8 Search Results for "Cannon, Sarah"

Document
RANDOM
DOI: 10.4230/LIPIcs.APPROX/RANDOM.2025.68
Rapid Mixing via Coupling Independence for Spin Systems with Unbounded Degree

Authors: Xiaoyu Chen and Weiming Feng

Published in: LIPIcs, Volume 353, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025)

Abstract
We develop a new framework to prove the mixing or relaxation time for the Glauber dynamics on spin systems with unbounded degree. It works for general spin systems including both 2-spin and multi-spin systems. As applications for this approach: - We prove the optimal O(n) relaxation time for the Glauber dynamics of random q-list-coloring on an n-vertices triangle-tree graph with maximum degree Δ such that q/Δ > α^⋆, where α^⋆ ≈ 1.763 is the unique positive solution of the equation α = exp(1/α). This improves the n^{1+o(1)} relaxation time for Glauber dynamics obtained by the previous work of Jain, Pham, and Vuong (2022). Besides, our framework can also give a near-linear time sampling algorithm under the same condition. - We prove the optimal O(n) relaxation time and near-optimal Õ(n) mixing time for the Glauber dynamics on hardcore models with parameter λ in balanced bipartite graphs such that λ < λ_c(Δ_L) for the max degree Δ_L in left part and the max degree Δ_R of right part satisfies Δ_R = O(Δ_L). This improves the previous result by Chen, Liu, and Yin (2023). At the heart of our proof is the notion of coupling independence which allows us to consider multiple vertices as a huge single vertex with exponentially large domain and do a "coarse-grained" local-to-global argument on spin systems. The technique works for general (multi) spin systems and helps us obtain some new comparison results for Glauber dynamics.
Cite as

Xiaoyu Chen and Weiming Feng. Rapid Mixing via Coupling Independence for Spin Systems with Unbounded Degree. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 353, pp. 68:1-68:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{chen_et_al:LIPIcs.APPROX/RANDOM.2025.68,
  author =	{Chen, Xiaoyu and Feng, Weiming},
  title =	{{Rapid Mixing via Coupling Independence for Spin Systems with Unbounded Degree}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025)},
  pages =	{68:1--68:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-397-3},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{353},
  editor =	{Ene, Alina and Chattopadhyay, Eshan},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2025.68},
  URN =		{urn:nbn:de:0030-drops-244345},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2025.68},
  annote =	{Keywords: coupling independence, Glauber dynamics, mixing times, relaxation times, spin systems}
}
Document
DOI: 10.4230/LIPIcs.WABI.2025.20
A k-mer-Based Estimator of the Substitution Rate Between Repetitive Sequences

Authors: Haonan Wu, Antonio Blanca, and Paul Medvedev

Published in: LIPIcs, Volume 344, 25th International Conference on Algorithms for Bioinformatics (WABI 2025)

Abstract
K-mer-based analysis of genomic data is ubiquitous, but the presence of repetitive k-mers continues to pose problems for the accuracy of many methods. For example, the Mash tool (Ondov et al. 2016) can accurately estimate the substitution rate between two low-repetitive sequences from their k-mer sketches; however, it is inaccurate on repetitive sequences such as the centromere of a human chromosome. Follow-up work by Blanca et al. (2021) has attempted to model how mutations affect k-mer sets based on strong assumptions that the sequence is non-repetitive and that mutations do not create spurious k-mer matches. However, the theoretical foundations for extending an estimator like Mash to work in the presence of repeat sequences have been lacking. In this work, we relax the non-repetitive assumption and propose a novel estimator for the mutation rate. We derive theoretical bounds on our estimator’s bias. Our experiments show that it remains accurate for repetitive genomic sequences, such as the alpha satellite higher order repeats in centromeres. We demonstrate our estimator’s robustness across diverse datasets and various ranges of the substitution rate and k-mer size. Finally, we show how sketching can be used to avoid dealing with large k-mer sets while retaining accuracy. Our software is available at https://github.com/medvedevgroup/Repeat-Aware_Substitution_Rate_Estimator.
Cite as

Haonan Wu, Antonio Blanca, and Paul Medvedev. A k-mer-Based Estimator of the Substitution Rate Between Repetitive Sequences. In 25th International Conference on Algorithms for Bioinformatics (WABI 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 344, pp. 20:1-20:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{wu_et_al:LIPIcs.WABI.2025.20,
  author =	{Wu, Haonan and Blanca, Antonio and Medvedev, Paul},
  title =	{{A k-mer-Based Estimator of the Substitution Rate Between Repetitive Sequences}},
  booktitle =	{25th International Conference on Algorithms for Bioinformatics (WABI 2025)},
  pages =	{20:1--20:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-386-7},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{344},
  editor =	{Brejov\'{a}, Bro\v{n}a and Patro, Rob},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.WABI.2025.20},
  URN =		{urn:nbn:de:0030-drops-239465},
  doi =		{10.4230/LIPIcs.WABI.2025.20},
  annote =	{Keywords: k-mers, sketching, mutation rates}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2025.41
Decomposing Multiparameter Persistence Modules

