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Documents authored by Langerman, Stefan

Document
DOI: 10.4230/LIPIcs.FUN.2022.6
Rolling Polyhedra on Tessellations

Authors: Akira Baes, Erik D. Demaine, Martin L. Demaine, Elizabeth Hartung, Stefan Langerman, Joseph O'Rourke, Ryuhei Uehara, Yushi Uno, and Aaron Williams

Published in: LIPIcs, Volume 226, 11th International Conference on Fun with Algorithms (FUN 2022)

Abstract
We study the space reachable by rolling a 3D convex polyhedron on a 2D periodic tessellation in the xy-plane, where at every step a face of the polyhedron must coincide exactly with a tile of the tessellation it rests upon, and the polyhedron rotates around one of the incident edges of that face until the neighboring face hits the xy plane. If the whole plane can be reached by a sequence of such rolls, we call the polyhedron a plane roller for the given tessellation. We further classify polyhedra that reach a constant fraction of the plane, an infinite area but vanishing fraction of the plane, or a bounded area as hollow-plane rollers, band rollers, and bounded rollers respectively. We present a polynomial-time algorithm to determine the set of tiles in a given periodic tessellation reachable by a given polyhedron from a given starting position, which in particular determines the roller type of the polyhedron and tessellation. Using this algorithm, we compute the reachability for every regular-faced convex polyhedron on every regular-tiled (≤ 4)-uniform tessellation.
Cite as

Akira Baes, Erik D. Demaine, Martin L. Demaine, Elizabeth Hartung, Stefan Langerman, Joseph O'Rourke, Ryuhei Uehara, Yushi Uno, and Aaron Williams. Rolling Polyhedra on Tessellations. In 11th International Conference on Fun with Algorithms (FUN 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 226, pp. 6:1-6:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{baes_et_al:LIPIcs.FUN.2022.6,
  author =	{Baes, Akira and Demaine, Erik D. and Demaine, Martin L. and Hartung, Elizabeth and Langerman, Stefan and O'Rourke, Joseph and Uehara, Ryuhei and Uno, Yushi and Williams, Aaron},
  title =	{{Rolling Polyhedra on Tessellations}},
  booktitle =	{11th International Conference on Fun with Algorithms (FUN 2022)},
  pages =	{6:1--6:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-232-7},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{226},
  editor =	{Fraigniaud, Pierre and Uno, Yushi},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FUN.2022.6},
  URN =		{urn:nbn:de:0030-drops-159761},
  doi =		{10.4230/LIPIcs.FUN.2022.6},
  annote =	{Keywords: polyhedra, tilings}
}
Document
DOI: 10.4230/LIPIcs.ESA.2018.38
Dynamic Trees with Almost-Optimal Access Cost

Authors: Mordecai Golin, John Iacono, Stefan Langerman, J. Ian Munro, and Yakov Nekrich

Published in: LIPIcs, Volume 112, 26th Annual European Symposium on Algorithms (ESA 2018)

Abstract
An optimal binary search tree for an access sequence on elements is a static tree that minimizes the total search cost. Constructing perfectly optimal binary search trees is expensive so the most efficient algorithms construct almost optimal search trees. There exists a long literature of constructing almost optimal search trees dynamically, i.e., when the access pattern is not known in advance. All of these trees, e.g., splay trees and treaps, provide a multiplicative approximation to the optimal search cost. In this paper we show how to maintain an almost optimal weighted binary search tree under access operations and insertions of new elements where the approximation is an additive constant. More technically, we maintain a tree in which the depth of the leaf holding an element e_i does not exceed min(log(W/w_i),log n)+O(1) where w_i is the number of times e_i was accessed and W is the total length of the access sequence. Our techniques can also be used to encode a sequence of m symbols with a dynamic alphabetic code in O(m) time so that the encoding length is bounded by m(H+O(1)), where H is the entropy of the sequence. This is the first efficient algorithm for adaptive alphabetic coding that runs in constant time per symbol.
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Mordecai Golin, John Iacono, Stefan Langerman, J. Ian Munro, and Yakov Nekrich. Dynamic Trees with Almost-Optimal Access Cost. In 26th Annual European Symposium on Algorithms (ESA 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 112, pp. 38:1-38:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{golin_et_al:LIPIcs.ESA.2018.38,
  author =	{Golin, Mordecai and Iacono, John and Langerman, Stefan and Munro, J. Ian and Nekrich, Yakov},
  title =	{{Dynamic Trees with Almost-Optimal Access Cost}},
  booktitle =	{26th Annual European Symposium on Algorithms (ESA 2018)},
  pages =	{38:1--38:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-081-1},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{112},
  editor =	{Azar, Yossi and Bast, Hannah 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.2018.38},
  URN =		{urn:nbn:de:0030-drops-95017},
  doi =		{10.4230/LIPIcs.ESA.2018.38},
  annote =	{Keywords: Data Structures, Binary Search Trees, Adaptive Alphabetic Coding}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2018.20
Subquadratic Encodings for Point Configurations

