We design fast algorithms for repeatedly sampling from strongly Rayleigh distributions, which include as special cases random spanning tree distributions and determinantal point processes. For a graph $G=(V, E)$, we show how to approximately sample uniformly random spanning trees from $G$ in $\widetilde{O}(\lvert V\rvert)$\footnote{Throughout, $\widetilde{O}(\cdot)$ hides polylogarithmic factors in $n$.} time per sample after an initial $\widetilde{O}(\lvert E\rvert)$ time preprocessing. This is the first nearlylinear runtime in the output size, which is clearly optimal. For a determinantal point process on $k$sized subsets of a ground set of $n$ elements, defined via an $n\times n$ kernel matrix, we show how to approximately sample in $\widetilde{O}(k^\omega)$ time after an initial $\widetilde{O}(nk^{\omega1})$ time preprocessing, where $\omega<2.372864$ is the matrix multiplication exponent. The time to compute just the weight of the output set is simply $\simeq k^\omega$, a natural barrier that suggests our runtime might be optimal for determinantal point processes as well. As a corollary, we even improve the state of the art for obtaining a single sample from a determinantal point process, from the prior runtime of $\widetilde{O}(\min\{nk^2, n^\omega\})$ to $\widetilde{O}(nk^{\omega1})$. In our main technical result, we achieve the optimal limit on domain sparsification for strongly Rayleigh distributions. In domain sparsification, samplingmore »
Sampling from Discrete Distributions in Combinational Hardware with Application to PostQuantum Cryptography
Random values from discrete distributions are typically generated from uniformlyrandom samples. A common technique is to use a cumulative distribution table (CDT) lookup for inversion sampling, but it is also possible to use Boolean functions to map a uniformlyrandom bit sequence into a value from a discrete distribution. This work presents a methodology for deriving such functions for any discrete distribution, encoding them in VHDL for implementation in combinational hardware, and (for moderate precision and sample space size) confirming the correctness of the produced distribution. The process is demonstrated using a discrete Gaussian distribution with a small sample space, but it is applicable to any discrete distribution with fixed parameters. Results are presented for sampling schemes from several submissions to the NIST PQC standardization process, comparing this method to CDT lookups on a Xilinx Artix7 FPGA. The process produces compact solutions for distributions up to moderate size and precision.
 Award ID(s):
 1801512
 Publication Date:
 NSFPAR ID:
 10174990
 Journal Name:
 2020 Design, Automation & Test in Europe Conference & Exhibition (DATE), Grenoble, France
 Page Range or eLocationID:
 610 to 613
 Sponsoring Org:
 National Science Foundation
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