18 Search Results for "Thorup, Mikkel"

Document
Invited Talk
DOI: 10.4230/LIPIcs.ESA.2024.1
Recent Progress on Correlation Clustering: From Local Algorithms to Better Approximation Algorithms and Back (Invited Talk)

Authors: Vincent Cohen-Addad

Published in: LIPIcs, Volume 308, 32nd Annual European Symposium on Algorithms (ESA 2024)

Abstract
Correlation clustering is a classic model for clustering problems arising in machine learning and data mining. Given a set of data elements represented as vertices of a graph and pairwise similarity represented as edges, the goal is to find a partition of the vertex set so as to minimize the total number of edges across the parts plus the total number of non-edges within the parts. Introduced in the early 2000s [Bansal et al., 2004], correlation clustering has received a large amount of attention through the years. A natural linear programming relaxation was shown to have an integrality gap of at least 2 and at most 2.5 [Ailon et al., 2008] in 2005, and in 2015 at most 2.06 [Chawla et al., 2015]. In 2021, motivated by large-scale application new structural insights allowed to derive a simple, practical algorithm that achieved an O(1)-approximation in a variety of models (Massively Parallel, Sublinear, Streaming or Differentially-private) [Vincent Cohen{-}Addad et al., 2021; Cohen-Addad et al., 2022]. These new insights turned out to be a key building block in designing better algorithms: It serves as a pre-clustering of the input graph that enables algorithm with approximation guarantees significantly better than 2 [Vincent Cohen{-}Addad et al., 2023; Vincent Cohen{-}Addad et al., 2022]. It is a key component in the new algorithm that achieves a 1.44-approximation [Nairen Cao et al., 2024] and in the new local-search based 1.84-approximation for the Massively Parallel, Sublinear, and Streaming models [Vincent Cohen{-}Addad et al., 2024]. This talk will review the above recent development and what are the main open research directions. A collection of joint works with Nairen Cao, Silvio Lattanzi, Euiwoong Lee, Shi Li, David Rasmussen Lolck, Slobodan Mitrovic, Alantha Newman, Ashkan Norouzi-Fard, Nikos Parotsidis, Marcin Pilipczuk, Jakub Tarnawski, Mikkel Thorup, Lukas Vogl, Shuyi Yan, Hanwen Zhang.
Cite as

Vincent Cohen-Addad. Recent Progress on Correlation Clustering: From Local Algorithms to Better Approximation Algorithms and Back (Invited Talk). In 32nd Annual European Symposium on Algorithms (ESA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 308, pp. 1:1-1:2, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{cohenaddad:LIPIcs.ESA.2024.1,
  author =	{Cohen-Addad, Vincent},
  title =	{{Recent Progress on Correlation Clustering: From Local Algorithms to Better Approximation Algorithms and Back}},
  booktitle =	{32nd Annual European Symposium on Algorithms (ESA 2024)},
  pages =	{1:1--1:2},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-338-6},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{308},
  editor =	{Chan, Timothy and Fischer, Johannes and Iacono, John 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.2024.1},
  URN =		{urn:nbn:de:0030-drops-210728},
  doi =		{10.4230/LIPIcs.ESA.2024.1},
  annote =	{Keywords: Approximation Algorithms, Clustering, Local Model}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2024.101
On the Space Usage of Approximate Distance Oracles with Sub-2 Stretch

Authors: Tsvi Kopelowitz, Ariel Korin, and Liam Roditty

Published in: LIPIcs, Volume 297, 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)

Abstract
For an undirected unweighted graph G = (V,E) with n vertices and m edges, let d(u,v) denote the distance from u ∈ V to v ∈ V in G. An (α,β)-stretch approximate distance oracle (ADO) for G is a data structure that given u,v ∈ V returns in constant (or near constant) time a value dˆ(u,v) such that d(u,v) ≤ dˆ(u,v) ≤ α⋅ d(u,v) + β, for some reals α > 1, β. Thorup and Zwick [Mikkel Thorup and Uri Zwick, 2005] showed that one cannot beat stretch 3 with subquadratic space (in terms of n) for general graphs. Pǎtraşcu and Roditty [Mihai Pǎtraşcu and Liam Roditty, 2010] showed that one can obtain stretch 2 using O(m^{1/3}n^{4/3}) space, and so if m is subquadratic in n then the space usage is also subquadratic. Moreover, Pǎtraşcu and Roditty [Mihai Pǎtraşcu and Liam Roditty, 2010] showed that one cannot beat stretch 2 with subquadratic space even for graphs where m = Õ(n), based on the set-intersection hypothesis. In this paper we explore the conditions for which an ADO can beat stretch 2 while using subquadratic space. In particular, we show that if the maximum degree in G is Δ_G ≤ O(n^{1/k-ε}) for some 0 < ε ≤ 1/k, then there exists an ADO for G that uses Õ(n^{2-(kε)/3) space and has a (2,1-k)-stretch. For k = 2 this result implies a subquadratic sub-2 stretch ADO for graphs with Δ_G ≤ O(n^{1/2-ε}). Moreover, we prove a conditional lower bound, based on the set intersection hypothesis, which states that for any positive integer k ≤ log n, obtaining a sub-(k+2)/k stretch for graphs with Δ_G = Θ(n^{1/k}) requires Ω̃(n²) space. Thus, for graphs with maximum degree Θ(n^{1/2}), obtaining a sub-2 stretch requires Ω̃(n²) space.
Cite as

