7 Search Results for "Allcock, Jonathan"

Document
DOI: 10.4230/LIPIcs.STACS.2026.57
Improving Lagarias-Odlyzko Algorithm for Average-Case Subset Sum: Modular Arithmetic Approach

Authors: Antoine Joux and Karol Węgrzycki

Published in: LIPIcs, Volume 364, 43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026)

Abstract
Lagarias and Odlyzko (J.ACM 1985) proposed a polynomial-time algorithm for solving "almost all" instances of the Subset Sum problem with n integers of size Ω(Γ_LO), where log₂(Γ_LO) > n² log₂(γ) and γ is a parameter of the lattice basis reduction (γ > √{4/3} for LLL). The algorithm of Lagarias and Odlyzko is a cornerstone of cryptography. However, the theoretical guarantee on the density of feasible instances has remained unimproved for almost 40 years. In this paper, we propose an algorithm that solves "almost all" instances of Subset Sum with integers of size Ω(√{Γ_LO}) after a single call to lattice reduction. Additionally, our approach allows solving the Subset Sum problem for multiple targets, whereas the previous method could handle only one target per call to lattice basis reduction. We introduce a modular arithmetic approach to the Subset Sum problem, leveraging lattice reduction to solve a linear system modulo a suitably large prime. By analyzing the lengths of the LLL-reduced basis vectors of both the primal and dual lattices simultaneously, we show that density guarantees can be improved.
Cite as

Antoine Joux and Karol Węgrzycki. Improving Lagarias-Odlyzko Algorithm for Average-Case Subset Sum: Modular Arithmetic Approach. In 43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 364, pp. 57:1-57:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{joux_et_al:LIPIcs.STACS.2026.57,
  author =	{Joux, Antoine and W\k{e}grzycki, Karol},
  title =	{{Improving Lagarias-Odlyzko Algorithm for Average-Case Subset Sum: Modular Arithmetic Approach}},
  booktitle =	{43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026)},
  pages =	{57:1--57:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-412-3},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{364},
  editor =	{Mahajan, Meena and Manea, Florin and McIver, Annabelle and Thắng, Nguy\~{ê}n Kim},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2026.57},
  URN =		{urn:nbn:de:0030-drops-255462},
  doi =		{10.4230/LIPIcs.STACS.2026.57},
  annote =	{Keywords: Average-Case Analysis, Subset Sum, Lattice Reduction, LLL}
}
Document
DOI: 10.4230/LIPIcs.ESA.2025.86
New Algorithms for Pigeonhole Equal Subset Sum

Authors: Ce Jin, Ryan Williams, and Stan Zhang

Published in: LIPIcs, Volume 351, 33rd Annual European Symposium on Algorithms (ESA 2025)

Abstract
We study the Pigeonhole Equal Subset Sum problem, which is a total-search variant of the Subset Sum problem introduced by Papadimitriou (1994): we are given a set of n positive integers {w₁,…,w_n} with the additional restriction that ∑_{i=1}^n w_i < 2ⁿ - 1, and want to find two different subsets A,B ⊆ [n] such that ∑_{i∈A} w_i = ∑_{i∈B} w_i. Very recently, Jin and Wu (ICALP 2024) gave a randomized algorithm solving Pigeonhole Equal Subset Sum in O^*(2^{0.4n}) time, beating the classical meet-in-the-middle algorithm with O^*(2^{n/2}) runtime. In this paper, we refine Jin and Wu’s techniques to improve the runtime even further to O^*(2^{n/3}).
Cite as

