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DOI: 10.4230/LIPIcs.AofA.2026.7

Ancestries and Descendants in a Random DAG

Authors: Fabian Burghart

Published in: LIPIcs, Volume 381, 37th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2026)

Abstract

We consider a random recursive DAG G_n on the vertex set [n] where every vertex i ≥ 2 has out-degree d, with the targets chosen uniformly at random among the earlier i-1 vertices. For this model, we propose a novel way to investigate the descendants of n (which have recently been studied in a paper by Janson) through what we call ancestry processes. The ancestor process a_i(n) of a vertex i is defined as the number of ancestors of i in G_n, and is closely related to the evolutions of multi-draw Pólya urns. Results on the descendants can then be obtained via asymptotic results on functionals of the ancestry processes, generally leading to technical integral expressions. We employ this method to make progress on two open problems posed by Janson, as well as to provide an alternative proof of a first-moment result contained in his work. We further prove limit theorems for the ancestor processes a_i(n) depending on i.

Cite as

Fabian Burghart. Ancestries and Descendants in a Random DAG. In 37th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 381, pp. 7:1-7:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{burghart:LIPIcs.AofA.2026.7,
  author =	{Burghart, Fabian},
  title =	{{Ancestries and Descendants in a Random DAG}},
  booktitle =	{37th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2026)},
  pages =	{7:1--7:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-435-2},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{381},
  editor =	{Panagiotou, Konstantinos},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.AofA.2026.7},
  URN =		{urn:nbn:de:0030-drops-262785},
  doi =		{10.4230/LIPIcs.AofA.2026.7},
  annote =	{Keywords: Random DAG, descendants, Markov process, Urn model, Limit theorems}
}

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DOI: 10.4230/LIPIcs.AofA.2026.19

Fringe Subtrees of Split Trees

Authors: Cecilia Holmgren, Jasper Ischebeck, and Svante Janson

Published in: LIPIcs, Volume 381, 37th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2026)

Abstract

We consider additive functionals X_n(ϕ) with small toll functions on split trees and a generalization of split trees, which we call fractional split trees, where the split vector does not need to sum up to 1. These additive functionals encompass e.g. the number of nodes, number of leaves and the number of fringe trees of a certain size. We show convergence of the first moment to a limit, which we can explicitly compute if all balls are distributed multinomially and for some models with Beta-distributed splitter. Generally, the first moment is given in terms of negative moments of a perpetuity and can often be approximated to arbitrary precision with known bounds. In split trees and certain fractional split trees, the standard deviation is of smaller order than the first moment, where we show a weak law of large numbers. In other fractional split trees, the standard deviation is of the same order and we show a distribution limit using the contraction method.

Cite as

Cecilia Holmgren, Jasper Ischebeck, and Svante Janson. Fringe Subtrees of Split Trees. In 37th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 381, pp. 19:1-19:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{holmgren_et_al:LIPIcs.AofA.2026.19,
  author =	{Holmgren, Cecilia and Ischebeck, Jasper and Janson, Svante},
  title =	{{Fringe Subtrees of Split Trees}},
  booktitle =	{37th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2026)},
  pages =	{19:1--19:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-435-2},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{381},
  editor =	{Panagiotou, Konstantinos},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.AofA.2026.19},
  URN =		{urn:nbn:de:0030-drops-262900},
  doi =		{10.4230/LIPIcs.AofA.2026.19},
  annote =	{Keywords: fringe tree, split tree, fractional split tree, limit theorem, additive functional, renewal theory, Fourier series}
}

Document

DOI: 10.4230/LIPIcs.CPM.2026.28

Periodicity Property Testing on Strings with Wildcards

Authors: Carl Barton, Panagiotis Charalampopoulos, Taha El Ghazi, Jonas Ellert, Oded Lachish, and Tatiana Starikovskaya

Published in: LIPIcs, Volume 369, 37th Annual Symposium on Combinatorial Pattern Matching (CPM 2026)

