27 research outputs found
Fine-Grained Complexity Theory: Conditional Lower Bounds for Computational Geometry
Fine-grained complexity theory is the area of theoretical computer sciencethat proves conditional lower bounds based on the Strong Exponential TimeHypothesis and similar conjectures. This area has been thriving in the lastdecade, leading to conditionally best-possible algorithms for a wide variety ofproblems on graphs, strings, numbers etc. This article is an introduction tofine-grained lower bounds in computational geometry, with a focus on lowerbounds for polynomial-time problems based on the Orthogonal Vectors Hypothesis.Specifically, we discuss conditional lower bounds for nearest neighbor searchunder the Euclidean distance and Fr\'echet distance.<br
Superlinear lower bounds based on ETH
Andras Z. Salamon acknowledges support from EPSRC grants EP/P015638/1 and EP/V027182/1.We introduce techniques for proving superlinear conditional lower bounds for polynomial time problems. In particular, we show that CircuitSAT for circuits with m gates and log(m) inputs (denoted by log-CircuitSAT) is not decidable in essentially-linear time unless the exponential time hypothesis (ETH) is false and k-Clique is decidable in essentially-linear time in terms of the graph's size for all fixed k. Such conditional lower bounds have previously only been demonstrated relative to the strong exponential time hypothesis (SETH). Our results therefore offer significant progress towards proving unconditional s uperlinear time complexity lower bounds for natural problems in polynomial time.Postprin
Reconstructing Words from Right-Bounded-Block Words
A reconstruction problem of words from scattered factors asks for the minimal
information, like multisets of scattered factors of a given length or the
number of occurrences of scattered factors from a given set, necessary to
uniquely determine a word. We show that a word can be
reconstructed from the number of occurrences of at most
scattered factors of the form . Moreover, we generalize the result to
alphabets of the form by showing that at most scattered factors suffices to reconstruct .
Both results improve on the upper bounds known so far. Complexity time bounds
on reconstruction algorithms are also considered here
Fine-Grained Complexity of Regular Path Queries
A regular path query (RPQ) is a regular expression q that returns all node pairs (u, v) from a graph database that are connected by an arbitrary path labelled with a word from L(q). The obvious algorithmic approach to RPQ evaluation (called PG-approach), i. e., constructing the product graph between an NFA for q and the graph database, is appealing due to its simplicity and also leads to efficient algorithms. However, it is unclear whether the PG-approach is optimal. We address this question by thoroughly investigating which upper complexity bounds can be achieved by the PG-approach, and we complement these with conditional lower bounds (in the sense of the fine-grained complexity framework). A special focus is put on enumeration and delay bounds, as well as the data complexity perspective. A main insight is that we can achieve optimal (or near optimal) algorithms with the PG-approach, but the delay for enumeration is rather high (linear in the database). We explore three successful approaches towards enumeration with sub-linear delay: super-linear preprocessing, approximations of the solution sets, and restricted classes of RPQs
Combinatorial Algorithms for Subsequence Matching: A Survey
In this paper we provide an overview of a series of recent results regarding
algorithms for searching for subsequences in words or for the analysis of the
sets of subsequences occurring in a word.Comment: This is a revised version of the paper with the same title which
appeared in the Proceedings of NCMA 2022, EPTCS 367, 2022, pp. 11-27 (DOI:
10.4204/EPTCS.367.2). The revision consists in citing a series of relevant
references which were not covered in the initial version, and commenting on
how they relate to the results we survey. arXiv admin note: text overlap with
arXiv:2206.1389
Matching Patterns with Variables Under Edit Distance
A pattern is a string of variables and terminal letters. We say that
matches a word , consisting only of terminal letters, if can be
obtained by replacing the variables of by terminal words. The matching
problem, i.e., deciding whether a given pattern matches a given word, was
heavily investigated: it is NP-complete in general, but can be solved
efficiently for classes of patterns with restricted structure. If we are
interested in what is the minimum Hamming distance between and any word
obtained by replacing the variables of by terminal words (so matching
under Hamming distance), one can devise efficient algorithms and matching
conditional lower bounds for the class of regular patterns (in which no
variable occurs twice), as well as for classes of patterns where we allow
unbounded repetitions of variables, but restrict the structure of the pattern,
i.e., the way the occurrences of different variables can be interleaved.
Moreover, under Hamming distance, if a variable occurs more than once and its
occurrences can be interleaved arbitrarily with those of other variables, even
if each of these occurs just once, the matching problem is intractable. In this
paper, we consider the problem of matching patterns with variables under edit
distance. We still obtain efficient algorithms and matching conditional lower
bounds for the class of regular patterns, but show that the problem becomes, in
this case, intractable already for unary patterns, consisting of repeated
occurrences of a single variable interleaved with terminals