A note on the random greedy triangle-packing algorithm

Bohman, Tom; Frieze, Alan; Lubetzky, Eyal

doi:10.4310/joc.2010.v1.n4.a5

Cited by 16 publications

(42 citation statements)

References 14 publications

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“…We are now ready to bound the conditional probability of A ij = 1 that is given by (6). 9 Note that we need some condition on the parameters in the statement of the theorem in order to rule out the case when, say, y b ≈ 2, but 1 xy −(a+b) < xy −a < 2, and u → ∞.…”

Section: Number-theoretic Factsmentioning

confidence: 99%

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The sharp threshold for making squares

2018

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Consider a random sequence of N integers, each chosen uniformly and independently from the set {1, . . . , x}. Motivated by applications to factorisation algorithms such as Dixon's algorithm, the quadratic sieve, and the number field sieve, Pomerance in 1994 posed the following problem: how large should N be so that, with high probability, this sequence contains a subsequence, the product of whose elements is a perfect square? Pomerance determined asymptotically the logarithm of the threshold for this event, and conjectured that it in fact exhibits a sharp threshold in N . More recently, Croot, Granville, Pemantle and Tetali determined the threshold up to a factor of 4/π + o(1) as x → ∞, and made a conjecture regarding the location of the sharp threshold.In this paper we prove both of these conjectures, by determining the sharp threshold for making squares. Our proof combines techniques from combinatorics, probability and analytic number theory; in particular, we use the so-called method of self-correcting martingales in order to control the size of the 2-core of the random hypergraph that encodes the prime factors of our random numbers. Our method also gives a new (and completely different) proof of the upper bound in the main theorem of Croot, Granville, Pemantle and Tetali.

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Section: Number-theoretic Factsmentioning

confidence: 99%

“…We will need the former property in order to obtain the self-correction (see below) that will play a crucial role in our proof, and the latter property in order to show that the bound (17) holds when k is reasonably large. 6 We shall prove the following theorem.…”

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The sharp threshold for making squares

2018

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“…Note that we have used the notation˜ (·): If f and g are functions of N such that f is bounded above by g times some poly-logarithmic factor we write f =˜ (g). Also note that D > N (see (4)) is used to get the last expression above. For our bound on d + (v) we apply Lemma 4.5 to the supermartingale Z + (v).…”

Section: Dynamic Concentrationmentioning

confidence: 99%

A note on the random greedy independent set algorithm

Bennett

Bohman

2016

Random Struct Algorithms

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Let r be a fixed constant and let scriptH be an r‐uniform, D‐regular hypergraph on N vertices. Assume further that D>Nϵ for some ϵ>0. Consider the random greedy algorithm for forming an independent set in scriptH. An independent set is chosen at random by iteratively choosing vertices at random to be in the independent set. At each step we chose a vertex uniformly at random from the collection of vertices that could be added to the independent set (i.e. the collection of vertices v with the property that v is not in the current independent set I and I∪{v} contains no edge in scriptH). Note that this process terminates at a maximal subset of vertices with the property that this set contains no edge of scriptH; that is, the process terminates at a maximal independent set. We prove that if scriptH satisfies certain degree and codegree conditions then there are Ωtrue(N·((logN)/D)1r−1true) vertices in the independent set produced by the random greedy algorithm with high probability. This result generalizes a lower bound on the number of steps in the H‐free process due to Bohman and Keevash and produces objects of interest in additive combinatorics. © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 49, 479–502, 2016

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“…Our improvement on previous analyses of this process exploits the self-correcting nature of key statistics of the process. For a treatment of self-correction in a simpler context see [6]. The methods that we use to establish self-correction of the triangle-free process build on the ideas used recently by Bohman, Frieze and Lubetzky [7] for an analysis of the triangle-removal process.…”

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confidence: 99%

Dynamic concentration of the triangle-free process

Bohman

Keevash

2013

The Seventh European Conference on Combinatorics, Graph Theory and Applications

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130

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The triangle-free process begins with an empty graph on n vertices and iteratively adds edges chosen uniformly at random subject to the constraint that no triangle is formed. We determine the asymptotic number of edges in the maximal triangle-free graph at which the triangle-free process terminates. We also bound the independence number of this graph, which gives an improved lower bound on the Ramsey numbers R(3, t): we show R(3, t) > (1/4 − o(1))t 2 / log t, which is within a 4 + o(1) factor of the best known upper bound. Our improvement on previous analyses of this process exploits the self-correcting nature of key statistics of the process. Furthermore, we determine which bounded size subgraphs are likely to appear in the maximal triangle-free graph produced by the triangle-free process: they are precisely those triangle-free graphs with 2-density at most 2.

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A note on the random greedy triangle-packing algorithm

Cited by 16 publications

References 14 publications

The sharp threshold for making squares

The sharp threshold for making squares

A note on the random greedy independent set algorithm

Dynamic concentration of the triangle-free process

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