Abstract

In this work, we study periodicity in strings with wildcards. A string T with at most k wildcards is called strongly (p,k)-periodic if the wildcards in T can be replaced with alphabet symbols to obtain a string with period p, and weakly (p,k)-periodic if T[i] matches T[i+p] for all i. Intuitively, both generalize to (≤ g, k)-periodicity, which is the property of being (p,k)-periodic for some p ∈ [1..g]. An ε-tester for a property 𝒫 is an algorithm that distinguishes between strings that satisfy 𝒫 and strings where one needs to change at least an ε-fraction of the symbols to obtain a string that satisfies 𝒫. We study one-sided error testers, where strings satisfying 𝒫 must always be accepted, while strings that are ε-far must be rejected with probability at least 2/3. The complexity of a tester is the worst-case number of symbols of an input of length n it must read to make the decision. We design the following testers for p,g ≤ n/2: 1) An ε-tester for strong (p,k)-periodicity with complexity Õ_ε(1) . 2) An ε-tester for strong (≤ g,k)-periodicity with complexity Õ_ε(√g). 3) An ε-tester for weak (p,k)-periodicity with complexity Õ_ε(min(k, n /(k+p))). 4) An ε-tester for weak (≤ g,k)-periodicity with complexity Õ_ε(min(k+ √{gk}, n/√k)). Additionally, we show a lower bound on the complexity of ε-testers for weak (≤ g,k)-periodicity, implying that our tester for weak (≤ g,k)-periodicity is optimal up to a multiplicative (ε^{-1} ln(gk))^O(1) factor for a wide range of g and k. Finally, our tester for strong (≤ g,k)-periodicity generalizes the one of [Lachish and Newman; Algorithmica 2011] for strings without wildcards, matching (up to polylogarithmic factors) the unconditional lower bound of ̃Ω(√g) in said work for constant ε.

Cite as

Carl Barton, Panagiotis Charalampopoulos, Taha El Ghazi, Jonas Ellert, Oded Lachish, and Tatiana Starikovskaya. Periodicity Property Testing on Strings with Wildcards. In 37th Annual Symposium on Combinatorial Pattern Matching (CPM 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 369, pp. 28:1-28:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{barton_et_al:LIPIcs.CPM.2026.28,
  author =	{Barton, Carl and Charalampopoulos, Panagiotis and Ghazi, Taha El and Ellert, Jonas and Lachish, Oded and Starikovskaya, Tatiana},
  title =	{{Periodicity Property Testing on Strings with Wildcards}},
  booktitle =	{37th Annual Symposium on Combinatorial Pattern Matching (CPM 2026)},
  pages =	{28:1--28:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-420-8},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{369},
  editor =	{Bille, Philip and Prezza, Nicola},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CPM.2026.28},
  URN =		{urn:nbn:de:0030-drops-259543},
  doi =		{10.4230/LIPIcs.CPM.2026.28},
  annote =	{Keywords: periodicity, property testing, wildcards}
}

Document

DOI: 10.4230/LIPIcs.STACS.2026.39

Planting and MCMC Sampling from the Potts Model

Authors: Andreas Galanis, Leslie Ann Goldberg, and Paulina Smolarova

Published in: LIPIcs, Volume 364, 43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026)

Abstract

We consider the problem of sampling from the ferromagnetic q-state Potts model on the random d-regular graph with parameter β > 0. A key difficulty that arises in sampling from the model is the existence of a "metastability" window β ∈ (β_u,β_u'), where roughly the distribution has two competing modes, the so-called disordered and ordered phases. This causes classical Markov-chain algorithms to be slow mixing from worst-case initialisations. Nevertheless, Helmuth, Jenssen and Perkins (SODA '19) designed a sampling algorithm that works for all β, when d ≥ 5 and q = d^{Ω(d)}, using polymers and cluster expansion methods; more recently, their analysis technique has been adapted to show that a Markov chain (random-cluster dynamics) mixes fast when initialised appropriately, in the same regime of q,d,β. Despite these positive algorithmic results, a well-known bottleneck behind cluster-expansion arguments is that they inherently only work for large q, whereas it is widely conjectured that sampling on the random d-regular graph is possible for all q,d ≥ 3. The only result so far that applies to general q,d ≥ 3 is by Blanca and Gheissari who showed that the random-cluster dynamics mixes fast in the "uniqueness" regime β < β_u where roughly only the disordered mode exists. For β ≥ β_u however, a second subdominant mode emerges creating bottlenecks and giving rise to correlations which have been hard to handle, especially for small values of q and d. Our main contribution is to perform a delicate analysis of the Potts distribution and the random-cluster dynamics that goes beyond the threshold β_u. We use planting as the main tool, a technique used in the analysis of random CSPs to capture how the space of solutions is correlated with the structure of the random instance. While planting arguments provide only weak sampling guarantees generically, here we instead combine planting with the analysis of random-cluster dynamics to obtain significantly stronger guarantees. We are thus able to show that the random-cluster dynamics initialised from all-out mixes fast for all integers q,d ≥ 3 beyond the uniqueness threshold β_u, all the way to the optimal threshold β_c ∈ (β_u,β_u') where the dominant mode switches from disordered to ordered. A more involved analysis also applies to the ordered regime β > β_c where we obtain an algorithm for all d ≥ 3 and q ≥ (5d)⁵, improving significantly upon the previous range of q,d by Carlson, Davies, Fraiman, Kolla, Potukuchi, and Yap (FOCS'22).

