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Approximate Shortest Path Queries in Graphs Using Voronoi Duals

Shinichi Honiden, Michael E. Houle, Christian Sommer, Martin Wolff
<span title="">2009</span> <i title="IEEE"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/u2ptsryv4bfktmfd7gausfyhwa" style="color: black;">2009 Sixth International Symposium on Voronoi Diagrams</a> </i> &nbsp;
We propose an approximation method to answer point-to-point shortest path queries in undirected graphs, based on random sampling and Voronoi duals.  ...  Edges are generated as the Voronoi dual of the original graph, using the selected nodes as Voronoi sites.  ...  ACKNOWLEDGMENTS We thank Georg Troxler (staila technologies) for providing us with a machine for our experiments, and Daniel Delling for supplying the public transportation graphs.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1109/isvd.2009.30">doi:10.1109/isvd.2009.30</a> <a target="_blank" rel="external noopener" href="https://dblp.org/rec/conf/isvd/HonidenHSW09.html">dblp:conf/isvd/HonidenHSW09</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/7n5gvkparvfm3mhr4xfinm5vea">fatcat:7n5gvkparvfm3mhr4xfinm5vea</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20170808172819/http://www.sommer.jp/voronoi.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/4c/61/4c61d8c5dbe58ab70d9c1833e96046a4f0e752fd.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1109/isvd.2009.30"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> ieee.com </button> </a>

Approximate Shortest Path Queries Using Voronoi Duals [chapter]

Shinichi Honiden, Michael E. Houle, Christian Sommer, Martin Wolff
<span title="">2010</span> <i title="Springer Berlin Heidelberg"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/2w3awgokqne6te4nvlofavy5a4" style="color: black;">Lecture Notes in Computer Science</a> </i> &nbsp;
We propose an approximation method to answer point-to-point shortest path queries in undirected edge-weighted graphs, based on random sampling and Voronoi duals.  ...  Edges are generated as the Voronoi dual of the original graph, using the selected nodes as Voronoi sites.  ...  We let the shortest path in the Voronoi dual guide the search for an approximate shortest path in the original graph.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/978-3-642-16007-3_2">doi:10.1007/978-3-642-16007-3_2</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/juzzyxzvyjbc5exsqddvuggise">fatcat:juzzyxzvyjbc5exsqddvuggise</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20170829123315/http://www.shortestpaths.org/voronoi_j.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/9e/1d/9e1d53c2398b422faed1004a2a2ce584274afecd.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/978-3-642-16007-3_2"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> springer.com </button> </a>

Better Tradeoffs for Exact Distance Oracles in Planar Graphs [chapter]

Pawel Gawrychowski, Shay Mozes, Oren Weimann, Christian Wulff-Nilsen
<span title="">2018</span> <i title="Society for Industrial and Applied Mathematics"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/ja7dpjgz2nhjlovlil3ij5m7ei" style="color: black;">Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms</a> </i> &nbsp;
We present an O(n 1.5 )-space distance oracle for directed planar graphs that answers distance queries in O(log n) time.  ...  We achieve this by designing an elegant and efficient point location data structure for Voronoi diagrams on planar graphs. We further show a smooth tradeoff between space and query-time.  ...  Approximate distance oracles for planar graphs with improved query time-space tradeoff. In SODA, pages 351-362, 2016.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1137/1.9781611975031.34">doi:10.1137/1.9781611975031.34</a> <a target="_blank" rel="external noopener" href="https://dblp.org/rec/conf/soda/GawrychowskiMWW18.html">dblp:conf/soda/GawrychowskiMWW18</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/wjg3bakp6rdu7kr6wqwelyfkgu">fatcat:wjg3bakp6rdu7kr6wqwelyfkgu</a> </span>
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Single-Source Shortest Paths and Strong Connectivity in Dynamic Planar Graphs