Ce Jin, Ryan Williams, and Stan Zhang. New Algorithms for Pigeonhole Equal Subset Sum. In 33rd Annual European Symposium on Algorithms (ESA 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 351, pp. 86:1-86:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{jin_et_al:LIPIcs.ESA.2025.86,
  author =	{Jin, Ce and Williams, Ryan and Zhang, Stan},
  title =	{{New Algorithms for Pigeonhole Equal Subset Sum}},
  booktitle =	{33rd Annual European Symposium on Algorithms (ESA 2025)},
  pages =	{86:1--86:12},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-395-9},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{351},
  editor =	{Benoit, Anne and Kaplan, Haim and Wild, Sebastian and Herman, Grzegorz},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2025.86},
  URN =		{urn:nbn:de:0030-drops-245541},
  doi =		{10.4230/LIPIcs.ESA.2025.86},
  annote =	{Keywords: pigeonhole principle, subset sums}
}
Document
DOI: 10.4230/LIPIcs.WADS.2025.24
Computational Geometry with Probabilistically Noisy Primitive Operations

Authors: David Eppstein, Michael T. Goodrich, and Vinesh Sridhar

Published in: LIPIcs, Volume 349, 19th International Symposium on Algorithms and Data Structures (WADS 2025)

Abstract
Much prior work has been done on designing computational geometry algorithms that handle input degeneracies, data imprecision, and arithmetic round-off errors. We take a new approach, inspired by the noisy sorting literature, and study computational geometry algorithms subject to noisy Boolean primitive operations in which, e.g., the comparison "is point q above line 𝓁?" returns the wrong answer with some fixed probability. We propose a novel technique called path-guided pushdown random walks that generalizes the results of noisy sorting. We apply this technique to solve point-location, plane-sweep, convex hulls in 2D and 3D, and Delaunay triangulations for noisy primitives in optimal time with high probability.
Cite as

David Eppstein, Michael T. Goodrich, and Vinesh Sridhar. Computational Geometry with Probabilistically Noisy Primitive Operations. In 19th International Symposium on Algorithms and Data Structures (WADS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 349, pp. 24:1-24:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{eppstein_et_al:LIPIcs.WADS.2025.24,
  author =	{Eppstein, David and Goodrich, Michael T. and Sridhar, Vinesh},
  title =	{{Computational Geometry with Probabilistically Noisy Primitive Operations}},
  booktitle =	{19th International Symposium on Algorithms and Data Structures (WADS 2025)},
  pages =	{24:1--24:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-398-0},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{349},
  editor =	{Morin, Pat and Oh, Eunjin},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.WADS.2025.24},
  URN =		{urn:nbn:de:0030-drops-242552},
  doi =		{10.4230/LIPIcs.WADS.2025.24},
  annote =	{Keywords: Computational geometry, noisy comparisons, random walks}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2025.9
On the Quantum Time Complexity of Divide and Conquer

Authors: Jonathan Allcock, Jinge Bao, Aleksandrs Belovs, Troy Lee, and Miklos Santha

Published in: LIPIcs, Volume 334, 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025)

Abstract
In this work, we initiate a systematic study of the time complexity of quantum divide and conquer (QD&C) algorithms for classical problems, and propose a general framework for their analysis. We establish generic conditions under which search and minimization problems with classical divide and conquer algorithms are amenable to quantum speedup, and apply these theorems to various problems involving strings, integers, and geometric objects. These include Longest Distinct Substring, Klee's Coverage, several optimization problems on stock transactions, and k-Increasing Subsequence. For most of these problems our quantum time upper bounds match the quantum query lower bounds, up to polylogarithmic factors. We give a structured framework for describing and classifying a wide variety of QD&C algorithms so that quantum speedups can be more easily identified and applied, and prove general statements on QD&C time complexity covering a range of cases, accounting for the time required for all operations. In particular, we explicitly account for memory access operations in the commonly used QRAM (read-only) and QRAG (read-write) models, which are assumed to take unit time in the query model, and which require careful analysis when involved in recursion. Our generic QD&C theorems have several nice features. 1) To apply them, it suffices to come up with a classical divide and conquer algorithm satisfying the conditions of the theorem. The quantization of the algorithm is then completely handled by the theorem. This can make it easier to find applications which admit a quantum speedup, and contrast with dynamic programming algorithms which can be difficult to quantize due to their highly sequential nature. 2) As these theorems give bounds on time complexity, they can be applied to a greater range of problems than those based on query complexity, e.g., where the best-known quantum algorithms require super-linear time. 3) It can handle minimization problems as well as boolean functions, which allows us to improve on the query complexity result of Childs et al. [Childs et al., 2025] for k-Increasing Subsequence by a logarithmic factor.
Cite as