Cite as

Andreas Galanis, Leslie Ann Goldberg, and Paulina Smolarova. Planting and MCMC Sampling from the Potts Model. In 43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 364, pp. 39:1-39:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{galanis_et_al:LIPIcs.STACS.2026.39,
  author =	{Galanis, Andreas and Goldberg, Leslie Ann and Smolarova, Paulina},
  title =	{{Planting and MCMC Sampling from the Potts Model}},
  booktitle =	{43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026)},
  pages =	{39:1--39:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-412-3},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{364},
  editor =	{Mahajan, Meena and Manea, Florin and McIver, Annabelle and Thắng, Nguy\~{ê}n Kim},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2026.39},
  URN =		{urn:nbn:de:0030-drops-255280},
  doi =		{10.4230/LIPIcs.STACS.2026.39},
  annote =	{Keywords: approximate sampling, Glauber dynamics, Potts model, random cluster model}
}

Document

DOI: 10.4230/LIPIcs.FSTTCS.2025.17

Cutoff for the Swendsen–Wang Dynamics on the Complete Graph

Authors: Antonio Blanca and Zhezheng Song

Published in: LIPIcs, Volume 360, 45th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2025)

Abstract

We study the speed of convergence of the Swendsen-Wang (SW) dynamics for the q-state ferromagnetic Potts model on the n-vertex complete graph, known as the mean-field model. The SW dynamics was introduced as an attractive alternative to the local Glauber dynamics, often offering faster convergence rates to stationarity in a variety of settings. A series of works have characterized the asymptotic behavior of the speed of convergence of the mean-field SW dynamics for all q ≥ 2 and all values of the inverse temperature parameter β > 0. In particular, it is known that when β > q the mixing time of the SW dynamics is Θ(log n). We strengthen this result by showing that for all β > q, there exists a constant c(β,q) > 0 such that the mixing time of the SW dynamics is c(β,q) log n + Θ(1). This implies that the mean-field SW dynamics exhibits the cutoff phenomenon in this temperature regime, demonstrating that this Markov chain undergoes a sharp transition from "far from stationarity" to "well-mixed" within a narrow Θ(1) time window. The presence of cutoff is algorithmically significant, as simulating the chain for fewer steps than its mixing time could lead to highly biased samples.

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Antonio Blanca and Zhezheng Song. Cutoff for the Swendsen–Wang Dynamics on the Complete Graph. In 45th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 360, pp. 17:1-17:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{blanca_et_al:LIPIcs.FSTTCS.2025.17,
  author =	{Blanca, Antonio and Song, Zhezheng},
  title =	{{Cutoff for the Swendsen–Wang Dynamics on the Complete Graph}},
  booktitle =	{45th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2025)},
  pages =	{17:1--17:21},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-406-2},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{360},
  editor =	{Aiswarya, C. and Mehta, Ruta and Roy, Subhajit},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2025.17},
  URN =		{urn:nbn:de:0030-drops-250987},
  doi =		{10.4230/LIPIcs.FSTTCS.2025.17},
  annote =	{Keywords: Markov chains, mixing times, cutoff phenomenon, Potts model, mean-field}
}

Document

DOI: 10.4230/LIPIcs.DISC.2025.14

Energy-Efficient Maximal Independent Sets in Radio Networks

Authors: Dominick Banasik, Varsha Dani, Fabien Dufoulon, Aayush Gupta, Thomas P. Hayes, and Gopal Pandurangan

Published in: LIPIcs, Volume 356, 39th International Symposium on Distributed Computing (DISC 2025)