Tsvi Kopelowitz, Ariel Korin, and Liam Roditty. On the Space Usage of Approximate Distance Oracles with Sub-2 Stretch. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 101:1-101:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{kopelowitz_et_al:LIPIcs.ICALP.2024.101,
  author =	{Kopelowitz, Tsvi and Korin, Ariel and Roditty, Liam},
  title =	{{On the Space Usage of Approximate Distance Oracles with Sub-2 Stretch}},
  booktitle =	{51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)},
  pages =	{101:1--101:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-322-5},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{297},
  editor =	{Bringmann, Karl and Grohe, Martin and Puppis, Gabriele and Svensson, Ola},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2024.101},
  URN =		{urn:nbn:de:0030-drops-202443},
  doi =		{10.4230/LIPIcs.ICALP.2024.101},
  annote =	{Keywords: Graph algorithms, Approximate distance oracle, data structures, shortest path}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2023.6
Optimal Decremental Connectivity in Non-Sparse Graphs

Authors: Anders Aamand, Adam Karczmarz, Jakub Łącki, Nikos Parotsidis, Peter M. R. Rasmussen, and Mikkel Thorup

Published in: LIPIcs, Volume 261, 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)

Abstract
We present a dynamic algorithm for maintaining the connected and 2-edge-connected components in an undirected graph subject to edge deletions. The algorithm is Monte-Carlo randomized and processes any sequence of edge deletions in O(m + n poly log n) total time. Interspersed with the deletions, it can answer queries whether any two given vertices currently belong to the same (2-edge-)connected component in constant time. Our result is based on a general Monte-Carlo randomized reduction from decremental c-edge-connectivity to a variant of fully-dynamic c-edge-connectivity on a sparse graph. For non-sparse graphs with Ω(n poly log n) edges, our connectivity and 2-edge-connectivity algorithms handle all deletions in optimal linear total time, using existing algorithms for the respective fully-dynamic problems. This improves upon an O(m log (n² / m) + n poly log n)-time algorithm of Thorup [J.Alg. 1999], which runs in linear time only for graphs with Ω(n²) edges. Our constant amortized cost for edge deletions in decremental connectivity in non-sparse graphs should be contrasted with an Ω(log n/log log n) worst-case time lower bound in the decremental setting [Alstrup, Husfeldt, and Rauhe FOCS'98] as well as an Ω(log n) amortized time lower-bound in the fully-dynamic setting [Patrascu and Demaine STOC'04].
Cite as

Anders Aamand, Adam Karczmarz, Jakub Łącki, Nikos Parotsidis, Peter M. R. Rasmussen, and Mikkel Thorup. Optimal Decremental Connectivity in Non-Sparse Graphs. In 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 261, pp. 6:1-6:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{aamand_et_al:LIPIcs.ICALP.2023.6,
  author =	{Aamand, Anders and Karczmarz, Adam and {\L}\k{a}cki, Jakub and Parotsidis, Nikos and Rasmussen, Peter M. R. and Thorup, Mikkel},
  title =	{{Optimal Decremental Connectivity in Non-Sparse Graphs}},
  booktitle =	{50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)},
  pages =	{6:1--6:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-278-5},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{261},
  editor =	{Etessami, Kousha and Feige, Uriel 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.2023.6},
  URN =		{urn:nbn:de:0030-drops-180581},
  doi =		{10.4230/LIPIcs.ICALP.2023.6},
  annote =	{Keywords: decremental connectivity, dynamic connectivity}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2023.76
A Sparse Johnson-Lindenstrauss Transform Using Fast Hashing

Authors: Jakob Bæk Tejs Houen and Mikkel Thorup

Published in: LIPIcs, Volume 261, 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)

Abstract
The Sparse Johnson-Lindenstrauss Transform of Kane and Nelson (SODA 2012) provides a linear dimensionality-reducing map A ∈ ℝ^{m × u} in 𝓁₂ that preserves distances up to distortion of 1 + ε with probability 1 - δ, where m = O(ε^{-2} log 1/δ) and each column of A has O(ε m) non-zero entries. The previous analyses of the Sparse Johnson-Lindenstrauss Transform all assumed access to a Ω(log 1/δ)-wise independent hash function. The main contribution of this paper is a more general analysis of the Sparse Johnson-Lindenstrauss Transform with less assumptions on the hash function. We also show that the Mixed Tabulation hash function of Dahlgaard, Knudsen, Rotenberg, and Thorup (FOCS 2015) satisfies the conditions of our analysis, thus giving us the first analysis of a Sparse Johnson-Lindenstrauss Transform that works with a practical hash function.
Cite as

