${\varepsilon}$-Cost Sharding: Scaling Hypergraph-Based Data Structures to Trillions of Keys

Abstract

We describe a simple and yet very scalable implementation of static functions (VFunc) and of static filters (VFilter) based on hypergraphs. We introduce the idea of $\varepsilon$-cost sharding, which allows us to build structures that can manage trillions of keys, at the same time increasing memory locality in hypergraph-based constructions. Contrarily to the commonly used HEM sharding method, $\varepsilon$-cost sharding does not require to store of additional information, and does not introduce dependencies in the computation chain; its only cost is that of few arithmetical instructions, and of a relative increase $\varepsilon$ in space usage. We apply $\varepsilon$-cost sharding to the classical MWHC construction, but we obtain the best result by combining Dietzfelbinger and Walzer's fuse graphs for large shards with lazy Gaussian elimination for small shards. We obtain large structures with an overhead of 10.5% with respect to the information-theoretical lower bound and with a query time that is a few nanoseconds away from the query time of the non-sharded version, which is the fastest currently available within the same space bounds. Besides comparing our structures with a non-sharded version, we contrast its tradeoffs with bumped ribbon constructions, a space-saving alternative to hypergraph-based static functions and filters, which provide optimum space consumption but slow construction and query time (though construction can be parallelized very efficiently). We build offline a trillion-key filter using commodity hardware in just 60 ns/key. Finally, we show that $\varepsilon$-cost sharding and fuse graphs carry over with no effort to data structures that reduce to static functions, such as monotone minimal perfect hash functions and compressed static functions.