Abstract

The maximal independent set (MIS) is one of the most fundamental problems in distributed computing, and it has been studied intensively for over four decades. This paper focuses on the MIS problem in the radio network model, a standard model widely used to model wireless networks, particularly ad hoc wireless and sensor networks. Energy is a premium resource in these networks, which are typically battery-powered. Hence, designing distributed algorithms that use as little energy as possible is crucial. We use the well-established energy model where a node can be sleeping or awake in a round, and only the awake rounds (when it can send or listen) determine the energy complexity of the algorithm, which we want to minimize. We present new, more energy-efficient MIS algorithms in radio networks with arbitrary and unknown graph topology. We present algorithms for two popular variants of the radio model - with collision detection (CD) and without collision detection (no-CD). Specifically, we obtain the following results: 1) CD model: We present a randomized distributed MIS algorithm with energy complexity O(log n), round complexity O(log² n), and failure probability 1 / poly(n), where n is the network size. We show that our energy complexity is optimal by showing a matching Ω(log n) lower bound. 2) no-CD model: In the more challenging no-CD model, we present a randomized distributed MIS algorithm with energy complexity O(log²n log log n), round complexity O(log³ n log Δ), and failure probability 1 / poly(n). The energy complexity of our algorithm is significantly lower than the round (and energy) complexity of O(log³ n) of the best known distributed MIS algorithm of Davies [PODC 2023] for arbitrary graph topology.

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Dominick Banasik, Varsha Dani, Fabien Dufoulon, Aayush Gupta, Thomas P. Hayes, and Gopal Pandurangan. Energy-Efficient Maximal Independent Sets in Radio Networks. In 39th International Symposium on Distributed Computing (DISC 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 356, pp. 14:1-14:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{banasik_et_al:LIPIcs.DISC.2025.14,
  author =	{Banasik, Dominick and Dani, Varsha and Dufoulon, Fabien and Gupta, Aayush and Hayes, Thomas P. and Pandurangan, Gopal},
  title =	{{Energy-Efficient Maximal Independent Sets in Radio Networks}},
  booktitle =	{39th International Symposium on Distributed Computing (DISC 2025)},
  pages =	{14:1--14:24},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-402-4},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{356},
  editor =	{Kowalski, Dariusz R.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.DISC.2025.14},
  URN =		{urn:nbn:de:0030-drops-248311},
  doi =		{10.4230/LIPIcs.DISC.2025.14},
  annote =	{Keywords: Distributed Computing, Energy Complexity, Sleeping Model, Radio Networks, Maximal Independent Set}
}

Document

DOI: 10.4230/LIPIcs.ESA.2025.85

A Simple yet Exact Analysis of the MultiQueue

Authors: Stefan Walzer and Marvin Williams

Published in: LIPIcs, Volume 351, 33rd Annual European Symposium on Algorithms (ESA 2025)

Abstract

The MultiQueue is a relaxed concurrent priority queue consisting of n internal priority queues, where an insertion uses a random queue and a deletion considers two random queues and deletes the minimum from the one with the smaller minimum. The rank error of the deletion is the number of smaller elements in the MultiQueue. Alistarh et al. [Alistarh et al., 2017] have demonstrated in a sophisticated potential argument that the expected rank error remains bounded by 𝒪(n) over long sequences of deletions. In this paper we present a simpler analysis by identifying the stable distribution of an underlying Markov chain and with it the long-term distribution of the rank error exactly. Simple calculations then reveal the expected long-term rank error to be (5/6)n-1+1/(6n). Our arguments generalize to deletion schemes where the probability to delete from a given queue depends only on the rank of the queue. Specifically, this includes deleting from the best of c randomly selected queues for any c > 1.

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Stefan Walzer and Marvin Williams. A Simple yet Exact Analysis of the MultiQueue. In 33rd Annual European Symposium on Algorithms (ESA 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 351, pp. 85:1-85:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{walzer_et_al:LIPIcs.ESA.2025.85,
  author =	{Walzer, Stefan and Williams, Marvin},
  title =	{{A Simple yet Exact Analysis of the MultiQueue}},
  booktitle =	{33rd Annual European Symposium on Algorithms (ESA 2025)},
  pages =	{85:1--85:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-395-9},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{351},
  editor =	{Benoit, Anne and Kaplan, Haim and Wild, Sebastian 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.2025.85},
  URN =		{urn:nbn:de:0030-drops-245533},
  doi =		{10.4230/LIPIcs.ESA.2025.85},
  annote =	{Keywords: MultiQueue, concurrent data structure, stochastic process, Markov chain}
}