Jonathan Allcock, Jinge Bao, Aleksandrs Belovs, Troy Lee, and Miklos Santha. On the Quantum Time Complexity of Divide and Conquer. In 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 334, pp. 9:1-9:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{allcock_et_al:LIPIcs.ICALP.2025.9,
  author =	{Allcock, Jonathan and Bao, Jinge and Belovs, Aleksandrs and Lee, Troy and Santha, Miklos},
  title =	{{On the Quantum Time Complexity of Divide and Conquer}},
  booktitle =	{52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025)},
  pages =	{9:1--9:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-372-0},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{334},
  editor =	{Censor-Hillel, Keren and Grandoni, Fabrizio and Ouaknine, Jo\"{e}l and Puppis, Gabriele},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2025.9},
  URN =		{urn:nbn:de:0030-drops-233863},
  doi =		{10.4230/LIPIcs.ICALP.2025.9},
  annote =	{Keywords: Quantum Computing, Quantum Algorithms, Divide and Conquer}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2025.51
Quantum Combine and Conquer and Its Applications to Sublinear Quantum Convex Hull and Maxima Set Construction

Authors: Shion Fukuzawa, Michael T. Goodrich, and Sandy Irani

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

Abstract
We introduce a quantum algorithm design paradigm called combine and conquer, which is a quantum version of the "marriage-before-conquest" technique of Kirkpatrick and Seidel. In a quantum combine-and-conquer algorithm, one performs the essential computation of the combine step of a quantum divide-and-conquer algorithm prior to the conquer step while avoiding recursion. This model is better suited for the quantum setting, due to its non-recursive nature. We show the utility of this approach by providing quantum algorithms for 2D maxima set and convex hull problems for sorted point sets running in Õ(√{nh}) time, w.h.p., where h is the size of the output.
Cite as

Shion Fukuzawa, Michael T. Goodrich, and Sandy Irani. Quantum Combine and Conquer and Its Applications to Sublinear Quantum Convex Hull and Maxima Set Construction. In 41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 51:1-51:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{fukuzawa_et_al:LIPIcs.SoCG.2025.51,
  author =	{Fukuzawa, Shion and Goodrich, Michael T. and Irani, Sandy},
  title =	{{Quantum Combine and Conquer and Its Applications to Sublinear Quantum Convex Hull and Maxima Set Construction}},
  booktitle =	{41st International Symposium on Computational Geometry (SoCG 2025)},
  pages =	{51:1--51:15},
  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.2025.51},
  URN =		{urn:nbn:de:0030-drops-232035},
  doi =		{10.4230/LIPIcs.SoCG.2025.51},
  annote =	{Keywords: quantum computing, computational geometry, divide and conquer, convex hulls, maxima sets}
}
Document
DOI: 10.4230/LIPIcs.STACS.2025.54
Multidimensional Quantum Walks, Recursion, and Quantum Divide & Conquer

Authors: Stacey Jeffery and Galina Pass

Published in: LIPIcs, Volume 327, 42nd International Symposium on Theoretical Aspects of Computer Science (STACS 2025)

Abstract
We introduce an object called a subspace graph that formalizes the technique of multidimensional quantum walks. Composing subspace graphs allows one to seamlessly combine quantum and classical reasoning, keeping a classical structure in mind, while abstracting quantum parts into subgraphs with simple boundaries as needed. As an example, we show how to combine a switching network with arbitrary quantum subroutines, to compute a composed function. As another application, we give a time-efficient implementation of quantum Divide & Conquer when the sub-problems are combined via a Boolean formula. We use this to quadratically speed up Savitch’s algorithm for directed st-connectivity.
Cite as

