10 Search Results for "D�rfler, Julian"

Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2023.50
Connected k-Center and k-Diameter Clustering

Authors: Lukas Drexler, Jan Eube, Kelin Luo, Heiko Röglin, Melanie Schmidt, and Julian Wargalla

Published in: LIPIcs, Volume 261, 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)

Abstract
Motivated by an application from geodesy, we study the connected k-center problem and the connected k-diameter problem. These problems arise from the classical k-center and k-diameter problems by adding a side constraint. For the side constraint, we are given an undirected connectivity graph G on the input points, and a clustering is now only feasible if every cluster induces a connected subgraph in G. Usually in clustering problems one assumes that the clusters are pairwise disjoint. We study this case but additionally also the case that clusters are allowed to be non-disjoint. This can help to satisfy the connectivity constraints. Our main result is an O(1)-approximation algorithm for the disjoint connected k-center and k-diameter problem for Euclidean spaces of low dimension (constant d) and for metrics with constant doubling dimension. For general metrics, we get an O(log²k)-approximation. Our algorithms work by computing a non-disjoint connected clustering first and transforming it into a disjoint connected clustering. We complement these upper bounds by several upper and lower bounds for variations and special cases of the model.
Cite as

Lukas Drexler, Jan Eube, Kelin Luo, Heiko Röglin, Melanie Schmidt, and Julian Wargalla. Connected k-Center and k-Diameter Clustering. In 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 261, pp. 50:1-50:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{drexler_et_al:LIPIcs.ICALP.2023.50,
  author =	{Drexler, Lukas and Eube, Jan and Luo, Kelin and R\"{o}glin, Heiko and Schmidt, Melanie and Wargalla, Julian},
  title =	{{Connected k-Center and k-Diameter Clustering}},
  booktitle =	{50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)},
  pages =	{50:1--50:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-278-5},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{261},
  editor =	{Etessami, Kousha and Feige, Uriel and Puppis, Gabriele},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2023.50},
  URN =		{urn:nbn:de:0030-drops-181024},
  doi =		{10.4230/LIPIcs.ICALP.2023.50},
  annote =	{Keywords: Approximation algorithms, Clustering, Connectivity constraints}
}
Document
DOI: 10.4230/LIPIcs.CCC.2021.29
On the Complexity of Evaluating Highest Weight Vectors

Authors: Markus Bläser, Julian Dörfler, and Christian Ikenmeyer

Published in: LIPIcs, Volume 200, 36th Computational Complexity Conference (CCC 2021)

Abstract
Geometric complexity theory (GCT) is an approach towards separating algebraic complexity classes through algebraic geometry and representation theory. Originally Mulmuley and Sohoni proposed (SIAM J Comput 2001, 2008) to use occurrence obstructions to prove Valiant’s determinant vs permanent conjecture, but recently Bürgisser, Ikenmeyer, and Panova (Journal of the AMS 2019) proved this impossible. However, fundamental theorems of algebraic geometry and representation theory grant that every lower bound in GCT can be proved by the use of so-called highest weight vectors (HWVs). In the setting of interest in GCT (namely in the setting of polynomials) we prove the NP-hardness of the evaluation of HWVs in general, and we give efficient algorithms if the treewidth of the corresponding Young-tableau is small, where the point of evaluation is concisely encoded as a noncommutative algebraic branching program! In particular, this gives a large new class of separating functions that can be efficiently evaluated at points with low (border) Waring rank. As a structural side result we prove that border Waring rank is bounded from above by the ABP width complexity.
Cite as