Document

Track A: Algorithms, Complexity and Games

DOI: 10.4230/LIPIcs.ICALP.2025.83

Low-Temperature Sampling on Sparse Random Graphs

Authors: Andreas Galanis, Leslie Ann Goldberg, and Paulina Smolarova

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

Abstract

We consider sampling in the so-called low-temperature regime, which is typically characterised by non-local behaviour and strong global correlations. Canonical examples include sampling independent sets on bipartite graphs and sampling from the ferromagnetic q-state Potts model. Low-temperature sampling is computationally intractable for general graphs, but recent advances based on the polymer method have made significant progress for graph families that exhibit certain expansion properties that reinforce the correlations, including for example expanders, lattices and dense graphs. One of the most natural graph classes that has so far escaped this algorithmic framework is the class of sparse Erdős-Rényi random graphs whose expansion only manifests for sufficiently large subsets of vertices; small sets of vertices on the other hand have vanishing expansion which makes them behave independently from the bulk of the graph and therefore weakens the correlations. At a more technical level, the expansion of small sets is crucial for establishing the Kotecky-Priess condition which underpins the applicability of the framework. Our main contribution is to develop the polymer method in the low-temperature regime for sparse random graphs. As our running example, we use the Potts and random-cluster models on G(n,d/n) for d = Θ(1), where we show a polynomial-time sampling algorithm for all sufficiently large q and d, at all temperatures. Our approach applies more generally for models that are monotone. Key to our result is a simple polymer definition that blends easily with the connectivity properties of the graph and allows us to show that polymers have size at most O(log n).

Cite as

Andreas Galanis, Leslie Ann Goldberg, and Paulina Smolarova. Low-Temperature Sampling on Sparse Random Graphs. In 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 334, pp. 83:1-83:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{galanis_et_al:LIPIcs.ICALP.2025.83,
  author =	{Galanis, Andreas and Goldberg, Leslie Ann and Smolarova, Paulina},
  title =	{{Low-Temperature Sampling on Sparse Random Graphs}},
  booktitle =	{52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025)},
  pages =	{83:1--83:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-372-0},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{334},
  editor =	{Censor-Hillel, Keren and Grandoni, Fabrizio and Ouaknine, Jo\"{e}l and Puppis, Gabriele},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2025.83},
  URN =		{urn:nbn:de:0030-drops-234606},
  doi =		{10.4230/LIPIcs.ICALP.2025.83},
  annote =	{Keywords: approximate counting, Glauber dynamics, random cluster model, approximate sampling, Erd\H{o}s-R\'{e}nyi Graphs}
}

Document

Track A: Algorithms, Complexity and Games

DOI: 10.4230/LIPIcs.ICALP.2025.58

A New Impossibility Result for Online Bipartite Matching Problems

Authors: Flavio Chierichetti, Mirko Giacchini, Alessandro Panconesi, and Andrea Vattani

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

Abstract

Online Bipartite Matching with random user arrival is a fundamental problem in the online advertisement ecosystem. Over the last 30 years, many algorithms and impossibility results have been developed for this problem. In particular, the latest impossibility result was established by Manshadi, Oveis Gharan and Saberi [Manshadi et al., 2011] in 2011. Since then, several algorithms have been published in an effort to narrow the gap between the upper and the lower bounds on the competitive ratio. In this paper we show that no algorithm can achieve a competitive ratio better than 1- e/(e^e) = 0.82062…, improving upon the 0.823 upper bound presented in [Manshadi et al., 2011]. Our construction is simple to state, accompanied by a fully analytic proof, and yields a competitive ratio bound intriguingly similar to 1 - 1/e, the optimal competitive ratio for the fully adversarial Online Bipartite Matching problem. Although the tightness of our upper bound remains an open question, we show that our construction is extremal in a natural class of instances.

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Flavio Chierichetti, Mirko Giacchini, Alessandro Panconesi, and Andrea Vattani. A New Impossibility Result for Online Bipartite Matching Problems. In 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 334, pp. 58:1-58:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{chierichetti_et_al:LIPIcs.ICALP.2025.58,
  author =	{Chierichetti, Flavio and Giacchini, Mirko and Panconesi, Alessandro and Vattani, Andrea},
  title =	{{A New Impossibility Result for Online Bipartite Matching Problems}},
  booktitle =	{52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025)},
  pages =	{58:1--58:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-372-0},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{334},
  editor =	{Censor-Hillel, Keren and Grandoni, Fabrizio and Ouaknine, Jo\"{e}l and Puppis, Gabriele},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2025.58},
  URN =		{urn:nbn:de:0030-drops-234354},
  doi =		{10.4230/LIPIcs.ICALP.2025.58},
  annote =	{Keywords: Bipartite Matching, Random Graphs, Competitive Ratio}
}

