Document

**Published in:** LIPIcs, Volume 258, 39th International Symposium on Computational Geometry (SoCG 2023)

The geodesic edge center of a polygon is a point c inside the polygon that minimizes the maximum geodesic distance from c to any edge of the polygon, where geodesic distance is the shortest path distance inside the polygon. We give a linear-time algorithm to find a geodesic edge center of a simple polygon. This improves on the previous O(n log n) time algorithm by Lubiw and Naredla [European Symposium on Algorithms, 2021]. The algorithm builds on an algorithm to find the geodesic vertex center of a simple polygon due to Pollack, Sharir, and Rote [Discrete & Computational Geometry, 1989] and an improvement to linear time by Ahn, Barba, Bose, De Carufel, Korman, and Oh [Discrete & Computational Geometry, 2016].
The geodesic edge center can easily be found from the geodesic farthest-edge Voronoi diagram of the polygon. Finding that Voronoi diagram in linear time is an open question, although the geodesic nearest edge Voronoi diagram (the medial axis) can be found in linear time. As a first step of our geodesic edge center algorithm, we give a linear-time algorithm to find the geodesic farthest-edge Voronoi diagram restricted to the polygon boundary.

Anna Lubiw and Anurag Murty Naredla. The Geodesic Edge Center of a Simple Polygon. In 39th International Symposium on Computational Geometry (SoCG 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 258, pp. 49:1-49:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{lubiw_et_al:LIPIcs.SoCG.2023.49, author = {Lubiw, Anna and Naredla, Anurag Murty}, title = {{The Geodesic Edge Center of a Simple Polygon}}, booktitle = {39th International Symposium on Computational Geometry (SoCG 2023)}, pages = {49:1--49:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-273-0}, ISSN = {1868-8969}, year = {2023}, volume = {258}, editor = {Chambers, Erin W. and Gudmundsson, Joachim}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2023.49}, URN = {urn:nbn:de:0030-drops-178994}, doi = {10.4230/LIPIcs.SoCG.2023.49}, annote = {Keywords: geodesic center of polygon, farthest edges, farthest-segment Voronoi diagram} }

Document

**Published in:** LIPIcs, Volume 244, 30th Annual European Symposium on Algorithms (ESA 2022)

Given a graph where every vertex has exactly one labeled token, how can we most quickly execute a given permutation on the tokens? In (sequential) token swapping, the goal is to use the shortest possible sequence of swaps, each of which exchanges the tokens at the two endpoints of an edge of the graph. In parallel token swapping, the goal is to use the fewest rounds, each of which consists of one or more swaps on the edges of a matching. We prove that both of these problems remain NP-hard when the graph is restricted to be a tree.
These token swapping problems have been studied by disparate groups of researchers in discrete mathematics, theoretical computer science, robot motion planning, game theory, and engineering. Previous work establishes NP-completeness on general graphs (for both problems), constant-factor approximation algorithms, and some poly-time exact algorithms for simple graph classes such as cliques, stars, paths, and cycles. Sequential and parallel token swapping on trees were first studied over thirty years ago (as "sorting with a transposition tree") and over twenty-five years ago (as "routing permutations via matchings"), yet their complexities were previously unknown.
We also show limitations on approximation of sequential token swapping on trees: we identify a broad class of algorithms that encompass all three known polynomial-time algorithms that achieve the best known approximation factor (which is 2) and show that no such algorithm can achieve an approximation factor less than 2.

Oswin Aichholzer, Erik D. Demaine, Matias Korman, Anna Lubiw, Jayson Lynch, Zuzana Masárová, Mikhail Rudoy, Virginia Vassilevska Williams, and Nicole Wein. Hardness of Token Swapping on Trees. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 3:1-3:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{aichholzer_et_al:LIPIcs.ESA.2022.3, author = {Aichholzer, Oswin and Demaine, Erik D. and Korman, Matias and Lubiw, Anna and Lynch, Jayson and Mas\'{a}rov\'{a}, Zuzana and Rudoy, Mikhail and Vassilevska Williams, Virginia and Wein, Nicole}, title = {{Hardness of Token Swapping on Trees}}, booktitle = {30th Annual European Symposium on Algorithms (ESA 2022)}, pages = {3:1--3:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-247-1}, ISSN = {1868-8969}, year = {2022}, volume = {244}, editor = {Chechik, Shiri and Navarro, Gonzalo and Rotenberg, Eva and Herman, Grzegorz}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2022.3}, URN = {urn:nbn:de:0030-drops-169413}, doi = {10.4230/LIPIcs.ESA.2022.3}, annote = {Keywords: Sorting, Token swapping, Trees, NP-hard, Approximation} }

