44 research outputs found
Fréchet Distance for Uncertain Curves
In this article, we study a wide range of variants for computing the (discrete and continuous) Fréchet distance between uncertain curves. An uncertain curve is a sequence of uncertainty regions, where each region is a disk, a line segment, or a set of points. A realisation of a curve is a polyline connecting one point from each region. Given an uncertain curve and a second (certain or uncertain) curve, we seek to compute the lower and upper bound Fréchet distance, which are the minimum and maximum Fréchet distance for any realisations of the curves. We prove that both problems are NP-hard for the Fréchet distance in several uncertainty models, and that the upper bound problem remains hard for the discrete Fréchet distance. In contrast, the lower bound (discrete [5] and continuous) Fréchet distance can be computed in polynomial time in some models. Furthermore, we show that computing the expected (discrete and continuous) Fréchet distance is #P-hard in some models.On the positive side, we present an FPTAS in constant dimension for the lower bound problem when Δ/δis polynomially bounded, where δis the Fréchet distance and Δbounds the diameter of the regions. We also show a near-linear-time 3-approximation for the decision problem on roughly δ-separated convex regions. Finally, we study the setting with Sakoe-Chiba time bands, where we restrict the alignment between the curves, and give polynomial-time algorithms for the upper bound and expected discrete and continuous Fréchet distance for uncertainty modelled as point sets.</p
Realizability of Free Spaces of Curves
The free space diagram is a popular tool to compute the well-known Fr\'echet
distance. As the Fr\'echet distance is used in many different fields, many
variants have been established to cover the specific needs of these
applications. Often, the question arises whether a certain pattern in the free
space diagram is "realizable", i.e., whether there exists a pair of polygonal
chains whose free space diagram corresponds to it. The answer to this question
may help in deciding the computational complexity of these distance measures,
as well as allowing to design more efficient algorithms for restricted input
classes that avoid certain free space patterns. Therefore, we study the inverse
problem: Given a potential free space diagram, do there exist curves that
generate this diagram?
Our problem of interest is closely tied to the classic Distance Geometry
problem. We settle the complexity of Distance Geometry in ,
showing -hardness. We use this to show that for curves in
, the realizability problem is
-complete, both for continuous and for discrete Fr\'echet
distance. We prove that the continuous case in is only weakly
NP-hard, and we provide a pseudo-polynomial time algorithm and show that it is
fixed-parameter tractable. Interestingly, for the discrete case in
, we show that the problem becomes solvable in polynomial time.Comment: 26 pages, 12 figures, 1 table, International Symposium on Algorithms
And Computations (ISAAC 2023
Fr\'echet Distance for Uncertain Curves
In this paper we study a wide range of variants for computing the (discrete
and continuous) Fr\'echet distance between uncertain curves. We define an
uncertain curve as a sequence of uncertainty regions, where each region is a
disk, a line segment, or a set of points. A realisation of a curve is a
polyline connecting one point from each region. Given an uncertain curve and a
second (certain or uncertain) curve, we seek to compute the lower and upper
bound Fr\'echet distance, which are the minimum and maximum Fr\'echet distance
for any realisations of the curves.
We prove that both the upper and lower bound problems are NP-hard for the
continuous Fr\'echet distance in several uncertainty models, and that the upper
bound problem remains hard for the discrete Fr\'echet distance. In contrast,
the lower bound (discrete and continuous) Fr\'echet distance can be computed in
polynomial time. Furthermore, we show that computing the expected discrete
Fr\'echet distance is #P-hard when the uncertainty regions are modelled as
point sets or line segments. The construction also extends to show #P-hardness
for computing the continuous Fr\'echet distance when regions are modelled as
point sets.
On the positive side, we argue that in any constant dimension there is a
FPTAS for the lower bound problem when is polynomially
bounded, where is the Fr\'echet distance and bounds the
diameter of the regions. We then argue there is a near-linear-time
3-approximation for the decision problem when the regions are convex and
roughly -separated. Finally, we also study the setting with
Sakoe--Chiba time bands, where we restrict the alignment between the two
curves, and give polynomial-time algorithms for upper bound and expected
discrete and continuous Fr\'echet distance for uncertainty regions modelled as
point sets.Comment: 48 pages, 11 figures. This is the full version of the paper to be
published in ICALP 202
LIPIcs, Volume 258, SoCG 2023, Complete Volume
LIPIcs, Volume 258, SoCG 2023, Complete Volum
Fine-Grained Complexity Analysis of Two Classic TSP Variants
We analyze two classic variants of the Traveling Salesman Problem using the
toolkit of fine-grained complexity. Our first set of results is motivated by
the Bitonic TSP problem: given a set of points in the plane, compute a
shortest tour consisting of two monotone chains. It is a classic
dynamic-programming exercise to solve this problem in time. While the
near-quadratic dependency of similar dynamic programs for Longest Common
Subsequence and Discrete Frechet Distance has recently been proven to be
essentially optimal under the Strong Exponential Time Hypothesis, we show that
bitonic tours can be found in subquadratic time. More precisely, we present an
algorithm that solves bitonic TSP in time and its bottleneck
version in time. Our second set of results concerns the popular
-OPT heuristic for TSP in the graph setting. More precisely, we study the
-OPT decision problem, which asks whether a given tour can be improved by a
-OPT move that replaces edges in the tour by new edges. A simple
algorithm solves -OPT in time for fixed . For 2-OPT, this is
easily seen to be optimal. For we prove that an algorithm with a runtime
of the form exists if and only if All-Pairs
Shortest Paths in weighted digraphs has such an algorithm. The results for
may suggest that the actual time complexity of -OPT is
. We show that this is not the case, by presenting an algorithm
that finds the best -move in time for
fixed . This implies that 4-OPT can be solved in time,
matching the best-known algorithm for 3-OPT. Finally, we show how to beat the
quadratic barrier for in two important settings, namely for points in the
plane and when we want to solve 2-OPT repeatedly.Comment: Extended abstract appears in the Proceedings of the 43rd
International Colloquium on Automata, Languages, and Programming (ICALP 2016