12 research outputs found
Perfect sampling algorithm for Schur processes
We describe random generation algorithms for a large class of random
combinatorial objects called Schur processes, which are sequences of random
(integer) partitions subject to certain interlacing conditions. This class
contains several fundamental combinatorial objects as special cases, such as
plane partitions, tilings of Aztec diamonds, pyramid partitions and more
generally steep domino tilings of the plane. Our algorithm, which is of
polynomial complexity, is both exact (i.e. the output follows exactly the
target probability law, which is either Boltzmann or uniform in our case), and
entropy optimal (i.e. it reads a minimal number of random bits as an input).
The algorithm encompasses previous growth procedures for special Schur
processes related to the primal and dual RSK algorithm, as well as the famous
domino shuffling algorithm for domino tilings of the Aztec diamond. It can be
easily adapted to deal with symmetric Schur processes and general Schur
processes involving infinitely many parameters. It is more concrete and easier
to implement than Borodin's algorithm, and it is entropy optimal.
At a technical level, it relies on unified bijective proofs of the different
types of Cauchy and Littlewood identities for Schur functions, and on an
adaptation of Fomin's growth diagram description of the RSK algorithm to that
setting. Simulations performed with this algorithm suggest interesting limit
shape phenomena for the corresponding tiling models, some of which are new.Comment: 26 pages, 19 figures (v3: final version, corrected a few misprints
present in v2
Sampling Geometric Inhomogeneous Random Graphs in Linear Time
Real-world networks, like social networks or the internet infrastructure,
have structural properties such as large clustering coefficients that can best
be described in terms of an underlying geometry. This is why the focus of the
literature on theoretical models for real-world networks shifted from classic
models without geometry, such as Chung-Lu random graphs, to modern
geometry-based models, such as hyperbolic random graphs.
With this paper we contribute to the theoretical analysis of these modern,
more realistic random graph models. Instead of studying directly hyperbolic
random graphs, we use a generalization that we call geometric inhomogeneous
random graphs (GIRGs). Since we ignore constant factors in the edge
probabilities, GIRGs are technically simpler (specifically, we avoid hyperbolic
cosines), while preserving the qualitative behaviour of hyperbolic random
graphs, and we suggest to replace hyperbolic random graphs by this new model in
future theoretical studies.
We prove the following fundamental structural and algorithmic results on
GIRGs. (1) As our main contribution we provide a sampling algorithm that
generates a random graph from our model in expected linear time, improving the
best-known sampling algorithm for hyperbolic random graphs by a substantial
factor O(n^0.5). (2) We establish that GIRGs have clustering coefficients in
{\Omega}(1), (3) we prove that GIRGs have small separators, i.e., it suffices
to delete a sublinear number of edges to break the giant component into two
large pieces, and (4) we show how to compress GIRGs using an expected linear
number of bits.Comment: 25 page
Geometric Inhomogeneous Random Graphs
Real-world networks, like social networks or the internet infrastructure, have structural properties such as their large clustering coefficient that can best be described in terms of an underlying geometry. This is why the focus of the literature on theoretical models for real-world networks shifted from classic models without geometry, such as Chung-Lu random graphs, to modern geometry-based models, such as hyperbolic random graphs. With this paper we contribute to the theoretical analysis of these modern, more realistic random graph models. However, we do not directly study hyperbolic random graphs, but replace them by a more general model that we call \emph{geometric inhomogeneous random graphs} (GIRGs). Since we ignore constant factors in the edge probabilities, our model is technically simpler (specifically, we avoid hyperbolic cosines), while preserving the qualitative behaviour of hyperbolic random graphs, and we suggest to replace hyperbolic random graphs by our new model in future theoretical studies. We prove the following fundamental structural and algorithmic results on GIRGs. (1) We provide a sampling algorithm that generates a random graph from our model in expected linear time, improving the best-known sampling algorithm for hyperbolic random graphs by a factor , (2) we establish that GIRGs have a constant clustering coefficient, (3) we show that GIRGs have small separators, i.e., it suffices to delete a sublinear number of edges to break the giant component into two large pieces, and (4) we show how to compress GIRGs using an expected linear number of bits
Internal DLA: Efficient Simulation of a Physical Growth Model
Abstract. The internal diffusion limited aggregation (IDLA) process places n particles on the two dimensional integer grid. The first particle is placed on the origin; every subsequent particle starts at the origin and performs an unbiased random walk until it reaches an unoccupied position. In this work we study the computational complexity of determining the subset that is generated after n particles have been placed. We develop the first algorithm that provably outperforms the naive step-by-step sim-ulation of all particles. Particularly, our algorithm has a running time of O(n log2 n) and a sublinear space requirement of O(n1/2 logn), both in expectation and with high probability. In contrast to some speedups proposed for similar models in the physics community, our algorithm samples from the exact distribution. To simulate a single particle fast we have to develop techniques for com-bining multiple steps of a random walk to large jumps without hitting a forbidden set of grid points. These techniques might be of independent interest for speeding up other problems based on random walks.
Optimal Approximate Sampling from Discrete Probability Distributions
This paper addresses a fundamental problem in random variate generation:
given access to a random source that emits a stream of independent fair bits,
what is the most accurate and entropy-efficient algorithm for sampling from a
discrete probability distribution , where the probabilities
of the output distribution of the sampling
algorithm must be specified using at most bits of precision? We present a
theoretical framework for formulating this problem and provide new techniques
for finding sampling algorithms that are optimal both statistically (in the
sense of sampling accuracy) and information-theoretically (in the sense of
entropy consumption). We leverage these results to build a system that, for a
broad family of measures of statistical accuracy, delivers a sampling algorithm
whose expected entropy usage is minimal among those that induce the same
distribution (i.e., is "entropy-optimal") and whose output distribution
is a closest approximation to the target
distribution among all entropy-optimal sampling algorithms
that operate within the specified -bit precision. This optimal approximate
sampler is also a closer approximation than any (possibly entropy-suboptimal)
sampler that consumes a bounded amount of entropy with the specified precision,
a class which includes floating-point implementations of inversion sampling and
related methods found in many software libraries. We evaluate the accuracy,
entropy consumption, precision requirements, and wall-clock runtime of our
optimal approximate sampling algorithms on a broad set of distributions,
demonstrating the ways that they are superior to existing approximate samplers
and establishing that they often consume significantly fewer resources than are
needed by exact samplers
Exact and Efficient Generation of Geometric Random Variates and Random Graphs
Abstract. The standard algorithm for fast generation of Erdős-Rényi random graphs only works in the Real RAM model. The critical point is the generation of geometric random variates Geo(p), for which there is no algorithm that is both exact and efficient in any bounded precision machine model. For a RAM model with word size w = Ω(log log(1/p)), we show that this is possible and present an exact algorithm for sampling Geo(p) in optimal expected time O(1 + log(1/p)/w). We also give an exact algorithm for sampling min{n, Geo(p)} in optimal expected time O(1+log(min{1/p, n})/w). This yields a new exact algorithm for sampling Erdős-Rényi and Chung-Lu random graphs of n vertices and m (expected) edges in optimal expected runtime O(n + m) on a RAM with word size w = Θ(log n).