Document

**Published in:** Dagstuhl Reports, Volume 12, Issue 2 (2022)

This report documents the program and the outcomes of Dagstuhl Seminar 22062 "Computation and Reconfiguration in Low-Dimensional Topological Spaces". The seminar consisted of a small collection of introductory talks, an open problem session, and then the seminar participants worked in small groups on problems on reconfiguration within the context of objects as diverse as elimination trees, morphings, curves on surfaces, translation surfaces and Delaunay triangulations.

Maike Buchin, Anna Lubiw, Arnaud de Mesmay, Saul Schleimer, and Florestan Brunck. Computation and Reconfiguration in Low-Dimensional Topological Spaces (Dagstuhl Seminar 22062). In Dagstuhl Reports, Volume 12, Issue 2, pp. 17-66, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@Article{buchin_et_al:DagRep.12.2.17, author = {Buchin, Maike and Lubiw, Anna and de Mesmay, Arnaud and Schleimer, Saul and Brunck, Florestan}, title = {{Computation and Reconfiguration in Low-Dimensional Topological Spaces (Dagstuhl Seminar 22062)}}, pages = {17--66}, journal = {Dagstuhl Reports}, ISSN = {2192-5283}, year = {2022}, volume = {12}, number = {2}, editor = {Buchin, Maike and Lubiw, Anna and de Mesmay, Arnaud and Schleimer, Saul and Brunck, Florestan}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/DagRep.12.2.17}, URN = {urn:nbn:de:0030-drops-169305}, doi = {10.4230/DagRep.12.2.17}, annote = {Keywords: Geometric Topology, Computational Geometry, Graph Drawing, Reconfiguration, Dagstuhl Seminar} }

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**Published in:** LIPIcs, Volume 204, 29th Annual European Symposium on Algorithms (ESA 2021)

The input to the distant representatives problem is a set of n objects in the plane and the goal is to find a representative point from each object while maximizing the distance between the closest pair of points. When the objects are axis-aligned rectangles, we give polynomial time constant-factor approximation algorithms for the L₁, L₂, and L_∞ distance measures. We also prove lower bounds on the approximation factors that can be achieved in polynomial time (unless P = NP).

Therese Biedl, Anna Lubiw, Anurag Murty Naredla, Peter Dominik Ralbovsky, and Graeme Stroud. Distant Representatives for Rectangles in the Plane. In 29th Annual European Symposium on Algorithms (ESA 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 204, pp. 17:1-17:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{biedl_et_al:LIPIcs.ESA.2021.17, author = {Biedl, Therese and Lubiw, Anna and Naredla, Anurag Murty and Ralbovsky, Peter Dominik and Stroud, Graeme}, title = {{Distant Representatives for Rectangles in the Plane}}, booktitle = {29th Annual European Symposium on Algorithms (ESA 2021)}, pages = {17:1--17:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-204-4}, ISSN = {1868-8969}, year = {2021}, volume = {204}, editor = {Mutzel, Petra and Pagh, Rasmus and Herman, Grzegorz}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2021.17}, URN = {urn:nbn:de:0030-drops-145982}, doi = {10.4230/LIPIcs.ESA.2021.17}, annote = {Keywords: Distant representatives, blocker shapes, matching, approximation algorithm, APX-hardness} }

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**Published in:** LIPIcs, Volume 204, 29th Annual European Symposium on Algorithms (ESA 2021)

We introduce the visibility center of a set of points inside a polygon - a point c_V such that the maximum geodesic distance from c_V to see any point in the set is minimized. For a simple polygon of n vertices and a set of m points inside it, we give an O((n+m) log (n+m)) time algorithm to find the visibility center. We find the visibility center of all points in a simple polygon in O(n log n) time.
Our algorithm reduces the visibility center problem to the problem of finding the geodesic center of a set of half-polygons inside a polygon, which is of independent interest. We give an O((n+k) log (n+k)) time algorithm for this problem, where k is the number of half-polygons.

