Computing the Fréchet distance with shortcuts is NP-hard

We study the shortcut Fréchet distance, a natural variant of the Fréchet distance that allows us to take shortcuts from and to any point along one of the curves. We show that, surprisingly, the problem of computing the shortcut Fréchet distance exactly is NP-hard. Furthermore, we give a 3-approximation algorithm for the decision version of the problem

Tight Bounds for Approximate Near Neighbor Searching for Time Series under the {F}r\'{e}chet Distance

We study the $c$-approximate near neighbor problem under the continuous Fr\'echet distance: Given a set of $n$ polygonal curves with $m$ vertices, a radius $\delta > 0$, and a parameter $k \leq m$, we want to preprocess the curves into a data structure that, given a query curve $q$ with $k$ vertices, either returns an input curve with Fr\'echet distance at most $c\cdot \delta$ to $q$, or returns that there exists no input curve with Fr\'echet distance at most $\delta$ to $q$. We focus on the case where the input and the queries are one-dimensional polygonal curves -- also called time series -- and we give a comprehensive analysis for this case. We obtain new upper bounds that provide different tradeoffs between approximation factor, preprocessing time, and query time. Our data structures improve upon the state of the art in several ways. We show that for any $0 < \varepsilon \leq 1$ an approximation factor of $(1+\varepsilon)$ can be achieved within the same asymptotic time bounds as the previously best result for $(2+\varepsilon)$. Moreover, we show that an approximation factor of $(2+\varepsilon)$ can be obtained by using preprocessing time and space $O(nm)$, which is linear in the input size, and query time in $O(\frac{1}{\varepsilon})^{k+2}$, where the previously best result used preprocessing time in $n \cdot O(\frac{m}{\varepsilon k})^k$ and query time in $O(1)^k$. We complement our upper bounds with matching conditional lower bounds based on the Orthogonal Vectors Hypothesis. Interestingly, some of our lower bounds already hold for any super-constant value of $k$. This is achieved by proving hardness of a one-sided sparse version of the Orthogonal Vectors problem as an intermediate problem, which we believe to be of independent interest

Computing the Similarity Between Moving Curves

In this paper we study similarity measures for moving curves which can, for example, model changing coastlines or retreating glacier termini. Points on a moving curve have two parameters, namely the position along the curve as well as time. We therefore focus on similarity measures for surfaces, specifically the Fr\'echet distance between surfaces. While the Fr\'echet distance between surfaces is not even known to be computable, we show for variants arising in the context of moving curves that they are polynomial-time solvable or NP-complete depending on the restrictions imposed on how the moving curves are matched. We achieve the polynomial-time solutions by a novel approach for computing a surface in the so-called free-space diagram based on max-flow min-cut duality

Seabed images from Southern Ocean shelf regions off the northern Antarctic Peninsula and in the southeastern Weddell Sea

Recent advances in underwater imaging technology allow for the gathering of invaluable scientific information on seafloor ecosystems, such as direct in situ views of seabed habitats and quantitative data on the composition, diversity, abundance, and distribution of epibenthic fauna. The imaging approach has been extensively used within the research project DynAMo (Dynamics of Antarctic Marine Shelf Ecosystems) at the Alfred Wegener Institute, Helmholtz Centre for Polar and Marine Research Bremerhaven (AWI), which aimed to comparatively assess the pace and quality of the dynamics of Southern Ocean benthos. Within this framework, epibenthic spatial distribution patterns have been comparatively investigated in two regions in the Atlantic sector of the Southern Ocean: the shelf areas off the northern tip of the Antarctic Peninsula, representing a region with above-average warming of surface waters and sea-ice reduction, and the shelves of the eastern Weddell Sea as an example of a stable high-Antarctic marine environment that is not (yet) affected by climate change. The AWI Ocean Floor Observation System (OFOS) was used to collect seabed imagery during two cruises of the German research vessel Polarstern, ANT-XXIX/3 (PS81) to the Antarctic Peninsula from January to March 2013 and ANT-XXXI/2 (PS96) to the Weddell Sea from December 2015 to February 2016. Here, we report on the image and data collections gathered during these cruises. During PS81, OFOS was successfully deployed at a total of 31 stations at water depths between 29 and 784 m. At most stations, series of 500 to 530 pictures ( >  15 000 in total, each depicting a seabed area of approximately 3.45 m2 or 2.3  ×  1.5 m) were taken along transects approximately 3.7 km in length. During PS96, OFOS was used at a total of 13 stations at water depths between 200 and 754 m, yielding series of 110 to 293 photos (2670 in total) along transects 0.9 to 2.6 km in length. All seabed images taken during the two cruises, including metadata, are available from the data publisher PANGAEA via the two persistent identifiers at https://doi.org/10.1594/PANGAEA.872719 (for PS81) and https://doi.org/10.1594/PANGAEA.862097 (for PS96)

