9 Search Results for "Khan, Asif"

Document
Invited Paper
DOI: 10.4230/OASIcs.RW.2024/2025.1
Rule-Based Knowledge Graph Completion (Invited Paper)

Authors: Patrick Betz, Christian Meilicke, and Heiner Stuckenschmidt

Published in: OASIcs, Volume 138, Joint Proceedings of the 20th and 21st Reasoning Web Summer Schools (RW 2024 & RW 2025)

Abstract
The field of knowledge graph completion is concerned with augmenting knowledge graphs with missing information. Symbolic rule-based approaches are not only efficient and interpretable but also competitive with embedding-based methods in regard to predictive quality. Rule-based knowledge graph completion can be separated into two stages, the learning stage and the application stage, which are both individually challenging. In the learning stage, horn rules are mined from a given knowledge graph. Given the vast size of the space of all possible rules, the mining approach must select relevant rules effectively. In the application stage, the mined rules are used to make new predictions which are assigned with plausibility scores. These scores need to be set by aggregating individual confidence values of rules that have the same consequence. This tutorial covers the fundamental aspects required to build a symbolic rule-based approach for knowledge graph completion. It will discuss the different rule types, mining strategies, and how to effectively apply the rules in different scenarios. Finally, we discuss practical examples for rule application by using the Python-based PyClause library.
Cite as

Patrick Betz, Christian Meilicke, and Heiner Stuckenschmidt. Rule-Based Knowledge Graph Completion (Invited Paper). In Joint Proceedings of the 20th and 21st Reasoning Web Summer Schools (RW 2024 & RW 2025). Open Access Series in Informatics (OASIcs), Volume 138, pp. 1:1-1:45, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{betz_et_al:OASIcs.RW.2024/2025.1,
  author =	{Betz, Patrick and Meilicke, Christian and Stuckenschmidt, Heiner},
  title =	{{Rule-Based Knowledge Graph Completion}},
  booktitle =	{Joint Proceedings of the 20th and 21st Reasoning Web Summer Schools (RW 2024 \& RW 2025)},
  pages =	{1:1--1:45},
  series =	{Open Access Series in Informatics (OASIcs)},
  ISBN =	{978-3-95977-405-5},
  ISSN =	{2190-6807},
  year =	{2025},
  volume =	{138},
  editor =	{Artale, Alessandro and Bienvenu, Meghyn and Garc{\'\i}a, Yazm{\'\i}n Ib\'{a}\~{n}ez and Murlak, Filip},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/OASIcs.RW.2024/2025.1},
  URN =		{urn:nbn:de:0030-drops-250461},
  doi =		{10.4230/OASIcs.RW.2024/2025.1},
  annote =	{Keywords: Knowledge Graph Completion, Rule Learning, Symbolic AI}
}
Document
DOI: 10.4230/LIPIcs.DNA.31.4
Differentiable Programming of Indexed Chemical Reaction Networks and Reaction-Diffusion Systems

Authors: Inhoo Lee, Salvador Buse, and Erik Winfree

Published in: LIPIcs, Volume 347, 31st International Conference on DNA Computing and Molecular Programming (DNA 31) (2025)