Authors: Jean Cardinal, Timothy M. Chan, John Iacono, Stefan Langerman, and Aurélien Ooms

Published in: LIPIcs, Volume 99, 34th International Symposium on Computational Geometry (SoCG 2018)

Abstract
For many algorithms dealing with sets of points in the plane, the only relevant information carried by the input is the combinatorial configuration of the points: the orientation of each triple of points in the set (clockwise, counterclockwise, or collinear). This information is called the order type of the point set. In the dual, realizable order types and abstract order types are combinatorial analogues of line arrangements and pseudoline arrangements. Too often in the literature we analyze algorithms in the real-RAM model for simplicity, putting aside the fact that computers as we know them cannot handle arbitrary real numbers without some sort of encoding. Encoding an order type by the integer coordinates of a realizing point set is known to yield doubly exponential coordinates in some cases. Other known encodings can achieve quadratic space or fast orientation queries, but not both. In this contribution, we give a compact encoding for abstract order types that allows efficient query of the orientation of any triple: the encoding uses O(n^2) bits and an orientation query takes O(log n) time in the word-RAM model with word size w >= log n. This encoding is space-optimal for abstract order types. We show how to shorten the encoding to O(n^2 {(log log n)}^2 / log n) bits for realizable order types, giving the first subquadratic encoding for those order types with fast orientation queries. We further refine our encoding to attain O(log n/log log n) query time at the expense of a negligibly larger space requirement. In the realizable case, we show that all those encodings can be computed efficiently. Finally, we generalize our results to the encoding of point configurations in higher dimension.
Cite as

Jean Cardinal, Timothy M. Chan, John Iacono, Stefan Langerman, and Aurélien Ooms. Subquadratic Encodings for Point Configurations. In 34th International Symposium on Computational Geometry (SoCG 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 99, pp. 20:1-20:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{cardinal_et_al:LIPIcs.SoCG.2018.20,
  author =	{Cardinal, Jean and Chan, Timothy M. and Iacono, John and Langerman, Stefan and Ooms, Aur\'{e}lien},
  title =	{{Subquadratic Encodings for Point Configurations}},
  booktitle =	{34th International Symposium on Computational Geometry (SoCG 2018)},
  pages =	{20:1--20:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-066-8},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{99},
  editor =	{Speckmann, Bettina and T\'{o}th, Csaba D.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2018.20},
  URN =		{urn:nbn:de:0030-drops-87337},
  doi =		{10.4230/LIPIcs.SoCG.2018.20},
  annote =	{Keywords: point configuration, order type, chirotope, succinct data structure}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2018.55
An Optimal Algorithm to Compute the Inverse Beacon Attraction Region

Authors: Irina Kostitsyna, Bahram Kouhestani, Stefan Langerman, and David Rappaport

Published in: LIPIcs, Volume 99, 34th International Symposium on Computational Geometry (SoCG 2018)

Abstract
The beacon model is a recent paradigm for guiding the trajectory of messages or small robotic agents in complex environments. A beacon is a fixed point with an attraction pull that can move points within a given polygon. Points move greedily towards a beacon: if unobstructed, they move along a straight line to the beacon, and otherwise they slide on the edges of the polygon. The Euclidean distance from a moving point to a beacon is monotonically decreasing. A given beacon attracts a point if the point eventually reaches the beacon. The problem of attracting all points within a polygon with a set of beacons can be viewed as a variation of the art gallery problem. Unlike most variations, the beacon attraction has the intriguing property of being asymmetric, leading to separate definitions of attraction region and inverse attraction region. The attraction region of a beacon is the set of points that it attracts. It is connected and can be computed in linear time for simple polygons. By contrast, it is known that the inverse attraction region of a point - the set of beacon positions that attract it - could have Omega(n) disjoint connected components. In this paper, we prove that, in spite of this, the total complexity of the inverse attraction region of a point in a simple polygon is linear, and present a O(n log n) time algorithm to construct it. This improves upon the best previous algorithm which required O(n^3) time and O(n^2) space. Furthermore we prove a matching Omega(n log n) lower bound for this task in the algebraic computation tree model of computation, even if the polygon is monotone.
Cite as