Document

Track A: Algorithms, Complexity and Games

DOI: 10.4230/LIPIcs.ICALP.2025.70

Tiling Random Regular Graphs Efficiently

Authors: Sahar Diskin, Ilay Hoshen, and Maksim Zhukovskii

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

Abstract

We show that for every ε > 0 there exists a sufficiently large d₀ ∈ ℕ such that for every d ≥ d₀, whp the random d-regular graph G(n,d) contains a T-factor for every tree T on at most (1-ε)d/log d vertices. This is best possible since, for large enough integer d, whp G(n,d) does not contain a ((1+ε)d)/(log d)-star-factor. Our method gives a randomised algorithm which whp finds said T-factor and whose expected running time is O(n^{1+o(1)}), as well as an efficient deterministic counterpart.

Cite as

Sahar Diskin, Ilay Hoshen, and Maksim Zhukovskii. Tiling Random Regular Graphs Efficiently. In 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 334, pp. 70:1-70:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{diskin_et_al:LIPIcs.ICALP.2025.70,
  author =	{Diskin, Sahar and Hoshen, Ilay and Zhukovskii, Maksim},
  title =	{{Tiling Random Regular Graphs Efficiently}},
  booktitle =	{52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025)},
  pages =	{70:1--70:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-372-0},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{334},
  editor =	{Censor-Hillel, Keren and Grandoni, Fabrizio and Ouaknine, Jo\"{e}l and Puppis, Gabriele},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2025.70},
  URN =		{urn:nbn:de:0030-drops-234477},
  doi =		{10.4230/LIPIcs.ICALP.2025.70},
  annote =	{Keywords: Random regular graphs, Tree tilings}
}

Document

DOI: 10.4230/LIPIcs.STACS.2025.75

Canonical Labeling of Sparse Random Graphs

Authors: Oleg Verbitsky and Maksim Zhukovskii

Published in: LIPIcs, Volume 327, 42nd International Symposium on Theoretical Aspects of Computer Science (STACS 2025)

Abstract

We show that if p = O(1/n), then the Erdős-Rényi random graph G(n,p) with high probability admits a canonical labeling computable in time O(nlog n). Combined with the previous results on the canonization of random graphs, this implies that G(n,p) with high probability admits a polynomial-time canonical labeling whatever the edge probability function p. Our algorithm combines the standard color refinement routine with simple post-processing based on the classical linear-time tree canonization. Noteworthy, our analysis of how well color refinement performs in this setting allows us to complete the description of the automorphism group of the 2-core of G(n,p).

Cite as

Oleg Verbitsky and Maksim Zhukovskii. Canonical Labeling of Sparse Random Graphs. In 42nd International Symposium on Theoretical Aspects of Computer Science (STACS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 327, pp. 75:1-75:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{verbitsky_et_al:LIPIcs.STACS.2025.75,
  author =	{Verbitsky, Oleg and Zhukovskii, Maksim},
  title =	{{Canonical Labeling of Sparse Random Graphs}},
  booktitle =	{42nd International Symposium on Theoretical Aspects of Computer Science (STACS 2025)},
  pages =	{75:1--75:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-365-2},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{327},
  editor =	{Beyersdorff, Olaf and Pilipczuk, Micha{\l} and Pimentel, Elaine and Thắng, Nguy\~{ê}n Kim},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2025.75},
  URN =		{urn:nbn:de:0030-drops-229003},
  doi =		{10.4230/LIPIcs.STACS.2025.75},
  annote =	{Keywords: Graph isomorphism, random graphs, canonical labeling, color refinement}
}

Document

DOI: 10.4230/LIPIcs.CSL.2025.12

First-Order Logic with Equicardinality in Random Graphs

Authors: Simi Haber, Tal Hershko, Mostafa Mirabi, and Saharon Shelah

Published in: LIPIcs, Volume 326, 33rd EACSL Annual Conference on Computer Science Logic (CSL 2025)

Abstract

We answer a question of Blass and Harary about the validity of the zero-one law in random graphs for extensions of first-order logic (FOL). For a given graph property P, the Lindström extension of FOL by P is defined as the minimal (regular) extension of FOL able to express P. For several graph properties P (e.g. Hamiltonicity), it is known that the Lindström extension by P is also able to interpret a segment of arithmetic, and thus strongly disobeys the zero-one law. Common to all these properties is the ability to express the Härtig quantifier, a natural extension of FOL testing if two definable sets are of the same size. We prove that the Härtig quantifier is sufficient for the interpretation of arithmetic, thus providing a general result which implies all known cases of Lindström extensions which are able to interpret a segment of arithmetic.

