19 Search Results for "Sauerwald, Thomas"


Document
Rumors with Changing Credibility

Authors: Charlotte Out, Nicolás Rivera, Thomas Sauerwald, and John Sylvester

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


Abstract
Randomized rumor spreading processes diffuse information on an undirected graph and have been widely studied. In this work, we present a generic framework for analyzing a broad class of such processes on regular graphs. Our analysis is protocol-agnostic, as it only requires the expected proportion of newly informed vertices in each round to be bounded, and a natural negative correlation property. This framework allows us to analyze various protocols, including PUSH, PULL, and PUSH-PULL, thereby extending prior research. Unlike previous work, our framework accommodates message failures at any time t ≥ 0 with a probability of 1-q(t), where the credibility q(t) is any function of time. This enables us to model real-world scenarios in which the transmissibility of rumors may fluctuate, as seen in the spread of "fake news" and viruses. Additionally, our framework is sufficiently broad to cover dynamic graphs.

Cite as

Charlotte Out, Nicolás Rivera, Thomas Sauerwald, and John Sylvester. Rumors with Changing Credibility. In 15th Innovations in Theoretical Computer Science Conference (ITCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 287, pp. 86:1-86:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)


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@InProceedings{out_et_al:LIPIcs.ITCS.2024.86,
  author =	{Out, Charlotte and Rivera, Nicol\'{a}s and Sauerwald, Thomas and Sylvester, John},
  title =	{{Rumors with Changing Credibility}},
  booktitle =	{15th Innovations in Theoretical Computer Science Conference (ITCS 2024)},
  pages =	{86:1--86:23},
  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.2024.86},
  URN =		{urn:nbn:de:0030-drops-196149},
  doi =		{10.4230/LIPIcs.ITCS.2024.86},
  annote =	{Keywords: Rumor spreading, epidemic algorithms, "fake news"}
}
Document
Track A: Algorithms, Complexity and Games
The Support of Open Versus Closed Random Walks

Authors: Thomas Sauerwald, He Sun, and Danny Vagnozzi

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


Abstract
A closed random walk of length 𝓁 on an undirected and connected graph G = (V,E) is a random walk that returns to the start vertex at step 𝓁, and its properties have been recently related to problems in different mathematical fields, e.g., geometry and combinatorics (Jiang et al., Annals of Mathematics '21) and spectral graph theory (McKenzie et al., STOC '21). For instance, in the context of analyzing the eigenvalue multiplicity of graph matrices, McKenzie et al. show that, with high probability, the support of a closed random walk of length 𝓁 ⩾ 1 is Ω(𝓁^{1/5}) on any bounded-degree graph, and leaves as an open problem whether a stronger bound of Ω(𝓁^{1/2}) holds for any regular graph. First, we show that the support of a closed random walk of length 𝓁 is at least Ω(𝓁^{1/2} / √{log n}) for any regular or bounded-degree graph on n vertices. Secondly, we prove for every 𝓁 ⩾ 1 the existence of a family of bounded-degree graphs, together with a start vertex such that the support is bounded by O(𝓁^{1/2}/√{log n}). Besides addressing the open problem of McKenzie et al., these two results also establish a subtle separation between closed random walks and open random walks, for which the support on any regular (or bounded-degree) graph is well-known to be Ω(𝓁^{1/2}) for all 𝓁 ⩾ 1. For irregular graphs, we prove that even if the start vertex is chosen uniformly, the support of a closed random walk may still be O(log 𝓁). This rules out a general polynomial lower bound in 𝓁 for all graphs. Finally, we apply our results on random walks to obtain new bounds on the multiplicity of the second largest eigenvalue of the adjacency matrices of graphs.

Cite as

Thomas Sauerwald, He Sun, and Danny Vagnozzi. The Support of Open Versus Closed Random Walks. In 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 261, pp. 103:1-103:21, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)


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@InProceedings{sauerwald_et_al:LIPIcs.ICALP.2023.103,
  author =	{Sauerwald, Thomas and Sun, He and Vagnozzi, Danny},
  title =	{{The Support of Open Versus Closed Random Walks}},
  booktitle =	{50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)},
  pages =	{103:1--103:21},
  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.2023.103},
  URN =		{urn:nbn:de:0030-drops-181556},
  doi =		{10.4230/LIPIcs.ICALP.2023.103},
  annote =	{Keywords: support of random walks, eigenvalue multiplicity}
}
Document
Tight Bounds for Repeated Balls-Into-Bins

