Search Results

Documents authored by Mütze, Torsten

Document
DOI: 10.4230/LIPIcs.GD.2024.30
Flips in Colorful Triangulations

Authors: Rohan Acharya, Torsten Mütze, and Francesco Verciani

Published in: LIPIcs, Volume 320, 32nd International Symposium on Graph Drawing and Network Visualization (GD 2024)

Abstract
The associahedron is the graph G_N that has as nodes all triangulations of a convex N-gon, and an edge between any two triangulations that differ in a flip operation. A flip removes an edge shared by two triangles and replaces it by the other diagonal of the resulting 4-gon. In this paper, we consider a large collection of induced subgraphs of G_N obtained by Ramsey-type colorability properties. Specifically, coloring the points of the N-gon red and blue alternatingly, we consider only colorful triangulations, namely triangulations in which every triangle has points in both colors, i.e., monochromatic triangles are forbidden. The resulting induced subgraph of G_N on colorful triangulations is denoted by F_N. We prove that F_N has a Hamilton cycle for all N ≥ 8, resolving a problem raised by Sagan, i.e., all colorful triangulations on N points can be listed so that any two cyclically consecutive triangulations differ in a flip. In fact, we prove that for an arbitrary fixed coloring pattern of the N points with at least 10 changes of color, the resulting subgraph of G_N on colorful triangulations (for that coloring pattern) admits a Hamilton cycle. We also provide an efficient algorithm for computing a Hamilton path in F_N that runs in time O(1) on average per generated node. This algorithm is based on a new and algorithmic construction of a tree rotation Gray code for listing all n-vertex k-ary trees that runs in time O(k) on average per generated tree.
Cite as

Rohan Acharya, Torsten Mütze, and Francesco Verciani. Flips in Colorful Triangulations. In 32nd International Symposium on Graph Drawing and Network Visualization (GD 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 320, pp. 30:1-30:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

Copy BibTex To Clipboard

@InProceedings{acharya_et_al:LIPIcs.GD.2024.30,
  author =	{Acharya, Rohan and M\"{u}tze, Torsten and Verciani, Francesco},
  title =	{{Flips in Colorful Triangulations}},
  booktitle =	{32nd International Symposium on Graph Drawing and Network Visualization (GD 2024)},
  pages =	{30:1--30:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-343-0},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{320},
  editor =	{Felsner, Stefan and Klein, Karsten},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.GD.2024.30},
  URN =		{urn:nbn:de:0030-drops-213143},
  doi =		{10.4230/LIPIcs.GD.2024.30},
  annote =	{Keywords: Flip graph, associahedron, triangulation, binary tree, vertex coloring, Hamilton cycle, Gray code}
}
Document
DOI: 10.4230/LIPIcs.ISAAC.2023.33
Pattern-Avoiding Binary Trees - Generation, Counting, and Bijections

Authors: Petr Gregor, Torsten Mütze, and Namrata

Published in: LIPIcs, Volume 283, 34th International Symposium on Algorithms and Computation (ISAAC 2023)

Abstract
In this paper we propose a notion of pattern avoidance in binary trees that generalizes the avoidance of contiguous tree patterns studied by Rowland and non-contiguous tree patterns studied by Dairyko, Pudwell, Tyner, and Wynn. Specifically, we propose algorithms for generating different classes of binary trees that are characterized by avoiding one or more of these generalized patterns. This is achieved by applying the recent Hartung-Hoang-Mütze-Williams generation framework, by encoding binary trees via permutations. In particular, we establish a one-to-one correspondence between tree patterns and certain mesh permutation patterns. We also conduct a systematic investigation of all tree patterns on at most 5 vertices, and we establish bijections between pattern-avoiding binary trees and other combinatorial objects, in particular pattern-avoiding lattice paths and set partitions.
Cite as

Petr Gregor, Torsten Mütze, and Namrata. Pattern-Avoiding Binary Trees - Generation, Counting, and Bijections. In 34th International Symposium on Algorithms and Computation (ISAAC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 283, pp. 33:1-33:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