Markus Bläser, Julian Dörfler, and Christian Ikenmeyer. On the Complexity of Evaluating Highest Weight Vectors. In 36th Computational Complexity Conference (CCC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 200, pp. 29:1-29:36, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{blaser_et_al:LIPIcs.CCC.2021.29,
  author =	{Bl\"{a}ser, Markus and D\"{o}rfler, Julian and Ikenmeyer, Christian},
  title =	{{On the Complexity of Evaluating Highest Weight Vectors}},
  booktitle =	{36th Computational Complexity Conference (CCC 2021)},
  pages =	{29:1--29:36},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-193-1},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{200},
  editor =	{Kabanets, Valentine},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2021.29},
  URN =		{urn:nbn:de:0030-drops-143036},
  doi =		{10.4230/LIPIcs.CCC.2021.29},
  annote =	{Keywords: Algebraic complexity theory, geometric complexity theory, algebraic branching program, Waring rank, border Waring rank, representation theory, highest weight vector, treewidth}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2021.60
A Parallel Batch-Dynamic Data Structure for the Closest Pair Problem

Authors: Yiqiu Wang, Shangdi Yu, Yan Gu, and Julian Shun

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

Abstract
We propose a theoretically-efficient and practical parallel batch-dynamic data structure for the closest pair problem. Our solution is based on a serial dynamic closest pair data structure by Golin et al., and supports batches of insertions and deletions in parallel. For a data set of size n, our data structure supports a batch of insertions or deletions of size m in O(m(1+log ((n+m)/m))) expected work and O(log (n+m)log^*(n+m)) depth with high probability, and takes linear space. The key techniques for achieving these bounds are a new work-efficient parallel batch-dynamic binary heap, and careful management of the computation across sets of points to minimize work and depth. We provide an optimized multicore implementation of our data structure using dynamic hash tables, parallel heaps, and dynamic k-d trees. Our experiments on a variety of synthetic and real-world data sets show that it achieves a parallel speedup of up to 38.57x (15.10x on average) on 48 cores with hyper-threading. In addition, we also implement and compare four parallel algorithms for static closest pair problem, for which we are not aware of any existing practical implementations. On 48 cores with hyper-threading, the static algorithms achieve up to 51.45x (29.42x on average) speedup, and Rabin’s algorithm performs the best on average. Comparing our dynamic algorithm to the fastest static algorithm, we find that it is advantageous to use the dynamic algorithm for batch sizes of up to 20% of the data set. As far as we know, our work is the first to experimentally evaluate parallel closest pair algorithms, in both the static and the dynamic settings.
Cite as

Yiqiu Wang, Shangdi Yu, Yan Gu, and Julian Shun. A Parallel Batch-Dynamic Data Structure for the Closest Pair Problem. In 37th International Symposium on Computational Geometry (SoCG 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 189, pp. 60:1-60:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{wang_et_al:LIPIcs.SoCG.2021.60,
  author =	{Wang, Yiqiu and Yu, Shangdi and Gu, Yan and Shun, Julian},
  title =	{{A Parallel Batch-Dynamic Data Structure for the Closest Pair Problem}},
  booktitle =	{37th International Symposium on Computational Geometry (SoCG 2021)},
  pages =	{60:1--60:16},
  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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2021.60},
  URN =		{urn:nbn:de:0030-drops-138594},
  doi =		{10.4230/LIPIcs.SoCG.2021.60},
  annote =	{Keywords: Closest Pair, Parallel Algorithms, Dynamic Algorithms, Experimental Algorithms}
}
Document
DOI: 10.4230/LIPIcs.ISAAC.2020.17
Complexity of Retrograde and Helpmate Chess Problems: Even Cooperative Chess Is Hard

Authors: Josh Brunner, Erik D. Demaine, Dylan Hendrickson, and Julian Wellman

Published in: LIPIcs, Volume 181, 31st International Symposium on Algorithms and Computation (ISAAC 2020)