Jakob Bæk Tejs Houen and Mikkel Thorup. A Sparse Johnson-Lindenstrauss Transform Using Fast Hashing. In 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 261, pp. 76:1-76:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{houen_et_al:LIPIcs.ICALP.2023.76,
  author =	{Houen, Jakob B{\ae}k Tejs and Thorup, Mikkel},
  title =	{{A Sparse Johnson-Lindenstrauss Transform Using Fast Hashing}},
  booktitle =	{50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)},
  pages =	{76:1--76:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-278-5},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{261},
  editor =	{Etessami, Kousha and Feige, Uriel 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.2023.76},
  URN =		{urn:nbn:de:0030-drops-181281},
  doi =		{10.4230/LIPIcs.ICALP.2023.76},
  annote =	{Keywords: dimensionality reduction, hashing, concentration bounds, moment bounds}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2022.74
Understanding the Moments of Tabulation Hashing via Chaoses

Authors: Jakob Bæk Tejs Houen and Mikkel Thorup

Published in: LIPIcs, Volume 229, 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)

Abstract
Simple tabulation hashing dates back to Zobrist in 1970 and is defined as follows: Each key is viewed as c characters from some alphabet Σ, we have c fully random hash functions h₀, …, h_{c - 1} : Σ → {{0, …, 2^l - 1}}, and a key x = (x₀, …, x_{c - 1}) is hashed to h(x) = h₀(x₀) ⊕ … ⊕ h_{c - 1}(x_{c - 1}) where ⊕ is the bitwise XOR operation. The previous results on tabulation hashing by Pǎtraşcu and Thorup [J.ACM'11] and by Aamand et al. [STOC'20] focused on proving Chernoff-style tail bounds on hash-based sums, e.g., the number keys hashing to a given value, for simple tabulation hashing, but their bounds do not cover the entire tail. Thus their results cannot bound moments. The paper Dahlgaard et al. [FOCS'15] provides a bound on the moments of certain hash-based sums, but their bound only holds for constant moments, and we need logarithmic moments. Chaoses are random variables of the form ∑ a_{i₀, …, i_{c - 1}} X_{i₀} ⋅ … ⋅ X_{i_{c - 1}} where X_i are independent random variables. Chaoses are a well-studied concept from probability theory, and tight analysis has been proven in several instances, e.g., when the independent random variables are standard Gaussian variables and when the independent random variables have logarithmically convex tails. We notice that hash-based sums of simple tabulation hashing can be seen as a sum of chaoses that are not independent. This motivates us to use techniques from the theory of chaoses to analyze hash-based sums of simple tabulation hashing. In this paper, we obtain bounds for all the moments of hash-based sums for simple tabulation hashing which are tight up to constants depending only on c. In contrast with the previous attempts, our approach will mostly be analytical and does not employ intricate combinatorial arguments. The improved analysis of simple tabulation hashing allows us to obtain bounds for the moments of hash-based sums for the mixed tabulation hashing introduced by Dahlgaard et al. [FOCS'15]. With simple tabulation hashing, there are certain inputs for which the concentration is much worse than with fully random hashing. However, with mixed tabulation, we get logarithmic moment bounds that are only a constant factor worse than those with fully random hashing for any possible input. This is a strong addition to other powerful probabilistic properties of mixed tabulation hashing proved by Dahlgaard et al.
Cite as

Jakob Bæk Tejs Houen and Mikkel Thorup. Understanding the Moments of Tabulation Hashing via Chaoses. In 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 229, pp. 74:1-74:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{houen_et_al:LIPIcs.ICALP.2022.74,
  author =	{Houen, Jakob B{\ae}k Tejs and Thorup, Mikkel},
  title =	{{Understanding the Moments of Tabulation Hashing via Chaoses}},
  booktitle =	{49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)},
  pages =	{74:1--74:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-235-8},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{229},
  editor =	{Boja\'{n}czyk, Miko{\l}aj and Merelli, Emanuela and Woodruff, David P.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2022.74},
  URN =		{urn:nbn:de:0030-drops-164154},
  doi =		{10.4230/LIPIcs.ICALP.2022.74},
  annote =	{Keywords: hashing, concentration bounds, moment bounds}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2022.107
Approximate Triangle Counting via Sampling and Fast Matrix Multiplication

Authors: Jakub Tětek

Published in: LIPIcs, Volume 229, 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)

Abstract
There is a simple O(n³/{ε²T}) time algorithm for 1±ε-approximate triangle counting where T is the number of triangles in the graph and n the number of vertices. At the same time, one may count triangles exactly using fast matrix multiplication in time Õ(n^ω). Is it possible to get a negative dependency on the number of triangles T while retaining the state-of-the-art n^ω dependency on n? We answer this question positively by providing an algorithm which runs in time O({n^ω}/T^{ω-2})⋅poly(n^o(1)/ε). This is optimal in the sense that as long as the exponent of T is independent of n, T, it cannot be improved while retaining the dependency on n. Our algorithm improves upon the state of the art when T ≫ 1 and T ≪ n. We also consider the problem of approximate triangle counting in sparse graphs, parameterized by the number of edges m. The best known algorithm runs in time Õ_ε(m^{3/2}/T) [Eden et al., SIAM Journal on Computing, 2017]. An algorithm by Alon et al. [JACM, 1995] for exact triangle counting that runs in time Õ(m^{2ω/(ω + 1)}). We again get an algorithm whose complexity has a state-of-the-art dependency on m while having negative dependency on T. Specifically, our algorithm runs in time O({m^{2ω/(ω+1)}}/{T^{2(ω-1)/(ω+1)}}) ⋅ poly(n^o(1)/ε). This is again optimal in the sense that no better constant exponent of T is possible without worsening the dependency on m. This algorithm improves upon the state of the art when T ≫ 1 and T ≪ √m. In both cases, algorithms with time complexity matching query complexity lower bounds were known on some range of parameters. While those algorithms have optimal query complexity for the whole range of T, the time complexity departs from the query complexity and is no longer optimal (as we show) for T ≪ n and T ≪ √m, respectively. We focus on the time complexity in this range of T. To the best of our knowledge, this is the first paper considering the discrepancy between query and time complexity in graph parameter estimation.
Cite as