Copy BibTex To Clipboard

@InProceedings{gregor_et_al:LIPIcs.ISAAC.2023.33,
  author =	{Gregor, Petr and M\"{u}tze, Torsten and Namrata},
  title =	{{Pattern-Avoiding Binary Trees - Generation, Counting, and Bijections}},
  booktitle =	{34th International Symposium on Algorithms and Computation (ISAAC 2023)},
  pages =	{33:1--33:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-289-1},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{283},
  editor =	{Iwata, Satoru and Kakimura, Naonori},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2023.33},
  URN =		{urn:nbn:de:0030-drops-193350},
  doi =		{10.4230/LIPIcs.ISAAC.2023.33},
  annote =	{Keywords: Generation, binary tree, pattern avoidance, permutation, bijection}
}
Document
DOI: 10.4230/LIPIcs.MFCS.2022.54
The Hamilton Compression of Highly Symmetric Graphs

Authors: Petr Gregor, Arturo Merino, and Torsten Mütze

Published in: LIPIcs, Volume 241, 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022)

Abstract
We say that a Hamilton cycle C = (x₁,…,x_n) in a graph G is k-symmetric, if the mapping x_i ↦ x_{i+n/k} for all i = 1,…,n, where indices are considered modulo n, is an automorphism of G. In other words, if we lay out the vertices x₁,…,x_n equidistantly on a circle and draw the edges of G as straight lines, then the drawing of G has k-fold rotational symmetry, i.e., all information about the graph is compressed into a 360^∘/k wedge of the drawing. We refer to the maximum k for which there exists a k-symmetric Hamilton cycle in G as the Hamilton compression of G. We investigate the Hamilton compression of four different families of vertex-transitive graphs, namely hypercubes, Johnson graphs, permutahedra and Cayley graphs of abelian groups. In several cases we determine their Hamilton compression exactly, and in other cases we provide close lower and upper bounds. The cycles we construct have a much higher compression than several classical Gray codes known from the literature. Our constructions also yield Gray codes for bitstrings, combinations and permutations that have few tracks and/or that are balanced.
Cite as

Petr Gregor, Arturo Merino, and Torsten Mütze. The Hamilton Compression of Highly Symmetric Graphs. In 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 241, pp. 54:1-54:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

Copy BibTex To Clipboard

@InProceedings{gregor_et_al:LIPIcs.MFCS.2022.54,
  author =	{Gregor, Petr and Merino, Arturo and M\"{u}tze, Torsten},
  title =	{{The Hamilton Compression of Highly Symmetric Graphs}},
  booktitle =	{47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022)},
  pages =	{54:1--54:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-256-3},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{241},
  editor =	{Szeider, Stefan and Ganian, Robert and Silva, Alexandra},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2022.54},
  URN =		{urn:nbn:de:0030-drops-168529},
  doi =		{10.4230/LIPIcs.MFCS.2022.54},
  annote =	{Keywords: Hamilton cycle, Gray code, hypercube, permutahedron, Johnson graph, Cayley graph, abelian group, vertex-transitive}
}
Document
DOI: 10.4230/LIPIcs.FUN.2022.22
All Your bases Are Belong to Us: Listing All Bases of a Matroid by Greedy Exchanges

Authors: Arturo Merino, Torsten Mütze, and Aaron Williams

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

Abstract
You provide us with a matroid and an initial base. We say that a subset of the bases "belongs to us" if we can visit each one via a sequence of base exchanges starting from the initial base. It is well-known that "All your base are belong to us". We refine this classic result by showing that it can be done by a simple greedy algorithm. For example, the spanning trees of a graph can be generated by edge exchanges using the following greedy rule: Minimize the larger label of an edge that enters or exits the current spanning tree and which creates a spanning tree that is new (i.e., hasn't been visited already). Amazingly, this works for any graph, for any labeling of its edges, for any initial spanning tree, and regardless of how you choose the edge with the smaller label in each exchange. Furthermore, by maintaining a small amount of information, we can generate each successive spanning tree without storing the previous trees. In general, for any matroid, we can greedily compute a listing of all its bases matroid such that consecutive bases differ by a base exchange. Our base exchange Gray codes apply a prefix-exchange on a prefix-minor of the matroid, and we can generate these orders using "history-free" iterative algorithms. More specifically, we store O(m) bits of data, and use O(m) time per base assuming O(1) time independence and coindependence oracles. Our work generalizes and extends a number of previous results. For example, the bases of the uniform matroid are combinations, and they belong to us using homogeneous transpositions via an Eades-McKay style order. Similarly, the spanning trees of fan graphs belong to us via face pivot Gray codes, which extends recent results of Cameron, Grubb, and Sawada [Pivot Gray Codes for the Spanning Trees of a Graph ft. the Fan, COCOON 2021].
Cite as