Irina Kostitsyna, Bahram Kouhestani, Stefan Langerman, and David Rappaport. An Optimal Algorithm to Compute the Inverse Beacon Attraction Region. In 34th International Symposium on Computational Geometry (SoCG 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 99, pp. 55:1-55:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{kostitsyna_et_al:LIPIcs.SoCG.2018.55,
  author =	{Kostitsyna, Irina and Kouhestani, Bahram and Langerman, Stefan and Rappaport, David},
  title =	{{An Optimal Algorithm to Compute the Inverse Beacon Attraction Region}},
  booktitle =	{34th International Symposium on Computational Geometry (SoCG 2018)},
  pages =	{55:1--55:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-066-8},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{99},
  editor =	{Speckmann, Bettina and T\'{o}th, Csaba D.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2018.55},
  URN =		{urn:nbn:de:0030-drops-87686},
  doi =		{10.4230/LIPIcs.SoCG.2018.55},
  annote =	{Keywords: beacon attraction, inverse attraction region, algorithm, optimal}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2017.13
Subquadratic Algorithms for Algebraic Generalizations of 3SUM

Authors: Luis Barba, Jean Cardinal, John Iacono, Stefan Langerman, Aurélien Ooms, and Noam Solomon

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

Abstract
The 3SUM problem asks if an input n-set of real numbers contains a triple whose sum is zero. We consider the 3POL problem, a natural generalization of 3SUM where we replace the sum function by a constant-degree polynomial in three variables. The motivations are threefold. Raz, Sharir, and de Zeeuw gave an O(n^{11/6}) upper bound on the number of solutions of trivariate polynomial equations when the solutions are taken from the cartesian product of three n-sets of real numbers. We give algorithms for the corresponding problem of counting such solutions. Grønlund and Pettie recently designed subquadratic algorithms for 3SUM. We generalize their results to 3POL. Finally, we shed light on the General Position Testing (GPT) problem: "Given n points in the plane, do three of them lie on a line?", a key problem in computational geometry. We prove that there exist bounded-degree algebraic decision trees of depth O(n^{12/7+e}) that solve 3POL, and that 3POL can be solved in O(n^2 (log log n)^{3/2} / (log n)^{1/2}) time in the real-RAM model. Among the possible applications of those results, we show how to solve GPT in subquadratic time when the input points lie on o((log n)^{1/6}/(log log n)^{1/2}) constant-degree polynomial curves. This constitutes the first step towards closing the major open question of whether GPT can be solved in subquadratic time. To obtain these results, we generalize important tools - such as batch range searching and dominance reporting - to a polynomial setting. We expect these new tools to be useful in other applications.
Cite as

Luis Barba, Jean Cardinal, John Iacono, Stefan Langerman, Aurélien Ooms, and Noam Solomon. Subquadratic Algorithms for Algebraic Generalizations of 3SUM. In 33rd International Symposium on Computational Geometry (SoCG 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 77, pp. 13:1-13:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{barba_et_al:LIPIcs.SoCG.2017.13,
  author =	{Barba, Luis and Cardinal, Jean and Iacono, John and Langerman, Stefan and Ooms, Aur\'{e}lien and Solomon, Noam},
  title =	{{Subquadratic Algorithms for Algebraic Generalizations of 3SUM}},
  booktitle =	{33rd International Symposium on Computational Geometry (SoCG 2017)},
  pages =	{13:1--13:15},
  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.2017.13},
  URN =		{urn:nbn:de:0030-drops-72214},
  doi =		{10.4230/LIPIcs.SoCG.2017.13},
  annote =	{Keywords: 3SUM, subquadratic algorithms, general position testing, range searching, dominance reporting, polynomial curves}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2017.21
Self-Approaching Paths in Simple Polygons

Authors: Prosenjit Bose, Irina Kostitsyna, and Stefan Langerman

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

Abstract
We study self-approaching paths that are contained in a simple polygon. A self-approaching path is a directed curve connecting two points such that the Euclidean distance between a point moving along the path and any future position does not increase, that is, for all points a, b, and c that appear in that order along the curve, |ac| >= |bc|. We analyze the properties, and present a characterization of shortest self-approaching paths. In particular, we show that a shortest self-approaching path connecting two points inside a polygon can be forced to follow a general class of non-algebraic curves. While this makes it difficult to design an exact algorithm, we show how to find a self-approaching path inside a polygon connecting two points under a model of computation which assumes that we can calculate involute curves of high order. Lastly, we provide an algorithm to test if a given simple polygon is self-approaching, that is, if there exists a self-approaching path for any two points inside the polygon.
Cite as