Anna Lubiw and Anurag Murty Naredla. The Visibility Center of a Simple Polygon. In 29th Annual European Symposium on Algorithms (ESA 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 204, pp. 65:1-65:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{lubiw_et_al:LIPIcs.ESA.2021.65, author = {Lubiw, Anna and Naredla, Anurag Murty}, title = {{The Visibility Center of a Simple Polygon}}, booktitle = {29th Annual European Symposium on Algorithms (ESA 2021)}, pages = {65:1--65:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-204-4}, ISSN = {1868-8969}, year = {2021}, volume = {204}, editor = {Mutzel, Petra and Pagh, Rasmus and Herman, Grzegorz}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2021.65}, URN = {urn:nbn:de:0030-drops-146466}, doi = {10.4230/LIPIcs.ESA.2021.65}, annote = {Keywords: Visibility, Shortest Paths, Simple Polygons, Facility Location} }

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**Published in:** LIPIcs, Volume 162, 17th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2020)

Motivated by the connectivity problem in wireless networks with directional antennas, we study bounded-angle spanning trees. Let P be a set of points in the plane and let α be an angle. An α-ST of P is a spanning tree of the complete Euclidean graph on P with the property that all edges incident to each point p ∈ P lie in a wedge of angle α centered at p. We study the following closely related problems for α = 120 degrees (however, our approximation ratios hold for any α ⩾ 120 degrees).
1) The α-minimum spanning tree problem asks for an α-ST of minimum sum of edge lengths. Among many interesting results, Aschner and Katz (ICALP 2014) proved the NP-hardness of this problem and presented a 6-approximation algorithm. Their algorithm finds an α-ST of length at most 6 times the length of the minimum spanning tree (MST). By adopting a somewhat similar approach and using different proof techniques we improve this ratio to 16/3.
2) To examine what is possible with non-uniform wedge angles, we define an ̅α-ST to be a spanning tree with the property that incident edges to all points lie in wedges of average angle α. We present an algorithm to find an ̅α-ST whose largest edge-length and sum of edge lengths are at most 2 and 1.5 times (respectively) those of the MST. These ratios are better than any achievable when all wedges have angle α. Our algorithm runs in linear time after computing the MST.

Ahmad Biniaz, Prosenjit Bose, Anna Lubiw, and Anil Maheshwari. Bounded-Angle Minimum Spanning Trees. In 17th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 162, pp. 14:1-14:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{biniaz_et_al:LIPIcs.SWAT.2020.14, author = {Biniaz, Ahmad and Bose, Prosenjit and Lubiw, Anna and Maheshwari, Anil}, title = {{Bounded-Angle Minimum Spanning Trees}}, booktitle = {17th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2020)}, pages = {14:1--14:22}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-150-4}, ISSN = {1868-8969}, year = {2020}, volume = {162}, editor = {Albers, Susanne}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SWAT.2020.14}, URN = {urn:nbn:de:0030-drops-122616}, doi = {10.4230/LIPIcs.SWAT.2020.14}, annote = {Keywords: bounded-angle MST, directional antenna, approximation algorithms} }

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**Published in:** Dagstuhl Reports, Volume 9, Issue 8 (2020)

This report documents the program and the outcomes of Dagstuhl Seminar 19352
``Computation in Low-Dimensional Geometry and Topology''.
The seminar participants investigated problems in: knot theory, trajectory
analysis, algorithmic topology, computational geometry of curves, and graph drawing,
with an emphasis on how low-dimensional structures change over time.