Jakub Tětek. Approximate Triangle Counting via Sampling and Fast Matrix Multiplication. In 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 229, pp. 107:1-107:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{tetek:LIPIcs.ICALP.2022.107,
  author =	{T\v{e}tek, Jakub},
  title =	{{Approximate Triangle Counting via Sampling and Fast Matrix Multiplication}},
  booktitle =	{49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)},
  pages =	{107:1--107:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-235-8},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{229},
  editor =	{Boja\'{n}czyk, Miko{\l}aj and Merelli, Emanuela and Woodruff, David P.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2022.107},
  URN =		{urn:nbn:de:0030-drops-164485},
  doi =		{10.4230/LIPIcs.ICALP.2022.107},
  annote =	{Keywords: Approximate triangle counting, Fast matrix multiplication, Sampling}
}
Document
Invited Paper
DOI: 10.4230/LIPIcs.SWAT.2022.3
Reconstructing the Tree of Life (Fitting Distances by Tree Metrics) (Invited Paper)

Authors: Mikkel Thorup

Published in: LIPIcs, Volume 227, 18th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2022)

Abstract
We consider the numerical taxonomy problem of fitting an S× S distance matrix D with a tree metric T. Here T is a weighted tree spanning S where the path lengths in T induce a metric on S. If there is a tree metric matching D exactly, then it is easily found. If there is no exact match, then for some k, we want to minimize the L_k norm of the errors, that is, pick T so as to minimize ‖D-T‖_k = (∑_{i,j ∈ S} |D(i,j)-T(i,j)|^k) ^{1/k}. This problem was raised in biology in the 1960s for k = 1,2. The biological interpretation is that T represents a possible evolution behind the species in S matching some measured distances in D. Sometimes, it is required that T is an ultrametric, meaning that all species are at the same distance from the root. An evolutionary tree induces a hierarchical classification of species and this is not just tied to biology. Medicine, ecology and linguistics are just some of the fields where this concept appears, and it is even an integral part of machine learning and data science. Fundamentally, if we can approximate distances with a tree, then they are much easier to reason about: many questions that are NP-hard for general metrics can be answered in linear time on tree metrics. In fact, humans have appreciated hierarchical classifications at least since Plato and Aristotle (350 BC). The numerical taxonomy problem is important in practice and many heuristics have been proposed. In this talk we will review the basic algorithmic theory, results and techniques, for the problem, including the most recent result from FOCS'21 [Vincent Cohen-Addad et al., 2021]. They paint a varied landscape with big differences between different moments, and with some very nice open problems remaining. - At STOC'93, Farach, Kannan, and Warnow [Martin Farach et al., 1995] proved that under L_∞, we can find the optimal ultrametric. Almost all other variants of the problem are APX-hard. - At SODA'96, Agarwala, Bafna, Farach, Paterson, and Thorup [Richa Agarwala et al., 1999] showed that for any norm L_k, k ≥ 1, if the best ultrametric can be α-approximated, then the best tree metric can be 3α-approximated. In particular, this implied a 3-approximation for tree metrics under L_∞. - At FOCS'05, Ailon and Charikar [Nir Ailon and Moses Charikar, 2011] showed that for any L_k, k ≥ 1, we can get an approximation factor of O(((log n)(log log n))^{1/k}) for both tree and ultrametrics. Their paper was focused on the L₁ norm, and they wrote "Determining whether an O(1) approximation can be obtained is a fascinating question". - At FOCS'21, Cohen-Addad, Das, Kipouridis, Parotsidis, and Thorup [Vincent Cohen-Addad et al., 2021] showed that indeed a constant factor is possible for L₁ for both tree and ultrametrics. This uses the special structure of L₁ in relation to hierarchies. - The status of L_k is wide open for 1 < k < ∞. All we know is that the approximation factor is between Ω(1) and O((log n)(log log n)).
Cite as

Mikkel Thorup. Reconstructing the Tree of Life (Fitting Distances by Tree Metrics) (Invited Paper). In 18th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 227, pp. 3:1-3:2, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{thorup:LIPIcs.SWAT.2022.3,
  author =	{Thorup, Mikkel},
  title =	{{Reconstructing the Tree of Life (Fitting Distances by Tree Metrics)}},
  booktitle =	{18th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2022)},
  pages =	{3:1--3:2},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-236-5},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{227},
  editor =	{Czumaj, Artur and Xin, Qin},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SWAT.2022.3},
  URN =		{urn:nbn:de:0030-drops-161631},
  doi =		{10.4230/LIPIcs.SWAT.2022.3},
  annote =	{Keywords: Numerical taxonomy, computational phylogenetics, hierarchical clustering}
}
Document
DOI: 10.4230/LIPIcs.ESA.2019.74
Hardness of Bichromatic Closest Pair with Jaccard Similarity

