Document

**Published in:** LIPIcs, Volume 302, 35th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2024)

We consider a synchronous process of particles moving on the vertices of a graph G, introduced by Cooper, McDowell, Radzik, Rivera and Shiraga (2018). Initially, M particles are placed on a vertex of G. In subsequent time steps, all particles that are located on a vertex inhabited by at least two particles jump independently to a neighbour chosen uniformly at random. The process ends at the first step when no vertex is inhabited by more than one particle; we call this (random) time step the dispersion time.
In this work we study the case where G is the complete graph on n vertices and the number of particles is M = n/2+α n^{1/2} + o(n^{1/2}), α ∈ ℝ. This choice of M corresponds to the critical window of the process, with respect to the dispersion time. We show that the dispersion time, if rescaled by n^{-1/2}, converges in p-th mean, as n → ∞ and for any p ∈ ℝ, to a continuous and almost surely positive random variable T_α. We find that T_α is the absorption time of a standard logistic branching process, thoroughly investigated by Lambert (2005), and we determine its expectation. In particular, in the middle of the critical window we show that 𝔼[T₀] = π^{3/2}/√7, and furthermore we formulate explicit asymptotics when |α| gets large that quantify the transition into and out of the critical window. We also study the random variable counting the total number of jumps that are performed by the particles until the dispersion time is reached and prove that, if rescaled by nln(n), it converges to 2/7 in probability.

Umberto De Ambroggio, Tamás Makai, Konstantinos Panagiotou, and Annika Steibel. Limit Laws for Critical Dispersion on Complete Graphs. In 35th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 302, pp. 26:1-26:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

Copy BibTex To Clipboard

@InProceedings{deambroggio_et_al:LIPIcs.AofA.2024.26, author = {De Ambroggio, Umberto and Makai, Tam\'{a}s and Panagiotou, Konstantinos and Steibel, Annika}, title = {{Limit Laws for Critical Dispersion on Complete Graphs}}, booktitle = {35th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2024)}, pages = {26:1--26:12}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-329-4}, ISSN = {1868-8969}, year = {2024}, volume = {302}, editor = {Mailler, C\'{e}cile and Wild, Sebastian}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.AofA.2024.26}, URN = {urn:nbn:de:0030-drops-204617}, doi = {10.4230/LIPIcs.AofA.2024.26}, annote = {Keywords: Random processes on graphs, diffusion processes, stochastic differential equations, martingale inequalities} }

Document

**Published in:** LIPIcs, Volume 187, 38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021)

Random factor graphs provide a powerful framework for the study of inference problems such as decoding problems or the stochastic block model. Information-theoretically the key quantity of interest is the mutual information between the observed factor graph and the underlying ground truth around which the factor graph was created; in the stochastic block model, this would be the planted partition. The mutual information gauges whether and how well the ground truth can be inferred from the observable data. For a very general model of random factor graphs we verify a formula for the mutual information predicted by physics techniques. As an application we prove a conjecture about low-density generator matrix codes from [Montanari: IEEE Transactions on Information Theory 2005]. Further applications include phase transitions of the stochastic block model and the mixed k-spin model from physics.

Amin Coja-Oghlan, Max Hahn-Klimroth, Philipp Loick, Noela Müller, Konstantinos Panagiotou, and Matija Pasch. Inference and Mutual Information on Random Factor Graphs. In 38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 187, pp. 24:1-24:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

Copy BibTex To Clipboard

@InProceedings{cojaoghlan_et_al:LIPIcs.STACS.2021.24, author = {Coja-Oghlan, Amin and Hahn-Klimroth, Max and Loick, Philipp and M\"{u}ller, Noela and Panagiotou, Konstantinos and Pasch, Matija}, title = {{Inference and Mutual Information on Random Factor Graphs}}, booktitle = {38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021)}, pages = {24:1--24:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-180-1}, ISSN = {1868-8969}, year = {2021}, volume = {187}, editor = {Bl\"{a}ser, Markus and Monmege, Benjamin}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2021.24}, URN = {urn:nbn:de:0030-drops-136692}, doi = {10.4230/LIPIcs.STACS.2021.24}, annote = {Keywords: Information theory, random factor graphs, inference problems, phase transitions} }

Document

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

In this work we consider three well-studied broadcast protocols: push, pull and push&pull. A key property of all these models, which is also an important reason for their popularity, is that they are presumed to be very robust, since they are simple, randomized, and, crucially, do not utilize explicitly the global structure of the underlying graph. While sporadic results exist, there has been no systematic theoretical treatment quantifying the robustness of these models. Here we investigate this question with respect to two orthogonal aspects: (adversarial) modifications of the underlying graph and message transmission failures.
We explore in particular the following notion of local resilience: beginning with a graph, we investigate up to which fraction of the edges an adversary may delete at each vertex, so that the protocols need significantly more rounds to broadcast the information. Our main findings establish a separation among the three models. On one hand pull is robust with respect to all parameters that we consider. On the other hand, push may slow down significantly, even if the adversary is allowed to modify the degrees of the vertices by an arbitrarily small positive fraction only. Finally, push&pull is robust when no message transmission failures are considered, otherwise it may be slowed down.
On the technical side, we develop two novel methods for the analysis of randomized rumour spreading protocols. First, we exploit the notion of self-bounding functions to facilitate significantly the round-based analysis: we show that for any graph the variance of the growth of informed vertices is bounded by its expectation, so that concentration results follow immediately. Second, in order to control adversarial modifications of the graph we make use of a powerful tool from extremal graph theory, namely Szemerédi’s Regularity Lemma.