Prosenjit Bose, Irina Kostitsyna, and Stefan Langerman. Self-Approaching Paths in Simple Polygons. In 33rd International Symposium on Computational Geometry (SoCG 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 77, pp. 21:1-21:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{bose_et_al:LIPIcs.SoCG.2017.21,
  author =	{Bose, Prosenjit and Kostitsyna, Irina and Langerman, Stefan},
  title =	{{Self-Approaching Paths in Simple Polygons}},
  booktitle =	{33rd International Symposium on Computational Geometry (SoCG 2017)},
  pages =	{21:1--21:15},
  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.2017.21},
  URN =		{urn:nbn:de:0030-drops-72166},
  doi =		{10.4230/LIPIcs.SoCG.2017.21},
  annote =	{Keywords: self-approaching path, simple polygon, shortest path, involute curve}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2016.15
Incremental Voronoi diagrams

Authors: Sarah R. Allen, Luis Barba, John Iacono, and Stefan Langerman

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

Abstract
We study the amortized number of combinatorial changes (edge insertions and removals) needed to update the graph structure of the Voronoi diagram VD(S) (and several variants thereof) of a set S of n sites in the plane as sites are added to the set. To that effect, we define a general update operation for planar graphs that can be used to model the incremental construction of several variants of Voronoi diagrams as well as the incremental construction of an intersection of halfspaces in R^3. We show that the amortized number of edge insertions and removals needed to add a new site to the Voronoi diagram is O(n^(1/2)). A matching Omega(n^(1/2)) combinatorial lower bound is shown, even in the case where the graph representing the Voronoi diagram is a tree. This contrasts with the O(log(n)) upper bound of Aronov et al. [Aronov et al., in proc. of LATIN, 2006] for farthest-point Voronoi diagrams in the special case where points are inserted in clockwise order along their convex hull. We then present a semi-dynamic data structure that maintains the Voronoi diagram of a set S of n sites in convex position. This data structure supports the insertion of a new site p (and hence the addition of its Voronoi cell) and finds the asymptotically minimal number K of edge insertions and removals needed to obtain the diagram of S U (p) from the diagram of S, in time O(K polylog n) worst case, which is O(n^(1/2) polylog n) amortized by the aforementioned combinatorial result. The most distinctive feature of this data structure is that the graph of the Voronoi diagram is maintained explicitly at all times and can be retrieved and traversed in the natural way; this contrasts with other known data structures supporting nearest neighbor queries. Our data structure supports general search operations on the current Voronoi diagram, which can, for example, be used to perform point location queries in the cells of the current Voronoi diagram in O(log n) time, or to determine whether two given sites are neighbors in the Delaunay triangulation.
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Sarah R. Allen, Luis Barba, John Iacono, and Stefan Langerman. Incremental Voronoi diagrams. In 32nd International Symposium on Computational Geometry (SoCG 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 51, pp. 15:1-15:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{allen_et_al:LIPIcs.SoCG.2016.15,
  author =	{Allen, Sarah R. and Barba, Luis and Iacono, John and Langerman, Stefan},
  title =	{{Incremental Voronoi diagrams}},
  booktitle =	{32nd International Symposium on Computational Geometry (SoCG 2016)},
  pages =	{15:1--15:16},
  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.2016.15},
  URN =		{urn:nbn:de:0030-drops-59079},
  doi =		{10.4230/LIPIcs.SoCG.2016.15},
  annote =	{Keywords: Voronoi diagrams, dynamic data structures, Delaunay triangulation}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2016.50
A Quasilinear-Time Algorithm for Tiling the Plane Isohedrally with a Polyomino

