prove that OVC is true in several computational models of interest: \n- For all sufficiently large n and d, OV for n vectors in {0,1}^d has branching program complexity Theta~(n * min(n,2^d)). In particular, the lower and upper bounds match up to polylog factors.\n- OV has Boolean formula complexity Theta~(n * min(n,2^d)), over all complete bases of O(1) fan-in.\n- OV requires Theta~(n * min(n,2^d)) wires, in formulas comprised of gates computing arbitrary symmetric functions of unbounded fan-in.\n Our lower bounds basically match the best known (quadratic) lower bounds for any explicit function in those models. Analogous lower bounds hold for many related problems shown to be hard under OVC, such as Batch Partial Match, Batch Subset Queries, and Batch Hamming Nearest Neighbors, all of which have very succinct reductions to OV.\nThe proofs use a certain kind of input restriction that is different from typical random restrictions where variables are assigned independently. We give a sense in which independent random restrictions cannot be used to show hardness, in that OVC is false in the \"average case\" even for AC^0 formulas: \nFor all p in (0,1) there is a delta_p > 0 such that for every n and d, OV instances with input bits independently set to 1 with probability p (and 0 otherwise) can be solved with AC^0 formulas of O(n^{2-delta_p}) size, on all but a o_n(1) fraction of instances. Moreover, lim_{p - > 1}delta_p = 1.","keywords":["fine-grained complexity","orthogonal vectors","branching programs","symmetric functions","Boolean formulas"],"author":[{"@type":"Person","name":"Kane, Daniel M.","givenName":"Daniel M.","familyName":"Kane","affiliation":"CSE and Mathematics, UC San Diego, La Jolla CA, USA","funding":"Supported by NSF Award CCF-1553288 (CAREER) and a Sloan Research Fellowship."},{"@type":"Person","name":"Williams, Richard Ryan","givenName":"Richard Ryan","familyName":"Williams","sameAs":"https:\/\/orcid.org\/0000-0003-2326-2233","affiliation":"EECS and CSAIL, MIT, 32 Vassar St., Cambridge MA, USA","funding":"Supported by NSF CCF-1741615 (CAREER: Common Links in Algorithms and Complexity). Any opinions, findings and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation."}],"position":48,"pageStart":"48:1","pageEnd":"48:15","dateCreated":"2019-01-08","datePublished":"2019-01-08","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"Kane, Daniel M.","givenName":"Daniel M.","familyName":"Kane","affiliation":"CSE and Mathematics, UC San Diego, La Jolla CA, USA","funding":"Supported by NSF Award CCF-1553288 (CAREER) and a Sloan Research Fellowship."},{"@type":"Person","name":"Williams, Richard Ryan","givenName":"Richard Ryan","familyName":"Williams","sameAs":"https:\/\/orcid.org\/0000-0003-2326-2233","affiliation":"EECS and CSAIL, MIT, 32 Vassar St., Cambridge MA, USA","funding":"Supported by NSF CCF-1741615 (CAREER: Common Links in Algorithms and Complexity). Any opinions, findings and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation."}],"copyrightYear":"2019","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/LIPIcs.ITCS.2019.48","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","citation":["https:\/\/cstheory.stackexchange.com\/questions\/37361\/pairwise-comparison-of-bit-vectors","http:\/\/eprint.iacr.org\/2017\/202","http:\/\/dx.doi.org\/10.1137\/0206054","http:\/\/dx.doi.org\/10.1137\/1.9781611974331.ch87","http:\/\/arxiv.org\/abs\/1609.08403","http:\/\/arxiv.org\/abs\/1703.00941","http:\/\/dx.doi.org\/10.1145\/322033.322037","http:\/\/michaelwehar.com\/documents\/TreeShaped.pdf","http:\/\/dx.doi.org\/10.4230\/LIPIcs.CCC.2016.2","http:\/\/dx.doi.org\/10.1137\/1.9781611973402.135"],"isPartOf":{"@type":"PublicationVolume","@id":"#volume6327","volumeNumber":124,"name":"10th Innovations in Theoretical Computer Science Conference (ITCS 2019)","dateCreated":"2019-01-08","datePublished":"2019-01-08","editor":{"@type":"Person","name":"Blum, Avrim","givenName":"Avrim","familyName":"Blum"},"isAccessibleForFree":true,"publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","hasPart":"#article11975","isPartOf":{"@type":"Periodical","@id":"#series116","name":"Leibniz International Proceedings in Informatics","issn":"1868-8969","isAccessibleForFree":true,"publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","hasPart":"#volume6327"}}}