Abstract
Many molecular systems are best understood in terms of prototypical species and reactions. The central dogma and related biochemistry are rife with examples: gene i is transcribed into RNA i, which is translated into protein i; kinase n phosphorylates substrate m; protein p dimerizes with protein q. Engineered nucleic acid systems also often have this form: oligonucleotide i hybridizes to complementary oligonucleotide j; signal strand n displaces the output of seesaw gate m; hairpin p triggers the opening of target q. When there are many variants of a small number of prototypes, it can be conceptually cleaner and computationally more efficient to represent the full system in terms of indexed species (e.g. for dimerization, M_p, D_pq) and indexed reactions (M_p + M_q → D_pq). Here, we formalize the Indexed Chemical Reaction Network (ICRN) model and describe a Python software package designed to simulate such systems in the well-mixed and reaction-diffusion settings, using a differentiable programming framework originally developed for large-scale neural network models, taking advantage of GPU acceleration when available. Notably, this framework makes it straightforward to train the models’ initial conditions and rate constants to optimize a target behavior, such as matching experimental data, performing a computation, or exhibiting spatial pattern formation. The natural map of indexed chemical reaction networks onto neural network formalisms provides a tangible yet general perspective for translating concepts and techniques from the theory and practice of neural computation into the design of biomolecular systems.
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Inhoo Lee, Salvador Buse, and Erik Winfree. Differentiable Programming of Indexed Chemical Reaction Networks and Reaction-Diffusion Systems. In 31st International Conference on DNA Computing and Molecular Programming (DNA 31). Leibniz International Proceedings in Informatics (LIPIcs), Volume 347, pp. 4:1-4:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{lee_et_al:LIPIcs.DNA.31.4,
  author =	{Lee, Inhoo and Buse, Salvador and Winfree, Erik},
  title =	{{Differentiable Programming of Indexed Chemical Reaction Networks and Reaction-Diffusion Systems}},
  booktitle =	{31st International Conference on DNA Computing and Molecular Programming (DNA 31)},
  pages =	{4:1--4:23},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-399-7},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{347},
  editor =	{Schaeffer, Josie and Zhang, Fei},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.DNA.31.4},
  URN =		{urn:nbn:de:0030-drops-238534},
  doi =		{10.4230/LIPIcs.DNA.31.4},
  annote =	{Keywords: Differentiable Programming, Chemical Reaction Networks, Reaction-Diffusion Systems}
}
Document
DOI: 10.4230/LIPIcs.DNA.31.10
Computing and Bounding Equilibrium Concentrations in Athermic Chemical Systems

Authors: Hamidreza Akef, Minki Hhan, and David Soloveichik

Published in: LIPIcs, Volume 347, 31st International Conference on DNA Computing and Molecular Programming (DNA 31) (2025)

Abstract
Computing equilibrium concentrations of molecular complexes is generally analytically intractable and requires numerical approaches. In this work we focus on the polymer-monomer level, where indivisible molecules (monomers) combine to form complexes (polymers). Rather than employing free-energy parameters for each polymer, we focus on the athermic setting where all interactions preserve enthalpy. This setting aligns with the strongly bonded (domain-based) regime in DNA nanotechnology when strands can bind in different ways, but always with maximum overall bonding - and is consistent with the saturated configurations in the Thermodynamic Binding Networks (TBNs) model. Within this context, we develop an iterative algorithm for assigning polymer concentrations to satisfy detailed-balance, where on-target (desired) polymers are in high concentrations and off-target (undesired) polymers are in low. Even if not directly executed, our algorithm provides effective insights into upper bounds on concentration of off-target polymers, connecting combinatorial arguments about discrete configurations such as those in the TBN model to real-valued concentrations. We conclude with an application of our method to decreasing leak in DNA logic and signal propagation. Our results offer a new framework for design and verification of equilibrium concentrations when configurations are distinguished by entropic forces.
Cite as

Hamidreza Akef, Minki Hhan, and David Soloveichik. Computing and Bounding Equilibrium Concentrations in Athermic Chemical Systems. In 31st International Conference on DNA Computing and Molecular Programming (DNA 31). Leibniz International Proceedings in Informatics (LIPIcs), Volume 347, pp. 10:1-10:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{akef_et_al:LIPIcs.DNA.31.10,
  author =	{Akef, Hamidreza and Hhan, Minki and Soloveichik, David},
  title =	{{Computing and Bounding Equilibrium Concentrations in Athermic Chemical Systems}},
  booktitle =	{31st International Conference on DNA Computing and Molecular Programming (DNA 31)},
  pages =	{10:1--10:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-399-7},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{347},
  editor =	{Schaeffer, Josie and Zhang, Fei},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.DNA.31.10},
  URN =		{urn:nbn:de:0030-drops-238595},
  doi =		{10.4230/LIPIcs.DNA.31.10},
  annote =	{Keywords: Equilibrium concentrations, Thermodynamic Binding Networks, Monomer-polymer model, Detailed balance}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2025.130
An Efficient Algorithm to Compute the Minimum Free Energy of Interacting Nucleic Acid Strands