Arturo Merino, Torsten Mütze, and Aaron Williams. All Your bases Are Belong to Us: Listing All Bases of a Matroid by Greedy Exchanges. In 11th International Conference on Fun with Algorithms (FUN 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 226, pp. 22:1-22:28, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

Copy BibTex To Clipboard

@InProceedings{merino_et_al:LIPIcs.FUN.2022.22,
  author =	{Merino, Arturo and M\"{u}tze, Torsten and Williams, Aaron},
  title =	{{All Your bases Are Belong to Us: Listing All Bases of a Matroid by Greedy Exchanges}},
  booktitle =	{11th International Conference on Fun with Algorithms (FUN 2022)},
  pages =	{22:1--22:28},
  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.2022.22},
  URN =		{urn:nbn:de:0030-drops-159928},
  doi =		{10.4230/LIPIcs.FUN.2022.22},
  annote =	{Keywords: Matroids, base exchange, Gray codes, combinatorial generation, greedy algorithms, spanning trees}
}
Document
DOI: 10.4230/LIPIcs.STACS.2022.34
Star Transposition Gray Codes for Multiset Permutations

Authors: Petr Gregor, Torsten Mütze, and Arturo Merino

Published in: LIPIcs, Volume 219, 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022)

Abstract
Given integers k ≥ 2 and a_1,…,a_k ≥ 1, let a: = (a_1,…,a_k) and n: = a_1+⋯+a_k. An a-multiset permutation is a string of length n that contains exactly a_i symbols i for each i = 1,…,k. In this work we consider the problem of exhaustively generating all a-multiset permutations by star transpositions, i.e., in each step, the first entry of the string is transposed with any other entry distinct from the first one. This is a far-ranging generalization of several known results. For example, it is known that permutations (a_1 = ⋯ = a_k = 1) can be generated by star transpositions, while combinations (k = 2) can be generated by these operations if and only if they are balanced (a_1 = a_2), with the positive case following from the middle levels theorem. To understand the problem in general, we introduce a parameter Δ(a): = n-2max{a_1,…,a_k} that allows us to distinguish three different regimes for this problem. We show that if Δ(a) < 0, then a star transposition Gray code for a-multiset permutations does not exist. We also construct such Gray codes for the case Δ(a) > 0, assuming that they exist for the case Δ(a) = 0. For the case Δ(a) = 0 we present some partial positive results. Our proofs establish Hamilton-connectedness or Hamilton-laceability of the underlying flip graphs, and they answer several cases of a recent conjecture of Shen and Williams. In particular, we prove that the middle levels graph is Hamilton-laceable.
Cite as

Petr Gregor, Torsten Mütze, and Arturo Merino. Star Transposition Gray Codes for Multiset Permutations. In 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 219, pp. 34:1-34:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

Copy BibTex To Clipboard

@InProceedings{gregor_et_al:LIPIcs.STACS.2022.34,
  author =	{Gregor, Petr and M\"{u}tze, Torsten and Merino, Arturo},
  title =	{{Star Transposition Gray Codes for Multiset Permutations}},
  booktitle =	{39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022)},
  pages =	{34:1--34:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-222-8},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{219},
  editor =	{Berenbrink, Petra and Monmege, Benjamin},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2022.34},
  URN =		{urn:nbn:de:0030-drops-158448},
  doi =		{10.4230/LIPIcs.STACS.2022.34},
  annote =	{Keywords: Gray code, permutation, combination, transposition, Hamilton cycle}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2021.54
Efficient Generation of Rectangulations via Permutation Languages

Authors: Arturo Merino and Torsten Mütze

Published in: LIPIcs, Volume 189, 37th International Symposium on Computational Geometry (SoCG 2021)