Rami Daknama, Konstantinos Panagiotou, and Simon Reisser. Robustness of Randomized Rumour Spreading. In 27th Annual European Symposium on Algorithms (ESA 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 144, pp. 36:1-36:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

Copy BibTex To Clipboard

@InProceedings{daknama_et_al:LIPIcs.ESA.2019.36, author = {Daknama, Rami and Panagiotou, Konstantinos and Reisser, Simon}, title = {{Robustness of Randomized Rumour Spreading}}, booktitle = {27th Annual European Symposium on Algorithms (ESA 2019)}, pages = {36:1--36:15}, 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.2019.36}, URN = {urn:nbn:de:0030-drops-111571}, doi = {10.4230/LIPIcs.ESA.2019.36}, annote = {Keywords: Rumour Spreading, Local Resilience, Robustness, Self-bounding Functions, Szemer\'{e}di’s Regularity Lemma} }

Document

Track A: Algorithms, Complexity and Games

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

In the last two decades the study of random instances of constraint satisfaction problems (CSPs) has flourished across several disciplines, including computer science, mathematics and physics. The diversity of the developed methods, on the rigorous and non-rigorous side, has led to major advances regarding both the theoretical as well as the applied viewpoints. The two most popular types of such CSPs are the Erdős-Rényi and the random regular CSPs.
Based on a ceteris paribus approach in terms of the density evolution equations known from statistical physics, we focus on a specific prominent class of problems of the latter type, the so-called occupation problems. The regular r-in-k occupation problems resemble a basis of this class. By now, out of these CSPs only the satisfiability threshold - the largest degree for which the problem admits asymptotically a solution - for the 1-in-k occupation problem has been rigorously established. In the present work we take a general approach towards a systematic analysis of occupation problems. In particular, we discover a surprising and explicit connection between the 2-in-k occupation problem satisfiability threshold and the determination of contraction coefficients, an important quantity in information theory measuring the loss of information that occurs when communicating through a noisy channel. We present methods to facilitate the computation of these coefficients and use them to establish explicitly the threshold for the 2-in-k occupation problem for k=4. Based on this result, for general k >= 5 we formulate a conjecture that pins down the exact value of the corresponding coefficient, which, if true, is shown to determine the threshold in all these cases.

Konstantinos Panagiotou and Matija Pasch. Satisfiability Thresholds for Regular Occupation Problems. In 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 132, pp. 90:1-90:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

Copy BibTex To Clipboard

@InProceedings{panagiotou_et_al:LIPIcs.ICALP.2019.90, author = {Panagiotou, Konstantinos and Pasch, Matija}, title = {{Satisfiability Thresholds for Regular Occupation Problems}}, booktitle = {46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)}, pages = {90:1--90:14}, 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.2019.90}, URN = {urn:nbn:de:0030-drops-106665}, doi = {10.4230/LIPIcs.ICALP.2019.90}, annote = {Keywords: Constraint satisfaction problem, replica symmetric, contraction coefficient, first moment, second moment, small subgraph conditioning} }

Document

**Published in:** LIPIcs, Volume 101, 16th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2018)

In k-ary cuckoo hashing, each of cn objects is associated with k random buckets in a hash table of size n. An l-orientation is an assignment of objects to associated buckets such that each bucket receives at most l objects. Several works have determined load thresholds c^* = c^*(k,l) for k-ary cuckoo hashing; that is, for c < c^* an l-orientation exists with high probability, and for c > c^* no l-orientation exists with high probability.
A natural variant of k-ary cuckoo hashing utilizes double hashing, where, when the buckets are numbered 0,1,...,n-1, the k choices of random buckets form an arithmetic progression modulo n. Double hashing simplifies implementation and requires less randomness, and it has been shown that double hashing has the same behavior as fully random hashing in several other data structures that similarly use multiple hashes for each object. Interestingly, previous work has come close to but has not fully shown that the load threshold for k-ary cuckoo hashing is the same when using double hashing as when using fully random hashing. Specifically, previous work has shown that the thresholds for both settings coincide, except that for double hashing it was possible that o(n) objects would have been left unplaced. Here we close this open question by showing the thresholds are indeed the same, by providing a combinatorial argument that reconciles this stubborn difference.

Michael Mitzenmacher, Konstantinos Panagiotou, and Stefan Walzer. Load Thresholds for Cuckoo Hashing with Double Hashing. In 16th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 101, pp. 29:1-29:9, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

Copy BibTex To Clipboard

@InProceedings{mitzenmacher_et_al:LIPIcs.SWAT.2018.29, author = {Mitzenmacher, Michael and Panagiotou, Konstantinos and Walzer, Stefan}, title = {{Load Thresholds for Cuckoo Hashing with Double Hashing}}, booktitle = {16th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2018)}, pages = {29:1--29:9}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-068-2}, ISSN = {1868-8969}, year = {2018}, volume = {101}, editor = {Eppstein, David}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SWAT.2018.29}, URN = {urn:nbn:de:0030-drops-88557}, doi = {10.4230/LIPIcs.SWAT.2018.29}, annote = {Keywords: Cuckoo Hashing, Double Hashing, Load Thresholds, Hypergraph Orientability} }

X

Feedback for Dagstuhl Publishing

Feedback submitted

Please try again later or send an E-mail