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 = ω(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. We prove that OVC is true in several computational models of interest: • For all sufficiently large n and d, OV for n vectors in {0, 1} d has branching program complexitỹ Θ(n · min(n, 2 d )). In particular, the lower bounds match the upper bounds up to polylog factors. wires, in formulas comprised of gates computing arbitrary symmetric functions of unbounded fan-in. 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. The 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: • For every fixed p ∈ (0, 1) there is an ε p > 0 such that for every n and d, OV instances where input bits are independently set to 1 with probability p (and 0 otherwise) can be solved with AC 0 formulas of size O(n 2−εp ), on all but a o n (1) fraction of instances. Moreover, ε p → 1 as p → 1. * Supported by an NSF CAREER.
Additionally, the OV conjecture was studied in restricted models, such as branching programs REF .
The Orthogonal Vectors Conjecture for Branching Programs and Formulas
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Mathematics Computer Science