Authors: Tamal K. Dey, Jan Jendrysiak, and Michael Kerber

Published in: LIPIcs, Volume 332, 41st International Symposium on Computational Geometry (SoCG 2025)

Abstract
Dey and Xin (J.Appl.Comput.Top., 2022) describe an algorithm to decompose finitely presented multiparameter persistence modules using a matrix reduction algorithm. Their algorithm only works for modules whose generators and relations are distinctly graded. We extend their approach to work on all finitely presented modules and introduce several improvements that lead to significant speed-ups in practice. Our algorithm is fixed-parameter tractable with respect to the maximal number of relations of the same degree and with further optimisation we obtain an O(n³) time algorithm for interval-decomposable modules. In particular, we can decide interval-decomposability in this time. As a by-product to the proofs of correctness we develop a theory of parameter restriction for persistence modules. Our algorithm is implemented as a software library aida, the first to enable the decomposition of large inputs. We show its capabilities via extensive experimental evaluation.
Cite as

Tamal K. Dey, Jan Jendrysiak, and Michael Kerber. Decomposing Multiparameter Persistence Modules. In 41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 41:1-41:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{dey_et_al:LIPIcs.SoCG.2025.41,
  author =	{Dey, Tamal K. and Jendrysiak, Jan and Kerber, Michael},
  title =	{{Decomposing Multiparameter Persistence Modules}},
  booktitle =	{41st International Symposium on Computational Geometry (SoCG 2025)},
  pages =	{41:1--41:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-370-6},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{332},
  editor =	{Aichholzer, Oswin and Wang, Haitao},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2025.41},
  URN =		{urn:nbn:de:0030-drops-231939},
  doi =		{10.4230/LIPIcs.SoCG.2025.41},
  annote =	{Keywords: Topological Data Analysis, Multiparameter Persistence Modules, Persistence, Decomposition}
}
Document
Invited Talk
DOI: 10.4230/LIPIcs.SAND.2024.3
Algorithmic Programmable Matter: From Local Markov Chains to "Dumb" Robots (Invited Talk)

Authors: Andréa Werneck Richa

Published in: LIPIcs, Volume 292, 3rd Symposium on Algorithmic Foundations of Dynamic Networks (SAND 2024)

Abstract
Many programmable matter systems have been developed, including modular and swarm robotics, synthetic biology, DNA tiling, and smart materials. We describe programmable matter as an abstract collection of simple computational elements (particles) with limited memory that each execute distributed, local algorithms to self-organize and solve system-wide problems, such as movement, reconfiguration, and coordination. Self-organizing particle systems (SOPS) have many interesting potential applications like coating objects for monitoring and repair purposes, and forming nano-scale devices for surgery and molecular-scale electronic structures. We describe some of our work on the algorithmic foundations of programmable matter, investigating how macro-scale system behaviors can naturally emerge from local micro-behaviors by individual particles: We utilize tools from statistical physics and Markov chain analysis to translate Markov chains defined at a system level into distributed, local algorithms for SOPS that drive the desired emergent collective behavior for the problems of compression, separation, and foraging, among others. We further establish the notion of algorithmic matter, where we leverage standard binary computation, as well as physical characteristics of the robots and interactions with the environment in order to implement our micro-level algorithms in actual testbeds composed of robots that are not capable of any standard computation. We conclude by addressing full concurrency and asynchrony in SOPS. This is joint work with Dana Randall and Dan Goldman (Georgia Tech), Michael Strano (MIT), Todd Murphey (Northwestern), Josh Daymude (Arizona State University), Sarah Cannon (Claremont McKenna), Christian Scheideler (University of Paderborn) and their research labs.
Cite as

Andréa Werneck Richa. Algorithmic Programmable Matter: From Local Markov Chains to "Dumb" Robots (Invited Talk). In 3rd Symposium on Algorithmic Foundations of Dynamic Networks (SAND 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 292, p. 3:1, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{richa:LIPIcs.SAND.2024.3,
  author =	{Richa, Andr\'{e}a Werneck},
  title =	{{Algorithmic Programmable Matter: From Local Markov Chains to "Dumb" Robots}},
  booktitle =	{3rd Symposium on Algorithmic Foundations of Dynamic Networks (SAND 2024)},
  pages =	{3:1--3:1},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-315-7},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{292},
  editor =	{Casteigts, Arnaud and Kuhn, Fabian},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SAND.2024.3},
  URN =		{urn:nbn:de:0030-drops-198811},
  doi =		{10.4230/LIPIcs.SAND.2024.3},
  annote =	{Keywords: Programmable matter, Self-organizing particle systems, Biologically-inspired distributed algorithms, Local Markov chains, Emergent collective behavior}
}
Document
RANDOM
DOI: 10.4230/LIPIcs.APPROX/RANDOM.2022.4
Fast and Perfect Sampling of Subgraphs and Polymer Systems

