Exact and Efficient Generation of Geometric Random Variates and Random Graphs

Abstract

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).

Internal DLA: Efficient Simulation of a Physical Growth Model

“…We improve upon this algorithm here and show that constant expected sampling time is possible, even without any preprocessing. We remark that this is in line with a similar result for geometric random variables [4].…”

“…In general, to sample a Bernoulli random variate Ber(p) it suffices to be able to compute an additive 2 −L -approximation of p in time L O(1) (see, e.g., [4]). Indeed, we can draw Ber(p) by taking a uniformly random number R ∈ [0, 1) and returning whether R ≤ p. We perform this check by computing 2 −L -approximationsp of p andR of R (by taking its first L random bits) and checking whether these approximations allow to decide whether R ≤ p (which is possible if |R −p| ≥ 2 1−L ).…”

Polynomial Algorithm for Minimal (1,2)-Dominating Set in Networks

“…The Minimal_12_Set algorithm and PowerOfNodes function were implemented and tested using the igraph library in R. First, we used random geometric graphs as models of a network. A geometric random graph is the model of a spatial network, namely an undirected graph, constructed by placing randomly a given number of nodes in a unit square and connecting two nodes by a link if and only if their distance is in a given range, see [21,22]. In our tests, we generated random geometric graphs of order from 55 to 190 nodes and the radius within which the nodes are connected by a link between 0.08 and 0.4.…”

Efficient Sampling Methods for Discrete Distributions

“…Additionally to the standard repertoire of operations, we assume that we can generate a uniformly random word in constant time. It is known that in this model Bernoulli and geometric random variates can be drawn in constant time [2] and the classic aliasing method for UnsortedPropor-…”