Cite as

Simi Haber, Tal Hershko, Mostafa Mirabi, and Saharon Shelah. First-Order Logic with Equicardinality in Random Graphs. In 33rd EACSL Annual Conference on Computer Science Logic (CSL 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 326, pp. 12:1-12:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{haber_et_al:LIPIcs.CSL.2025.12,
  author =	{Haber, Simi and Hershko, Tal and Mirabi, Mostafa and Shelah, Saharon},
  title =	{{First-Order Logic with Equicardinality in Random Graphs}},
  booktitle =	{33rd EACSL Annual Conference on Computer Science Logic (CSL 2025)},
  pages =	{12:1--12:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-362-1},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{326},
  editor =	{Endrullis, J\"{o}rg and Schmitz, Sylvain},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2025.12},
  URN =		{urn:nbn:de:0030-drops-227694},
  doi =		{10.4230/LIPIcs.CSL.2025.12},
  annote =	{Keywords: finite model theory, first-order logic, monadic second-order logic, random graphs, zero-one laws, generalized quantifiers, equicardinality}
}

Document

DOI: 10.4230/LIPIcs.OPODIS.2024.37

Almost Time-Optimal Loosely-Stabilizing Leader Election on Arbitrary Graphs Without Identifiers in Population Protocols

Authors: Haruki Kanaya, Ryota Eguchi, Taisho Sasada, and Michiko Inoue

Published in: LIPIcs, Volume 324, 28th International Conference on Principles of Distributed Systems (OPODIS 2024)

Abstract

The population protocol model is a computational model for passive mobile agents. We address the leader election problem, which determines a unique leader on arbitrary communication graphs starting from any configuration. Unfortunately, self-stabilizing leader election is impossible to be solved without knowing the exact number of agents; thus, we consider loosely-stabilizing leader election, which converges to safe configurations in a relatively short time, and holds the specification (maintains a unique leader) for a relatively long time. When agents have unique identifiers, Sudo {et al. }(2019) proposed a protocol that, given an upper bound N for the number of agents n, converges in O(mNlog n) expected steps, where m is the number of edges. When unique identifiers are not required, they also proposed a protocol that, using random numbers and given N, converges in O(mN²log{N}) expected steps. Both protocols have a holding time of Ω(e^{2N}) expected steps and use O(log{N}) bits of memory. They also showed that the lower bound of the convergence time is Ω(mN) expected steps for protocols with a holding time of Ω(e^N) expected steps given N. In this paper, we propose protocols that do not require unique identifiers. These protocols achieve convergence times close to the lower bound with increasing memory usage. Specifically, given N and an upper bound Δ for the maximum degree, we propose two protocols whose convergence times are O(mNlog n) and O(mNlog N) both in expectation and with high probability. The former protocol uses random numbers, while the latter does not require them. Both protocols utilize O(Δ log N) bits of memory and hold the specification for Ω(e^{2N}) expected steps.

Cite as

Haruki Kanaya, Ryota Eguchi, Taisho Sasada, and Michiko Inoue. Almost Time-Optimal Loosely-Stabilizing Leader Election on Arbitrary Graphs Without Identifiers in Population Protocols. In 28th International Conference on Principles of Distributed Systems (OPODIS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 324, pp. 37:1-37:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{kanaya_et_al:LIPIcs.OPODIS.2024.37,
  author =	{Kanaya, Haruki and Eguchi, Ryota and Sasada, Taisho and Inoue, Michiko},
  title =	{{Almost Time-Optimal Loosely-Stabilizing Leader Election on Arbitrary Graphs Without Identifiers in Population Protocols}},
  booktitle =	{28th International Conference on Principles of Distributed Systems (OPODIS 2024)},
  pages =	{37:1--37:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-360-7},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{324},
  editor =	{Bonomi, Silvia and Galletta, Letterio and Rivi\`{e}re, Etienne and Schiavoni, Valerio},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.OPODIS.2024.37},
  URN =		{urn:nbn:de:0030-drops-225734},
  doi =		{10.4230/LIPIcs.OPODIS.2024.37},
  annote =	{Keywords: Population protocols, Leader election, Loose-stabilization, Self-stabilization}
}

