Parallel Peeling Algorithms

The analysis of several algorithms and data structures can be framed as a peeling process on a random hypergraph: vertices with degree less than k are removed until there are no vertices of degree less than k left. The remaining hypergraph is known as the k-core. In this paper, we analyze parallel peeling processes, where in each round, all vertices of degree less than k are removed. It is known that, below a specific edge density threshold, the k-core is empty with high probability. We show that, with high probability, below this threshold, only (log log n)/log(k-1)(r-1) + O(1) rounds of peeling are needed to obtain the empty k-core for r-uniform hypergraphs. Interestingly, we show that above this threshold, Omega(log n) rounds of peeling are required to find the non-empty k-core. Since most algorithms and data structures aim to peel to an empty k-core, this asymmetry appears fortunate. We verify the theoretical results both with simulation and with a parallel implementation using graphics processing units (GPUs). Our implementation provides insights into how to structure parallel peeling algorithms for efficiency in practice.Comment: Appears in SPAA 2014. Minor typo corrections relative to previous versio

Cache-Oblivious Peeling of Random Hypergraphs

The computation of a peeling order in a randomly generated hypergraph is the most time-consuming step in a number of constructions, such as perfect hashing schemes, random $r$-SAT solvers, error-correcting codes, and approximate set encodings. While there exists a straightforward linear time algorithm, its poor I/O performance makes it impractical for hypergraphs whose size exceeds the available internal memory. We show how to reduce the computation of a peeling order to a small number of sequential scans and sorts, and analyze its I/O complexity in the cache-oblivious model. The resulting algorithm requires $O(\mathrm{sort}(n))$ I/Os and $O(n \log n)$ time to peel a random hypergraph with $n$ edges. We experimentally evaluate the performance of our implementation of this algorithm in a real-world scenario by using the construction of minimal perfect hash functions (MPHF) as our test case: our algorithm builds a MPHF of $7.6$ billion keys in less than $21$ hours on a single machine. The resulting data structure is both more space-efficient and faster than that obtained with the current state-of-the-art MPHF construction for large-scale key sets

Dense peelable random uniform hypergraphs

We describe a new family of k-uniform hypergraphs with independent random edges. The hypergraphs have a high probability of being peelable, i.e. to admit no sub-hypergraph of minimum degree 2, even when the edge density (number of edges over vertices) is close to 1. In our construction, the vertex set is partitioned into linearly arranged segments and each edge is incident to random vertices of k consecutive segments. Quite surprisingly, the linear geometry allows our graphs to be peeled "from the outside in". The density thresholds f_k for peelability of our hypergraphs (f_3 ~ 0.918, f_4 ~ 0.977, f_5 ~ 0.992, ...) are well beyond the corresponding thresholds (c_3 ~ 0.818, c_4 ~ 0.772, c_5 ~ 0.702, ...) of standard k-uniform random hypergraphs. To get a grip on f_k, we analyse an idealised peeling process on the random weak limit of our hypergraph family. The process can be described in terms of an operator on [0,1]^Z and f_k can be linked to thresholds relating to the operator. These thresholds are then tractable with numerical methods. Random hypergraphs underlie the construction of various data structures based on hashing, for instance invertible Bloom filters, perfect hash functions, retrieval data structures, error correcting codes and cuckoo hash tables, where inputs are mapped to edges using hash functions. Frequently, the data structures rely on peelability of the hypergraph or peelability allows for simple linear time algorithms. Memory efficiency is closely tied to edge density while worst and average case query times are tied to maximum and average edge size. To demonstrate the usefulness of our construction, we used our 3-uniform hypergraphs as a drop-in replacement for the standard 3-uniform hypergraphs in a retrieval data structure by Botelho et al. [Fabiano Cupertino Botelho et al., 2013]. This reduces memory usage from 1.23m bits to 1.12m bits (m being the input size) with almost no change in running time. Using k > 3 attains, at small sacrifices in running time, further improvements to memory usage

Simple Set Sketching

Imagine handling collisions in a hash table by storing, in each cell, the bit-wise exclusive-or of the set of keys hashing there. This appears to be a terrible idea: For $\alpha n$ keys and $n$ buckets, where $\alpha$ is constant, we expect that a constant fraction of the keys will be unrecoverable due to collisions. We show that if this collision resolution strategy is repeated three times independently the situation reverses: If $\alpha$ is below a threshold of $\approx 0.81$ then we can recover the set of all inserted keys in linear time with high probability. Even though the description of our data structure is simple, its analysis is nontrivial. Our approach can be seen as a variant of the Invertible Bloom Filter (IBF) of Eppstein and Goodrich. While IBFs involve an explicit checksum per bucket to decide whether the bucket stores a single key, we exploit the idea of quotienting, namely that some bits of the key are implicit in the location where it is stored. We let those serve as an implicit checksum. These bits are not quite enough to ensure that no errors occur and the main technical challenge is to show that decoding can recover from these errors.Comment: To be published at SIAM Symposium on Simplicity in Algorithms (SOSA23

Invertible Bloom Lookup Tables with Listing Guarantees

The Invertible Bloom Lookup Table (IBLT) is a probabilistic concise data structure for set representation that supports a listing operation as the recovery of the elements in the represented set. Its applications can be found in network synchronization and traffic monitoring as well as in error-correction codes. IBLT can list its elements with probability affected by the size of the allocated memory and the size of the represented set, such that it can fail with small probability even for relatively small sets. While previous works only studied the failure probability of IBLT, this work initiates the worst case analysis of IBLT that guarantees successful listing for all sets of a certain size. The worst case study is important since the failure of IBLT imposes high overhead. We describe a novel approach that guarantees successful listing when the set satisfies a tunable upper bound on its size. To allow that, we develop multiple constructions that are based on various coding techniques such as stopping sets and the stopping redundancy of error-correcting codes, Steiner systems, and covering arrays as well as new methodologies we develop. We analyze the sizes of IBLTs with listing guarantees obtained by the various methods as well as their mapping memory consumption. Lastly, we study lower bounds on the achievable sizes of IBLT with listing guarantees and verify the results in the paper by simulations