198 research outputs found

### Why walking the dog takes time: Frechet distance has no strongly subquadratic algorithms unless SETH fails

The Frechet distance is a well-studied and very popular measure of similarity
of two curves. Many variants and extensions have been studied since Alt and
Godau introduced this measure to computational geometry in 1991. Their original
algorithm to compute the Frechet distance of two polygonal curves with n
vertices has a runtime of O(n^2 log n). More than 20 years later, the state of
the art algorithms for most variants still take time more than O(n^2 / log n),
but no matching lower bounds are known, not even under reasonable complexity
theoretic assumptions.
To obtain a conditional lower bound, in this paper we assume the Strong
Exponential Time Hypothesis or, more precisely, that there is no
O*((2-delta)^N) algorithm for CNF-SAT for any delta > 0. Under this assumption
we show that the Frechet distance cannot be computed in strongly subquadratic
time, i.e., in time O(n^{2-delta}) for any delta > 0. This means that finding
faster algorithms for the Frechet distance is as hard as finding faster CNF-SAT
algorithms, and the existence of a strongly subquadratic algorithm can be
considered unlikely.
Our result holds for both the continuous and the discrete Frechet distance.
We extend the main result in various directions. Based on the same assumption
we (1) show non-existence of a strongly subquadratic 1.001-approximation, (2)
present tight lower bounds in case the numbers of vertices of the two curves
are imbalanced, and (3) examine realistic input assumptions (c-packed curves)

### Improved approximation for Fr\'echet distance on c-packed curves matching conditional lower bounds

The Fr\'echet distance is a well-studied and very popular measure of
similarity of two curves. The best known algorithms have quadratic time
complexity, which has recently been shown to be optimal assuming the Strong
Exponential Time Hypothesis (SETH) [Bringmann FOCS'14].
To overcome the worst-case quadratic time barrier, restricted classes of
curves have been studied that attempt to capture realistic input curves. The
most popular such class are c-packed curves, for which the Fr\'echet distance
has a $(1+\epsilon)$-approximation in time $\tilde{O}(c n /\epsilon)$ [Driemel
et al. DCG'12]. In dimension $d \ge 5$ this cannot be improved to
$O((cn/\sqrt{\epsilon})^{1-\delta})$ for any $\delta > 0$ unless SETH fails
[Bringmann FOCS'14].
In this paper, exploiting properties that prevent stronger lower bounds, we
present an improved algorithm with runtime $\tilde{O}(cn/\sqrt{\epsilon})$.
This is optimal in high dimensions apart from lower order factors unless SETH
fails. Our main new ingredients are as follows: For filling the classical
free-space diagram we project short subcurves onto a line, which yields
one-dimensional separated curves with roughly the same pairwise distances
between vertices. Then we tackle this special case in near-linear time by
carefully extending a greedy algorithm for the Fr\'echet distance of
one-dimensional separated curves

### A Note on Hardness of Diameter Approximation

We revisit the hardness of approximating the diameter of a network. In the
CONGEST model of distributed computing, $\tilde \Omega (n)$ rounds are
necessary to compute the diameter [Frischknecht et al. SODA'12], where $\tilde
\Omega (\cdot)$ hides polylogarithmic factors. Abboud et al. [DISC 2016]
extended this result to sparse graphs and, at a more fine-grained level, showed
that, for any integer $1 \leq \ell \leq \operatorname{polylog} (n)$,
distinguishing between networks of diameter $4 \ell + 2$ and $6 \ell + 1$
requires $\tilde \Omega (n)$ rounds. We slightly tighten this result by
showing that even distinguishing between diameter $2 \ell + 1$ and $3 \ell +
1$ requires $\tilde \Omega (n)$ rounds. The reduction of Abboud et al. is
inspired by recent conditional lower bounds in the RAM model, where the
orthogonal vectors problem plays a pivotal role. In our new lower bound, we
make the connection to orthogonal vectors explicit, leading to a conceptually
more streamlined exposition.Comment: Accepted to Information Processing Letter

