2 Search Results for "Huang, Shengyu"

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DOI: 10.4230/LIPIcs.SEA.2024.27

Improved Cut Strategy for Tensor Network Contraction Orders

Authors: Christoph Staudt, Mark Blacher, Julien Klaus, Farin Lippmann, and Joachim Giesen

Published in: LIPIcs, Volume 301, 22nd International Symposium on Experimental Algorithms (SEA 2024)

Abstract

In the field of quantum computing, simulating quantum systems on classical computers is crucial. Tensor networks are fundamental in simulating quantum systems. A tensor network is a collection of tensors, that need to be contracted into a result tensor. Tensor contraction is a generalization of matrix multiplication to higher order tensors. The contractions can be performed in different orders, and the order has a significant impact on the number of floating point operations (flops) needed to get the result tensor. It is known that finding an optimal contraction order is NP-hard. The current state-of-the-art approach for finding efficient contraction orders is to combinine graph partitioning with a greedy strategy. Although heavily used in practice, the current approach ignores so-called free indices, chooses node weights without regarding previous computations, and requires numerous hyperparameters that need to be tuned at runtime. In this paper, we address these shortcomings by developing a novel graph cut strategy. The proposed modifications yield contraction orders that significantly reduce the number of flops in the tensor contractions compared to the current state of the art. Moreover, by removing the need for hyperparameter tuning at runtime, our approach converges to an efficient solution faster, which reduces the required optimization time by at least an order of magnitude.

Cite as

Christoph Staudt, Mark Blacher, Julien Klaus, Farin Lippmann, and Joachim Giesen. Improved Cut Strategy for Tensor Network Contraction Orders. In 22nd International Symposium on Experimental Algorithms (SEA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 301, pp. 27:1-27:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{staudt_et_al:LIPIcs.SEA.2024.27,
  author =	{Staudt, Christoph and Blacher, Mark and Klaus, Julien and Lippmann, Farin and Giesen, Joachim},
  title =	{{Improved Cut Strategy for Tensor Network Contraction Orders}},
  booktitle =	{22nd International Symposium on Experimental Algorithms (SEA 2024)},
  pages =	{27:1--27:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-325-6},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{301},
  editor =	{Liberti, Leo},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SEA.2024.27},
  URN =		{urn:nbn:de:0030-drops-203924},
  doi =		{10.4230/LIPIcs.SEA.2024.27},
  annote =	{Keywords: tensor network, contraction order, graph partitioniong, quantum simulation}
}

Document

DOI: 10.4230/LIPIcs.STACS.2023.37

Approximate Selection with Unreliable Comparisons in Optimal Expected Time

Authors: Shengyu Huang, Chih-Hung Liu, and Daniel Rutschmann

Published in: LIPIcs, Volume 254, 40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023)

Abstract

Given n elements, an integer k ≤ n/2 and a parameter ε ≥ 1/n, we study the problem of selecting an element with rank in (k-nε, k+nε] using unreliable comparisons where the outcome of each comparison is incorrect independently with a constant error probability, and multiple comparisons between the same pair of elements are independent. In this fault model, the fundamental problems of finding the minimum, selecting the k-th smallest element and sorting have been shown to require Θ(n log 1/Q), Θ(n log k/Q) and Θ(n log n/Q) comparisons, respectively, to achieve success probability 1-Q [Uriel Feige et al., 1994]. Considering the increasing complexity of modern computing, it is of great interest to develop approximation algorithms that enable a trade-off between the solution quality and the number of comparisons. In particular, approximation algorithms would even be able to attain a sublinear number of comparisons. Very recently, Leucci and Liu [Stefano Leucci and Chih-Hung Liu, 2022] proved that the approximate minimum selection problem, which covers the case that k ≤ nε, requires expected Θ(ε^{-1} log 1/Q) comparisons, but the general case, i.e., for nε < k ≤ n/2, is still open. We develop a randomized algorithm that performs expected O(k/n ε^{-2} log 1/Q) comparisons to achieve success probability at least 1-Q. For k = n ε, the number of comparisons is O(ε^{-1} log 1/Q), matching Leucci and Liu’s result [Stefano Leucci and Chih-Hung Liu, 2022], whereas for k = n/2 (i.e., approximating the median), the number of comparisons is O(ε^{-2} log 1/Q). We also prove that even in the absence of comparison faults, any randomized algorithm with success probability at least 1-Q performs expected Ω(min{n, k/n ε^{-2} log 1/Q}) comparisons. As long as n is large enough, i.e., when n = Ω(k/n ε^{-2} log 1/Q), our lower bound demonstrates the optimality of our algorithm, which covers the possible range of attaining a sublinear number of comparisons. Surprisingly, for constant Q, our algorithm performs expected O(k/n ε^{-2}) comparisons, matching the best possible approximation algorithm in the absence of computation faults. In contrast, for the exact selection problem, the expected number of comparisons is Θ(n log k) with faults versus Θ(n) without faults. Our results also indicate a clear distinction between approximating the minimum and approximating the k-th smallest element, which holds even for the high probability guarantee, e.g., if k = n/2, Q = 1/n and ε = n^{-α} for α ∈ (0, 1/2), the asymptotic difference is almost quadratic, i.e., Θ̃(n^α) versus Θ̃(n^{2α}).

Cite as

Shengyu Huang, Chih-Hung Liu, and Daniel Rutschmann. Approximate Selection with Unreliable Comparisons in Optimal Expected Time. In 40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 254, pp. 37:1-37:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{huang_et_al:LIPIcs.STACS.2023.37,
  author =	{Huang, Shengyu and Liu, Chih-Hung and Rutschmann, Daniel},
  title =	{{Approximate Selection with Unreliable Comparisons in Optimal Expected Time}},
  booktitle =	{40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023)},
  pages =	{37:1--37:23},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-266-2},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{254},
  editor =	{Berenbrink, Petra and Bouyer, Patricia and Dawar, Anuj and Kant\'{e}, Mamadou Moustapha},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2023.37},
  URN =		{urn:nbn:de:0030-drops-176898},
  doi =		{10.4230/LIPIcs.STACS.2023.37},
  annote =	{Keywords: Approximate Selection, Unreliable Comparisons, Independent Faults}
}

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