Abstract
We prove PSPACE-completeness of two classic types of Chess problems when generalized to n × n boards. A "retrograde" problem asks whether it is possible for a position to be reached from a natural starting position, i.e., whether the position is "valid" or "legal" or "reachable". Most real-world retrograde Chess problems ask for the last few moves of such a sequence; we analyze the decision question which gets at the existence of an exponentially long move sequence. A "helpmate" problem asks whether it is possible for a player to become checkmated by any sequence of moves from a given position. A helpmate problem is essentially a cooperative form of Chess, where both players work together to cause a particular player to win; it also arises in regular Chess games, where a player who runs out of time (flags) loses only if they could ever possibly be checkmated from the current position (i.e., the helpmate problem has a solution). Our PSPACE-hardness reductions are from a variant of a puzzle game called Subway Shuffle.
Cite as

Josh Brunner, Erik D. Demaine, Dylan Hendrickson, and Julian Wellman. Complexity of Retrograde and Helpmate Chess Problems: Even Cooperative Chess Is Hard. In 31st International Symposium on Algorithms and Computation (ISAAC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 181, pp. 17:1-17:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{brunner_et_al:LIPIcs.ISAAC.2020.17,
  author =	{Brunner, Josh and Demaine, Erik D. and Hendrickson, Dylan and Wellman, Julian},
  title =	{{Complexity of Retrograde and Helpmate Chess Problems: Even Cooperative Chess Is Hard}},
  booktitle =	{31st International Symposium on Algorithms and Computation (ISAAC 2020)},
  pages =	{17:1--17:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-173-3},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{181},
  editor =	{Cao, Yixin and Cheng, Siu-Wing and Li, Minming},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2020.17},
  URN =		{urn:nbn:de:0030-drops-133618},
  doi =		{10.4230/LIPIcs.ISAAC.2020.17},
  annote =	{Keywords: hardness, board games, PSPACE}
}
Document
DOI: 10.4230/LIPIcs.FUN.2021.7
1 X 1 Rush Hour with Fixed Blocks Is PSPACE-Complete

Authors: Josh Brunner, Lily Chung, Erik D. Demaine, Dylan Hendrickson, Adam Hesterberg, Adam Suhl, and Avi Zeff

Published in: LIPIcs, Volume 157, 10th International Conference on Fun with Algorithms (FUN 2021) (2020)

Abstract
Consider n²-1 unit-square blocks in an n × n square board, where each block is labeled as movable horizontally (only), movable vertically (only), or immovable - a variation of Rush Hour with only 1 × 1 cars and fixed blocks. We prove that it is PSPACE-complete to decide whether a given block can reach the left edge of the board, by reduction from Nondeterministic Constraint Logic via 2-color oriented Subway Shuffle. By contrast, polynomial-time algorithms are known for deciding whether a given block can be moved by one space, or when each block either is immovable or can move both horizontally and vertically. Our result answers a 15-year-old open problem by Tromp and Cilibrasi, and strengthens previous PSPACE-completeness results for Rush Hour with vertical 1 × 2 and horizontal 2 × 1 movable blocks and 4-color Subway Shuffle.
Cite as

Josh Brunner, Lily Chung, Erik D. Demaine, Dylan Hendrickson, Adam Hesterberg, Adam Suhl, and Avi Zeff. 1 X 1 Rush Hour with Fixed Blocks Is PSPACE-Complete. In 10th International Conference on Fun with Algorithms (FUN 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 157, pp. 7:1-7:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{brunner_et_al:LIPIcs.FUN.2021.7,
  author =	{Brunner, Josh and Chung, Lily and Demaine, Erik D. and Hendrickson, Dylan and Hesterberg, Adam and Suhl, Adam and Zeff, Avi},
  title =	{{1 X 1 Rush Hour with Fixed Blocks Is PSPACE-Complete}},
  booktitle =	{10th International Conference on Fun with Algorithms (FUN 2021)},
  pages =	{7:1--7:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-145-0},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{157},
  editor =	{Farach-Colton, Martin and Prencipe, Giuseppe and Uehara, Ryuhei},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.FUN.2021.7},
  URN =		{urn:nbn:de:0030-drops-127681},
  doi =		{10.4230/LIPIcs.FUN.2021.7},
  annote =	{Keywords: puzzles, sliding blocks, PSPACE-hardness}
}
Document
DOI: 10.4230/LIPIcs.MFCS.2019.26
Counting Induced Subgraphs: An Algebraic Approach to #W[1]-hardness

