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Documents authored by Grønlund, Allan

Document
DOI: 10.4230/LIPIcs.MFCS.2024.8
Sublinear Time Shortest Path in Expander Graphs

Authors: Noga Alon, Allan Grønlund, Søren Fuglede Jørgensen, and Kasper Green Larsen

Published in: LIPIcs, Volume 306, 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024)

Abstract
Computing a shortest path between two nodes in an undirected unweighted graph is among the most basic algorithmic tasks. Breadth first search solves this problem in linear time, which is clearly also a lower bound in the worst case. However, several works have shown how to solve this problem in sublinear time in expectation when the input graph is drawn from one of several classes of random graphs. In this work, we extend these results by giving sublinear time shortest path (and short path) algorithms for expander graphs. We thus identify a natural deterministic property of a graph (that is satisfied by typical random regular graphs) which suffices for sublinear time shortest paths. The algorithms are very simple, involving only bidirectional breadth first search and short random walks. We also complement our new algorithms by near-matching lower bounds.
Cite as

Noga Alon, Allan Grønlund, Søren Fuglede Jørgensen, and Kasper Green Larsen. Sublinear Time Shortest Path in Expander Graphs. In 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 306, pp. 8:1-8:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{alon_et_al:LIPIcs.MFCS.2024.8,
  author =	{Alon, Noga and Gr{\o}nlund, Allan and J{\o}rgensen, S{\o}ren Fuglede and Larsen, Kasper Green},
  title =	{{Sublinear Time Shortest Path in Expander Graphs}},
  booktitle =	{49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024)},
  pages =	{8:1--8:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-335-5},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{306},
  editor =	{Kr\'{a}lovi\v{c}, Rastislav and Ku\v{c}era, Anton{\'\i}n},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2024.8},
  URN =		{urn:nbn:de:0030-drops-205646},
  doi =		{10.4230/LIPIcs.MFCS.2024.8},
  annote =	{Keywords: Shortest Path, Expanders, Breadth First Search, Graph Algorithms}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2024.22
The NFA Acceptance Hypothesis: Non-Combinatorial and Dynamic Lower Bounds

Authors: Karl Bringmann, Allan Grønlund, Marvin Künnemann, and Kasper Green Larsen

Published in: LIPIcs, Volume 287, 15th Innovations in Theoretical Computer Science Conference (ITCS 2024)

Abstract
We pose the fine-grained hardness hypothesis that the textbook algorithm for the NFA Acceptance problem is optimal up to subpolynomial factors, even for dense NFAs and fixed alphabets. We show that this barrier appears in many variations throughout the algorithmic literature by introducing a framework of Colored Walk problems. These yield fine-grained equivalent formulations of the NFA Acceptance problem as problems concerning detection of an s-t-walk with a prescribed color sequence in a given edge- or node-colored graph. For NFA Acceptance on sparse NFAs (or equivalently, Colored Walk in sparse graphs), a tight lower bound under the Strong Exponential Time Hypothesis has been rediscovered several times in recent years. We show that our hardness hypothesis, which concerns dense NFAs, has several interesting implications: - It gives a tight lower bound for Context-Free Language Reachability. This proves conditional optimality for the class of 2NPDA-complete problems, explaining the cubic bottleneck of interprocedural program analysis. - It gives a tight (n+nm^{1/3})^{1-o(1)} lower bound for the Word Break problem on strings of length n and dictionaries of total size m. - It implies the popular OMv hypothesis. Since the NFA acceptance problem is a static (i.e., non-dynamic) problem, this provides a static reason for the hardness of many dynamic problems. Thus, a proof of the NFA Acceptance hypothesis would resolve several interesting barriers. Conversely, a refutation of the NFA Acceptance hypothesis may lead the way to attacking the current barriers observed for Context-Free Language Reachability, the Word Break problem and the growing list of dynamic problems proven hard under the OMv hypothesis.
Cite as

