Local Access to Huge Random Objects through Partial Sampling

TL;DR

This paper develops methods for efficiently accessing and sampling large random objects, such as graphs and combinatorial structures, through local queries without generating the entire object upfront.

Contribution

It introduces novel local-access algorithms for random graphs, Catalan objects, and graph colorings, enabling on-the-fly sampling and query answering for complex random structures.

Findings

Efficient local-access implementations for Erdős-Rényi and Stochastic Block models.

New query types for Dyck paths and random trees.

Sub-linear time algorithms for graph colorings with q > 9Δ.

Abstract

Consider an algorithm performing a computation on a huge random object. Is it necessary to generate the entire object up front, or is it possible to provide query access to the object and sample it incrementally "on-the-fly"? Such an implementation should emulate the object by answering queries in a manner consistent with a random instance sampled from the true distribution. Our first set of results focus on undirected graphs with independent edge probabilities, under certain assumptions. Then, we use this to obtain the first efficient implementations for the Erdos-Renyi model and the Stochastic Block model. As in previous local-access implementations for random graphs, we support Vertex-Pair and Next-Neighbor queries. We also introduce a new Random-Neighbor query. Next, we show how to implement random Catalan objects, specifically focusing on Dyck paths (always positive random walks on the integer line). Here, we support Height queries to find the position of the walk, and First-Return queries to find the time when the walk returns to a specified height. This in turn can be used to implement Next-Neighbor queries on random rooted/binary trees, and Matching-Bracket queries on random well bracketed expressions. Finally, we define a new model that: (1) allows multiple independent simultaneous instantiations of the same implementation to be consistent with each other without communication (2) allows us to generate a richer class of random objects that do not have a succinct description. Specifically, we study uniformly random valid $q$-colorings of an input graph $G$ with max degree $\Delta$. The distribution over valid colorings is specified via a "huge" underlying graph $G$, that is far too large to be read in sub-linear time. Instead, we access $G$ through local neighborhood probes. We are able to answer queries to the color of any vertex in sub-linear time for $q > 9\Delta$.