### False theta functions and companions to Capparelli's identities

Capparelli conjectured two modular identities for partitions whose parts
satisfy certain gap conditions, where were motivated by the calculation of
characters for the standard modules of certain affine Lie algebras and by
vertex operator theory. These identities were subsequently proved and refined
by Andrews, who related them to Jacobi theta functions, and also by
Alladi-Andrews-Gordon, Capparelli, and Tamba-Xie. In this paper we prove two
new companions to Capparelli's identities, where the evaluations are expressed
in terms of Jacobi theta functions and false theta functions.Comment: 17 pages; references update

### Multivariate Fine-Grained Complexity of Longest Common Subsequence

We revisit the classic combinatorial pattern matching problem of finding a
longest common subsequence (LCS). For strings $x$ and $y$ of length $n$, a
textbook algorithm solves LCS in time $O(n^2)$, but although much effort has
been spent, no $O(n^{2-\varepsilon})$-time algorithm is known. Recent work
indeed shows that such an algorithm would refute the Strong Exponential Time
Hypothesis (SETH) [Abboud, Backurs, Vassilevska Williams + Bringmann,
K\"unnemann FOCS'15].
Despite the quadratic-time barrier, for over 40 years an enduring scientific
interest continued to produce fast algorithms for LCS and its variations.
Particular attention was put into identifying and exploiting input parameters
that yield strongly subquadratic time algorithms for special cases of interest,
e.g., differential file comparison. This line of research was successfully
pursued until 1990, at which time significant improvements came to a halt. In
this paper, using the lens of fine-grained complexity, our goal is to (1)
justify the lack of further improvements and (2) determine whether some special
cases of LCS admit faster algorithms than currently known.
To this end, we provide a systematic study of the multivariate complexity of
LCS, taking into account all parameters previously discussed in the literature:
the input size $n:=\max\{|x|,|y|\}$, the length of the shorter string
$m:=\min\{|x|,|y|\}$, the length $L$ of an LCS of $x$ and $y$, the numbers of
deletions $\delta := m-L$ and $\Delta := n-L$, the alphabet size, as well as
the numbers of matching pairs $M$ and dominant pairs $d$. For any class of
instances defined by fixing each parameter individually to a polynomial in
terms of the input size, we prove a SETH-based lower bound matching one of
three known algorithms. Specifically, we determine the optimal running time for
LCS under SETH as $(n+\min\{d, \delta \Delta, \delta m\})^{1\pm o(1)}$.
[...]Comment: Presented at SODA'18. Full Version. 66 page

### Remarks on Category-Based Routing in Social Networks

It is well known that individuals can route messages on short paths through
social networks, given only simple information about the target and using only
local knowledge about the topology. Sociologists conjecture that people find
routes greedily by passing the message to an acquaintance that has more in
common with the target than themselves, e.g. if a dentist in Saarbr\"ucken
wants to send a message to a specific lawyer in Munich, he may forward it to
someone who is a lawyer and/or lives in Munich. Modelling this setting,
Eppstein et al. introduced the notion of category-based routing. The goal is to
assign a set of categories to each node of a graph such that greedy routing is
possible. By proving bounds on the number of categories a node has to be in we
can argue about the plausibility of the underlying sociological model. In this
paper we substantially improve the upper bounds introduced by Eppstein et al.
and prove new lower bounds.Comment: 21 page

### Double series representations for Schur's partition function and related identities

We prove new double summation hypergeometric $q$-series representations for
several families of partitions, including those that appear in the famous
product identities of G\"ollnitz, Gordon, and Schur. We give several different
proofs for our results, using bijective partitions mappings and modular
diagrams, the theory of $q$-difference equations and recurrences, and the
theories of summation and transformation for $q$-series. We also consider a
general family of similar double series and highlight a number of other
interesting special cases.Comment: 19 page

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