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1. Efficient Usage of One-Sided RDMA for Linear Probing

   Wang, T; Yang, S; Kimura, H; Swart, G; Blanas, S (September 2020, Eleventh International Workshop on Accelerating Analytics and Data Management Systems Using Modern Processor and Storage Architectures (AMDS'20))

   null (Ed.)

   RDMA has been an important building block for many high-performance distributed key-value stores in recent prior work. To sustain millions of operations per second per node, many of these works use hashing schemes, such as cuckoo hashing, that guarantee that an existing key can be found in a small, constant number of RDMA operations. In this paper, we explore whether linear probing is a compelling design alternative to cuckoo-based hashing schemes. Instead of reading a fixed number of bytes per RDMA request, this paper introduces a mathematical model that picks the optimal read size by balancing the cost of performing an RDMA operation with the probability of encountering a probe sequence of a certain length. The model can further leverage optimization hints about the location of clusters of keys, which commonly occur in skewed key distributions. We extensively evaluate the performance of linear probing with a variable read size in a modern cluster to understand the trade-offs between cuckoo hashing and linear probing. We find that cuckoo hashing outperforms linear probing only in very highly loaded hash tables (load factors at 90% or higher) that would be prohibitively expensive to maintain in practice. Overall, linear probing with variable-sized reads using our model has up to 2.8× higher throughput than cuckoo hashing, with throughput as high as 50M lookup operations per second per node. 

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2. Analyzing and Implementing GPU Hash Tables

   https://doi.org/10.1137/1.9781611977578.ch3  

   Awad, Muhammad A.; Ashkiani, Saman; Porumbescu, Serban D.; Farach-Colton, Martín; Owens, John D. (January 2023, SIAM Symposium on Algorithmic Principles of Computer Systems)

   We revisit the problem of building static hash tables on the GPU and present an efficient implementation of bucketed hash tables. By decoupling the probing scheme from the hash table in-memory representation, we offer an implementation where the number of probes and the bucket size are the only factors limiting performance. Our analysis sweeps through the hash table parameter space for two probing schemes: cuckoo and iceberg hashing. We show that a bucketed cuckoo hash table (BCHT) that uses three hash functions outperforms alternative methods that use iceberg hashing and a cuckoo hash table that uses a bucket size of one. At load factors as high as 0.99, BCHT enjoys an average probe count of 1.43 during insertion. Using three hash functions only, positive and negative queries require at most 1.39 and 2.8 average probes per key, respectively. 

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3. Cuckoo Feature Hashing: Dynamic Weight Sharing for Sparse Analytics

   https://doi.org/10.24963/ijcai.2018/295  

   Gao, Jinyang; Ooi, Beng Chin; Shen, Yanyan; Lee, Wang-Chien (July 2018, Proceedings of the Twenty-Seventh International Joint Conference on Artificial Intelligence)

   Feature hashing is widely used to process large scale sparse features for learning of predictive models. Collisions inherently happen in the hashing process and hurt the model performance. In this paper, we develop a feature hashing scheme called Cuckoo Feature Hashing(CCFH) based on the principle behind Cuckoo hashing, a hashing scheme designed to resolve collisions. By providing multiple possible hash locations for each feature, CCFH prevents the collisions between predictive features by dynamically hashing them into alternative locations during model training. Experimental results on prediction tasks with hundred-millions of features demonstrate that CCFH can achieve the same level of performance by using only 15%-25% parameters compared with conventional feature hashing. 

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4. Linear Probing Revisited: Tombstones Mark the Demise of Primary Clustering

   Bender, M.; Kuszmaul, B.; Kuszmaul, W. (January 2021, Annual Symposium on Foundations of Computer Science)

   First introduced in 1954, the linear-probing hash table is among the oldest data structures in computer science, and thanks to its unrivaled data locality, linear probing continues to be one of the fastest hash tables in practice. It is widely believed and taught, however, that linear probing should never be used at high load factors; this is because of an effect known as primary clustering which causes insertions at a load factor of 1 - 1/x to take expected time O(x2) (rather than the intuitive running time of ⇥(x)). The dangers of primary clustering, first discovered by Knuth in 1963, have now been taught to generations of computer scientists, and have influenced the design of some of the most widely used hash tables in production. We show that primary clustering is not the foregone conclusion that it is reputed to be. We demonstrate that seemingly small design decisions in how deletions are implemented have dramatic effects on the asymptotic performance of insertions: if these design decisions are made correctly, then even if a hash table operates continuously at a load factor of 1 - (1/x), the expected amortized cost per insertion/deletion is O(x). This is because the tombstones left behind by deletions can actually cause an anti-clustering effect that combats primary clustering. Interestingly, these design decisions, despite their remarkable effects, have historically been viewed as simply implementation-level engineering choices. We also present a new variant of linear probing (which we call graveyard hashing) that completely eliminates primary clustering on any sequence of operations: if, when an operation is performed, the current load factor is 1 1/x for some x, then the expected cost of the operation is O(x). Thus we can achieve the data locality of traditional linear probing without any of the disadvantages of primary clustering. One corollary is that, in the external-memory model with a data blocks of size B, graveyard hashing offers the following remarkably strong guarantee: at any load factor 1 1/x satisfying x = o(B), graveyard hashing achieves 1 + o(1) expected block transfers per operation. In contrast, past external-memory hash tables have only been able to offer a 1 + o(1) guarantee when the block size B is at least O(x2). Our results come with actionable lessons for both theoreticians and practitioners, in particular, that well- designed use of tombstones can completely change the asymptotic landscape of how the linear probing behaves (and even in workloads without deletions). 

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5. ASA: A ccelerating S parse A ccumulation in Column-wise SpGEMM

   https://doi.org/10.1145/3543068  

   Zhang, Chao; Bremer, Maximilian; Chan, Cy; Shalf, John; Guo, Xiaochen (December 2022, ACM Transactions on Architecture and Code Optimization)

   Sparse linear algebra is an important kernel in many different applications. Among various sparse general matrix-matrix multiplication (SpGEMM) algorithms, Gustavson’s column-wise SpGEMM has good locality when reading input matrix and can be easily parallelized by distributing the computation of different columns of an output matrix to different processors. However, the sparse accumulation (SPA) step in column-wise SpGEMM, which merges partial sums from each of the multiplications by the row indices, is still a performance bottleneck. The state-of-the-art software implementation uses a hash table for partial sum search in the SPA, which makes SPA the largest contributor to the execution time of SpGEMM. There are three reasons that cause the SPA to become the bottleneck: (1) hash probing requires data-dependent branches that are difficult for a branch predictor to predict correctly; (2) the accumulation of partial sum is dependent on the results of the hash probing, which makes it difficult to hide the hash probing latency; and (3) hash collision requires time-consuming linear search and optimizations to reduce these collisions require an accurate estimation of the number of non-zeros in each column of the output matrix. This work proposes ASA architecture to accelerate the SPA. ASA overcomes the challenges of SPA by (1) executing the partial sum search and accumulate with a single instruction through ISA extension to eliminate data-dependent branches in hash probing, (2) using a dedicated on-chip cache to perform the search and accumulation in a pipelined fashion, (3) relying on the parallel search capability of a set-associative cache to reduce search latency, and (4) delaying the merging of overflowed entries. As a result, ASA achieves an average of 2.25× and 5.05× speedup as compared to the state-of-the-art software implementation of a Markov clustering application and its SpGEMM kernel, respectively. As compared to a state-of-the-art hashing accelerator design, ASA achieves an average of 1.95× speedup in the SpGEMM kernel. 

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