Authors: Stefan Langerman and Andrew Winslow

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

Abstract
A plane tiling consisting of congruent copies of a shape is isohedral provided that for any pair of copies, there exists a symmetry of the tiling mapping one copy to the other. We give a O(n*log^2(n))-time algorithm for deciding if a polyomino with n edges can tile the plane isohedrally. This improves on the O(n^{18})-time algorithm of Keating and Vince and generalizes recent work by Brlek, Provençal, Fédou, and the second author.
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Stefan Langerman and Andrew Winslow. A Quasilinear-Time Algorithm for Tiling the Plane Isohedrally with a Polyomino. In 32nd International Symposium on Computational Geometry (SoCG 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 51, pp. 50:1-50:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{langerman_et_al:LIPIcs.SoCG.2016.50,
  author =	{Langerman, Stefan and Winslow, Andrew},
  title =	{{A Quasilinear-Time Algorithm for Tiling the Plane Isohedrally with a Polyomino}},
  booktitle =	{32nd International Symposium on Computational Geometry (SoCG 2016)},
  pages =	{50:1--50:15},
  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.2016.50},
  URN =		{urn:nbn:de:0030-drops-59423},
  doi =		{10.4230/LIPIcs.SoCG.2016.50},
  annote =	{Keywords: Plane tiling, polyomino, boundary word, isohedral}
}
Document
DOI: 10.4230/LIPIcs.FUN.2016.22
Threes!, Fives, 1024!, and 2048 are Hard

Authors: Stefan Langerman and Yushi Uno

Published in: LIPIcs, Volume 49, 8th International Conference on Fun with Algorithms (FUN 2016)

Abstract
We analyze the computational complexity of the popular computer games Threes!, 1024!, 2048 and many of their variants. For most known versions expanded to an m*n board, we show that it is NP-hard to decide whether a given starting position can be played to reach a specific (constant) tile value.
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Stefan Langerman and Yushi Uno. Threes!, Fives, 1024!, and 2048 are Hard. In 8th International Conference on Fun with Algorithms (FUN 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 49, pp. 22:1-22:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{langerman_et_al:LIPIcs.FUN.2016.22,
  author =	{Langerman, Stefan and Uno, Yushi},
  title =	{{Threes!, Fives, 1024!, and 2048 are Hard}},
  booktitle =	{8th International Conference on Fun with Algorithms (FUN 2016)},
  pages =	{22:1--22:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-005-7},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{49},
  editor =	{Demaine, Erik D. and Grandoni, Fabrizio},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FUN.2016.22},
  URN =		{urn:nbn:de:0030-drops-58661},
  doi =		{10.4230/LIPIcs.FUN.2016.22},
  annote =	{Keywords: algorithmic combinatorial game theory}
}
Document
DOI: 10.4230/LIPIcs.STACS.2013.281
Space-Time Trade-offs for Stack-Based Algorithms

Authors: Luis Barba, Matias Korman, Stefan Langerman, Rodrigo I. Silveira, and Kunihiko Sadakane

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

Abstract
In memory-constrained algorithms we have read-only access to the input, and the number of additional variables is limited. In this paper we introduce the compressed stack technique, a method that allows to transform algorithms whose space bottleneck is a stack into memory-constrained algorithms. Given an algorithm A that runs in O(n) time using a stack of length Theta(n), we can modify it so that it runs in O(n^2/2^s) time using a workspace of O(s) variables (for any s \in o(log n)) or O(n log n/log p)$ time using O(p log n/log p) variables (for any 2 <= p <= n). We also show how the technique can be applied to solve various geometric problems, namely computing the convex hull of a simple polygon, a triangulation of a monotone polygon, the shortest path between two points inside a monotone polygon, 1-dimensional pyramid approximation of a 1-dimensional vector, and the visibility profile of a point inside a simple polygon. Our approach exceeds or matches the best-known results for these problems in constant-workspace models (when they exist), and gives a trade-off between the size of the workspace and running time. To the best of our knowledge, this is the first general framework for obtaining memory-constrained algorithms.
Cite as

Luis Barba, Matias Korman, Stefan Langerman, Rodrigo I. Silveira, and Kunihiko Sadakane. Space-Time Trade-offs for Stack-Based Algorithms. In 30th International Symposium on Theoretical Aspects of Computer Science (STACS 2013). Leibniz International Proceedings in Informatics (LIPIcs), Volume 20, pp. 281-292, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2013)

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@InProceedings{barba_et_al:LIPIcs.STACS.2013.281,
  author =	{Barba, Luis and Korman, Matias and Langerman, Stefan and Silveira, Rodrigo I. and Sadakane, Kunihiko},
  title =	{{Space-Time Trade-offs for Stack-Based Algorithms}},
  booktitle =	{30th International Symposium on Theoretical Aspects of Computer Science (STACS 2013)},
  pages =	{281--292},
  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.2013.281},
  URN =		{urn:nbn:de:0030-drops-39411},
  doi =		{10.4230/LIPIcs.STACS.2013.281},
  annote =	{Keywords: space-time trade-off, constant workspace, stack algorithms}
}
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