Authors: Ahmed Shalaby and Damien Woods

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

Abstract
The information-encoding molecules RNA and DNA bind via base pairing to form an exponentially large set of secondary structures. Practitioners need algorithms to predict the most favoured structures, called minimum free energy (MFE) structures, or to compute a partition function that allows assigning a probability to any structure. MFE prediction is NP-hard in the presence pseudoknots - base pairings that violate a restricted planarity condition. However, for single-stranded unpseudoknotted structures, there are polynomial time dynamic programming algorithms. For multiple strands, the problem is significantly more complicated: Codon, Hajiaghayi and Thachuk [DNA27, 2021] proved it NP-hard for N bases and 𝒪(N) strands. Dirks, Bois, Schaeffer, Winfree and Pierce [SIAM Review, 2007] gave a polynomial time partition function algorithm for multiple (𝒪(1)) strands, now widely-used, however their technique did not generalise to MFE which they left open. We give an 𝒪(N⁴) time algorithm for unpseudoknotted multiple (𝒪(1)) strand MFE prediction, answering the open problem from Dirks et al. The challenge lies in considering the rotational symmetry of secondary structures, a global feature not immediately amenable to local subproblem decomposition used in dynamic programming. Our proof has two main technical contributions: First, a characterisation of symmetric secondary structures implying only quadratically many need to be considered when computing the rotational symmetry penalty. Second, that bound is leveraged by a backtracking algorithm to efficiently find the MFE in an exponential space of contenders.
Cite as

Ahmed Shalaby and Damien Woods. An Efficient Algorithm to Compute the Minimum Free Energy of Interacting Nucleic Acid Strands. In 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 334, pp. 130:1-130:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{shalaby_et_al:LIPIcs.ICALP.2025.130,
  author =	{Shalaby, Ahmed and Woods, Damien},
  title =	{{An Efficient Algorithm to Compute the Minimum Free Energy of Interacting Nucleic Acid Strands}},
  booktitle =	{52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025)},
  pages =	{130:1--130:20},
  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.2025.130},
  URN =		{urn:nbn:de:0030-drops-235071},
  doi =		{10.4230/LIPIcs.ICALP.2025.130},
  annote =	{Keywords: Minimum free energy, MFE, partition function, nucleic acid, DNA, RNA, secondary structure, computational complexity, algorithm analysis and design, dynamic programming}
}
Document
DOI: 10.4230/LIPIcs.FSTTCS.2024.4
The Parallel Dynamic Complexity of the Abelian Cayley Group Membership Problem

Authors: V. Arvind, Samir Datta, Asif Khan, Shivdutt Sharma, Yadu Vasudev, and Shankar Ram Vasudevan

Published in: LIPIcs, Volume 323, 44th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2024)

Abstract
Let G be a finite group given as input by its multiplication table. For a subset S ⊆ G and an element g ∈ G the Cayley Group Membership Problem (CGM) is to check if g belongs to the subgroup generated by S. While this problem is easily seen to be in polynomial time, pinpointing its parallel complexity has been of research interest over the years. Barrington et al [Barrington et al., 2001] have shown that for abelian groups the CGM problem can be solved in O(log log |G|) parallel time. In this paper we further explore the parallel complexity of the abelian CGM problem, with focus on the dynamic setting: the generating set S changes with insertions and deletions and the goal is to maintain a data structure that supports efficient membership queries to the subgroup ⟨S⟩. Though the static version of the CGM problem can be easily reduced to digraph reachability, the reduction does not carry over to the dynamic setting. We obtain the following results: 1) First, we consider the more general problem of Monoid Membership, where G is a monoid input by its multiplication table. When G is a commutative monoid we show there is a deterministic dynamic AC⁰ algorithm for membership testing that supports O(1) insertions and deletions in each step. 2) Building on the previous result we show that there is a dynamic randomized AC⁰ algorithm for abelian CGM that supports polylog(|G|) insertions/deletions to S in each step. 3) If the number of insertions/deletions is at most O(log n/log log n) then we obtain a deterministic dynamic AC⁰ algorithm for abelian CGM. 4) Applying these algorithms we obtain analogous results for the dynamic abelian Group Isomorphism. We can also handle sub-linearly many changes to the multiplication table for G, utilizing the hamming distance between multiplication tables of any two distinct groups.
Cite as