Authors: Rasmus Pagh, Nina Mesing Stausholm, and Mikkel Thorup

Published in: LIPIcs, Volume 144, 27th Annual European Symposium on Algorithms (ESA 2019)

Abstract
Consider collections A and B of red and blue sets, respectively. Bichromatic Closest Pair is the problem of finding a pair from A x B that has similarity higher than a given threshold according to some similarity measure. Our focus here is the classic Jaccard similarity |a cap b|/|a cup b| for (a,b) in A x B. We consider the approximate version of the problem where we are given thresholds j_1 > j_2 and wish to return a pair from A x B that has Jaccard similarity higher than j_2 if there exists a pair in A x B with Jaccard similarity at least j_1. The classic locality sensitive hashing (LSH) algorithm of Indyk and Motwani (STOC '98), instantiated with the MinHash LSH function of Broder et al., solves this problem in Õ(n^(2-delta)) time if j_1 >= j_2^(1-delta). In particular, for delta=Omega(1), the approximation ratio j_1/j_2 = 1/j_2^delta increases polynomially in 1/j_2. In this paper we give a corresponding hardness result. Assuming the Orthogonal Vectors Conjecture (OVC), we show that there cannot be a general solution that solves the Bichromatic Closest Pair problem in O(n^(2-Omega(1))) time for j_1/j_2 = 1/j_2^o(1). Specifically, assuming OVC, we prove that for any delta>0 there exists an epsilon>0 such that Bichromatic Closest Pair with Jaccard similarity requires time Omega(n^(2-delta)) for any choice of thresholds j_2 < j_1 < 1-delta, that satisfy j_1 <= j_2^(1-epsilon).
Cite as

Rasmus Pagh, Nina Mesing Stausholm, and Mikkel Thorup. Hardness of Bichromatic Closest Pair with Jaccard Similarity. In 27th Annual European Symposium on Algorithms (ESA 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 144, pp. 74:1-74:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{pagh_et_al:LIPIcs.ESA.2019.74,
  author =	{Pagh, Rasmus and Stausholm, Nina Mesing and Thorup, Mikkel},
  title =	{{Hardness of Bichromatic Closest Pair with Jaccard Similarity}},
  booktitle =	{27th Annual European Symposium on Algorithms (ESA 2019)},
  pages =	{74:1--74:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-124-5},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{144},
  editor =	{Bender, Michael A. and Svensson, Ola 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.2019.74},
  URN =		{urn:nbn:de:0030-drops-111951},
  doi =		{10.4230/LIPIcs.ESA.2019.74},
  annote =	{Keywords: fine-grained complexity, set similarity search, bichromatic closest pair, jaccard similarity}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2019.95
Dynamic Ordered Sets with Approximate Queries, Approximate Heaps and Soft Heaps

Authors: Mikkel Thorup, Or Zamir, and Uri Zwick

Published in: LIPIcs, Volume 132, 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)

Abstract
We consider word RAM data structures for maintaining ordered sets of integers whose select and rank operations are allowed to return approximate results, i.e., ranks, or items whose rank, differ by less than Delta from the exact answer, where Delta=Delta(n) is an error parameter. Related to approximate select and rank is approximate (one-dimensional) nearest-neighbor. A special case of approximate select queries are approximate min queries. Data structures that support approximate min operations are known as approximate heaps (priority queues). Related to approximate heaps are soft heaps, which are approximate heaps with a different notion of approximation. We prove the optimality of all the data structures presented, either through matching cell-probe lower bounds, or through equivalences to well studied static problems. For approximate select, rank, and nearest-neighbor operations we get matching cell-probe lower bounds. We prove an equivalence between approximate min operations, i.e., approximate heaps, and the static partitioning problem. Finally, we prove an equivalence between soft heaps and the classical sorting problem, on a smaller number of items. Our results have many interesting and unexpected consequences. It turns out that approximation greatly speeds up some of these operations, while others are almost unaffected. In particular, while select and rank have identical operation times, both in comparison-based and word RAM implementations, an interesting separation emerges between the approximate versions of these operations in the word RAM model. Approximate select is much faster than approximate rank. It also turns out that approximate min is exponentially faster than the more general approximate select. Next, we show that implementing soft heaps is harder than implementing approximate heaps. The relation between them corresponds to the relation between sorting and partitioning. Finally, as an interesting byproduct, we observe that a combination of known techniques yields a deterministic word RAM algorithm for (exactly) sorting n items in O(n log log_w n) time, where w is the word length. Even for the easier problem of finding duplicates, the best previous deterministic bound was O(min{n log log n,n log_w n}). Our new unifying bound is an improvement when w is sufficiently large compared with n.
Cite as

