arXiv subjects

Nicholas Harvey

Publications and source records attributed to Nicholas Harvey.

4 recordsLinked to original sources

arXiv2024

Explicit Orthogonal Arrays and Universal Hashing with Arbitrary Parameters

Orthogonal arrays are a type of combinatorial design that were developed in the 1940s in the design of statistical experiments. In 1947, Rao proved a lower bound on the size of any orthogonal array, and raised the problem of constructing arrays of minimum size. Kuperberg, Lovett and Peled (2017) gave a non-constructive existence proof of orthogonal arrays whose size is near-optimal (i.e., within a polynomial of Rao's lower bound), leaving open the question of an algorithmic construction. We give the first explicit, deterministic, algorithmic construction of orthogonal arrays achieving near-optimal size for all parameters. Our construction uses algebraic geometry codes. In pseudorandomness, the notions of $t$-independent generators or $t$-independent hash functions are equivalent to orthogonal arrays. Classical constructions of $t$-independent hash functions are known when the size of the codomain is a prime power, but very few constructions are known for an arbitrary codomain. Our construction yields algorithmically efficient $t$-independent hash functions for arbitrary domain and codomain.

cs.DS↗

arXiv2020

LSQT: Low-Stretch Quasi-Trees for Bundling and Layout

We introduce low-stretch trees to the visualization community with LSQT, our novel technique that uses quasi-trees for both layout and edge bundling. Our method offers strong computational speed and complexity guarantees by leveraging the convenient properties of low-stretch trees, which accurately reflect the topological structure of arbitrary graphs with superior fidelity compared to arbitrary spanning trees. Low-stretch quasi-trees also have provable sparseness guarantees, providing algorithmic support for aggressive de-cluttering of hairball graphs. LSQT does not rely on previously computed vertex positions and computes bundles based on topological structure before any geometric layout occurs. Edge bundles are computed efficiently and stored in an explicit data structure that supports sophisticated visual encoding and interaction techniques, including dynamic layout adjustment and interactive bundle querying. Our unoptimized implementation handles graphs of over 100,000 edges in eight seconds, providing substantially higher performance than previous approaches.

cs.HC↗

arXiv2017

Matroids Hitting Sets and Unsupervised Dependency Grammar Induction

This paper formulates a novel problem on graphs: find the minimal subset of edges in a fully connected graph, such that the resulting graph contains all spanning trees for a set of specifed sub-graphs. This formulation is motivated by an un-supervised grammar induction problem from computational linguistics. We present a reduction to some known problems and algorithms from graph theory, provide computational complexity results, and describe an approximation algorithm.

cs.DM↗

arXiv2015

An Algorithmic Proof of the Lovasz Local Lemma via Resampling Oracles

The Lovasz Local Lemma is a seminal result in probabilistic combinatorics. It gives a sufficient condition on a probability space and a collection of events for the existence of an outcome that simultaneously avoids all of those events. Finding such an outcome by an efficient algorithm has been an active research topic for decades. Breakthrough work of Moser and Tardos (2009) presented an efficient algorithm for a general setting primarily characterized by a product structure on the probability space. In this work we present an efficient algorithm for a much more general setting. Our main assumption is that there exist certain functions, called resampling oracles, that can be invoked to address the undesired occurrence of the events. We show that, in all scenarios to which the original Lovasz Local Lemma applies, there exist resampling oracles, although they are not necessarily efficient. Nevertheless, for essentially all known applications of the Lovasz Local Lemma and its generalizations, we have designed efficient resampling oracles. As applications of these techniques, we present new results for packings of Latin transversals, rainbow matchings and rainbow spanning trees.

cs.DS↗