Authors: Antonio Blanca, Sarah Cannon, and Will Perkins

Published in: LIPIcs, Volume 245, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2022)

Abstract
We give an efficient perfect sampling algorithm for weighted, connected induced subgraphs (or graphlets) of rooted, bounded degree graphs. Our algorithm utilizes a vertex-percolation process with a carefully chosen rejection filter and works under a percolation subcriticality condition. We show that this condition is optimal in the sense that the task of (approximately) sampling weighted rooted graphlets becomes impossible in finite expected time for infinite graphs and intractable for finite graphs when the condition does not hold. We apply our sampling algorithm as a subroutine to give near linear-time perfect sampling algorithms for polymer models and weighted non-rooted graphlets in finite graphs, two widely studied yet very different problems. This new perfect sampling algorithm for polymer models gives improved sampling algorithms for spin systems at low temperatures on expander graphs and unbalanced bipartite graphs, among other applications.
Cite as

Antonio Blanca, Sarah Cannon, and Will Perkins. Fast and Perfect Sampling of Subgraphs and Polymer Systems. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 245, pp. 4:1-4:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{blanca_et_al:LIPIcs.APPROX/RANDOM.2022.4,
  author =	{Blanca, Antonio and Cannon, Sarah and Perkins, Will},
  title =	{{Fast and Perfect Sampling of Subgraphs and Polymer Systems}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2022)},
  pages =	{4:1--4:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-249-5},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{245},
  editor =	{Chakrabarti, Amit and Swamy, Chaitanya},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2022.4},
  URN =		{urn:nbn:de:0030-drops-171261},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2022.4},
  annote =	{Keywords: Random Sampling, perfect sampling, graphlets, polymer models, spin systems, percolation}
}
Document
RANDOM
DOI: 10.4230/LIPIcs.APPROX-RANDOM.2019.54
A Local Stochastic Algorithm for Separation in Heterogeneous Self-Organizing Particle Systems

Authors: Sarah Cannon, Joshua J. Daymude, Cem Gökmen, Dana Randall, and Andréa W. Richa

Published in: LIPIcs, Volume 145, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019)

Abstract
We present and rigorously analyze the behavior of a distributed, stochastic algorithm for separation and integration in self-organizing particle systems, an abstraction of programmable matter. Such systems are composed of individual computational particles with limited memory, strictly local communication abilities, and modest computational power. We consider heterogeneous particle systems of two different colors and prove that these systems can collectively separate into different color classes or integrate, indifferent to color. We accomplish both behaviors with the same fully distributed, local, stochastic algorithm. Achieving separation or integration depends only on a single global parameter determining whether particles prefer to be next to other particles of the same color or not; this parameter is meant to represent external, environmental influences on the particle system. The algorithm is a generalization of a previous distributed, stochastic algorithm for compression (PODC '16) that can be viewed as a special case of separation where all particles have the same color. It is significantly more challenging to prove that the desired behavior is achieved in the heterogeneous setting, however, even in the bichromatic case we focus on. This requires combining several new techniques, including the cluster expansion from statistical physics, a new variant of the bridging argument of Miracle, Pascoe and Randall (RANDOM '11), the high-temperature expansion of the Ising model, and careful probabilistic arguments.
Cite as

Sarah Cannon, Joshua J. Daymude, Cem Gökmen, Dana Randall, and Andréa W. Richa. A Local Stochastic Algorithm for Separation in Heterogeneous Self-Organizing Particle Systems. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 145, pp. 54:1-54:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{cannon_et_al:LIPIcs.APPROX-RANDOM.2019.54,
  author =	{Cannon, Sarah and Daymude, Joshua J. and G\"{o}kmen, Cem and Randall, Dana and Richa, Andr\'{e}a W.},
  title =	{{A Local Stochastic Algorithm for Separation in Heterogeneous Self-Organizing Particle Systems}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019)},
  pages =	{54:1--54:22},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-125-2},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{145},
  editor =	{Achlioptas, Dimitris and V\'{e}gh, L\'{a}szl\'{o} A.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2019.54},
  URN =		{urn:nbn:de:0030-drops-112696},
  doi =		{10.4230/LIPIcs.APPROX-RANDOM.2019.54},
  annote =	{Keywords: Markov chains, Programmable matter, Cluster expansion}
}
Document
DOI: 10.4230/LIPIcs.APPROX-RANDOM.2017.34
Polynomial Mixing of the Edge-Flip Markov Chain for Unbiased Dyadic Tilings