Mikkel Thorup, Or Zamir, and Uri Zwick. Dynamic Ordered Sets with Approximate Queries, Approximate Heaps and Soft Heaps. In 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 132, pp. 95:1-95:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{thorup_et_al:LIPIcs.ICALP.2019.95,
  author =	{Thorup, Mikkel and Zamir, Or and Zwick, Uri},
  title =	{{Dynamic Ordered Sets with Approximate Queries, Approximate Heaps and Soft Heaps}},
  booktitle =	{46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)},
  pages =	{95:1--95:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-109-2},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{132},
  editor =	{Baier, Christel and Chatzigiannakis, Ioannis and Flocchini, Paola and Leonardi, Stefano},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2019.95},
  URN =		{urn:nbn:de:0030-drops-106712},
  doi =		{10.4230/LIPIcs.ICALP.2019.95},
  annote =	{Keywords: Order queries, word RAM, lower bounds}
}
Document
DOI: 10.4230/LIPIcs.ICALP.2018.5
Power of d Choices with Simple Tabulation

Authors: Anders Aamand, Mathias Bæk Tejs Knudsen, and Mikkel Thorup

Published in: LIPIcs, Volume 107, 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)

Abstract
We consider the classic d-choice paradigm of Azar et al. [STOC'94] in which m balls are put into n bins sequentially as follows: For each ball we are given a choice of d bins chosen according to d hash functions and the ball is placed in the least loaded of these bins, breaking ties arbitrarily. The interest is in the number of balls in the fullest bin after all balls have been placed. In this paper we suppose that the d hash functions are simple tabulation hash functions which are easy to implement and can be evaluated in constant time. Generalising a result by Dahlgaard et al. [SODA'16] we show that for an arbitrary constant d >= 2 the expected maximum load is at most (lg lg n)/(lg d) + O(1). We further show that by using a simple tie-breaking algorithm introduced by Vöcking [J.ACM'03] the expected maximum load is reduced to (lg lg n)/(d lg phi_d) + O(1) where phi_d is the rate of growth of the d-ary Fibonacci numbers. Both of these expected bounds match those known from the fully random setting. The analysis by Dahlgaard et al. relies on a proof by Patrascu and Thorup [J.ACM'11] concerning the use of simple tabulation for cuckoo hashing. We require a generalisation to d>2 hash functions, but the original proof is an 8-page tour de force of ad-hoc arguments that do not appear to generalise. Our main technical contribution is a shorter, simpler and more accessible proof of the result by Patrascu and Thorup, where the relevant parts generalise nicely to the analysis of d choices.
Cite as

Anders Aamand, Mathias Bæk Tejs Knudsen, and Mikkel Thorup. Power of d Choices with Simple Tabulation. In 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 107, pp. 5:1-5:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{aamand_et_al:LIPIcs.ICALP.2018.5,
  author =	{Aamand, Anders and B{\ae}k Tejs Knudsen, Mathias and Thorup, Mikkel},
  title =	{{Power of d Choices with Simple Tabulation}},
  booktitle =	{45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)},
  pages =	{5:1--5:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-076-7},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{107},
  editor =	{Chatzigiannakis, Ioannis and Kaklamanis, Christos and Marx, D\'{a}niel and Sannella, Donald},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2018.5},
  URN =		{urn:nbn:de:0030-drops-90096},
  doi =		{10.4230/LIPIcs.ICALP.2018.5},
  annote =	{Keywords: Hashing, Load Balancing, Balls and Bins, Simple Tabulation}
}
Document
Invited Talk
DOI: 10.4230/LIPIcs.ICALP.2017.4
Fast and Powerful Hashing Using Tabulation (Invited Talk)

Authors: Mikkel Thorup

Published in: LIPIcs, Volume 80, 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)

Abstract
Randomized algorithms are often enjoyed for their simplicity, but the hash functions employed to yield the desired probabilistic guarantees are often too complicated to be practical. Here we survey recent results on how simple hashing schemes based on tabulation provide unexpectedly strong guarantees. Simple tabulation hashing dates back to Zobrist [1970]. Keys are viewed as consisting of c characters and we have precomputed character tables h_1,...,h_q mapping characters to random hash values. A key x=(x_1,...,x_c) is hashed to h_1[x_1] xor h_2[x_2]..... xor h_c[x_c]. This schemes is very fast with character tables in cache. While simple tabulation is not even 4-independent, it does provide many of the guarantees that are normally obtained via higher independence, e.g., linear probing and Cuckoo hashing. Next we consider twisted tabulation where one character is "twisted" with some simple operations. The resulting hash function has powerful distributional properties: Chernoff-Hoeffding type tail bounds and a very small bias for min-wise hashing. Finally, we consider double tabulation where we compose two simple tabulation functions, applying one to the output of the other, and show that this yields very high independence in the classic framework of Carter and Wegman [1977]. In fact, w.h.p., for a given set of size proportional to that of the space consumed, double tabulation gives fully-random hashing. While these tabulation schemes are all easy to implement and use, their analysis is not. This keynote talk surveys results from the papers in the reference list.
Cite as