Panagiotis Charalampopoulos, Adam Karczmarz, Peter Sanders, Grzegorz Herman, Fabrizio Grandoni
<span title="2020-08-26">2020</span> <i title="Schloss Dagstuhl - Leibniz-Zentrum für Informatik"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/uot5qmdbz5g6rasdd5kuoonkzu" style="color: black;">European Symposium on Algorithms</a> </i> &nbsp;
In this paper, we show how Voronoi diagrams can be useful in obtaining dynamic planar graph algorithms and apply them to classical problems such as dynamic single-source shortest paths and dynamic strongly  ...  Somewhat surprisingly, the Voronoi diagram based approach we take for single-source shortest paths can also be used in the fully dynamic strongly connected components problem.  ...  On dynamic approximate shortest paths for planar graphs with worst-case costs.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.4230/lipics.esa.2020.31">doi:10.4230/lipics.esa.2020.31</a> <a target="_blank" rel="external noopener" href="https://dblp.org/rec/conf/esa/Charalampopoulos20.html">dblp:conf/esa/Charalampopoulos20</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/5so72hnvf5emrkngtb6rxxznb4">fatcat:5so72hnvf5emrkngtb6rxxznb4</a> </span>
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Better Tradeoffs for Exact Distance Oracles in Planar Graphs [article]

Paweł Gawrychowski, Shay Mozes, Oren Weimann, Christian Wulff-Nilsen
<span title="2017-08-04">2017</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
We present an O(n^1.5)-space distance oracle for directed planar graphs that answers distance queries in O( n) time.  ...  We achieve this by designing an elegant and efficient point location data structure for Voronoi diagrams on planar graphs. We further show a smooth tradeoff between space and query-time.  ...  Figure 1 : 1 A planar graph (black edges) with four sites on the infinite face together with the dual Voronoi diagram VD * (in blue).  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1708.01386v1">arXiv:1708.01386v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/226uixbxvjd3zeofowwtraj5au">fatcat:226uixbxvjd3zeofowwtraj5au</a> </span>
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Literature Review

I. Youngster, M. Green
<span title="2013-11-24">2013</span> <i title="Oxford University Press (OUP)"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/44gnyxptifd25gwn2cxqvrtzze" style="color: black;">Journal of the Pediatric Infectious Diseases Society</a> </i> &nbsp;
In this paper the authors propose an approximation method to answer shortest path queries in graphs, based on hierarchical random sampling and Voronoi duals.  ...  Edges are generated as the Voronoi dual within the lower level, using the selected nodes as Voronoi sites. This hierarchy allows for fast computation of approximate shortest paths for general graphs.  ...  We can also find the k shortest paths from a given source s to each vertex in the graph, in total time O (m+n log n + kn).  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1093/jpids/pit078">doi:10.1093/jpids/pit078</a> <a target="_blank" rel="external noopener" href="https://www.ncbi.nlm.nih.gov/pubmed/26619506">pmid:26619506</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/opm7qy3d3behjm6e36szpqzlim">fatcat:opm7qy3d3behjm6e36szpqzlim</a> </span>
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Almost Optimal Distance Oracles for Planar Graphs [article]

Panagiotis Charalampopoulos, Paweł Gawrychowski, Shay Mozes, Oren Weimann
<span title="2018-11-05">2018</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
We present new tradeoffs between space and query-time for exact distance oracles in directed weighted planar graphs.  ...  These tradeoffs are almost optimal in the sense that they are within polylogarithmic, sub-polynomial or arbitrarily small polynomial factors from the naïve linear space, constant query-time lower bound  ...  The multiple-source shortest paths (MSSP) data structure [30] represents all shortest path trees rooted at the vertices of a single face f in a planar graph using a persistent dynamic tree.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1811.01551v1">arXiv:1811.01551v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/zdfayqwq3ffszhojt4uudnamsu">fatcat:zdfayqwq3ffszhojt4uudnamsu</a> </span>
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Chapter 7 A survey of computational geometry [chapter]