Authors: Dimitrios Los and Thomas Sauerwald

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


Abstract
We study the repeated balls-into-bins process introduced by Becchetti, Clementi, Natale, Pasquale and Posta (2019). This process starts with m balls arbitrarily distributed across n bins. At each round t = 1,2,…, one ball is selected from each non-empty bin, and then placed it into a bin chosen independently and uniformly at random. We prove the following results: - For any n ⩽ m ⩽ poly(n), we prove a lower bound of Ω(m/n ⋅ log n) on the maximum load. For the special case m = n, this matches the upper bound of 𝒪(log n), as shown in [Luca Becchetti et al., 2019]. It also provides a positive answer to the conjecture in [Luca Becchetti et al., 2019] that for m = n the maximum load is ω(log n/ log log n) at least once in a polynomially large time interval. For m ∈ [ω(n), n log n], our new lower bound disproves the conjecture in [Luca Becchetti et al., 2019] that the maximum load remains 𝒪(log n). - For any n ⩽ m ⩽ poly(n), we prove an upper bound of 𝒪(m/n ⋅ log n) on the maximum load for all steps of a polynomially large time interval. This matches our lower bound up to multiplicative constants. - For any m ⩾ n, our analysis also implies an 𝒪(m²/n) waiting time to reach a configuration with a 𝒪(m/n ⋅ log m) maximum load, even for worst-case initial distributions. - For m ⩾ n, we show that every ball visits every bin in 𝒪(m log m) rounds. For m = n, this improves the previous upper bound of 𝒪(n log² n) in [Luca Becchetti et al., 2019]. We also prove that the upper bound is tight up to multiplicative constants for any n ⩽ m ⩽ poly(n).

Cite as

Dimitrios Los and Thomas Sauerwald. Tight Bounds for Repeated Balls-Into-Bins. In 40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 254, pp. 45:1-45:22, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)


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@InProceedings{los_et_al:LIPIcs.STACS.2023.45,
  author =	{Los, Dimitrios and Sauerwald, Thomas},
  title =	{{Tight Bounds for Repeated Balls-Into-Bins}},
  booktitle =	{40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023)},
  pages =	{45:1--45:22},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-266-2},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{254},
  editor =	{Berenbrink, Petra and Bouyer, Patricia and Dawar, Anuj and Kant\'{e}, Mamadou Moustapha},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2023.45},
  URN =		{urn:nbn:de:0030-drops-176975},
  doi =		{10.4230/LIPIcs.STACS.2023.45},
  annote =	{Keywords: Repeated balls-into-bins, self-stabilizing systems, balanced allocations, potential functions, random walks}
}
Document
Balanced Allocations with Incomplete Information: The Power of Two Queries

Authors: Dimitrios Los and Thomas Sauerwald

Published in: LIPIcs, Volume 215, 13th Innovations in Theoretical Computer Science Conference (ITCS 2022)


Abstract
We consider the allocation of m balls into n bins with incomplete information. In the classical Two-Choice process a ball first queries the load of two randomly chosen bins and is then placed in the least loaded bin. In our setting, each ball also samples two random bins but can only estimate a bin’s load by sending binary queries of the form "Is the load at least the median?" or "Is the load at least 100?". For the lightly loaded case m = 𝒪(n), Feldheim and Gurel-Gurevich (2021) showed that with one query it is possible to achieve a maximum load of 𝒪(√{log n/log log n}), and they also pose the question whether a maximum load of m/n+𝒪(√{log n/log log n}) is possible for any m = Ω(n). In this work, we resolve this open problem by proving a lower bound of m/n+Ω(√{log n}) for a fixed m = Θ(n √{log n}), and a lower bound of m/n+Ω(log n/log log n) for some m depending on the used strategy. We complement this negative result by proving a positive result for multiple queries. In particular, we show that with only two binary queries per chosen bin, there is an oblivious strategy which ensures a maximum load of m/n+𝒪(√{log n}) for any m ≥ 1. Further, for any number of k = 𝒪(log log n) binary queries, the upper bound on the maximum load improves to m/n + 𝒪(k(log n)^{1/k}) for any m ≥ 1. This result for k queries has several interesting consequences: (i) it implies new bounds for the (1+β)-process introduced by Peres, Talwar and Wieder (2015), (ii) it leads to new bounds for the graphical balanced allocation process on dense expander graphs, and (iii) it recovers and generalizes the bound of m/n+𝒪(log log n) on the maximum load achieved by the Two-Choice process, including the heavily loaded case m = Ω(n) which was derived in previous works by Berenbrink et al. (2006) as well as Talwar and Wieder (2014). One novel aspect of our proofs is the use of multiple super-exponential potential functions, which might be of use in future work.