Mikkel Thorup. Fast and Powerful Hashing Using Tabulation (Invited Talk). In 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 80, pp. 4:1-4:2, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{thorup:LIPIcs.ICALP.2017.4,
  author =	{Thorup, Mikkel},
  title =	{{Fast and Powerful Hashing Using Tabulation}},
  booktitle =	{44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)},
  pages =	{4:1--4:2},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-041-5},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{80},
  editor =	{Chatzigiannakis, Ioannis and Indyk, Piotr and Kuhn, Fabian and Muscholl, Anca},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2017.4},
  URN =		{urn:nbn:de:0030-drops-75074},
  doi =		{10.4230/LIPIcs.ICALP.2017.4},
  annote =	{Keywords: Hashing, Randomized Algorithms}
}
Document
DOI: 10.4230/LIPIcs.ICALP.2017.128
Near-Optimal Induced Universal Graphs for Bounded Degree Graphs

Authors: Mikkel Abrahamsen, Stephen Alstrup, Jacob Holm, Mathias Bæk Tejs Knudsen, and Morten Stöckel

Published in: LIPIcs, Volume 80, 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)

Abstract
A graph U is an induced universal graph for a family F of graphs if every graph in F is a vertex-induced subgraph of U. We give upper and lower bounds for the size of induced universal graphs for the family of graphs with n vertices of maximum degree D. Our new bounds improve several previous results except for the special cases where D is either near-constant or almost n/2. For constant even D Butler [Graphs and Combinatorics 2009] has shown O(n^(D/2)) and recently Alon and Nenadov [SODA 2017] showed the same bound for constant odd D. For constant D Butler also gave a matching lower bound. For generals graphs, which corresponds to D = n, Alon [Geometric and Functional Analysis, to appear] proved the existence of an induced universal graph with (1+o(1)) \cdot 2^((n-1)/2) vertices, leading to a smaller constant than in the previously best known bound of 16 * 2^(n/2) by Alstrup, Kaplan, Thorup, and Zwick [STOC 2015]. In this paper we give the following lower and upper bound of binom(floor(n/2))(floor(D/2)) * n^(-O(1)) and binom(floor(n/2))(floor(D/2)) * 2^(O(sqrt(D log D) * log(n/D))), respectively, where the upper bound is the main contribution. The proof that it is an induced universal graph relies on a randomized argument. We also give a deterministic upper bound of O(n^k / (k-1)!). These upper bounds are the best known when D <= n/2 - tilde-Omega(n^(3/4)) and either D is even and D = omega(1) or D is odd and D = omega(log n/log log n). In this range we improve asymptotically on the previous best known results by Butler [Graphs and Combinatorics 2009], Esperet, Arnaud and Ochem [IPL 2008], Adjiashvili and Rotbart [ICALP 2014], Alon and Nenadov [SODA 2017], and Alon [Geometric and Functional Analysis, to appear].
Cite as

Mikkel Abrahamsen, Stephen Alstrup, Jacob Holm, Mathias Bæk Tejs Knudsen, and Morten Stöckel. Near-Optimal Induced Universal Graphs for Bounded Degree Graphs. In 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 80, pp. 128:1-128:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{abrahamsen_et_al:LIPIcs.ICALP.2017.128,
  author =	{Abrahamsen, Mikkel and Alstrup, Stephen and Holm, Jacob and Knudsen, Mathias B{\ae}k Tejs and St\"{o}ckel, Morten},
  title =	{{Near-Optimal Induced Universal Graphs for Bounded Degree Graphs}},
  booktitle =	{44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)},
  pages =	{128:1--128:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-041-5},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{80},
  editor =	{Chatzigiannakis, Ioannis and Indyk, Piotr and Kuhn, Fabian and Muscholl, Anca},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2017.128},
  URN =		{urn:nbn:de:0030-drops-74114},
  doi =		{10.4230/LIPIcs.ICALP.2017.128},
  annote =	{Keywords: Adjacency labeling schemes, Bounded degree graphs, Induced universal graphs, Distributed computing}
}
Document
Invited Talk
DOI: 10.4230/LIPIcs.FSTTCS.2016.1
Fast and Powerful Hashing Using Tabulation (Invited Talk)

Authors: Mikkel Thorup

Published in: LIPIcs, Volume 65, 36th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2016)

Abstract
Randomized algorithms are often enjoyed for their simplicity, but the hash functions employed to yield the desired probabilistic guarantees are often too complicated to be practical. Here we survey recent results on how simple hashing schemes based on tabulation provide unexpectedly strong guarantees. Simple tabulation hashing dates back to Zobrist [1970]. Keys are viewed as consisting of c characters and we have precomputed character tables h_1, ... , h_q mapping characters to random hash values. A key x = (x_1 , ..., x_c) is hashed to h_1[x_1] + h_2[x_2] ... + h_c[x_c]. This scheme is very fast with character tables in cache. While simple tabulation is not even 4-independent, it does provide many of the guarantees that are normally obtained via higher independence, e.g., linear probing and Cuckoo hashing. Next we consider twisted tabulation where one character is "twisted" with some simple operations. The resulting hash function has powerful distributional properties: Chernoff-Hoeffding type tail bounds and a very small bias for min-wise hashing. Finally, we consider double tabulation where we compose two simple tabulation functions, applying one to the output of the other, and show that this yields very high independence in the classic framework of Carter and Wegman [1977]. In fact, w.h.p., for a given set of size proportional to that of the space consumed, double tabulation gives fully-random hashing. While these tabulation schemes are all easy to implement and use, their analysis is not.
Cite as