Joseph S.B. Mitchell, Subhash Suri
<span title="">1995</span> <i title="Elsevier"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/7u3dpsmdbvchtoxn3h5yhwxoju" style="color: black;">Handbooks in Operations Research and Management Science</a> </i> &nbsp;
The topic of path planning is a vast one, with problems ranging from nding shortest paths in a discrete graph to deciding the feasible motion of a complex robot in an environment full of complex obstacles  ...  We brie y mention most of the major developments in path planning research over the last two decades, but to a large extent limit ourselves to issues related to shortest paths in a planar domain.  ...  For instance, if Voronoi diagrams are used for answering \nearest neighbor" queries, one needs to locate the Voronoi polyhedron containing the query point.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/s0927-0507(05)80124-0">doi:10.1016/s0927-0507(05)80124-0</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/nogmdqmv6bflni3waox5dxrsiy">fatcat:nogmdqmv6bflni3waox5dxrsiy</a> </span>
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Fast and Compact Exact Distance Oracle for Planar Graphs [article]

Vincent Cohen-Addad, Søren Dahlgaard, Christian Wulff-Nilsen
<span title="2017-05-02">2017</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
All previous distance oracles for planar graphs with truly subquadratic space i.e., space O(n^2 - ϵ) for some constant ϵ > 0) either required query time polynomial in n or could only answer approximate  ...  For a given a graph, a distance oracle is a data structure that answers distance queries between pairs of vertices.  ...  The Distance Query on input u, v returns the length of the shortest path between vertices u and v in the graph. Proof.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1702.03259v3">arXiv:1702.03259v3</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/g74ouv5jfnhujg747kwirtyiwa">fatcat:g74ouv5jfnhujg747kwirtyiwa</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20191023160907/https://arxiv.org/pdf/1702.03259v3.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/ce/8d/ce8de354e81d62fac546be15491989fc06523b6d.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1702.03259v3" title="arxiv.org access"> <button class="ui compact blue labeled icon button serp-button"> <i class="file alternate outline icon"></i> arxiv.org </button> </a>

Planning short paths with clearance using explicit corridors

Roland Geraerts
<span title="">2010</span> <i title="IEEE"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/nytnrt3gtzbixld5r6a4talbky" style="color: black;">2010 IEEE International Conference on Robotics and Automation</a> </i> &nbsp;
We introduce a new data structure, i.e. the Explicit Corridor Map, which allows creating the shortest path, the path that has the largest amount of clearance, or any path in between.  ...  Since environments and their characters are growing more realistic, a character's path needs to be visually convincing, meaning that the path is smooth, short, has some clearance to the obstacles in the  ...  Then, the dual graph of this triangulation is determined. In this graph, a vertex is placed in the face of each triangle.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1109/robot.2010.5509263">doi:10.1109/robot.2010.5509263</a> <a target="_blank" rel="external noopener" href="https://dblp.org/rec/conf/icra/Geraerts10.html">dblp:conf/icra/Geraerts10</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/ig7cjccenvgkhm4zlu74h65tne">fatcat:ig7cjccenvgkhm4zlu74h65tne</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20170809080433/http://www.cs.uu.nl/docs/vakken/mpp/papers/2.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/d2/03/d2033135eb5231c46725a3ee35ac09f1cb0a9fff.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1109/robot.2010.5509263"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> ieee.com </button> </a>

Planar Distance Oracles with Better Time-Space Tradeoffs [article]

Yaowei Long, Seth Pettie
<span title="2020-07-16">2020</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
In a recent breakthrough, Charalampopoulos, Gawrychowski, Mozes, and Weimann (STOC 2019) showed that exact distance queries on planar graphs could be answered in n^o(1) time by a data structure occupying  ...  In this paper we present a new way to do point-location in planar Voronoi diagrams, which leads to a new exact distance oracle.  ...  We thank Danny Sleator and Bob Tarjan for discussing update/query time tradeoffs for dynamic trees.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/2007.08585v1">arXiv:2007.08585v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/64zpneogfjdqzcgwrpjco3a7ru">fatcat:64zpneogfjdqzcgwrpjco3a7ru</a> </span>
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K Nearest Neighbor Search in Navigation Systems

