2 Search Results for "Miracle, Sarah"

Document
RANDOM
DOI: 10.4230/LIPIcs.APPROX/RANDOM.2020.3
Iterated Decomposition of Biased Permutations via New Bounds on the Spectral Gap of Markov Chains

Authors: Sarah Miracle, Amanda Pascoe Streib, and Noah Streib

Published in: LIPIcs, Volume 176, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020)

Abstract
In this paper, we address a conjecture of Fill [Fill03] about the spectral gap of a nearest-neighbor transposition Markov chain ℳ_nn over biased permutations of [n]. Suppose we are given a set of input probabilities 𝒫 = {p_{i,j}} for all 1 ≤ i, j ≤ n with p_{i, j} = 1-p_{j, i}. The Markov chain ℳ_nn operates by uniformly choosing a pair of adjacent elements, i and j, and putting i ahead of j with probability p_{i,j} and j ahead of i with probability p_{j,i}, independent of their current ordering. We build on previous work [S. Miracle and A.P. Streib, 2018] that analyzed the spectral gap of ℳ_nn when the particles in [n] fall into k classes. There, the authors iteratively decomposed ℳ_nn into simpler chains, but incurred a multiplicative penalty of n^-2 for each application of the decomposition theorem of [Martin and Randall, 2000], leading to an exponentially small lower bound on the gap. We make progress by introducing a new complementary decomposition theorem. We introduce the notion of ε-orthogonality, and show that for ε-orthogonal chains, the complementary decomposition theorem may be iterated O(1/√ε) times while only giving away a constant multiplicative factor on the overall spectral gap. We show the decomposition given in [S. Miracle and A.P. Streib, 2018] of a related Markov chain ℳ_pp over k-class particle systems is 1/n²-orthogonal when the number of particles in each class is at least C log n, where C is a constant not depending on n. We then apply the complementary decomposition theorem iteratively n times to prove nearly optimal bounds on the spectral gap of ℳ_pp and to further prove the first inverse-polynomial bound on the spectral gap of ℳ_nn when k is as large as Θ(n/log n). The previous best known bound assumed k was at most a constant.
Cite as

Sarah Miracle, Amanda Pascoe Streib, and Noah Streib. Iterated Decomposition of Biased Permutations via New Bounds on the Spectral Gap of Markov Chains. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 176, pp. 3:1-3:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{miracle_et_al:LIPIcs.APPROX/RANDOM.2020.3,
  author =	{Miracle, Sarah and Streib, Amanda Pascoe and Streib, Noah},
  title =	{{Iterated Decomposition of Biased Permutations via New Bounds on the Spectral Gap of Markov Chains}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020)},
  pages =	{3:1--3:21},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-164-1},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{176},
  editor =	{Byrka, Jaros{\l}aw and Meka, Raghu},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2020.3},
  URN =		{urn:nbn:de:0030-drops-126069},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2020.3},
  annote =	{Keywords: Markov chains, Permutations, Decomposition, Spectral Gap, Iterated Decomposition}
}
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}
}
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