Maarten Löffler, Anna Lubiw, Saul Schleimer, and Erin Moriarty Wolf Chambers. Computation in Low-Dimensional Geometry and Topology (Dagstuhl Seminar 19352). In Dagstuhl Reports, Volume 9, Issue 8, pp. 84-112, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@Article{loffler_et_al:DagRep.9.8.84, author = {L\"{o}ffler, Maarten and Lubiw, Anna and Schleimer, Saul and Wolf Chambers, Erin Moriarty}, title = {{Computation in Low-Dimensional Geometry and Topology (Dagstuhl Seminar 19352)}}, pages = {84--112}, journal = {Dagstuhl Reports}, ISSN = {2192-5283}, year = {2019}, volume = {9}, number = {8}, editor = {L\"{o}ffler, Maarten and Lubiw, Anna and Schleimer, Saul and Wolf Chambers, Erin Moriarty}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/DagRep.9.8.84}, URN = {urn:nbn:de:0030-drops-117734}, doi = {10.4230/DagRep.9.8.84}, annote = {Keywords: Geometric topology, Graph Drawing, Computational Geometry} }

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**Published in:** LIPIcs, Volume 107, 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)

A rollercoaster is a sequence of real numbers for which every maximal contiguous subsequence - increasing or decreasing - has length at least three. By translating this sequence to a set of points in the plane, a rollercoaster can be defined as an x-monotone polygonal path for which every maximal sub-path, with positive- or negative-slope edges, has at least three vertices. Given a sequence of distinct real numbers, the rollercoaster problem asks for a maximum-length (not necessarily contiguous) subsequence that is a rollercoaster. It was conjectured that every sequence of n distinct real numbers contains a rollercoaster of length at least ceil[n/2] for n>7, while the best known lower bound is Omega(n/log n). In this paper we prove this conjecture. Our proof is constructive and implies a linear-time algorithm for computing a rollercoaster of this length. Extending the O(n log n)-time algorithm for computing a longest increasing subsequence, we show how to compute a maximum-length rollercoaster within the same time bound. A maximum-length rollercoaster in a permutation of {1,...,n} can be computed in O(n log log n) time.
The search for rollercoasters was motivated by orthogeodesic point-set embedding of caterpillars. A caterpillar is a tree such that deleting the leaves gives a path, called the spine. A top-view caterpillar is one of maximum degree 4 such that the two leaves adjacent to each vertex lie on opposite sides of the spine. As an application of our result on rollercoasters, we are able to find a planar drawing of every n-vertex top-view caterpillar on every set of 25/3(n+4) points in the plane, such that each edge is an orthogonal path with one bend. This improves the previous best known upper bound on the number of required points, which is O(n log n). We also show that such a drawing can be obtained in linear time, when the points are given in sorted order.

Therese Biedl, Ahmad Biniaz, Robert Cummings, Anna Lubiw, Florin Manea, Dirk Nowotka, and Jeffrey Shallit. Rollercoasters and Caterpillars. In 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 107, pp. 18:1-18:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{biedl_et_al:LIPIcs.ICALP.2018.18, author = {Biedl, Therese and Biniaz, Ahmad and Cummings, Robert and Lubiw, Anna and Manea, Florin and Nowotka, Dirk and Shallit, Jeffrey}, title = {{Rollercoasters and Caterpillars}}, booktitle = {45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)}, pages = {18:1--18:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-076-7}, ISSN = {1868-8969}, year = {2018}, volume = {107}, editor = {Chatzigiannakis, Ioannis and Kaklamanis, Christos and Marx, D\'{a}niel and Sannella, Donald}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2018.18}, URN = {urn:nbn:de:0030-drops-90224}, doi = {10.4230/LIPIcs.ICALP.2018.18}, annote = {Keywords: sequences, alternating runs, patterns in permutations, caterpillars} }