Maytham Safar
<span title="">2005</span> <i title="Hindawi Limited"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/khqygtfojnby3jyh5obuwd7ko4" style="color: black;">Mobile Information Systems</a> </i> &nbsp;
., shortest paths) between objects. In this paper, we propose a novel approach to efficiently and accurately evaluatekNN queries in a mobile information system that uses spatial network databases.  ...  A frequent type of query in a car navigation system is to find theknearest neighbors (kNN) of a given query object (e.g., car) using the actual road network maps.  ...  Their approach is based on partitioning a large graph into layers of smaller sub graphs and pushing up the pre-computed shortest paths between the borders of the sub graphs in a hierarchical manner to  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1155/2005/692568">doi:10.1155/2005/692568</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/tth2lhnvkndglk2r7qnqnrpnje">fatcat:tth2lhnvkndglk2r7qnqnrpnje</a> </span>
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Link Distance and Shortest Path Problems in the Plane [chapter]

Atlas F. Cook, Carola Wenk
<span title="">2009</span> <i title="Springer Berlin Heidelberg"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/2w3awgokqne6te4nvlofavy5a4" style="color: black;">Lecture Notes in Computer Science</a> </i> &nbsp;
This paper shows how to compute new variations of these metrics using shortest paths or link-based paths that avoid polygonal obstacles in the plane.  ...  The shortest path maps in this paper support queries from any source point on a fixed line segment.  ...  We are also grateful to Valentin Polishchuk and the anonymous reviewers for a logarithmic improvement in the time bound of Lemma 13.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/978-3-642-02158-9_13">doi:10.1007/978-3-642-02158-9_13</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/oc5rnreumrgmhnyyypu2qitgji">fatcat:oc5rnreumrgmhnyyypu2qitgji</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20121114153827/http://www.my.cs.utsa.edu:80/~acook/papers/CGTA_Link_Shortest_Plane.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/f7/39/f7397cf81603cd842dba32b96ab1254c0d2144f5.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/978-3-642-02158-9_13"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> springer.com </button> </a>

Link distance and shortest path problems in the plane

Atlas F. Cook, Carola Wenk
<span title="">2011</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/is2k7576zbherpepqy3y7zkzuy" style="color: black;">Computational geometry</a> </i> &nbsp;
This paper shows how to compute new variations of these metrics using shortest paths or link-based paths that avoid polygonal obstacles in the plane.  ...  The shortest path maps in this paper support queries from any source point on a fixed line segment.  ...  We are also grateful to Valentin Polishchuk and the anonymous reviewers for a logarithmic improvement in the time bound of Lemma 13.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/j.comgeo.2011.04.004">doi:10.1016/j.comgeo.2011.04.004</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/jalqon4cdbafhlx47cfn3cf25m">fatcat:jalqon4cdbafhlx47cfn3cf25m</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20121114153827/http://www.my.cs.utsa.edu:80/~acook/papers/CGTA_Link_Shortest_Plane.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/f7/39/f7397cf81603cd842dba32b96ab1254c0d2144f5.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/j.comgeo.2011.04.004"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> elsevier.com </button> </a>

Exact Distance Oracles for Planar Graphs with Failing Vertices [article]

Panagiotis Charalampopoulos, Shay Mozes, Benjamin Tebeka
<span title="2021-08-30">2021</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
Given a source vertex u, a target vertex v and a set X of k failed vertices, such an oracle returns the length of a shortest u-to-v path that avoids all vertices in X.  ...  In particular, for a directed weighted planar graph with n vertices and any constant k, we show an 𝒪̃(n)-size, 𝒪̃(√(n))-query-time oracle.  ...  for solving the all-pairs shortest path problems in weighted (general) graphs [1] .  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1807.05968v3">arXiv:1807.05968v3</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/5cwp7ftnozcedbmwi5lav6ggme">fatcat:5cwp7ftnozcedbmwi5lav6ggme</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20210909052138/https://arxiv.org/pdf/1807.05968v3.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/5d/61/5d617bb4ed74693ab7f4ceeb1926cbceecbdb1bc.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1807.05968v3" title="arxiv.org access"> <button class="ui compact blue labeled icon button serp-button"> <i class="file alternate outline icon"></i> arxiv.org </button> </a>
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