Abstract
A generic rectangulation is a partition of a rectangle into finitely many interior-disjoint rectangles, such that no four rectangles meet in a point. In this work we present a versatile algorithmic framework for exhaustively generating a large variety of different classes of generic rectangulations. Our algorithms work under very mild assumptions, and apply to a large number of rectangulation classes known from the literature, such as generic rectangulations, diagonal rectangulations, 1-sided/area-universal, block-aligned rectangulations, and their guillotine variants. They also apply to classes of rectangulations that are characterized by avoiding certain patterns, and in this work we initiate a systematic investigation of pattern avoidance in rectangulations. Our generation algorithms are efficient, in some cases even loopless or constant amortized time, i.e., each new rectangulation is generated in constant time in the worst case or on average, respectively. Moreover, the Gray codes we obtain are cyclic, and sometimes provably optimal, in the sense that they correspond to a Hamilton cycle on the skeleton of an underlying polytope. These results are obtained by encoding rectangulations as permutations, and by applying our recently developed permutation language framework.
Cite as

Arturo Merino and Torsten Mütze. Efficient Generation of Rectangulations via Permutation Languages. In 37th International Symposium on Computational Geometry (SoCG 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 189, pp. 54:1-54:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

Copy BibTex To Clipboard

@InProceedings{merino_et_al:LIPIcs.SoCG.2021.54,
  author =	{Merino, Arturo and M\"{u}tze, Torsten},
  title =	{{Efficient Generation of Rectangulations via Permutation Languages}},
  booktitle =	{37th International Symposium on Computational Geometry (SoCG 2021)},
  pages =	{54:1--54:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-184-9},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{189},
  editor =	{Buchin, Kevin and Colin de Verdi\`{e}re, \'{E}ric},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2021.54},
  URN =		{urn:nbn:de:0030-drops-138534},
  doi =		{10.4230/LIPIcs.SoCG.2021.54},
  annote =	{Keywords: Exhaustive generation, Gray code, flip graph, polytope, generic rectangulation, diagonal rectangulation, cartogram, floorplan, permutation pattern}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2020.60
On the Central Levels Problem

Authors: Petr Gregor, Ondřej Mička, and Torsten Mütze

Published in: LIPIcs, Volume 168, 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)

Abstract
The central levels problem asserts that the subgraph of the (2m+1)-dimensional hypercube induced by all bitstrings with at least m+1-𝓁 many 1s and at most m+𝓁 many 1s, i.e., the vertices in the middle 2𝓁 levels, has a Hamilton cycle for any m ≥ 1 and 1 ≤ 𝓁 ≤ m+1. This problem was raised independently by Savage, by Gregor and Škrekovski, and by Shen and Williams, and it is a common generalization of the well-known middle levels problem, namely the case 𝓁 = 1, and classical binary Gray codes, namely the case 𝓁 = m+1. In this paper we present a general constructive solution of the central levels problem. Our results also imply the existence of optimal cycles through any sequence of 𝓁 consecutive levels in the n-dimensional hypercube for any n ≥ 1 and 1 ≤ 𝓁 ≤ n+1. Moreover, extending an earlier construction by Streib and Trotter, we construct a Hamilton cycle through the n-dimensional hypercube, n≥ 2, that contains the symmetric chain decomposition constructed by Greene and Kleitman in the 1970s, and we provide a loopless algorithm for computing the corresponding Gray code.
Cite as

Petr Gregor, Ondřej Mička, and Torsten Mütze. On the Central Levels Problem. In 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 168, pp. 60:1-60:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

Copy BibTex To Clipboard

@InProceedings{gregor_et_al:LIPIcs.ICALP.2020.60,
  author =	{Gregor, Petr and Mi\v{c}ka, Ond\v{r}ej and M\"{u}tze, Torsten},
  title =	{{On the Central Levels Problem}},
  booktitle =	{47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)},
  pages =	{60:1--60:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-138-2},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{168},
  editor =	{Czumaj, Artur and Dawar, Anuj and Merelli, Emanuela},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2020.60},
  URN =		{urn:nbn:de:0030-drops-124678},
  doi =		{10.4230/LIPIcs.ICALP.2020.60},
  annote =	{Keywords: Gray code, Hamilton cycle, hypercube, middle levels, symmetric chain decomposition}
}
Document
DOI: 10.4230/LIPIcs.ICALP.2018.66
Gray Codes and Symmetric Chains