Document

**Published in:** LIPIcs, Volume 77, 33rd International Symposium on Computational Geometry (SoCG 2017)

Given a triangulation of a point set in the plane, a flip deletes an edge e whose removal leaves a convex quadrilateral, and replaces e by the opposite diagonal of the quadrilateral. It is well known that any triangulation of a point set can be reconfigured to any other triangulation by some sequence of flips. We explore this question in the setting where each edge of a triangulation has a label, and a flip transfers the label of the removed edge to the new edge. It is not true that every labelled triangulation of a point set can be reconfigured to every other labelled triangulation via a sequence of flips, but we characterize when this is possible. There is an obvious necessary condition: for each label l, if edge e has label l in the first triangulation and edge f has label l in the second triangulation, then there must be some sequence of flips that moves label l from e to f, ignoring all other labels. Bose, Lubiw, Pathak and Verdonschot formulated the Orbit Conjecture, which states that this necessary condition is also sufficient, i.e. that all labels can be simultaneously mapped to their destination if and only if each label individually can be mapped to its destination. We prove this conjecture. Furthermore, we give a polynomial-time algorithm to find a sequence of flips to reconfigure one labelled triangulation to another, if such a sequence exists, and we prove an upper bound of O(n^7) on the length of the flip sequence.
Our proof uses the topological result that the sets of pairwise non-crossing edges on a planar point set form a simplicial complex that is homeomorphic to a high-dimensional ball (this follows from a result of Orden and Santos; we give a different proof based on a shelling argument). The dual cell complex of this simplicial ball, called the flip complex, has the usual flip graph as its 1-skeleton. We use properties of the 2-skeleton of the flip complex to prove the Orbit Conjecture.

Anna Lubiw, Zuzana Masárová, and Uli Wagner. A Proof of the Orbit Conjecture for Flipping Edge-Labelled Triangulations. In 33rd International Symposium on Computational Geometry (SoCG 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 77, pp. 49:1-49:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{lubiw_et_al:LIPIcs.SoCG.2017.49, author = {Lubiw, Anna and Mas\'{a}rov\'{a}, Zuzana and Wagner, Uli}, title = {{A Proof of the Orbit Conjecture for Flipping Edge-Labelled Triangulations}}, booktitle = {33rd International Symposium on Computational Geometry (SoCG 2017)}, pages = {49:1--49:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-038-5}, ISSN = {1868-8969}, year = {2017}, volume = {77}, editor = {Aronov, Boris and Katz, Matthew J.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2017.49}, URN = {urn:nbn:de:0030-drops-72078}, doi = {10.4230/LIPIcs.SoCG.2017.49}, annote = {Keywords: triangulations, reconfiguration, flip, constrained triangulations, Delaunay triangulation, shellability, piecewise linear balls} }

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**Published in:** LIPIcs, Volume 66, 34th Symposium on Theoretical Aspects of Computer Science (STACS 2017)

A rectifier network is a directed acyclic graph with distinguished sources and sinks; it is said to compute a Boolean matrix M that has a 1 in the entry (i,j) iff there is a path from the j-th source to the i-th sink. The smallest number of edges in a rectifier network that computes M is a classic complexity measure on matrices, which has been studied for more than half a century.
We explore two techniques that have hitherto found little to no applications in this theory. They build upon a basic fact that depth-2 rectifier networks are essentially weighted coverings of Boolean matrices with rectangles. Using fractional and greedy coverings (defined in the standard way), we obtain new results in this area.
First, we show that all fractional coverings of the so-called full triangular matrix have cost at least n log n. This provides (a fortiori) a new proof of the tight lower bound on its depth-2 complexity (the exact value has been known since 1965, but previous proofs are based on different arguments). Second, we show that the greedy heuristic is instrumental in tightening the upper bound on the depth-2 complexity of the Kneser-Sierpinski (disjointness) matrix. The previous upper bound is O(n^{1.28}), and we improve it to O(n^{1.17}), while the best known lower bound is Omega(n^{1.16}). Third, using fractional coverings, we obtain a form of direct product theorem that gives a lower bound on unbounded-depth complexity of Kronecker (tensor) products of matrices. In this case, the greedy heuristic shows (by an argument due to Lovász) that our result is only a logarithmic factor away from the "full" direct product theorem. Our second and third results constitute progress on open problem 7.3 and resolve, up to a logarithmic factor, open problem 7.5 from a recent book by Jukna and Sergeev (in Foundations and Trends in Theoretical Computer Science (2013)).

