Conditional Indexing Lower Bounds Through Self-Reducibility
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We provide a general technique to turn a conditional lower bound result based on the Orthogonal Vectors Hypothesis (OVH) into one for the indexed version of the same problem. This consists of two results that hold under OVH: (i) the Orthogonal Vectors (OV) problem cannot be indexed in polynomial time to support sub-quadratic time queries, and (ii) any problem to which OV reduces with a natural property also cannot be indexed in polynomial time to support fast queries. We demonstrate the power of this technique on the problem of exact string matching on graphs, deriving the first, and tight, conditional lower bound for its indexed version. The result has an interesting corollary for automata theory: Unless OVH is false, there is no polynomial determinisation algorithm of a finite automaton, even for an acyclic non-deterministic automata whose only non-deterministic transitions are from the start state. For another example of the use of the technique, we apply it to the reduction by Backurs and Indyk (STOC 2015) on edit distance. This results into the first tight conditional indexing lower bound for approximate string matching. This strengthens the recent tailored reduction by Cohen-Addad, Feuilloley and Starikovskaya (SODA 2019), but with a slightly different boundary condition.
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