Cite as

Dimitrios Los and Thomas Sauerwald. Balanced Allocations with Incomplete Information: The Power of Two Queries. In 13th Innovations in Theoretical Computer Science Conference (ITCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 215, pp. 103:1-103:23, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)


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@InProceedings{los_et_al:LIPIcs.ITCS.2022.103,
  author =	{Los, Dimitrios and Sauerwald, Thomas},
  title =	{{Balanced Allocations with Incomplete Information: The Power of Two Queries}},
  booktitle =	{13th Innovations in Theoretical Computer Science Conference (ITCS 2022)},
  pages =	{103:1--103:23},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-217-4},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{215},
  editor =	{Braverman, Mark},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2022.103},
  URN =		{urn:nbn:de:0030-drops-156994},
  doi =		{10.4230/LIPIcs.ITCS.2022.103},
  annote =	{Keywords: power-of-two-choices, balanced allocations, potential functions, thinning}
}
Document
The Impact of Geometry on Monochrome Regions in the Flip Schelling Process

Authors: Thomas Bläsius, Tobias Friedrich, Martin S. Krejca, and Louise Molitor

Published in: LIPIcs, Volume 212, 32nd International Symposium on Algorithms and Computation (ISAAC 2021)


Abstract
Schelling’s classical segregation model gives a coherent explanation for the wide-spread phenomenon of residential segregation. We introduce an agent-based saturated open-city variant, the Flip Schelling Process (FSP), in which agents, placed on a graph, have one out of two types and, based on the predominant type in their neighborhood, decide whether to change their types; similar to a new agent arriving as soon as another agent leaves the vertex. We investigate the probability that an edge {u,v} is monochrome, i.e., that both vertices u and v have the same type in the FSP, and we provide a general framework for analyzing the influence of the underlying graph topology on residential segregation. In particular, for two adjacent vertices, we show that a highly decisive common neighborhood, i.e., a common neighborhood where the absolute value of the difference between the number of vertices with different types is high, supports segregation and, moreover, that large common neighborhoods are more decisive. As an application, we study the expected behavior of the FSP on two common random graph models with and without geometry: (1) For random geometric graphs, we show that the existence of an edge {u,v} makes a highly decisive common neighborhood for u and v more likely. Based on this, we prove the existence of a constant c > 0 such that the expected fraction of monochrome edges after the FSP is at least 1/2 + c. (2) For Erdős-Rényi graphs we show that large common neighborhoods are unlikely and that the expected fraction of monochrome edges after the FSP is at most 1/2 + o(1). Our results indicate that the cluster structure of the underlying graph has a significant impact on the obtained segregation strength.

Cite as

Thomas Bläsius, Tobias Friedrich, Martin S. Krejca, and Louise Molitor. The Impact of Geometry on Monochrome Regions in the Flip Schelling Process. In 32nd International Symposium on Algorithms and Computation (ISAAC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 212, pp. 29:1-29:17, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)


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@InProceedings{blasius_et_al:LIPIcs.ISAAC.2021.29,
  author =	{Bl\"{a}sius, Thomas and Friedrich, Tobias and Krejca, Martin S. and Molitor, Louise},
  title =	{{The Impact of Geometry on Monochrome Regions in the Flip Schelling Process}},
  booktitle =	{32nd International Symposium on Algorithms and Computation (ISAAC 2021)},
  pages =	{29:1--29:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-214-3},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{212},
  editor =	{Ahn, Hee-Kap and Sadakane, Kunihiko},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2021.29},
  URN =		{urn:nbn:de:0030-drops-154623},
  doi =		{10.4230/LIPIcs.ISAAC.2021.29},
  annote =	{Keywords: Agent-based Model, Schelling Segregation, Spin System}
}
Document
Track A: Algorithms, Complexity and Games
Multiple Random Walks on Graphs: Mixing Few to Cover Many

Authors: Nicolás Rivera, Thomas Sauerwald, and John Sylvester

Published in: LIPIcs, Volume 198, 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)


