我们提出了一种通用的数据结构CORoBTS，用于以缓存遗忘的方式动态存储类B树搜索树，它将van Emde Boas内存布局与压缩内存数组相结合。迄今为止，在使用vEB布局时，人们主要考虑的是搜索复杂度。我们展示了子树和连续内存区域的深度优先搜索复杂度，并更深入地揭示了树中顶点位置与内存中位置之间的关系。我们描述了如果能够模拟任意树的深度优先搜索，如何以vEB布局构建该树。类似地，我们考察了压缩内存数组的批量更新。在CORoBTS中，所存储的搜索树必须满足所有叶子处于相同深度，且顶点的分支数介于所选常数$a$和$b$之间。该数据结构支持以最优I/O复杂度$\mathcal{O}(\log_B{N})$进行搜索，并以线性空间存储。它提供插入和删除子树的操作；两者的摊销I/O复杂度均为$\mathcal{O}(S\cdot(\log^2 N)/B + \log_B N\cdot\log\log S + 1)$，摊销时间复杂度均为$\mathcal{O}(S\cdot\log^2 N)$，其中$S$为子树的大小，$N$为整个所存储树的大小。如果各树层上的顶点数量不变，重建现有子树可在两种复杂度中省去乘性的$\mathcal{O}(\log^2 N)$因子；否则，仅为被插入/删除的顶点付出该代价。修改Davoodi等人[ESA, pages 296-308. Springer, 2014]提出的缓存遗忘部分持久数组以使用CORoBTS，可将其空间复杂度从$\mathcal{O}(U^{\log_2 3} + V \log U)$改进为$\mathcal{O}(U + V \log U)$，其中$U$是数组的最大规模，$V$是版本数；当前读取和持久读取的数据局部性与I/O复杂度保持不变；写入的I/O复杂度会因一个多对数因子而变差。
英文摘要：
We propose a general data structure CORoBTS for storing B-tree-like search trees dynamically in a cache-oblivious way combining the van Emde Boas memory layout with packed memory array.
In the use of the vEB layout mostly search complexity was considered, so far. We show the complexity of depth-first search of a subtree and contiguous memory area and provide better insight into the relationship between positions of vertices in tree and in memory. We describe how to build an arbitrary tree in vEB layout if we can simulate its depth-first search. Similarly, we examine batch updates of packed memory array.
In CORoBTS, the stored search tree has to satisfy that all leaves are at the same depth and vertices have arity between the chosen constants $a$ and $b$. The data structure allows searching with an optimal I/O complexity $\mathcal{O}(\log_B{N})$ and is stored in linear space. It provides operations for inserting and removing a subtree; both have an amortized I/O complexity $\mathcal{O}(S\cdot(\log^2 N)/B + \log_B N\cdot\log\log S + 1)$ and amortized time complexity $\mathcal{O}(S\cdot\log^2 N)$, where $S$ is the size of the subtree and $N$ the size of the whole stored tree. Rebuilding an existing subtree saves the multiplicative $\mathcal{O}(\log^2 N)$ in both complexities if the number of vertices on individual tree levels is not changed; it is paid only for the inserted/removed vertices otherwise.
Modifying cache-oblivious partially persistent array proposed by Davoodi et al. [ESA, pages 296-308. Springer, 2014] to use CORoBTS improves its space complexity from $\mathcal{O}(U^{\log_2 3} + V \log U)$ to $\mathcal{O}(U + V \log U)$, where $U$ is the maximal size of the array and $V$ is the number of versions; the data locality and I/O complexity of both present and persistent reads are kept unchanged; I/O complexity of writes is worsened by a polylogarithmic factor.
Partial persistence is a general transformation that takes a data structure and allows queries to be executed on any past state of the structure. The cache-oblivious model is the leading model of a modern multi-level memory hierarchy.We present the first general transformation for making cache-oblivious model data structures partially persistent.
This paper proposes a general framework for generating cache-oblivious layouts for binary search trees. A cache-oblivious layout attempts to minimize cache misses on any hierarchical memory, independent of the number of memory levels and attributes at each level such as cache size, line size, and replacement policy. Recursively partitioning a tree into contiguous subtrees and prescribing an ordering amongst the subtrees, Hierarchical Layouts generalize many commonly used layouts for trees such as in-order, pre-order and breadth-first. They also generalize the various flavors of the van Emde Bo…
The VAT-model (virtual address translation model) extends the EM-model (external memory model) and takes the cost of address translation in virtual memories into account. In this model, the cost of a single memory access may be logarithmic in the largest address used. We show that the VAT-cost of cache-oblivious algorithms is only by a constant factor larger than their EM-cost; this requires a somewhat more stringent tall cache assumption as for the EM-model.