V. Arvind, Samir Datta, Asif Khan, Shivdutt Sharma, Yadu Vasudev, and Shankar Ram Vasudevan. The Parallel Dynamic Complexity of the Abelian Cayley Group Membership Problem. In 44th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 323, pp. 4:1-4:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{arvind_et_al:LIPIcs.FSTTCS.2024.4,
  author =	{Arvind, V. and Datta, Samir and Khan, Asif and Sharma, Shivdutt and Vasudev, Yadu and Vasudevan, Shankar Ram},
  title =	{{The Parallel Dynamic Complexity of the Abelian Cayley Group Membership Problem}},
  booktitle =	{44th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2024)},
  pages =	{4:1--4:23},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-355-3},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{323},
  editor =	{Barman, Siddharth and Lasota, S{\l}awomir},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2024.4},
  URN =		{urn:nbn:de:0030-drops-221939},
  doi =		{10.4230/LIPIcs.FSTTCS.2024.4},
  annote =	{Keywords: Dynamic Complexity, Group Theory, Cayley Group Membership, Abelian Group Isomorphism}
}
Document
DOI: 10.4230/LIPIcs.MFCS.2024.46
Query Maintenance Under Batch Changes with Small-Depth Circuits

Authors: Samir Datta, Asif Khan, Anish Mukherjee, Felix Tschirbs, Nils Vortmeier, and Thomas Zeume

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

Abstract
Which dynamic queries can be maintained efficiently? For constant-size changes, it is known that constant-depth circuits or, equivalently, first-order updates suffice for maintaining many important queries, among them reachability, tree isomorphism, and the word problem for context-free languages. In other words, these queries are in the dynamic complexity class DynFO. We show that most of the existing results for constant-size changes can be recovered for batch changes of polylogarithmic size if one allows circuits of depth 𝒪(log log n) or, equivalently, first-order updates that are iterated 𝒪(log log n) times.
Cite as

Samir Datta, Asif Khan, Anish Mukherjee, Felix Tschirbs, Nils Vortmeier, and Thomas Zeume. Query Maintenance Under Batch Changes with Small-Depth Circuits. In 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 306, pp. 46:1-46:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{datta_et_al:LIPIcs.MFCS.2024.46,
  author =	{Datta, Samir and Khan, Asif and Mukherjee, Anish and Tschirbs, Felix and Vortmeier, Nils and Zeume, Thomas},
  title =	{{Query Maintenance Under Batch Changes with Small-Depth Circuits}},
  booktitle =	{49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024)},
  pages =	{46:1--46:16},
  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.2024.46},
  URN =		{urn:nbn:de:0030-drops-206027},
  doi =		{10.4230/LIPIcs.MFCS.2024.46},
  annote =	{Keywords: Dynamic complexity theory, parallel computation, dynamic algorithms}
}
Document
Position
DOI: 10.4230/TGDK.1.1.2
Large Language Models and Knowledge Graphs: Opportunities and Challenges

Authors: Jeff Z. Pan, Simon Razniewski, Jan-Christoph Kalo, Sneha Singhania, Jiaoyan Chen, Stefan Dietze, Hajira Jabeen, Janna Omeliyanenko, Wen Zhang, Matteo Lissandrini, Russa Biswas, Gerard de Melo, Angela Bonifati, Edlira Vakaj, Mauro Dragoni, and Damien Graux

Published in: TGDK, Volume 1, Issue 1 (2023): Special Issue on Trends in Graph Data and Knowledge. Transactions on Graph Data and Knowledge, Volume 1, Issue 1