Abstract
Random walks on graphs are an essential primitive for many randomised algorithms and stochastic processes. It is natural to ask how much can be gained by running k multiple random walks independently and in parallel. Although the cover time of multiple walks has been investigated for many natural networks, the problem of finding a general characterisation of multiple cover times for worst-case start vertices (posed by Alon, Avin, Koucký, Kozma, Lotker, and Tuttle in 2008) remains an open problem. First, we improve and tighten various bounds on the stationary cover time when k random walks start from vertices sampled from the stationary distribution. For example, we prove an unconditional lower bound of Ω((n/k) log n) on the stationary cover time, holding for any n-vertex graph G and any 1 ≤ k = o(nlog n). Secondly, we establish the stationary cover times of multiple walks on several fundamental networks up to constant factors. Thirdly, we present a framework characterising worst-case cover times in terms of stationary cover times and a novel, relaxed notion of mixing time for multiple walks called the partial mixing time. Roughly speaking, the partial mixing time only requires a specific portion of all random walks to be mixed. Using these new concepts, we can establish (or recover) the worst-case cover times for many networks including expanders, preferential attachment graphs, grids, binary trees and hypercubes.

Cite as

Nicolás Rivera, Thomas Sauerwald, and John Sylvester. Multiple Random Walks on Graphs: Mixing Few to Cover Many. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 107:1-107:16, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)


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@InProceedings{rivera_et_al:LIPIcs.ICALP.2021.107,
  author =	{Rivera, Nicol\'{a}s and Sauerwald, Thomas and Sylvester, John},
  title =	{{Multiple Random Walks on Graphs: Mixing Few to Cover Many}},
  booktitle =	{48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)},
  pages =	{107:1--107:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-195-5},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{198},
  editor =	{Bansal, Nikhil and Merelli, Emanuela and Worrell, James},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.107},
  URN =		{urn:nbn:de:0030-drops-141764},
  doi =		{10.4230/LIPIcs.ICALP.2021.107},
  annote =	{Keywords: Multiple Random walks, Markov Chains, Random Walks, Cover Time}
}
Document
Spread of Information and Diseases via Random Walks in Sparse Graphs

Authors: George Giakkoupis, Hayk Saribekyan, and Thomas Sauerwald

Published in: LIPIcs, Volume 179, 34th International Symposium on Distributed Computing (DISC 2020)


Abstract
We consider a natural network diffusion process, modeling the spread of information or infectious diseases. Multiple mobile agents perform independent simple random walks on an n-vertex connected graph G. The number of agents is linear in n and the walks start from the stationary distribution. Initially, a single vertex has a piece of information (or a virus). An agent becomes informed (or infected) the first time it visits some vertex with the information (or virus); thereafter, the agent informs (infects) all vertices it visits. Giakkoupis et al. [George Giakkoupis et al., 2019] have shown that the spreading time, i.e., the time before all vertices are informed, is asymptotically and w.h.p. the same as in the well-studied randomized rumor spreading process, on any d-regular graph with d = Ω(log n). The case of sub-logarithmic degree was left open, and is the main focus of this paper. First, we observe that the equivalence shown in [George Giakkoupis et al., 2019] does not hold for small d: We give an example of a 3-regular graph with logarithmic diameter for which the expected spreading time is Ω(log² n/log log n), whereas randomized rumor spreading is completed in time Θ(log n), w.h.p. Next, we show a general upper bound of Õ(d ⋅ diam(G) + log³ n /d), w.h.p., for the spreading time on any d-regular graph. We also provide a version of the bound based on the average degree, for non-regular graphs. Next, we give tight analyses for specific graph families. We show that the spreading time is O(log n), w.h.p., for constant-degree regular expanders. For the binary tree, we show an upper bound of O(log n⋅ log log n), w.h.p., and prove that this is tight, by giving a matching lower bound for the cover time of the tree by n random walks. Finally, we show a bound of O(diam(G)), w.h.p., for k-dimensional grids (k ≥ 1 is constant), by adapting a technique by Kesten and Sidoravicius [Kesten and Sidoravicius, 2003; Kesten and Sidoravicius, 2005].