Authors: Petr Gregor, Sven Jäger, Torsten Mütze, Joe Sawada, and Kaja Wille

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

Abstract
We consider the problem of constructing a cyclic listing of all bitstrings of length 2n+1 with Hamming weights in the interval [n+1-l,n+l], where 1 <= l <= n+1, by flipping a single bit in each step. This is a far-ranging generalization of the well-known middle two levels problem (l=1). We provide a solution for the case l=2 and solve a relaxed version of the problem for general values of l, by constructing cycle factors for those instances. Our proof uses symmetric chain decompositions of the hypercube, a concept known from the theory of posets, and we present several new constructions of such decompositions. In particular, we construct four pairwise edge-disjoint symmetric chain decompositions of the n-dimensional hypercube for any n >= 12.
Cite as

Petr Gregor, Sven Jäger, Torsten Mütze, Joe Sawada, and Kaja Wille. Gray Codes and Symmetric Chains. In 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 107, pp. 66:1-66:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

Copy BibTex To Clipboard

@InProceedings{gregor_et_al:LIPIcs.ICALP.2018.66,
  author =	{Gregor, Petr and J\"{a}ger, Sven and M\"{u}tze, Torsten and Sawada, Joe and Wille, Kaja},
  title =	{{Gray Codes and Symmetric Chains}},
  booktitle =	{45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)},
  pages =	{66:1--66:14},
  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.2018.66},
  URN =		{urn:nbn:de:0030-drops-90703},
  doi =		{10.4230/LIPIcs.ICALP.2018.66},
  annote =	{Keywords: Gray code, Hamilton cycle, hypercube, poset, symmetric chain}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2018.38
Rainbow Cycles in Flip Graphs

Authors: Stefan Felsner, Linda Kleist, Torsten Mütze, and Leon Sering

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

Abstract
The flip graph of triangulations has as vertices all triangulations of a convex n-gon, and an edge between any two triangulations that differ in exactly one edge. An r-rainbow cycle in this graph is a cycle in which every inner edge of the triangulation appears exactly r times. This notion of a rainbow cycle extends in a natural way to other flip graphs. In this paper we investigate the existence of r-rainbow cycles for three different flip graphs on classes of geometric objects: the aforementioned flip graph of triangulations of a convex n-gon, the flip graph of plane spanning trees on an arbitrary set of n points, and the flip graph of non-crossing perfect matchings on a set of n points in convex position. In addition, we consider two flip graphs on classes of non-geometric objects: the flip graph of permutations of {1,2,...,n } and the flip graph of k-element subsets of {1,2,...,n }. In each of the five settings, we prove the existence and non-existence of rainbow cycles for different values of r, n and k.
Cite as

Stefan Felsner, Linda Kleist, Torsten Mütze, and Leon Sering. Rainbow Cycles in Flip Graphs. In 34th International Symposium on Computational Geometry (SoCG 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 99, pp. 38:1-38:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

Copy BibTex To Clipboard

@InProceedings{felsner_et_al:LIPIcs.SoCG.2018.38,
  author =	{Felsner, Stefan and Kleist, Linda and M\"{u}tze, Torsten and Sering, Leon},
  title =	{{Rainbow Cycles in Flip Graphs}},
  booktitle =	{34th International Symposium on Computational Geometry (SoCG 2018)},
  pages =	{38:1--38:14},
  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.2018.38},
  URN =		{urn:nbn:de:0030-drops-87514},
  doi =		{10.4230/LIPIcs.SoCG.2018.38},
  annote =	{Keywords: flip graph, cycle, rainbow, Gray code, triangulation, spanning tree, matching, permutation, subset, combination}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2018.51
Distance-Preserving Graph Contractions

Authors: Aaron Bernstein, Karl Däubel, Yann Disser, Max Klimm, Torsten Mütze, and Frieder Smolny

Published in: LIPIcs, Volume 94, 9th Innovations in Theoretical Computer Science Conference (ITCS 2018)

