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1. Write-Optimized Skip Lists

   https://doi.org/10.1145/3034786.3056117  

   Bender, Michael ; Farach-Colton, Martin ; Johnson, Rob ; Mauras, Simon ; Mayer, Tyler ; Phillips, Cynthia ; Xu, Helen ( January 2017 , PODS '17 Proceedings of the 36th ACM SIGMOD-SIGACT-SIGAI Symposium on Principles of Database Systems)

   The skip list is an elegant dictionary data structure that is com- monly deployed in RAM. A skip list with N elements supports searches, inserts, and deletes in O(logN) operations with high probability (w.h.p.) and range queries returning K elements in O(log N + K) operations w.h.p. A seemingly natural way to generalize the skip list to external memory with block size B is to “promote” with probability 1/B, rather than 1/2. However, there are practical and theoretical obsta- cles to getting the skip list to retain its efficient performance, space bounds, and high-probability guarantees. We give an external-memory skip list that achieves write- optimized bounds. That is, for 0 < ε < 1, range queries take O(logBε N + K/B) I/Os w.h.p. and insertions and deletions take O((logBε N)/B1−ε) amortized I/Os w.h.p. Our write-optimized skip list inherits the virtue of simplicity from RAM skip lists. Moreover, it matches or beats the asymptotic bounds of prior write-optimized data structures such as Bε trees or LSM trees. These data structures are deployed in high-performance databases and file systems. 

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2. 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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3. Cross-Referenced Dictionaries and the Limits of Write Optimization

   https://doi.org/10.1137/1.9781611974782.99  

   Afshani, Peyman ; Bender, Michael ; Farach-Colton, Martin ; Fineman, Jeremy ; Goswami, Mayank ; Tsai, Meng-Tsung Tsai ( January 2017 , Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms)

   Dictionaries remain the most well studied class of data structures. A dictionary supports insertions, deletions, membership queries, and usually successor, predecessor, and extract-min. In a RAM, all such operations take O(log n) time on n elements. Dictionaries are often cross-referenced as follows. Consider a set of tuples {〈ai,bi,ci…〉}. A database might include more than one dictionary on such a set, for example, one indexed on the a ‘s, another on the b‘s, and so on. Once again, in a RAM, inserting into a set of L cross-referenced dictionaries takes O(L log n) time, as does deleting. The situation is more interesting in external memory. On a Disk Access Machine (DAM), B-trees achieve O(logB N) I/Os for insertions and deletions on a single dictionary and K-element range queries take optimal O(logB N + K/B) I/Os. These bounds are also achievable by a B-tree on cross-referenced dictionaries, with a slowdown of an L factor on insertion and deletions. In recent years, both the theory and practice of external- memory dictionaries has been revolutionized by write- optimization techniques. A dictionary is write optimized if it is close to a B-tree for query time while beating B-trees on insertions. The best (and optimal) dictionaries achieve a substantially improved insertion and deletion cost of amortized I/Os on a single dictionary while maintaining optimal O(log1+B∊ N + K/B)- I/O range queries. Although write optimization still helps for insertions into cross-referenced dictionaries, its value for deletions would seem to be greatly reduced. A deletion into a cross- referenced dictionary only specifies a key a. It seems to be necessary to look up the associated values b, c … in order to delete them from the other dictionaries. This takes Ω(logB N) I/Os, well above the per-dictionary write-optimization budget of So the total deletion cost is In short, for deletions, write optimization offers an advantage over B-trees in that L multiplies a lower order term, but when L = 2, write optimization seems to offer no asymptotic advantage over B-trees. That is, no known query- optimal solution for pairs of cross-referenced dictionaries seem to beat B-trees for deletions. In this paper, we show a lower bound establishing that a pair of cross-referenced dictionaries that are optimal for range queries and that supports deletions cannot match the write optimization bound available to insert-only dictionaries. This result thus establishes a limit to the applicability of write-optimization techniques on which many new databases and file systems are based. Read More: http://epubs.siam.org/doi/10.1137/1.9781611974782.99 

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4. Wormhole: A Fast Ordered Index for In-memory Data Management

   https://doi.org/10.1145/3302424.3303955  

   Wu, Xingbo ; Ni, Fan ; Jiang, Song ( March 2019 , EuroSys '19 Proceedings of the Fourteenth EuroSys Conference 2019)

   In-memory data management systems, such as key-value stores, have become an essential infrastructure in today's big-data processing and cloud computing. They rely on efficient index structures to access data. While unordered indexes, such as hash tables, can perform point search with O(1) time, they cannot be used in many scenarios where range queries must be supported. Many ordered indexes, such as B+ tree and skip list, have a O(log N) lookup cost, where N is number of keys in an index. For an ordered index hosting billions of keys, it may take more than 30 key-comparisons in a lookup, which is an order of magnitude more expensive than that on a hash table. With availability of large memory and fast network in today's data centers, this O(log N) time is taking a heavy toll on applications that rely on ordered indexes. In this paper we introduce a new ordered index structure, named Wormhole, that takes O(log L) worst-case time for looking up a key with a length of L. The low cost is achieved by simultaneously leveraging strengths of three indexing structures, namely hash table, prefix tree, and B+ tree, to orchestrate a single fast ordered index. Wormhole's range operations can be performed by a linear scan of a list after an initial lookup. This improvement of access efficiency does not come at a price of compromised space efficiency. Instead, Wormhole's index space is comparable to those of B+ tree and skip list. Experiment results show that Wormhole outperforms skip list, B+ tree, ART, and Masstree by up to 8.4x, 4.9x, 4.3x, and 6.6x in terms of key lookup throughput, respectively. 

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5. On the Optimal Time/Space Tradeoff for Hash Tables

   Bender, Michael ; Farach-Colton, Martin ; Kuszmaul, John ; Kuszmaul, William ; Liu, Mingmou ( January 2022 , STOC)

   For nearly six decades, the central open question in the study of hash tables has been to determine the optimal achievable tradeoff curve between time and space. State-of-the-art hash tables offer the following guarantee: If keys/values are Θ(logn) bits each, then it is possible to achieve constant-time insertions/deletions/queries while wasting only O(loglogn) bits of space per key when compared to the information-theoretic optimum. Even prior to this bound being achieved, the target of O(log log n) wasted bits per key was known to be a natural end goal, and was proven to be optimal for a number of closely related problems (e.g., stable hashing, dynamic retrieval, and dynamically-resized filters). This paper shows that O(log log n) wasted bits per key is not the end of the line for hashing. In fact, for any k ∈ [log∗ n], it is possible to achieve O(k)-time insertions/deletions, O(1)-time queries, and O(log(k) n) = Ologlog···logn 􏰟 􏰞􏰝 􏰠 k wasted bits per key (all with high probability in n). This means that, each time we increase inser- tion/deletion time by an additive constant, we reduce the wasted bits per key exponentially. We further show that this tradeoff curve is the best achievable by any of a large class of hash tables, including any hash table designed using the current framework for making constant-time hash tables succinct. Our results hold not just for fixed-capacity hash tables, but also for hash tables that are dynamically resized (this is a fundamental departure from what is possible for filters); and for hash tables that store very large keys/values, each of which can be up to no(1) bits (this breaks with the conventional wisdom that larger keys/values should lead to more wasted bits per key). For very small keys/values, we are able to tighten our bounds to o(1) wasted bits per key, even when k = O(1). Building on this, we obtain a constant-time dynamic filter that uses n􏰕logε−1􏰖+nloge+o(n) bits of space for a wide choice of 

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