Cite as

George Giakkoupis, Hayk Saribekyan, and Thomas Sauerwald. Spread of Information and Diseases via Random Walks in Sparse Graphs. In 34th International Symposium on Distributed Computing (DISC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 179, pp. 9:1-9:17, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)


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@InProceedings{giakkoupis_et_al:LIPIcs.DISC.2020.9,
  author =	{Giakkoupis, George and Saribekyan, Hayk and Sauerwald, Thomas},
  title =	{{Spread of Information and Diseases via Random Walks in Sparse Graphs}},
  booktitle =	{34th International Symposium on Distributed Computing (DISC 2020)},
  pages =	{9:1--9:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-168-9},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{179},
  editor =	{Attiya, Hagit},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.DISC.2020.9},
  URN =		{urn:nbn:de:0030-drops-130873},
  doi =		{10.4230/LIPIcs.DISC.2020.9},
  annote =	{Keywords: parallel random walks, information dissemination, infectious diseases}
}
Document
Track A: Algorithms, Complexity and Games
Dynamic Averaging Load Balancing on Cycles

Authors: Dan Alistarh, Giorgi Nadiradze, and Amirmojtaba Sabour

Published in: LIPIcs, Volume 168, 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)


Abstract
We consider the following dynamic load-balancing process: given an underlying graph G with n nodes, in each step t≥ 0, one unit of load is created, and placed at a randomly chosen graph node. In the same step, the chosen node picks a random neighbor, and the two nodes balance their loads by averaging them. We are interested in the expected gap between the minimum and maximum loads at nodes as the process progresses, and its dependence on n and on the graph structure. Variants of the above graphical balanced allocation process have been studied previously by Peres, Talwar, and Wieder [Peres et al., 2015], and by Sauerwald and Sun [Sauerwald and Sun, 2015]. These authors left as open the question of characterizing the gap in the case of cycle graphs in the dynamic case, where weights are created during the algorithm’s execution. For this case, the only known upper bound is of 𝒪(n log n), following from a majorization argument due to [Peres et al., 2015], which analyzes a related graphical allocation process. In this paper, we provide an upper bound of 𝒪 (√n log n) on the expected gap of the above process for cycles of length n. We introduce a new potential analysis technique, which enables us to bound the difference in load between k-hop neighbors on the cycle, for any k ≤ n/2. We complement this with a "gap covering" argument, which bounds the maximum value of the gap by bounding its value across all possible subsets of a certain structure, and recursively bounding the gaps within each subset. We provide analytical and experimental evidence that our upper bound on the gap is tight up to a logarithmic factor.

Cite as

Dan Alistarh, Giorgi Nadiradze, and Amirmojtaba Sabour. Dynamic Averaging Load Balancing on Cycles. In 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 168, pp. 7:1-7:16, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)


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@InProceedings{alistarh_et_al:LIPIcs.ICALP.2020.7,
  author =	{Alistarh, Dan and Nadiradze, Giorgi and Sabour, Amirmojtaba},
  title =	{{Dynamic Averaging Load Balancing on Cycles}},
  booktitle =	{47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)},
  pages =	{7:1--7:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-138-2},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{168},
  editor =	{Czumaj, Artur and Dawar, Anuj and Merelli, Emanuela},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2020.7},
  URN =		{urn:nbn:de:0030-drops-124142},
  doi =		{10.4230/LIPIcs.ICALP.2020.7},
  annote =	{Keywords: Algorithms, Load Balancing}
}
Document
Choice and Bias in Random Walks

Authors: Agelos Georgakopoulos, John Haslegrave, Thomas Sauerwald, and John Sylvester

Published in: LIPIcs, Volume 151, 11th Innovations in Theoretical Computer Science Conference (ITCS 2020)


Abstract
We analyse the following random walk process inspired by the power-of-two-choice paradigm: starting from a given vertex, at each step, unlike the simple random walk (SRW) that always moves to a randomly chosen neighbour, we have the choice between two uniformly and independently chosen neighbours. We call this process the choice random walk (CRW). We first prove that for any graph, there is a strategy for the CRW that visits any given vertex in expected time ?(|E|). Then we introduce a general tool that quantifies by how much the probability of a rare event in the simple random walk can be boosted under a suitable CRW strategy. We believe this result to be of independent interest, and apply it here to derive an almost optimal ?(n log log n) bound for the cover time of bounded-degree expanders. This tool also applies to so-called biased walks, and allows us to make progress towards a conjecture of Azar et al. [STOC 1992]. Finally, we prove the following dichotomy: computing an optimal strategy to minimise the hitting time of a vertex takes polynomial time, whereas computing one to minimise the cover time is NP-hard.

