Load thresholds for cuckoo hashing with overlapping blocks

Dietzfelbinger and Weidling [DW07] proposed a natural variation of cuckoo hashing where each of $cn$ objects is assigned $k = 2$ intervals of size $\ell$ in a linear (or cyclic) hash table of size $n$ and both start points are chosen independently and uniformly at random. Each object must be placed into a table cell within its intervals, but each cell can only hold one object. Experiments suggested that this scheme outperforms the variant with blocks in which intervals are aligned at multiples of $\ell$. In particular, the load threshold is higher, i.e. the load $c$ that can be achieved with high probability. For instance, Lehman and Panigrahy [LP09] empirically observed the threshold for $\ell = 2$ to be around $96.5\%$ as compared to roughly $89.7\%$ using blocks. They managed to pin down the asymptotics of the thresholds for large $\ell$, but the precise values resisted rigorous analysis. We establish a method to determine these load thresholds for all $\ell \geq 2$, and, in fact, for general $k \geq 2$. For instance, for $k = \ell = 2$ we get $\approx 96.4995\%$. The key tool we employ is an insightful and general theorem due to Leconte, Lelarge, and Massouli\'e [LLM13], which adapts methods from statistical physics to the world of hypergraph orientability. In effect, the orientability thresholds for our graph families are determined by belief propagation equations for certain graph limits. As a side note we provide experimental evidence suggesting that placements can be constructed in linear time with loads close to the threshold using an adapted version of an algorithm by Khosla [Kho13]

Fast Lightweight Accurate Xenograft Sorting

Motivation: With an increasing number of patient-derived xenograft (PDX) models being created and subsequently sequenced to study tumor heterogeneity and to guide therapy decisions, there is a similarly increasing need for methods to separate reads originating from the graft (human) tumor and reads originating from the host species\u27 (mouse) surrounding tissue. Two kinds of methods are in use: On the one hand, alignment-based tools require that reads are mapped and aligned (by an external mapper/aligner) to the host and graft genomes separately first; the tool itself then processes the resulting alignments and quality metrics (typically BAM files) to assign each read or read pair. On the other hand, alignment-free tools work directly on the raw read data (typically FASTQ files). Recent studies compare different approaches and tools, with varying results. Results: We show that alignment-free methods for xenograft sorting are superior concerning CPU time usage and equivalent in accuracy. We improve upon the state of the art by presenting a fast lightweight approach based on three-way bucketed quotiented Cuckoo hashing. Our hash table requires memory comparable to an FM index typically used for read alignment and less than other alignment-free approaches. It allows extremely fast lookups and uses less CPU time than other alignment-free methods and alignment-based methods at similar accuracy

Fast Gapped k-mer Counting with Subdivided Multi-Way Bucketed Cuckoo Hash Tables

Motivation. In biological sequence analysis, alignment-free (also known as k-mer-based) methods are increasingly replacing mapping- and alignment-based methods for various applications. A basic step of such methods consists of building a table of all k-mers of a given set of sequences (a reference genome or a dataset of sequenced reads) and their counts. Over the past years, efficient methods and tools for k-mer counting have been developed. In a different line of work, the use of gapped k-mers has been shown to offer advantages over the use of the standard contiguous k-mers. However, no tool seems to be available that is able to count gapped k-mers with the same efficiency as contiguous k-mers. One reason is that the most efficient k-mer counters use minimizers (of a length m < k) to group k-mers into buckets, such that many consecutive k-mers are classified into the same bucket. This approach leads to cache-friendly (and hence extremely fast) algorithms, but the approach does not transfer easily to gapped k-mers. Consequently, the existing efficient k-mer counters cannot be trivially modified to count gapped k-mers with the same efficiency. Results. We present a different approach that is equally applicable to contiguous k-mers and gapped k-mers. We use multi-way bucketed Cuckoo hash tables to efficiently store (gapped) k-mers and their counts. We also describe a method to parallelize counting over multiple threads without using locks: We subdivide the hash table into independent subtables, and use a producer-consumer model, such that each thread serves one subtable. This requires designing Cuckoo hash functions with the property that all alternative locations for each k-mer are located in the same subtable. Compared to some of the fastest contiguous k-mer counters, our approach is of comparable speed, or even faster, on large datasets, and it is the only one that supports gapped k-mers

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

LIPIcs, Volume 244, ESA 2022, Complete Volume

LIPIcs, Volume 244, ESA 2022, Complete Volum