Authors: Julian Dörfler, Marc Roth, Johannes Schmitt, and Philip Wellnitz

Published in: LIPIcs, Volume 138, 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)

Abstract
We study the problem #IndSub(Phi) of counting all induced subgraphs of size k in a graph G that satisfy the property Phi. This problem was introduced by Jerrum and Meeks and shown to be #W[1]-hard when parameterized by k for some families of properties Phi including, among others, connectivity [JCSS 15] and even- or oddness of the number of edges [Combinatorica 17]. Very recently [IPEC 18], two of the authors introduced a novel technique for the complexity analysis of #IndSub(Phi), inspired by the "topological approach to evasiveness" of Kahn, Saks and Sturtevant [FOCS 83] and the framework of graph motif parameters due to Curticapean, Dell and Marx [STOC 17], allowing them to prove hardness of a wide range of properties Phi. In this work, we refine this technique for graph properties that are non-trivial on edge-transitive graphs with a prime power number of edges. In particular, we fully classify the case of monotone bipartite graph properties: It is shown that, given any graph property Phi that is closed under the removal of vertices and edges, and that is non-trivial for bipartite graphs, the problem #IndSub(Phi) is #W[1]-hard and cannot be solved in time f(k)* n^{o(k)} for any computable function f, unless the Exponential Time Hypothesis fails. This holds true even if the input graph is restricted to be bipartite and counting is done modulo a fixed prime. A similar result is shown for properties that are closed under the removal of edges only.
Cite as

Julian Dörfler, Marc Roth, Johannes Schmitt, and Philip Wellnitz. Counting Induced Subgraphs: An Algebraic Approach to #W[1]-hardness. In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, pp. 26:1-26:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{dorfler_et_al:LIPIcs.MFCS.2019.26,
  author =	{D\"{o}rfler, Julian and Roth, Marc and Schmitt, Johannes and Wellnitz, Philip},
  title =	{{Counting Induced Subgraphs: An Algebraic Approach to #W\lbrack1\rbrack-hardness}},
  booktitle =	{44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)},
  pages =	{26:1--26:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-117-7},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{138},
  editor =	{Rossmanith, Peter and Heggernes, Pinar and Katoen, Joost-Pieter},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2019.26},
  URN =		{urn:nbn:de:0030-drops-109703},
  doi =		{10.4230/LIPIcs.MFCS.2019.26},
  annote =	{Keywords: counting complexity, edge-transitive graphs, graph homomorphisms, induced subgraphs, parameterized complexity}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2019.51
On Geometric Complexity Theory: Multiplicity Obstructions Are Stronger Than Occurrence Obstructions

Authors: Julian Dörfler, Christian Ikenmeyer, and Greta Panova

Published in: LIPIcs, Volume 132, 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)

Abstract
Geometric Complexity Theory as initiated by Mulmuley and Sohoni in two papers (SIAM J Comput 2001, 2008) aims to separate algebraic complexity classes via representation theoretic multiplicities in coordinate rings of specific group varieties. We provide the first toy setting in which a separation can be achieved for a family of polynomials via these multiplicities. Mulmuley and Sohoni’s papers also conjecture that the vanishing behavior of multiplicities would be sufficient to separate complexity classes (so-called occurrence obstructions). The existence of such strong occurrence obstructions has been recently disproven in 2016 in two successive papers, Ikenmeyer-Panova (Adv. Math.) and Bürgisser-Ikenmeyer-Panova (J. AMS). This raises the question whether separating group varieties via representation theoretic multiplicities is stronger than separating them via occurrences. We provide first finite settings where a separation via multiplicities can be achieved, while the separation via occurrences is provably impossible. These settings are surprisingly simple and natural: We study the variety of products of homogeneous linear forms (the so-called Chow variety) and the variety of polynomials of bounded border Waring rank (i.e. a higher secant variety of the Veronese variety). As a side result we prove a slight generalization of Hermite’s reciprocity theorem, which proves Foulkes' conjecture for a new infinite family of cases.
Cite as