Authors: Sarah Cannon, David A. Levin, and Alexandre Stauffer

Published in: LIPIcs, Volume 81, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017)

Abstract
We give the first polynomial upper bound on the mixing time of the edge-flip Markov chain for unbiased dyadic tilings, resolving an open problem originally posed by Janson, Randall, and Spencer in 2002. A dyadic tiling of size n is a tiling of the unit square by n non-overlapping dyadic rectangles, each of area 1/n, where a dyadic rectangle is any rectangle that can be written in the form [a2^{-s}, (a+1)2^{-s}] x [b2^{-t}, (b+1)2^{-t}] for a,b,s,t nonnegative integers. The edge-flip Markov chain selects a random edge of the tiling and replaces it with its perpendicular bisector if doing so yields a valid dyadic tiling. Specifically, we show that the relaxation time of the edge-flip Markov chain for dyadic tilings is at most O(n^{4.09}), which implies that the mixing time is at most O(n^{5.09}). We complement this by showing that the relaxation time is at least Omega(n^{1.38}), improving upon the previously best lower bound of Omega(n*log n) coming from the diameter of the chain.
Cite as

Sarah Cannon, David A. Levin, and Alexandre Stauffer. Polynomial Mixing of the Edge-Flip Markov Chain for Unbiased Dyadic Tilings. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 81, pp. 34:1-34:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{cannon_et_al:LIPIcs.APPROX-RANDOM.2017.34,
  author =	{Cannon, Sarah and Levin, David A. and Stauffer, Alexandre},
  title =	{{Polynomial Mixing of the Edge-Flip Markov Chain for Unbiased Dyadic Tilings}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017)},
  pages =	{34:1--34:21},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-044-6},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{81},
  editor =	{Jansen, Klaus and Rolim, Jos\'{e} D. P. and Williamson, David P. and Vempala, Santosh S.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2017.34},
  URN =		{urn:nbn:de:0030-drops-75830},
  doi =		{10.4230/LIPIcs.APPROX-RANDOM.2017.34},
  annote =	{Keywords: Random dyadic tilings, spectral gap, rapid mixing}
}
Document
DOI: 10.4230/LIPIcs.STACS.2013.172
Two Hands Are Better Than One (up to constant factors): Self-Assembly In The 2HAM vs. aTAM

Authors: Sarah Cannon, Erik D. Demaine, Martin L. Demaine, Sarah Eisenstat, Matthew J. Patitz, Robert T. Schweller, Scott M Summers, and Andrew Winslow

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

Abstract
We study the difference between the standard seeded model (aTAM) of tile self-assembly, and the "seedless" two-handed model of tile self-assembly (2HAM). Most of our results suggest that the two-handed model is more powerful. In particular, we show how to simulate any seeded system with a two-handed system that is essentially just a constant factor larger. We exhibit finite shapes with a busy-beaver separation in the number of distinct tiles required by seeded versus two-handed, and exhibit an infinite shape that can be constructed two-handed but not seeded. Finally, we show that verifying whether a given system uniquely assembles a desired supertile is co-NP-complete in the two-handed model, while it was known to be polynomially solvable in the seeded model.
Cite as

Sarah Cannon, Erik D. Demaine, Martin L. Demaine, Sarah Eisenstat, Matthew J. Patitz, Robert T. Schweller, Scott M Summers, and Andrew Winslow. Two Hands Are Better Than One (up to constant factors): Self-Assembly In The 2HAM vs. aTAM. In 30th International Symposium on Theoretical Aspects of Computer Science (STACS 2013). Leibniz International Proceedings in Informatics (LIPIcs), Volume 20, pp. 172-184, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2013)

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@InProceedings{cannon_et_al:LIPIcs.STACS.2013.172,
  author =	{Cannon, Sarah and Demaine, Erik D. and Demaine, Martin L. and Eisenstat, Sarah and Patitz, Matthew J. and Schweller, Robert T. and Summers, Scott M and Winslow, Andrew},
  title =	{{Two Hands Are Better Than One (up to constant factors): Self-Assembly In The 2HAM vs. aTAM}},
  booktitle =	{30th International Symposium on Theoretical Aspects of Computer Science (STACS 2013)},
  pages =	{172--184},
  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.2013.172},
  URN =		{urn:nbn:de:0030-drops-39321},
  doi =		{10.4230/LIPIcs.STACS.2013.172},
  annote =	{Keywords: abstract tile assembly model, hierarchical tile assembly model, two-handed tile assembly model, algorithmic self-assembly, DNA computing, biocomputing}
}
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