Karl Bringmann, Allan Grønlund, Marvin Künnemann, and Kasper Green Larsen. The NFA Acceptance Hypothesis: Non-Combinatorial and Dynamic Lower Bounds. In 15th Innovations in Theoretical Computer Science Conference (ITCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 287, pp. 22:1-22:25, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{bringmann_et_al:LIPIcs.ITCS.2024.22,
  author =	{Bringmann, Karl and Gr{\o}nlund, Allan and K\"{u}nnemann, Marvin and Larsen, Kasper Green},
  title =	{{The NFA Acceptance Hypothesis: Non-Combinatorial and Dynamic Lower Bounds}},
  booktitle =	{15th Innovations in Theoretical Computer Science Conference (ITCS 2024)},
  pages =	{22:1--22:25},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-309-6},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{287},
  editor =	{Guruswami, Venkatesan},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2024.22},
  URN =		{urn:nbn:de:0030-drops-195500},
  doi =		{10.4230/LIPIcs.ITCS.2024.22},
  annote =	{Keywords: Fine-grained complexity theory, non-deterministic finite automata}
}
Document
DOI: 10.4230/LIPIcs.STACS.2018.22
Upper and Lower Bounds for Dynamic Data Structures on Strings

Authors: Raphaël Clifford, Allan Grønlund, Kasper Green Larsen, and Tatiana Starikovskaya

Published in: LIPIcs, Volume 96, 35th Symposium on Theoretical Aspects of Computer Science (STACS 2018)

Abstract
We consider a range of simply stated dynamic data structure problems on strings. An update changes one symbol in the input and a query asks us to compute some function of the pattern of length m and a substring of a longer text. We give both conditional and unconditional lower bounds for variants of exact matching with wildcards, inner product, and Hamming distance computation via a sequence of reductions. As an example, we show that there does not exist an O(m^{1/2-epsilon}) time algorithm for a large range of these problems unless the online Boolean matrix-vector multiplication conjecture is false. We also provide nearly matching upper bounds for most of the problems we consider.
Cite as

Raphaël Clifford, Allan Grønlund, Kasper Green Larsen, and Tatiana Starikovskaya. Upper and Lower Bounds for Dynamic Data Structures on Strings. In 35th Symposium on Theoretical Aspects of Computer Science (STACS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 96, pp. 22:1-22:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{clifford_et_al:LIPIcs.STACS.2018.22,
  author =	{Clifford, Rapha\"{e}l and Gr{\o}nlund, Allan and Larsen, Kasper Green and Starikovskaya, Tatiana},
  title =	{{Upper and Lower Bounds for Dynamic Data Structures on Strings}},
  booktitle =	{35th Symposium on Theoretical Aspects of Computer Science (STACS 2018)},
  pages =	{22:1--22:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-062-0},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{96},
  editor =	{Niedermeier, Rolf and Vall\'{e}e, Brigitte},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2018.22},
  URN =		{urn:nbn:de:0030-drops-85088},
  doi =		{10.4230/LIPIcs.STACS.2018.22},
  annote =	{Keywords: exact pattern matching with wildcards, hamming distance, inner product, conditional lower bounds}
}
Document
DOI: 10.4230/LIPIcs.ICALP.2016.92
Towards Tight Lower Bounds for Range Reporting on the RAM

Authors: Allan Grønlund and Kasper Green Larsen

Published in: LIPIcs, Volume 55, 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)

Abstract
In the orthogonal range reporting problem, we are to preprocess a set of n points with integer coordinates on a UxU grid. The goal is to support reporting all k points inside an axis-aligned query rectangle. This is one of the most fundamental data structure problems in databases and computational geometry. Despite the importance of the problem its complexity remains unresolved in the word-RAM. On the upper bound side, three best tradeoffs exist, all derived by reducing range reporting to a ball-inheritance problem. Ball-inheritance is a problem that essentially encapsulates all previous attempts at solving range reporting in the word-RAM. In this paper we make progress towards closing the gap between the upper and lower bounds for range reporting by proving cell probe lower bounds for ball-inheritance. Our lower bounds are tight for a large range of parameters, excluding any further progress for range reporting using the ball-inheritance reduction.
Cite as

Allan Grønlund and Kasper Green Larsen. Towards Tight Lower Bounds for Range Reporting on the RAM. In 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 55, pp. 92:1-92:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{grnlund_et_al:LIPIcs.ICALP.2016.92,
  author =	{Gr{\o}nlund, Allan and Larsen, Kasper Green},
  title =	{{Towards Tight Lower Bounds for Range Reporting on the RAM}},
  booktitle =	{43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)},
  pages =	{92:1--92:12},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-013-2},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{55},
  editor =	{Chatzigiannakis, Ioannis and Mitzenmacher, Michael and Rabani, Yuval and Sangiorgi, Davide},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2016.92},
  URN =		{urn:nbn:de:0030-drops-61936},
  doi =		{10.4230/LIPIcs.ICALP.2016.92},
  annote =	{Keywords: Data Structures, Lower Bounds, Cell Probe Model, Range Reporting}
}
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