Mikkel Thorup. Fast and Powerful Hashing Using Tabulation (Invited Talk). In 36th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 65, pp. 1:1-1:2, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{thorup:LIPIcs.FSTTCS.2016.1,
  author =	{Thorup, Mikkel},
  title =	{{Fast and Powerful Hashing Using Tabulation}},
  booktitle =	{36th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2016)},
  pages =	{1:1--1:2},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-027-9},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{65},
  editor =	{Lal, Akash and Akshay, S. and Saurabh, Saket and Sen, Sandeep},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2016.1},
  URN =		{urn:nbn:de:0030-drops-68869},
  doi =		{10.4230/LIPIcs.FSTTCS.2016.1},
  annote =	{Keywords: Hashing, Randomized Algorithms}
}
Document
DOI: 10.4230/LIPIcs.ESA.2016.46
Incremental Exact Min-Cut in Poly-logarithmic Amortized Update Time

Authors: Gramoz Goranci, Monika Henzinger, and Mikkel Thorup

Published in: LIPIcs, Volume 57, 24th Annual European Symposium on Algorithms (ESA 2016)

Abstract
We present a deterministic incremental algorithm for exactly maintaining the size of a minimum cut with ~O(1) amortized time per edge insertion and O(1) query time. This result partially answers an open question posed by Thorup [Combinatorica 2007]. It also stays in sharp contrast to a polynomial conditional lower-bound for the fully-dynamic weighted minimum cut problem. Our algorithm is obtained by combining a recent sparsification technique of Kawarabayashi and Thorup [STOC 2015] and an exact incremental algorithm of Henzinger [J. of Algorithm 1997]. We also study space-efficient incremental algorithms for the minimum cut problem. Concretely, we show that there exists an O(n log n/epsilon^2) space Monte-Carlo algorithm that can process a stream of edge insertions starting from an empty graph, and with high probability, the algorithm maintains a (1+epsilon)-approximation to the minimum cut. The algorithm has ~O(1) amortized update-time and constant query-time.
Cite as

Gramoz Goranci, Monika Henzinger, and Mikkel Thorup. Incremental Exact Min-Cut in Poly-logarithmic Amortized Update Time. In 24th Annual European Symposium on Algorithms (ESA 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 57, pp. 46:1-46:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{goranci_et_al:LIPIcs.ESA.2016.46,
  author =	{Goranci, Gramoz and Henzinger, Monika and Thorup, Mikkel},
  title =	{{Incremental Exact Min-Cut in Poly-logarithmic Amortized Update Time}},
  booktitle =	{24th Annual European Symposium on Algorithms (ESA 2016)},
  pages =	{46:1--46:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-015-6},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{57},
  editor =	{Sankowski, Piotr and Zaroliagis, Christos},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2016.46},
  URN =		{urn:nbn:de:0030-drops-63584},
  doi =		{10.4230/LIPIcs.ESA.2016.46},
  annote =	{Keywords: Dynamic Graph Algorithms, Minimum Cut, Edge Connectivity}
}
Document
DOI: 10.4230/LIPIcs.ESA.2016.53
Faster Worst Case Deterministic Dynamic Connectivity

Authors: Casper Kejlberg-Rasmussen, Tsvi Kopelowitz, Seth Pettie, and Mikkel Thorup

Published in: LIPIcs, Volume 57, 24th Annual European Symposium on Algorithms (ESA 2016)

Abstract
We present a deterministic dynamic connectivity data structure for undirected graphs with worst case update time O(sqrt{(n(log(log(n)))^2)/log(n)}) and constant query time. This improves on the previous best deterministic worst case algorithm of Frederickson (SIAM J. Comput., 1985) and Eppstein Galil, Italiano, and Nissenzweig (J. ACM, 1997), which had update time O(sqrt{n}). All other algorithms for dynamic connectivity are either randomized (Monte Carlo) or have only amortized performance guarantees.
Cite as

Casper Kejlberg-Rasmussen, Tsvi Kopelowitz, Seth Pettie, and Mikkel Thorup. Faster Worst Case Deterministic Dynamic Connectivity. In 24th Annual European Symposium on Algorithms (ESA 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 57, pp. 53:1-53:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{kejlbergrasmussen_et_al:LIPIcs.ESA.2016.53,
  author =	{Kejlberg-Rasmussen, Casper and Kopelowitz, Tsvi and Pettie, Seth and Thorup, Mikkel},
  title =	{{Faster Worst Case Deterministic Dynamic Connectivity}},
  booktitle =	{24th Annual European Symposium on Algorithms (ESA 2016)},
  pages =	{53:1--53:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-015-6},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{57},
  editor =	{Sankowski, Piotr and Zaroliagis, Christos},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2016.53},
  URN =		{urn:nbn:de:0030-drops-63953},
  doi =		{10.4230/LIPIcs.ESA.2016.53},
  annote =	{Keywords: dynamic graph, spanning tree}
}
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