Peeling Close to the Orientability Threshold: Spatial Coupling in Hashing-Based Data Structures
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Hypergraphs with random hyperedges underlie various data structures where hash functions map inputs to hyperedges, e.g. cuckoo hash tables, invertible Bloom lookup tables, retrieval data structures and perfect hash functions. High memory efficiency and quick query times call for high hyperedge density and small hyperedge size. Moreover, orientability or even peelability of the hypergraph is required or advantageous. For k-uniform fully random hypergraphs, the thresholds c_k,ℓ^* for ℓ-orientability significantly exceed the thresholds for ℓ-peelability. In this paper, for every k ≥ 2 and ℓ≥ 1 with (k,ℓ) ≠ (2,1) and every z > 0, we construct a new family of random k-uniform hypergraphs with i.i.d. random hyperedges such that both the ℓ-peelability and the ℓ-orientability thresholds approach c_k,ℓ^* as z →∞. Our construction is simple: The N vertices are linearly ordered and each hyperedge selects its k elements uniformly at random from a random range of N/z consecutive vertices. We thus exploit the phenomenon of threshold saturation via spatial coupling discovered in the context of low density parity check codes. Once the connection to data structures is in plain sight, we employ a framework by Kudekar, Richardson and Urbanke (2015) to do the heavy lifting in our proof. We demonstrate the usefulness of our construction, using our hypergraphs as a drop-in replacement in a retrieval data structure by Botelho et al. (2013). This reduces memory usage from 1.23m bits to 1.12m bits (for input size m) with no downsides.
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