@InProceedings{kanaya_et_al:LIPIcs.OPODIS.2024.37,
  author =	{Kanaya, Haruki and Eguchi, Ryota and Sasada, Taisho and Inoue, Michiko},
  title =	{{Almost Time-Optimal Loosely-Stabilizing Leader Election on Arbitrary Graphs Without Identifiers in Population Protocols}},
  booktitle =	{28th International Conference on Principles of Distributed Systems (OPODIS 2024)},
  pages =	{37:1--37:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-360-7},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{324},
  editor =	{Bonomi, Silvia and Galletta, Letterio and Rivi\`{e}re, Etienne and Schiavoni, Valerio},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.OPODIS.2024.37},
  URN =		{urn:nbn:de:0030-drops-225734},
  doi =		{10.4230/LIPIcs.OPODIS.2024.37},
  annote =	{Keywords: Population protocols, Leader election, Loose-stabilization, Self-stabilization}
}

Document

DOI: 10.4230/LIPIcs.AofA.2024.1

Fringe Trees for Random Trees with Given Vertex Degrees

Authors: Gabriel Berzunza Ojeda, Cecilia Holmgren, and Svante Janson

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

Abstract

We prove that the number of fringe subtrees, isomorphic to a given tree, in uniformly random trees with given vertex degrees, asymptotically follows a normal distribution. As an application, we establish the same asymptotic normality for random simply generated trees (conditioned Galton-Watson trees). Our approach relies on an extension of Gao and Wormald’s (2004) theorem to the multivariate setting.

Cite as

Gabriel Berzunza Ojeda, Cecilia Holmgren, and Svante Janson. Fringe Trees for Random Trees with Given Vertex Degrees. 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. 1:1-1:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{berzunzaojeda_et_al:LIPIcs.AofA.2024.1,
  author =	{Berzunza Ojeda, Gabriel and Holmgren, Cecilia and Janson, Svante},
  title =	{{Fringe Trees for Random Trees with Given Vertex Degrees}},
  booktitle =	{35th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2024)},
  pages =	{1:1--1:13},
  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.2024.1},
  URN =		{urn:nbn:de:0030-drops-204369},
  doi =		{10.4230/LIPIcs.AofA.2024.1},
  annote =	{Keywords: Conditioned Galton-Watson trees, fringe trees, simply generated trees, uniformly random trees with given vertex degrees}
}

Document

DOI: 10.4230/LIPIcs.AofA.2024.5

Bit-Array-Based Alternatives to HyperLogLog

Authors: Svante Janson, Jérémie Lumbroso, and Robert Sedgewick

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

Abstract

We present a family of algorithms for the problem of estimating the number of distinct items in an input stream that are simple to implement and are appropriate for practical applications. Our algorithms are a logical extension of the series of algorithms developed by Flajolet and his coauthors starting in 1983 that culminated in the widely used HyperLogLog algorithm. These algorithms divide the input stream into M substreams and lead to a time-accuracy tradeoff where a constant number of bits per substream are saved to achieve a relative accuracy proportional to 1/√M. Our algorithms use just one or two bits per substream. Their effectiveness is demonstrated by a proof of approximate normality, with explicit expressions for standard errors that inform parameter settings and allow proper quantitative comparisons with other methods. Hypotheses about performance are validated through experiments using a realistic input stream, with the conclusion that our algorithms are more accurate than HyperLogLog when using the same amount of memory, and they use two-thirds as much memory as HyperLogLog to achieve a given accuracy.

Cite as

Svante Janson, Jérémie Lumbroso, and Robert Sedgewick. Bit-Array-Based Alternatives to HyperLogLog. 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. 5:1-5:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{janson_et_al:LIPIcs.AofA.2024.5,
  author =	{Janson, Svante and Lumbroso, J\'{e}r\'{e}mie and Sedgewick, Robert},
  title =	{{Bit-Array-Based Alternatives to HyperLogLog}},
  booktitle =	{35th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2024)},
  pages =	{5:1--5:19},
  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.2024.5},
  URN =		{urn:nbn:de:0030-drops-204402},
  doi =		{10.4230/LIPIcs.AofA.2024.5},
  annote =	{Keywords: Cardinality estimation, sketching, Hyperloglog}
}

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