Cite as

Agelos Georgakopoulos, John Haslegrave, Thomas Sauerwald, and John Sylvester. Choice and Bias in Random Walks. In 11th Innovations in Theoretical Computer Science Conference (ITCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 151, pp. 76:1-76:19, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)


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@InProceedings{georgakopoulos_et_al:LIPIcs.ITCS.2020.76,
  author =	{Georgakopoulos, Agelos and Haslegrave, John and Sauerwald, Thomas and Sylvester, John},
  title =	{{Choice and Bias in Random Walks}},
  booktitle =	{11th Innovations in Theoretical Computer Science Conference (ITCS 2020)},
  pages =	{76:1--76:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-134-4},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{151},
  editor =	{Vidick, Thomas},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2020.76},
  URN =		{urn:nbn:de:0030-drops-117612},
  doi =		{10.4230/LIPIcs.ITCS.2020.76},
  annote =	{Keywords: Power of Two Choices, Markov Chains, Random Walks, Cover Time, Markov Decision Processes}
}
Document
Topological Data Analysis Reveals Principles of Chromosome Structure in Cellular Differentiation

Authors: Natalie Sauerwald, Yihang Shen, and Carl Kingsford

Published in: LIPIcs, Volume 143, 19th International Workshop on Algorithms in Bioinformatics (WABI 2019)


Abstract
Topological data analysis (TDA) is a mathematically well-founded set of methods to derive robust information about the structure and topology of data. It has been applied successfully in several biological contexts. Derived primarily from algebraic topology, TDA rigorously identifies persistent features in complex data, making it well-suited to better understand the key features of three-dimensional chromosome structure. Chromosome structure has a significant influence in many diverse genomic processes and has recently been shown to relate to cellular differentiation. While there exist many methods to study specific substructures of chromosomes, we are still missing a global view of all geometric features of chromosomes. By applying TDA to the study of chromosome structure through differentiation across three cell lines, we provide insight into principles of chromosome folding and looping. We identify persistent connected components and one-dimensional topological features of chromosomes and characterize them across cell types and stages of differentiation. Availability: Scripts to reproduce the results from this study can be found at https://github.com/Kingsford-Group/hictda

Cite as

Natalie Sauerwald, Yihang Shen, and Carl Kingsford. Topological Data Analysis Reveals Principles of Chromosome Structure in Cellular Differentiation. In 19th International Workshop on Algorithms in Bioinformatics (WABI 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 143, pp. 23:1-23:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


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@InProceedings{sauerwald_et_al:LIPIcs.WABI.2019.23,
  author =	{Sauerwald, Natalie and Shen, Yihang and Kingsford, Carl},
  title =	{{Topological Data Analysis Reveals Principles of Chromosome Structure in Cellular Differentiation}},
  booktitle =	{19th International Workshop on Algorithms in Bioinformatics (WABI 2019)},
  pages =	{23:1--23:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-123-8},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{143},
  editor =	{Huber, Katharina T. and Gusfield, Dan},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.WABI.2019.23},
  URN =		{urn:nbn:de:0030-drops-110537},
  doi =		{10.4230/LIPIcs.WABI.2019.23},
  annote =	{Keywords: topological data analysis, chromosome structure, Hi-C, topologically associating domains}
}
Document
Deleting Edges to Restrict the Size of an Epidemic in Temporal Networks

Authors: Jessica Enright, Kitty Meeks, George B. Mertzios, and Viktor Zamaraev

Published in: LIPIcs, Volume 138, 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)


Abstract
Spreading processes on graphs are a natural model for a wide variety of real-world phenomena, including information or behaviour spread over social networks, biological diseases spreading over contact or trade networks, and the potential flow of goods over logistical infrastructure. Often, the networks over which these processes spread are dynamic in nature, and can be modeled with graphs whose structure is subject to discrete changes over time, i.e. with temporal graphs. Here, we consider temporal graphs in which edges are available at specified timesteps, and study the problem of deleting edges from a given temporal graph in order to reduce the number of vertices (temporally) reachable from a given starting point. This could be used to control the spread of a disease, rumour, etc. in a temporal graph. In particular, our aim is to find a temporal subgraph in which a process starting at any single vertex can be transferred to only a limited number of other vertices using a temporally-feasible path (i.e. a path, along which the times of the edge availabilities increase). We introduce a natural deletion problem for temporal graphs and we provide positive and negative results on its computational complexity, both in the traditional and the parameterised sense (subject to various natural parameters), as well as addressing the approximability of this problem.

