Probabilistic Methods for Algorithmic Discrete Mathematics

Habib, M.

doi:10.1007/978-3-662-12788-9

Cited by 92 publications

(2 citation statements)

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“…In the proof of the lower bound we are going to use the Azuma-Hoeffding inequality for random permutations (see, e.g., Lemma 11 in [16] or Section 3.2 in [20]). Let us recall that Π n stands for a random permutation selected uniformly from all (2n)!…”

Section: In Random Matchingsmentioning

confidence: 99%

Ordered Unavoidable Sub-Structures in Matchings and Random Matchings

Dudek,

Grytczuk,

Ruciński

2024

Electron. J. Combin.

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An ordered matching of size $n$ is a graph on a linearly ordered vertex set $V$, $|V|=2n$, consisting of $n$ pairwise disjoint edges. There are three different ordered matchings of size two on $V=\{1,2,3,4\}$: an alignment $\{1,2\},\{3,4\}$, a nesting $\{1,4\},\{2,3\}$, and a crossing $\{1,3\},\{2,4\}$. Accordingly, there are three basic homogeneous types of ordered matchings (with all pairs of edges arranged in the same way) which we call, respectively, lines, stacks, and waves. We prove an Erdős-Szekeres type result guaranteeing in every ordered matching of size $n$ the presence of one of the three basic sub-structures of a given size. In particular, one of them must be of size at least $n^{1/3}$. We also investigate the size of each of the three sub-structures in a random ordered matching. Additionally, the former result is generalized to $3$-uniform ordered matchings. Another type of unavoidable patterns we study are twins, that is, pairs of order-isomorphic, disjoint sub-matchings. By relating to a similar problem for permutations, we prove that the maximum size of twins that occur in every ordered matching of size $n$ is $O\left(n^{2/3}\right)$ and $\Omega\left(n^{3/5}\right)$. We conjecture that the upper bound is the correct order of magnitude and confirm it for almost all matchings. In fact, our results for twins are proved more generally for $r$-multiple twins, $r\ge2$.

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Section: In Random Matchingsmentioning

confidence: 99%

Ordered Unavoidable Sub-Structures in Matchings and Random Matchings

Dudek,

Grytczuk,

Ruciński

2024

Electron. J. Combin.

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“…Proof. Edges in the stationary measure L are simply independent Bernoulli variables and so by a standard Chernoff bound [23,Theorem 2.3(a)]…”

Section: Lower Boundsmentioning

confidence: 99%

Voter models on subcritical scale‐free random graphs

Fernley

Ortgiese

2022

Random Struct Algorithms

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The voter model is a classical interacting particle system modelling how consensus is formed across a network. We analyse the time to consensus for the voter model when the underlying graph is a subcritical scale-free random graph. Moreover, we generalise the model to include a 'temperature' parameter. The interplay between the temperature and the structure of the random graph leads to a very rich phase diagram, where in the different phases different parts of the underlying geometry dominate the time to consensus. Finally, we also consider a discursive voter model, where voters discuss their opinions with their neighbours. Our proofs rely on the well-known duality to coalescing random walks and a detailed understanding of the structure of the random graphs.

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2001

Average Case Analysis of Algorithms on Sequences

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Probabilistic Methods for Algorithmic Discrete Mathematics

Cited by 92 publications

References 0 publications

Ordered Unavoidable Sub-Structures in Matchings and Random Matchings

Ordered Unavoidable Sub-Structures in Matchings and Random Matchings

Voter models on subcritical scale‐free random graphs

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