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This extended abstract describes and analyses a near-optimal probabilistic algorithm, HYPERLOGLOG, dedicated to estimating the number of distinct elements (the cardinality) of very large data ensembles. Using an auxiliary memory of m units (typically, “short bytes”), HYPERLOGLOG performs a single pass over the data and produces an estimate of the… (More)

- Éric Fusy
- 2005

This extended abstract introduces a new algorithm for the random generation of labelled planar graphs. Its principles rely on Boltzmann samplers as recently developed by Duchon, Flajolet, Louchard, and Schaeffer. It combines the Boltzmann framework, a judicious use of rejection, a new combinatorial bijection found by Fusy, Poulalhon and Schaeffer, as well… (More)

- Éric Fusy
- 2007

- Olivier Bernardi, Éric Fusy
- J. Comb. Theory, Ser. A
- 2012

A d-angulation is a planar map with faces of degree d. We present for each integer d ≥ 3 a bijection between the class of d-angulations of girth d and a class of decorated plane trees. Each of the bijections is obtained by specializing a “master bijection” which extends an earlier construction of the first author. Bijections already existed for… (More)

- Éric Fusy
- Discrete Mathematics
- 2009

This article focuses on a combinatorial structure specific to triangulated plane graphs with quadrangular outer face and no separating triangle, which are called irreducible triangulations. The structure has been introduced by Xin He under the name of regular edge-labelling and consists of two bipolar orientations that are transversal. For this reason, the… (More)

Boltzmann models from statistical physics combined with methods from analytic combinatorics give rise to efficient algorithms for the random generation of unlabelled objects. The resulting algorithms generate in an unbiased manner discrete configurations that may have nontrivial symmetries, and they do so by means of real-arithmetic computations. We present… (More)

- Éric Fusy, Dominique Poulalhon, Gilles Schaeffer
- SODA
- 2005

We present a bijection between some quadrangular dissections of an hexagon and unrooted binary trees. This correspondence has interesting consequences for enumeration, mesh compression and random graph sampling.It yields a succinct representation for the set <i>P(n)</i> of <i>n</i>-edge 3-connected planar graphs matching the entropy bound 1/<i>n</i> log… (More)

- Éric Fusy, Frédéric Giroire
- ANALCO
- 2007

A new algorithm is introduced to estimate the number of distinct flows (or connections) in a data stream. The algorithm maintains an accurate estimate of the number of distinct flows over a sliding window. It is simple to implement, parallelizes optimally, and has a very good tradeoff between auxiliary memory and accuracy of the estimate: a relative… (More)

- Éric Fusy
- Graph Drawing
- 2005

We define and investigate so-called transversal structures related to triangulations without separating triangles, which are equivalent to the regular edge labelings discovered by Kant and He. We study other properties of transversal structures and show that they give rise to a new straight-line drawing algorithm for triangulations without separating… (More)

- Mireille Bousquet-Mélou, Éric Fusy, Louis-François Préville-Ratelle
- Electr. J. Comb.
- 2012

MIREILLE BOUSQUET-MÉLOU, ÉRIC FUSY, AND LOUIS-FRANÇOIS PRÉVILLE-RATELLE Abstra t. An m-ballot path of size n is a path on the square grid onsisting of north and east steps, starting at (0, 0), ending at (mn, n), and never going below the line {x = my}. The set of these paths an be equipped with a latti e stru ture, alled the m-Tamari latti e and denoted by… (More)