Cite as

Jessica Enright, Kitty Meeks, George B. Mertzios, and Viktor Zamaraev. Deleting Edges to Restrict the Size of an Epidemic in Temporal Networks. In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, pp. 57:1-57:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


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@InProceedings{enright_et_al:LIPIcs.MFCS.2019.57,
  author =	{Enright, Jessica and Meeks, Kitty and Mertzios, George B. and Zamaraev, Viktor},
  title =	{{Deleting Edges to Restrict the Size of an Epidemic in Temporal Networks}},
  booktitle =	{44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)},
  pages =	{57:1--57:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-117-7},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{138},
  editor =	{Rossmanith, Peter and Heggernes, Pinar and Katoen, Joost-Pieter},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2019.57},
  URN =		{urn:nbn:de:0030-drops-110010},
  doi =		{10.4230/LIPIcs.MFCS.2019.57},
  annote =	{Keywords: Temporal networks, spreading processes, graph modification, parameterised complexity}
}
Document
Track A: Algorithms, Complexity and Games
Random Walks on Dynamic Graphs: Mixing Times, Hitting Times, and Return Probabilities

Authors: Thomas Sauerwald and Luca Zanetti

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


Abstract
We establish and generalise several bounds for various random walk quantities including the mixing time and the maximum hitting time. Unlike previous analyses, our derivations are based on rather intuitive notions of local expansion properties which allow us to capture the progress the random walk makes through t-step probabilities. We apply our framework to dynamically changing graphs, where the set of vertices is fixed while the set of edges changes in each round. For random walks on dynamic connected graphs for which the stationary distribution does not change over time, we show that their behaviour is in a certain sense similar to static graphs. For example, we show that the mixing and hitting times of any sequence of d-regular connected graphs is O(n^2), generalising a well-known result for static graphs. We also provide refined bounds depending on the isoperimetric dimension of the graph, matching again known results for static graphs. Finally, we investigate properties of random walks on dynamic graphs that are not always connected: we relate their convergence to stationarity to the spectral properties of an average of transition matrices and provide some examples that demonstrate strong discrepancies between static and dynamic graphs.

Cite as

Thomas Sauerwald and Luca Zanetti. Random Walks on Dynamic Graphs: Mixing Times, Hitting Times, and Return Probabilities. In 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 132, pp. 93:1-93:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2019)


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@InProceedings{sauerwald_et_al:LIPIcs.ICALP.2019.93,
  author =	{Sauerwald, Thomas and Zanetti, Luca},
  title =	{{Random Walks on Dynamic Graphs: Mixing Times, Hitting Times, and Return Probabilities}},
  booktitle =	{46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)},
  pages =	{93:1--93:15},
  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.2019.93},
  URN =		{urn:nbn:de:0030-drops-106696},
  doi =		{10.4230/LIPIcs.ICALP.2019.93},
  annote =	{Keywords: random walks, dynamic graphs, hitting times}
}
Document
Bounds on the Satisfiability Threshold for Power Law Distributed Random SAT

Authors: Tobias Friedrich, Anton Krohmer, Ralf Rothenberger, Thomas Sauerwald, and Andrew M. Sutton

Published in: LIPIcs, Volume 87, 25th Annual European Symposium on Algorithms (ESA 2017)


Abstract
Propositional satisfiability (SAT) is one of the most fundamental problems in computer science. The worst-case hardness of SAT lies at the core of computational complexity theory. The average-case analysis of SAT has triggered the development of sophisticated rigorous and non-rigorous techniques for analyzing random structures. Despite a long line of research and substantial progress, nearly all theoretical work on random SAT assumes a uniform distribution on the variables. In contrast, real-world instances often exhibit large fluctuations in variable occurrence. This can be modeled by a scale-free distribution of the variables, which results in distributions closer to industrial SAT instances. We study random k-SAT on n variables, m = Theta(n) clauses, and a power law distribution on the variable occurrences with exponent beta. We observe a satisfiability threshold at beta = (2k-1)/(k-1). This threshold is tight in the sense that instances with beta <= (2k-1)/(k-1)-epsilon for any constant epsilon > 0 are unsatisfiable with high probability (w.h.p.). For beta >= (2k-1)/(k-1)+epsilon, the picture is reminiscent of the uniform case: instances are satisfiable w.h.p. for sufficiently small constant clause-variable ratios m/n; they are unsatisfiable above a ratio m/n that depends on beta.