Julian Dörfler, Christian Ikenmeyer, and Greta Panova. On Geometric Complexity Theory: Multiplicity Obstructions Are Stronger Than Occurrence Obstructions. In 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 132, pp. 51:1-51:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{dorfler_et_al:LIPIcs.ICALP.2019.51,
  author =	{D\"{o}rfler, Julian and Ikenmeyer, Christian and Panova, Greta},
  title =	{{On Geometric Complexity Theory: Multiplicity Obstructions Are Stronger Than Occurrence Obstructions}},
  booktitle =	{46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)},
  pages =	{51:1--51:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-109-2},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{132},
  editor =	{Baier, Christel and Chatzigiannakis, Ioannis and Flocchini, Paola and Leonardi, Stefano},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2019.51},
  URN =		{urn:nbn:de:0030-drops-106276},
  doi =		{10.4230/LIPIcs.ICALP.2019.51},
  annote =	{Keywords: Algebraic complexity theory, geometric complexity theory, Waring rank, plethysm coefficients, occurrence obstructions, multiplicity obstructions}
}
Document
DOI: 10.4230/DagSemProc.04351.12
Deadlocks and Dihomotopy in Mutual Exclusion Models

Authors: Martin Raussen

Published in: Dagstuhl Seminar Proceedings, Volume 4351, Spatial Representation: Discrete vs. Continuous Computational Models (2005)

Abstract
Parallel processes in concurrency theory can be modelled in a geometric framework. A convenient model are the Higher Dimensional Automata of V. Pratt and E. Goubault with cubical complexes as their mathematical description. More abstract models are given by (locally) partially ordered topological spaces, the directed ($d$-spaces) of M.Grandis and the flows of P. Gaucher. All models invite to use or modify ideas from algebraic topology, notably homotopy. In specific semaphore models for mutual exclusion, we have developed methods and algorithms that can detect deadlocks and unsafe regions and give information about essentially different schedules using higher dimensional ``geometric'' representations of the state space and executions (directed paths) along it.
Cite as

Martin Raussen. Deadlocks and Dihomotopy in Mutual Exclusion Models. In Spatial Representation: Discrete vs. Continuous Computational Models. Dagstuhl Seminar Proceedings, Volume 4351, pp. 1-8, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2005)

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@InProceedings{raussen:DagSemProc.04351.12,
  author =	{Raussen, Martin},
  title =	{{Deadlocks and Dihomotopy in Mutual Exclusion Models}},
  booktitle =	{Spatial Representation: Discrete vs. Continuous Computational Models},
  pages =	{1--8},
  series =	{Dagstuhl Seminar Proceedings (DagSemProc)},
  ISSN =	{1862-4405},
  year =	{2005},
  volume =	{4351},
  editor =	{Ralph Kopperman and Michael B. Smyth and Dieter Spreen and Julian Webster},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/DagSemProc.04351.12},
  URN =		{urn:nbn:de:0030-drops-1364},
  doi =		{10.4230/DagSemProc.04351.12},
  annote =	{Keywords: Mutual exclusion , deadlock detection , dihomotopy}
}
Document
DOI: 10.4230/DagSemProc.04351.13
Dihomotopy Classes of Dipaths in the Geometric Realization of a Cubical Set: from Discrete to Continuous and back again

Authors: Lisbeth Fajstrup

Published in: Dagstuhl Seminar Proceedings, Volume 4351, Spatial Representation: Discrete vs. Continuous Computational Models (2005)