Abstract
Large Language Models (LLMs) have taken Knowledge Representation - and the world - by storm. This inflection point marks a shift from explicit knowledge representation to a renewed focus on the hybrid representation of both explicit knowledge and parametric knowledge. In this position paper, we will discuss some of the common debate points within the community on LLMs (parametric knowledge) and Knowledge Graphs (explicit knowledge) and speculate on opportunities and visions that the renewed focus brings, as well as related research topics and challenges.
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Jeff Z. Pan, Simon Razniewski, Jan-Christoph Kalo, Sneha Singhania, Jiaoyan Chen, Stefan Dietze, Hajira Jabeen, Janna Omeliyanenko, Wen Zhang, Matteo Lissandrini, Russa Biswas, Gerard de Melo, Angela Bonifati, Edlira Vakaj, Mauro Dragoni, and Damien Graux. Large Language Models and Knowledge Graphs: Opportunities and Challenges. In Special Issue on Trends in Graph Data and Knowledge. Transactions on Graph Data and Knowledge (TGDK), Volume 1, Issue 1, pp. 2:1-2:38, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@Article{pan_et_al:TGDK.1.1.2,
  author =	{Pan, Jeff Z. and Razniewski, Simon and Kalo, Jan-Christoph and Singhania, Sneha and Chen, Jiaoyan and Dietze, Stefan and Jabeen, Hajira and Omeliyanenko, Janna and Zhang, Wen and Lissandrini, Matteo and Biswas, Russa and de Melo, Gerard and Bonifati, Angela and Vakaj, Edlira and Dragoni, Mauro and Graux, Damien},
  title =	{{Large Language Models and Knowledge Graphs: Opportunities and Challenges}},
  journal =	{Transactions on Graph Data and Knowledge},
  pages =	{2:1--2:38},
  year =	{2023},
  volume =	{1},
  number =	{1},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/TGDK.1.1.2},
  URN =		{urn:nbn:de:0030-drops-194766},
  doi =		{10.4230/TGDK.1.1.2},
  annote =	{Keywords: Large Language Models, Pre-trained Language Models, Knowledge Graphs, Ontology, Retrieval Augmented Language Models}
}
Document
Survey
DOI: 10.4230/TGDK.1.1.4
Knowledge Graph Embeddings: Open Challenges and Opportunities

Authors: Russa Biswas, Lucie-Aimée Kaffee, Michael Cochez, Stefania Dumbrava, Theis E. Jendal, Matteo Lissandrini, Vanessa Lopez, Eneldo Loza Mencía, Heiko Paulheim, Harald Sack, Edlira Kalemi Vakaj, and Gerard de Melo

Published in: TGDK, Volume 1, Issue 1 (2023): Special Issue on Trends in Graph Data and Knowledge. Transactions on Graph Data and Knowledge, Volume 1, Issue 1