Cite as

Tobias Friedrich, Anton Krohmer, Ralf Rothenberger, Thomas Sauerwald, and Andrew M. Sutton. Bounds on the Satisfiability Threshold for Power Law Distributed Random SAT. In 25th Annual European Symposium on Algorithms (ESA 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 87, pp. 37:1-37:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2017)


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@InProceedings{friedrich_et_al:LIPIcs.ESA.2017.37,
  author =	{Friedrich, Tobias and Krohmer, Anton and Rothenberger, Ralf and Sauerwald, Thomas and Sutton, Andrew M.},
  title =	{{Bounds on the Satisfiability Threshold for Power Law Distributed Random SAT}},
  booktitle =	{25th Annual European Symposium on Algorithms (ESA 2017)},
  pages =	{37:1--37:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-049-1},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{87},
  editor =	{Pruhs, Kirk and Sohler, Christian},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2017.37},
  URN =		{urn:nbn:de:0030-drops-78356},
  doi =		{10.4230/LIPIcs.ESA.2017.37},
  annote =	{Keywords: satisfiability, random structures, random SAT, power law distribution, scale-freeness, phase transitions}
}
Document
Randomized Load Balancing on Networks with Stochastic Inputs

Authors: Leran Cai and Thomas Sauerwald

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


Abstract
Iterative load balancing algorithms for indivisible tokens have been studied intensively in the past. Complementing previous worst-case analyses, we study an average-case scenario where the load inputs are drawn from a fixed probability distribution. For cycles, tori, hypercubes and expanders, we obtain almost matching upper and lower bounds on the discrepancy, the difference between the maximum and the minimum load. Our bounds hold for a variety of probability distributions including the uniform and binomial distribution but also distributions with unbounded range such as the Poisson and geometric distribution. For graphs with slow convergence like cycles and tori, our results demonstrate a substantial difference between the convergence in the worst- and average-case. An important ingredient in our analysis is a new upper bound on the t-step transition probability of a general Markov chain, which is derived by invoking the evolving set process.

Cite as

Leran Cai and Thomas Sauerwald. Randomized Load Balancing on Networks with Stochastic Inputs. In 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 80, pp. 139:1-139:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2017)


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@InProceedings{cai_et_al:LIPIcs.ICALP.2017.139,
  author =	{Cai, Leran and Sauerwald, Thomas},
  title =	{{Randomized Load Balancing on Networks with Stochastic Inputs}},
  booktitle =	{44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)},
  pages =	{139:1--139:14},
  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.2017.139},
  URN =		{urn:nbn:de:0030-drops-74839},
  doi =		{10.4230/LIPIcs.ICALP.2017.139},
  annote =	{Keywords: random walks, randomized algorithms, parallel computing}
}
Document
Multiple Random Walks on Paths and Grids

Authors: Andrej Ivaskovic, Adrian Kosowski, Dominik Pajak, and Thomas Sauerwald

Published in: LIPIcs, Volume 66, 34th Symposium on Theoretical Aspects of Computer Science (STACS 2017)


Abstract
We derive several new results on multiple random walks on "low dimensional" graphs. First, inspired by an example of a weighted random walk on a path of three vertices given by Efremenko and Reingold, we prove the following dichotomy: as the path length n tends to infinity, we have a super-linear speed-up w.r.t. the cover time if and only if the number of walks k is equal to 2. An important ingredient of our proofs is the use of a continuous-time analogue of multiple random walks, which might be of independent interest. Finally, we also present the first tight bounds on the speed-up of the cover time for any d-dimensional grid with d >= 2 being an arbitrary constant, and reveal a sharp transition between linear and logarithmic speed-up.

Cite as

Andrej Ivaskovic, Adrian Kosowski, Dominik Pajak, and Thomas Sauerwald. Multiple Random Walks on Paths and Grids. In 34th Symposium on Theoretical Aspects of Computer Science (STACS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 66, pp. 44:1-44:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2017)


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@InProceedings{ivaskovic_et_al:LIPIcs.STACS.2017.44,
  author =	{Ivaskovic, Andrej and Kosowski, Adrian and Pajak, Dominik and Sauerwald, Thomas},
  title =	{{Multiple Random Walks on Paths and Grids}},
  booktitle =	{34th Symposium on Theoretical Aspects of Computer Science (STACS 2017)},
  pages =	{44:1--44:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-028-6},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{66},
  editor =	{Vollmer, Heribert 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.2017.44},
  URN =		{urn:nbn:de:0030-drops-69897},
  doi =		{10.4230/LIPIcs.STACS.2017.44},
  annote =	{Keywords: random walks, randomized algorithms, parallel computing}
}
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