Stacey Jeffery and Galina Pass. Multidimensional Quantum Walks, Recursion, and Quantum Divide & Conquer. In 42nd International Symposium on Theoretical Aspects of Computer Science (STACS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 327, pp. 54:1-54:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{jeffery_et_al:LIPIcs.STACS.2025.54,
  author =	{Jeffery, Stacey and Pass, Galina},
  title =	{{Multidimensional Quantum Walks, Recursion, and Quantum Divide \& Conquer}},
  booktitle =	{42nd International Symposium on Theoretical Aspects of Computer Science (STACS 2025)},
  pages =	{54:1--54:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-365-2},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{327},
  editor =	{Beyersdorff, Olaf and Pilipczuk, Micha{\l} and Pimentel, Elaine and Thắng, Nguy\~{ê}n Kim},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2025.54},
  URN =		{urn:nbn:de:0030-drops-228791},
  doi =		{10.4230/LIPIcs.STACS.2025.54},
  annote =	{Keywords: Quantum Divide \& Conquer, Time-Efficient, Subspace Graphs, Quantum Walks, Switching Networks, Directed st-Connectivity}
}
Document
DOI: 10.4230/LIPIcs.ESA.2022.6
Classical and Quantum Algorithms for Variants of Subset-Sum via Dynamic Programming

Authors: Jonathan Allcock, Yassine Hamoudi, Antoine Joux, Felix Klingelhöfer, and Miklos Santha

Published in: LIPIcs, Volume 244, 30th Annual European Symposium on Algorithms (ESA 2022)

Abstract
Subset-Sum is an NP-complete problem where one must decide if a multiset of n integers contains a subset whose elements sum to a target value m. The best known classical and quantum algorithms run in time Õ(2^{n/2}) and Õ(2^{n/3}), respectively, based on the well-known meet-in-the-middle technique. Here we introduce a novel classical dynamic-programming-based data structure with applications to Subset-Sum and a number of variants, including Equal-Sums (where one seeks two disjoint subsets with the same sum), 2-Subset-Sum (a relaxed version of Subset-Sum where each item in the input set can be used twice in the summation), and Shifted-Sums, a generalization of both of these variants, where one seeks two disjoint subsets whose sums differ by some specified value. Given any modulus p, our data structure can be constructed in time O(np), after which queries can be made in time O(n) to the lists of subsets summing to any value modulo p. We use this data structure in combination with variable-time amplitude amplification and a new quantum pair finding algorithm, extending the quantum claw finding algorithm to the multiple solutions case, to give an O(2^{0.504n}) quantum algorithm for Shifted-Sums. This provides a notable improvement on the best known O(2^{0.773n}) classical running time established by Mucha et al. [Mucha et al., 2019]. We also study Pigeonhole Equal-Sums, a variant of Equal-Sums where the existence of a solution is guaranteed by the pigeonhole principle. For this problem we give faster classical and quantum algorithms with running time Õ(2^{n/2}) and Õ(2^{2n/5}), respectively.
Cite as

Jonathan Allcock, Yassine Hamoudi, Antoine Joux, Felix Klingelhöfer, and Miklos Santha. Classical and Quantum Algorithms for Variants of Subset-Sum via Dynamic Programming. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 6:1-6:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{allcock_et_al:LIPIcs.ESA.2022.6,
  author =	{Allcock, Jonathan and Hamoudi, Yassine and Joux, Antoine and Klingelh\"{o}fer, Felix and Santha, Miklos},
  title =	{{Classical and Quantum Algorithms for Variants of Subset-Sum via Dynamic Programming}},
  booktitle =	{30th Annual European Symposium on Algorithms (ESA 2022)},
  pages =	{6:1--6:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-247-1},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{244},
  editor =	{Chechik, Shiri and Navarro, Gonzalo and Rotenberg, Eva and Herman, Grzegorz},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2022.6},
  URN =		{urn:nbn:de:0030-drops-169444},
  doi =		{10.4230/LIPIcs.ESA.2022.6},
  annote =	{Keywords: Quantum algorithm, classical algorithm, dynamic programming, representation technique, subset-sum, equal-sum, shifted-sum}
}
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