[Paper Review] Conditional Indexing Lower Bounds Through Self-Reducibility

Massimo Equi, Veli Mäkinen+1 more|arXiv (Cornell University)|Feb 3, 2020

semigroups and automata theory4 citations

TL;DR

This paper introduces a general technique to derive conditional lower bounds for indexed versions of problems based on the Orthogonal Vectors Hypothesis (OVH). It proves that orthogonal vectors and problems reducible to it cannot be indexed in polynomial time to support sub-quadratic query time, establishing the first tight conditional lower bound for indexed exact string matching on graphs and implying no polynomial-time determinization for certain acyclic NFAs.

ABSTRACT

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.

Motivation & Objective

• To establish conditional lower bounds for the indexed version of the Orthogonal Vectors (OV) problem under the Orthogonal Vectors Hypothesis (OVH).
• To extend these lower bounds to problems that reduce to OV with a natural reduction property.
• To apply the technique to derive the first tight conditional lower bound for indexed exact string matching on graphs.
• To demonstrate implications for automata theory, particularly on the existence of polynomial-time determinization for acyclic NFAs with limited non-determinism.
• To strengthen prior results on approximate string matching by deriving the first tight conditional indexing lower bound for edit distance via a generalized reduction.

Proposed method

• Leverages self-reducibility to transform conditional lower bounds from standard problems into indexed problem variants.
• Applies the OVH to show that no polynomial-time indexing structure can support sub-quadratic query time for the OV problem.
• Uses a natural reduction property to extend the lower bound to any problem that reduces to OV under this condition.
• Applies the framework to exact string matching on graphs, deriving a tight indexing lower bound.
• Transfers the result to automata theory, showing no polynomial determinization exists for acyclic NFAs with only start-state non-determinism.
• Extends the technique to approximate string matching via the Backurs-Indyk reduction, yielding a tight lower bound for edit distance indexing.

Experimental results

Research questions

• RQ1 Can conditional lower bounds for the standard orthogonal vectors problem be transferred to its indexed variant under OVH?
• RQ2 What are the implications of such indexing lower bounds for exact string matching on graphs?
• RQ3 Is there a polynomial-time determinization algorithm for acyclic non-deterministic finite automata with only start-state non-determinism, assuming OVH?
• RQ4 Can the technique be applied to derive tight lower bounds for approximate string matching via edit distance?
• RQ5 How does this framework compare to prior tailored reductions in the context of edit distance indexing?

Key findings

• The indexed version of the Orthogonal Vectors problem cannot be solved with polynomial preprocessing and sub-quadratic query time under OVH.
• Any problem that reduces to OV via a natural reduction also inherits this indexing lower bound.
• The first tight conditional lower bound for indexed exact string matching on graphs is established.
• Unless OVH is false, no polynomial-time determinization algorithm exists for acyclic NFAs whose only non-deterministic transitions originate from the start state.
• The technique yields the first tight conditional lower bound for indexing approximate string matching via edit distance, improving upon prior tailored reductions.
• The framework generalizes and strengthens previous results by Cohen-Addad, Feuilloley, and Starikovskaya, with a slightly different boundary condition.