Computing the Fréchet Distance with a Retractable Leash

All known algorithms for the Fréchet distance between curves proceed in two steps: first, they construct an efficient oracle for the decision version; second, they use this oracle to find the optimum from a finite set of critical values. We present a novel approach that avoids the detour through the decision version. This gives the first quadratic time algorithm for the Fréchet distance between polygonal curves in (Formula presented.) under polyhedral distance functions (e.g., (Formula presented.) and (Formula presented.)). We also get a (Formula presented.)-approximation of the Fréchet distance under the Euclidean metric, in quadratic time for any fixed (Formula presented.). For the exact Euclidean case, our framework currently yields an algorithm with running time (Formula presented.). However, we conjecture that it may eventually lead to a faster exact algorithm

FRESH: Fréchet similarity with hashing

This paper studies the r-range search problem for curves under the continuous Fréchet distance: given a dataset S of n polygonal curves and a threshold >0 , construct a data structure that, for any query curve q, efficiently returns all entries in S with distance at most r from q. We propose FRESH, an approximate and randomized approach for r-range search, that leverages on a locality sensitive hashing scheme for detecting candidate near neighbors of the query curve, and on a subsequent pruning step based on a cascade of curve simplifications. We experimentally compare FRESH to exact and deterministic solutions, and we show that high performance can be reached by suitably relaxing precision and recall

Oral acantholytic squamous cell carcinoma shares clinical and histological features with angiosarcoma

<p>Abstract</p> <p>Background</p> <p>acantholytic squamous cell carcinomas (ASCC) and intraoral angiosarcoma share similar histopathological features. Aim of this study was to find marker for a clear distinction.</p> <p>Methods</p> <p>Four oral acantholytic squamous cell carcinomas and one intraoral angiosarcoma are used to compare the eruptive intraoral growth-pattern, age-peak, unfavourable prognosis and slit-like intratumorous spaces in common histological staining as identical clinical and histopathological features. Immunohistochemical staining for pancytokeratin, cytokeratin, collagen type IV, γ2-chain of laminin-5, endothelial differentiation marker CD31 and CD34, F VIII-associated antigen, Ki 67-antigen, β-catenin, E-cadherin, α-smooth-muscle-actin and Fli-1 were done.</p> <p>Results</p> <p>Cytokeratin-immunoreactive cells can be identified in both lesions. The large vascularization of ASCC complicates the interpretation of vascular differential markers being characteristic for angiosarcoma. Loss of cell-cell-adhesion, monitored by loss of E-cadherin and β-catenin membrane-staining, are indetified as reasons for massive expression of invasion-factor ln-5 in ASCC and considered responsible for unfavourable prognosis of ASCC. Expression of Fli-1 in angiosarcoma and cellular immunoreaction for ln-5 in ASCC are worked out as distinguishing features of both entities.</p> <p>Conclusion</p> <p>Fli-1 in angiosarcoma and ln-5 in ASCC are distinguishing features.</p