Dmitry Chistikov, Szabolcs Iván, Anna Lubiw, and Jeffrey Shallit. Fractional Coverings, Greedy Coverings, and Rectifier Networks. In 34th Symposium on Theoretical Aspects of Computer Science (STACS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 66, pp. 23:1-23:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{chistikov_et_al:LIPIcs.STACS.2017.23, author = {Chistikov, Dmitry and Iv\'{a}n, Szabolcs and Lubiw, Anna and Shallit, Jeffrey}, title = {{Fractional Coverings, Greedy Coverings, and Rectifier Networks}}, booktitle = {34th Symposium on Theoretical Aspects of Computer Science (STACS 2017)}, pages = {23:1--23:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-028-6}, ISSN = {1868-8969}, year = {2017}, volume = {66}, editor = {Vollmer, Heribert and Vall\'{e}e, Brigitte}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2017.23}, URN = {urn:nbn:de:0030-drops-70107}, doi = {10.4230/LIPIcs.STACS.2017.23}, annote = {Keywords: rectifier network, OR-circuit, biclique covering, fractional covering, greedy covering} }

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Complete Volume

**Published in:** LIPIcs, Volume 51, 32nd International Symposium on Computational Geometry (SoCG 2016)

LIPIcs, Volume 51, SoCG'16, Complete Volume

32nd International Symposium on Computational Geometry (SoCG 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 51, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@Proceedings{fekete_et_al:LIPIcs.SoCG.2016, title = {{LIPIcs, Volume 51, SoCG'16, Complete Volume}}, booktitle = {32nd International Symposium on Computational Geometry (SoCG 2016)}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-009-5}, ISSN = {1868-8969}, year = {2016}, volume = {51}, editor = {Fekete, S\'{a}ndor and Lubiw, Anna}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2016}, URN = {urn:nbn:de:0030-drops-60580}, doi = {10.4230/LIPIcs.SoCG.2016}, annote = {Keywords: Analysis of Algorithms and Problem Complexity, Nonnumerical Algorithms and Problems – Geometrical problems and computations, Discrete Mathematics, Combinatorics, Computer Graphics, Computational Geometry and Object Modeling} }

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Front Matter

**Published in:** LIPIcs, Volume 51, 32nd International Symposium on Computational Geometry (SoCG 2016)

Front Matter, Table of Contents, Foreword, Conference Organization, External Reviewers, Sponsors

32nd International Symposium on Computational Geometry (SoCG 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 51, pp. 0:i-0:xviii, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{fekete_et_al:LIPIcs.SoCG.2016.0, author = {Fekete, S\'{a}ndor and Lubiw, Anna}, title = {{Front Matter, Table of Contents, Foreword, Conference Organization, External Reviewers, Sponsors}}, booktitle = {32nd International Symposium on Computational Geometry (SoCG 2016)}, pages = {0:i--0:xviii}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-009-5}, ISSN = {1868-8969}, year = {2016}, volume = {51}, editor = {Fekete, S\'{a}ndor and Lubiw, Anna}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2016.0}, URN = {urn:nbn:de:0030-drops-59658}, doi = {10.4230/LIPIcs.SoCG.2016.0}, annote = {Keywords: Front Matter, Table of Contents, Foreword, Conference Organization, External Reviewers, Sponsors} }

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**Published in:** LIPIcs, Volume 34, 31st International Symposium on Computational Geometry (SoCG 2015)

We give an algorithm to compute a morph between any two convex drawings of the same plane graph. The morph preserves the convexity of the drawing at any time instant and moves each vertex along a piecewise linear curve with linear complexity. The linear bound is asymptotically optimal in the worst case.