Abstract
The geometric models of concurrency - Dijkstra's PV-models and V. Pratt's Higher Dimensional Automata - rely on a translation of discrete or algebraic information to geometry. In both these cases, the translation is the geometric realisation of a semi cubical complex, which is then a locally partially ordered space, an lpo space. The aim is to use the algebraic topology machinery, suitably adapted to the fact that there is a preferred time direction. Then the results - for instance dihomotopy classes of dipaths, which model the number of inequivalent computations should be used on the discrete model and give the corresponding discrete objects. We prove that this is in fact the case for the models considered: Each dipath is dihomottopic to a combinatorial dipath and if two combinatorial dipaths are dihomotopic, then they are combinatorially equivalent. Moreover, the notions of dihomotopy (LF., E. Goubault, M. Raussen) and d-homotopy (M. Grandis) are proven to be equivalent for these models - hence the Van Kampen theorem is available for dihomotopy. Finally we give an idea of how many spaces have a local po-structure given by cubes. The answer is, that any cubicalized space has such a structure after at most one subdivision. In particular, all triangulable spaces have a cubical local po-structure.
Cite as

Lisbeth Fajstrup. Dihomotopy Classes of Dipaths in the Geometric Realization of a Cubical Set: from Discrete to Continuous and back again. In Spatial Representation: Discrete vs. Continuous Computational Models. Dagstuhl Seminar Proceedings, Volume 4351, pp. 1-3, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2005)

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@InProceedings{fajstrup:DagSemProc.04351.13,
  author =	{Fajstrup, Lisbeth},
  title =	{{Dihomotopy Classes of Dipaths in the Geometric Realization of a Cubical Set: from Discrete to Continuous and back again}},
  booktitle =	{Spatial Representation: Discrete vs. Continuous Computational Models},
  pages =	{1--3},
  series =	{Dagstuhl Seminar Proceedings (DagSemProc)},
  ISSN =	{1862-4405},
  year =	{2005},
  volume =	{4351},
  editor =	{Ralph Kopperman and Michael B. Smyth and Dieter Spreen and Julian Webster},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/DagSemProc.04351.13},
  URN =		{urn:nbn:de:0030-drops-1328},
  doi =		{10.4230/DagSemProc.04351.13},
  annote =	{Keywords: Cubical Complex , Higher Dimensional Automaton , Ditopology}
}
Document
DOI: 10.4230/DagSemProc.04351.17
On Maximality of Compact Topologies

Authors: Martin Kovar

Published in: Dagstuhl Seminar Proceedings, Volume 4351, Spatial Representation: Discrete vs. Continuous Computational Models (2005)

Abstract
Using some advanced properties of the de Groot dual and some generalization of the Hofmann-Mislove theorem, we solve in the positive the question of D. E. Cameron: Is every compact topology contained in some maximal compact topology?
Cite as

Martin Kovar. On Maximality of Compact Topologies. In Spatial Representation: Discrete vs. Continuous Computational Models. Dagstuhl Seminar Proceedings, Volume 4351, pp. 1-10, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2005)

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@InProceedings{kovar:DagSemProc.04351.17,
  author =	{Kovar, Martin},
  title =	{{On Maximality of Compact Topologies}},
  booktitle =	{Spatial Representation: Discrete vs. Continuous Computational Models},
  pages =	{1--10},
  series =	{Dagstuhl Seminar Proceedings (DagSemProc)},
  ISSN =	{1862-4405},
  year =	{2005},
  volume =	{4351},
  editor =	{Ralph Kopperman and Michael B. Smyth and Dieter Spreen and Julian Webster},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/DagSemProc.04351.17},
  URN =		{urn:nbn:de:0030-drops-1182},
  doi =		{10.4230/DagSemProc.04351.17},
  annote =	{Keywords: de Groot dual , compact saturated set , wide Scott open filter , maximal compact topology}
}
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