Abstract
Compression and sparsification algorithms are frequently applied in a preprocessing step before analyzing or optimizing large networks/graphs. In this paper we propose and study a new framework contracting edges of a graph (merging vertices into super-vertices) with the goal of preserving pairwise distances as accurately as possible. Formally, given an edge-weighted graph, the contraction should guarantee that for any two vertices at distance d, the corresponding super-vertices remain at distance at least \varphi(d) in the contracted graph, where \varphi is a tolerance function bounding the permitted distance distortion. We present a comprehensive picture of the algorithmic complexity of the contraction problem for affine tolerance functions \varphi(x)=x/\alpha-\beta, where \alpha \geq 1 and \beta \geq 0 are arbitrary real-valued parameters. Specifically, we present polynomial-time algorithms for trees as well as hardness and inapproximability results for different graph classes, precisely separating easy and hard cases. Further we analyze the asymptotic behavior of the size of contractions, and find efficient algorithms to compute (non-optimal) contractions despite our hardness results.
Cite as

Aaron Bernstein, Karl Däubel, Yann Disser, Max Klimm, Torsten Mütze, and Frieder Smolny. Distance-Preserving Graph Contractions. In 9th Innovations in Theoretical Computer Science Conference (ITCS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 94, pp. 51:1-51:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

Copy BibTex To Clipboard

@InProceedings{bernstein_et_al:LIPIcs.ITCS.2018.51,
  author =	{Bernstein, Aaron and D\"{a}ubel, Karl and Disser, Yann and Klimm, Max and M\"{u}tze, Torsten and Smolny, Frieder},
  title =	{{Distance-Preserving Graph Contractions}},
  booktitle =	{9th Innovations in Theoretical Computer Science Conference (ITCS 2018)},
  pages =	{51:1--51:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-060-6},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{94},
  editor =	{Karlin, Anna R.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2018.51},
  URN =		{urn:nbn:de:0030-drops-83427},
  doi =		{10.4230/LIPIcs.ITCS.2018.51},
  annote =	{Keywords: distance oracle, contraction, spanner}
}
Document
DOI: 10.4230/LIPIcs.STACS.2017.40
Trimming and Gluing Gray Codes

Authors: Petr Gregor and Torsten Mütze

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

Abstract
We consider the algorithmic problem of generating each subset of [n]:={1,2,...,n} whose size is in some interval [k,l], 0 <= k <= l <= n, exactly once (cyclically) by repeatedly adding or removing a single element, or by exchanging a single element. For k=0 and l=n this is the classical problem of generating all 2^n subsets of [n] by element additions/removals, and for k=l this is the classical problem of generating all n over k subsets of [n] by element exchanges. We prove the existence of such cyclic minimum-change enumerations for a large range of values n, k, and l, improving upon and generalizing several previous results. For all these existential results we provide optimal algorithms to compute the corresponding Gray codes in constant time O(1) per generated set and space O(n). Rephrased in terms of graph theory, our results establish the existence of (almost) Hamilton cycles in the subgraph of the n-dimensional cube Q_n induced by all levels [k,l]. We reduce all remaining open cases to a generalized version of the middle levels conjecture, which asserts that the subgraph of Q_(2k+1) induced by all levels [k-c,k+1+c], c in {0, 1, ... , k}, has a Hamilton cycle. We also prove an approximate version of this conjecture, showing that this graph has a cycle that visits a (1-o(1))-fraction of all vertices.
Cite as

Petr Gregor and Torsten Mütze. Trimming and Gluing Gray Codes. In 34th Symposium on Theoretical Aspects of Computer Science (STACS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 66, pp. 40:1-40:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

Copy BibTex To Clipboard

@InProceedings{gregor_et_al:LIPIcs.STACS.2017.40,
  author =	{Gregor, Petr and M\"{u}tze, Torsten},
  title =	{{Trimming and Gluing Gray Codes}},
  booktitle =	{34th Symposium on Theoretical Aspects of Computer Science (STACS 2017)},
  pages =	{40:1--40:14},
  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.2017.40},
  URN =		{urn:nbn:de:0030-drops-69930},
  doi =		{10.4230/LIPIcs.STACS.2017.40},
  annote =	{Keywords: Gray code, subset, combination, loopless algorithm}
}
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail
© 2023-2024 Schloss Dagstuhl – LZI GmbH Imprint Privacy Contact