Patrizio Angelini, Giordano Da Lozzo, Fabrizio Frati, Anna Lubiw, Maurizio Patrignani, and Vincenzo Roselli. Optimal Morphs of Convex Drawings. In 31st International Symposium on Computational Geometry (SoCG 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 34, pp. 126-140, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{angelini_et_al:LIPIcs.SOCG.2015.126, author = {Angelini, Patrizio and Da Lozzo, Giordano and Frati, Fabrizio and Lubiw, Anna and Patrignani, Maurizio and Roselli, Vincenzo}, title = {{Optimal Morphs of Convex Drawings}}, booktitle = {31st International Symposium on Computational Geometry (SoCG 2015)}, pages = {126--140}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-83-5}, ISSN = {1868-8969}, year = {2015}, volume = {34}, editor = {Arge, Lars and Pach, J\'{a}nos}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SOCG.2015.126}, URN = {urn:nbn:de:0030-drops-51333}, doi = {10.4230/LIPIcs.SOCG.2015.126}, annote = {Keywords: Convex Drawings, Planar Graphs, Morphing, Geometric Representations} }

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**Published in:** LIPIcs, Volume 34, 31st International Symposium on Computational Geometry (SoCG 2015)

There are two known ways to unfold a convex polyhedron without overlap: the star unfolding and the source unfolding, both of which use shortest paths from vertices to a source point on the surface of the polyhedron. Non-overlap of the source unfolding is straightforward; non-overlap of the star unfolding was proved by Aronov and O'Rourke in 1992. Our first contribution is a much simpler proof of non-overlap of the star unfolding.
Both the source and star unfolding can be generalized to use a simple geodesic curve instead of a source point. The star unfolding from a geodesic curve cuts the geodesic curve and a shortest path from each vertex to the geodesic curve. Demaine and Lubiw conjectured that the star unfolding from a geodesic curve does not overlap. We prove a special case of the conjecture. Our special case includes the previously known case of unfolding from a geodesic loop. For the general case we prove that the star unfolding from a geodesic curve can be separated into at most two non-overlapping pieces.

Stephen Kiazyk and Anna Lubiw. Star Unfolding from a Geodesic Curve. In 31st International Symposium on Computational Geometry (SoCG 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 34, pp. 390-404, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{kiazyk_et_al:LIPIcs.SOCG.2015.390, author = {Kiazyk, Stephen and Lubiw, Anna}, title = {{Star Unfolding from a Geodesic Curve}}, booktitle = {31st International Symposium on Computational Geometry (SoCG 2015)}, pages = {390--404}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-83-5}, ISSN = {1868-8969}, year = {2015}, volume = {34}, editor = {Arge, Lars and Pach, J\'{a}nos}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SOCG.2015.390}, URN = {urn:nbn:de:0030-drops-51380}, doi = {10.4230/LIPIcs.SOCG.2015.390}, annote = {Keywords: unfolding, convex polyhedra, geodesic curve} }

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**Published in:** LIPIcs, Volume 20, 30th International Symposium on Theoretical Aspects of Computer Science (STACS 2013)

We prove that every simple polygon can be made as a (2D) pop-up card/book that opens to any desired angle between 0 and 360°.
More precisely, given a simple polygon attached to the two walls of the open pop-up, our polynomial-time algorithm subdivides the polygon into a single-degree-of-freedom linkage structure, such that closing the pop-up flattens the linkage without collision. This result solves an open problem of Hara and Sugihara from 2009. We also show how to obtain a more efficient construction for the special case of orthogonal polygons, and how to make 3D orthogonal polyhedra, from pop-ups that open to 90°, 180°, 270°, or 360°.

Zachary Abel, Erik D. Demaine, Martin L. Demaine, Sarah Eisenstat, Anna Lubiw, André Schulz, Diane L. Souvaine, Giovanni Viglietta, and Andrew Winslow. Algorithms for Designing Pop-Up Cards. In 30th International Symposium on Theoretical Aspects of Computer Science (STACS 2013). Leibniz International Proceedings in Informatics (LIPIcs), Volume 20, pp. 269-280, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2013)