Abstract
While Knowledge Graphs (KGs) have long been used as valuable sources of structured knowledge, in recent years, KG embeddings have become a popular way of deriving numeric vector representations from them, for instance, to support knowledge graph completion and similarity search. This study surveys advances as well as open challenges and opportunities in this area. For instance, the most prominent embedding models focus primarily on structural information. However, there has been notable progress in incorporating further aspects, such as semantics, multi-modal, temporal, and multilingual features. Most embedding techniques are assessed using human-curated benchmark datasets for the task of link prediction, neglecting other important real-world KG applications. Many approaches assume a static knowledge graph and are unable to account for dynamic changes. Additionally, KG embeddings may encode data biases and lack interpretability. Overall, this study provides an overview of promising research avenues to learn improved KG embeddings that can address a more diverse range of use cases.
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Russa Biswas, Lucie-Aimée Kaffee, Michael Cochez, Stefania Dumbrava, Theis E. Jendal, Matteo Lissandrini, Vanessa Lopez, Eneldo Loza Mencía, Heiko Paulheim, Harald Sack, Edlira Kalemi Vakaj, and Gerard de Melo. Knowledge Graph Embeddings: Open Challenges and Opportunities. In Special Issue on Trends in Graph Data and Knowledge. Transactions on Graph Data and Knowledge (TGDK), Volume 1, Issue 1, pp. 4:1-4:32, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@Article{biswas_et_al:TGDK.1.1.4,
  author =	{Biswas, Russa and Kaffee, Lucie-Aim\'{e}e and Cochez, Michael and Dumbrava, Stefania and Jendal, Theis E. and Lissandrini, Matteo and Lopez, Vanessa and Menc{\'\i}a, Eneldo Loza and Paulheim, Heiko and Sack, Harald and Vakaj, Edlira Kalemi and de Melo, Gerard},
  title =	{{Knowledge Graph Embeddings: Open Challenges and Opportunities}},
  journal =	{Transactions on Graph Data and Knowledge},
  pages =	{4:1--4:32},
  year =	{2023},
  volume =	{1},
  number =	{1},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/TGDK.1.1.4},
  URN =		{urn:nbn:de:0030-drops-194783},
  doi =		{10.4230/TGDK.1.1.4},
  annote =	{Keywords: Knowledge Graphs, KG embeddings, Link prediction, KG applications}
}
Document
DOI: 10.4230/LIPIcs.MFCS.2023.39
Dynamic Planar Embedding Is in DynFO

Authors: Samir Datta, Asif Khan, and Anish Mukherjee

Published in: LIPIcs, Volume 272, 48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023)

Abstract
Planar Embedding is a drawing of a graph on the plane such that the edges do not intersect each other except at the vertices. We know that testing the planarity of a graph and computing its embedding (if it exists), can efficiently be computed, both sequentially [John E. Hopcroft and Robert Endre Tarjan, 1974] and in parallel [Vijaya Ramachandran and John H. Reif, 1994], when the entire graph is presented as input. In the dynamic setting, the input graph changes one edge at a time through insertion and deletions and planarity testing/embedding has to be updated after every change. By storing auxilliary information we can improve the complexity of dynamic planarity testing/embedding over the obvious recomputation from scratch. In the sequential dynamic setting, there has been a series of works [David Eppstein et al., 1996; Giuseppe F. Italiano et al., 1993; Jacob Holm et al., 2018; Jacob Holm and Eva Rotenberg, 2020], culminating in the breakthrough result of polylog(n) sequential time (amortized) planarity testing algorithm of Holm and Rotenberg [Jacob Holm and Eva Rotenberg, 2020]. In this paper we study planar embedding through the lens of DynFO, a parallel dynamic complexity class introduced by Patnaik et al [Sushant Patnaik and Neil Immerman, 1997] (also [Guozhu Dong et al., 1995]). We show that it is possible to dynamically maintain whether an edge can be inserted to a planar graph without causing non-planarity in DynFO. We extend this to show how to maintain an embedding of a planar graph under both edge insertions and deletions, while rejecting edge insertions that violate planarity. Our main idea is to maintain embeddings of only the triconnected components and a special two-colouring of separating pairs that enables us to side-step cascading flips when embedding of a biconnected planar graph changes, a major issue for sequential dynamic algorithms [Jacob Holm and Eva Rotenberg, 2020; Jacob Holm and Eva Rotenberg, 2020].
Cite as

Samir Datta, Asif Khan, and Anish Mukherjee. Dynamic Planar Embedding Is in DynFO. In 48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 272, pp. 39:1-39:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{datta_et_al:LIPIcs.MFCS.2023.39,
  author =	{Datta, Samir and Khan, Asif and Mukherjee, Anish},
  title =	{{Dynamic Planar Embedding Is in DynFO}},
  booktitle =	{48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023)},
  pages =	{39:1--39:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-292-1},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{272},
  editor =	{Leroux, J\'{e}r\^{o}me and Lombardy, Sylvain and Peleg, David},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2023.39},
  URN =		{urn:nbn:de:0030-drops-185736},
  doi =		{10.4230/LIPIcs.MFCS.2023.39},
  annote =	{Keywords: Dynamic Complexity, Planar graphs, Planar embedding}
}
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