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@InProceedings{abel_et_al:LIPIcs.STACS.2013.269, author = {Abel, Zachary and Demaine, Erik D. and Demaine, Martin L. and Eisenstat, Sarah and Lubiw, Anna and Schulz, Andr\'{e} and Souvaine, Diane L. and Viglietta, Giovanni and Winslow, Andrew}, title = {{Algorithms for Designing Pop-Up Cards}}, booktitle = {30th International Symposium on Theoretical Aspects of Computer Science (STACS 2013)}, pages = {269--280}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-50-7}, ISSN = {1868-8969}, year = {2013}, volume = {20}, editor = {Portier, Natacha and Wilke, Thomas}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2013.269}, URN = {urn:nbn:de:0030-drops-39407}, doi = {10.4230/LIPIcs.STACS.2013.269}, annote = {Keywords: geometric folding, linkages, universality} }

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**Published in:** LIPIcs, Volume 3, 26th International Symposium on Theoretical Aspects of Computer Science (2009)

In this paper we study a variant of the shortest path problem in graphs: given a weighted graph $G$ and vertices $s$ and $t$, and given a set $X$ of forbidden paths in $G$, find a shortest $s$-$t$ path $P$ such that no path in $X$ is a subpath of $P$. Path $P$ is allowed to repeat vertices and edges. We call each path in $X$ an \emph{exception}, and our desired path a \emph{shortest exception avoiding path}. We formulate a new version of the problem where the algorithm has no a priori knowledge of $X$, and finds out about an exception $x \in X$ only when a path containing $x$ fails. This situation arises in computing shortest paths in optical networks. We give an algorithm that finds a shortest exception avoiding path in time polynomial in $|G|$ and $|X|$. The main idea is to run Dijkstra's algorithm incrementally after replicating vertices when an exception is discovered.

Mustaq Ahmed and Anna Lubiw. Shortest Paths Avoiding Forbidden Subpaths. In 26th International Symposium on Theoretical Aspects of Computer Science. Leibniz International Proceedings in Informatics (LIPIcs), Volume 3, pp. 63-74, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2009)

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@InProceedings{ahmed_et_al:LIPIcs.STACS.2009.1831, author = {Ahmed, Mustaq and Lubiw, Anna}, title = {{Shortest Paths Avoiding Forbidden Subpaths}}, booktitle = {26th International Symposium on Theoretical Aspects of Computer Science}, pages = {63--74}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-09-5}, ISSN = {1868-8969}, year = {2009}, volume = {3}, editor = {Albers, Susanne and Marion, Jean-Yves}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2009.1831}, URN = {urn:nbn:de:0030-drops-18318}, doi = {10.4230/LIPIcs.STACS.2009.1831}, annote = {Keywords: Algorithms and data structures, Graph algorithms, Optical networks} }

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**Published in:** Dagstuhl Seminar Proceedings, Volume 8191, Graph Drawing with Applications to Bioinformatics and Social Sciences (2008)

We considered the following problem: Given a set of vertices V and a set of paths
P, where each path is a sequence of vertices, represent these paths somehow.
We explored representations in different dimensions and with different conditions on the paths.

Stephen Borgatti, Ulrik Brandes, Michael Kaufmann, Stephen Kobourov, Anna Lubiw, and Dorothea Wagner. 08191 Working Group Report – Visualization of Trajectories. In Graph Drawing with Applications to Bioinformatics and Social Sciences. Dagstuhl Seminar Proceedings, Volume 8191, pp. 1-3, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2008)

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@InProceedings{borgatti_et_al:DagSemProc.08191.4, author = {Borgatti, Stephen and Brandes, Ulrik and Kaufmann, Michael and Kobourov, Stephen and Lubiw, Anna and Wagner, Dorothea}, title = {{08191 Working Group Report – Visualization of Trajectories}}, booktitle = {Graph Drawing with Applications to Bioinformatics and Social Sciences}, pages = {1--3}, series = {Dagstuhl Seminar Proceedings (DagSemProc)}, ISSN = {1862-4405}, year = {2008}, volume = {8191}, editor = {Stephen P. Borgatti and Stephen Kobourov and Oliver Kohlbacher and Petra Mutzel}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.08191.4}, URN = {urn:nbn:de:0030-drops-15558}, doi = {10.4230/DagSemProc.08191.4}, annote = {Keywords: Graph drawing, trajectories, paths} }