Let X[0..n-1] and Y[0..m-1] be two sorted arrays, and define the mxn matrix A by A[j][i]=X[i]+Y[j]. Frederickson and Johnson gave an efficient algorithm for selecting the k-th smallest element from A. We show how to make this algorithm IO-efficient. Our cache-oblivious algorithm performs O((m+n)/B) IOs, where B is the block size of memory transfers.
In this paper we present an implicit dynamic dictionary with the working-set property, supporting insert(e) and delete(e) in O(log n) time, predecessor(e) in O(log l_{p(e)}) time, successor(e) in O(log l_{s(e)}) time and search(e) in O(log min(l_{p(e)},l_{e}, l_{s(e)})) time, where n is the number of elements stored in the dictionary, l_{e} is the number of distinct elements searched for since element e was last searched for and p(e) and s(e) are the predecessor and successor of e, respectively. The time-bounds are all worst-case. The dictionary stores the elements in an array of size n using …
We investigate effects of ordering in blocked matrix--matrix multiplication. We find that submatrices do not have to be stored contiguously in memory to achieve near optimal performance. Instead it is the choice of execution order of the submatrix multiplications that leads to a speedup of up to four times for small block sizes. This is in contrast to results for single matrix elements showing that contiguous memory allocation quickly becomes irrelevant as the blocksize increases.
We present priority queues in the cache-oblivious external memory model with block size $B$ and main memory size $M$ that support on $N$ elements, operation \textsc{UPDATE} (combination of \textsc{INSERT} and \textsc{DECREASEKEY}) in $O \left(\frac{1}{B}\log_{\fracλ{B}} \frac{N}{B}\right)$ amortized I/Os and operations \textsc{EXTRACT-MIN} and \textsc{DELETE} in $O \left(\lceil \frac{λ^{\varepsilon}}{B} \log_{\fracλ{B}} \frac{N}{B} \rceil \log_{\fracλ{B}} \frac{N}{B}\right)$ amortized I/Os, using $O \left(\frac{N}{B}\log_{\fracλ{B}} \frac{N}{B}\right)$ blocks, for a user-defined parameter $λ\i…
Frigo et al. proposed an ideal cache model and a recursive technique to design sequential cache-efficient algorithms in a cache-oblivious fashion. Ballard et al. pointed out that it is a fundamental open problem to extend the technique to an arbitrary architecture. Ballard et al. raised another open question on how to parallelize Strassen's algorithm exactly and efficiently on an arbitrary number of processors. We propose a novel way of partitioning a cache-oblivious algorithm to achieve perfect strong scaling on an arbitrary number, even a prime number, of processors within a certain range in…
We study the problem of computing the triplet distance between two rooted unordered trees with $n$ labeled leafs. Introduced by Dobson 1975, the triplet distance is the number of leaf triples that induce different topologies in the two trees. The current theoretically best algorithm is an $\mathrm{O}(n \log n)$ time algorithm by Brodal et al. (SODA 2013). Recently Jansson and Rajaby proposed a new algorithm that, while slower in theory, requiring $\mathrm{O}(n \log^3 n)$ time, in practice it outperforms the theoretically faster $\mathrm{O}(n \log n)$ algorithm. Both algorithms do not scale to …
Emerging non-volatile main memory (NVRAM) technologies provide byte-addressability, low idle power, and improved memory-density, and are likely to be a key component in the future memory hierarchy. However, a critical challenge in achieving high performance is in accounting for the asymmetry that NVRAM writes can be significantly more expensive than NVRAM reads. In this paper, we consider a large class of cache-oblivious algorithms for dynamic programming (DP) and linear algebra, and try to reduce the writes in the asymmetric setting while maintaining high parallelism. To achieve that, our key…
While a lot of work in theoretical computer science has gone into optimizing the runtime and space usage of data structures, such work very often neglects a very important component of modern computers: the cache. In doing so, very often, data structures are developed that achieve theoretically-good runtimes but are slow in practice due to a large number of cache misses. In 1999, Frigo et al. introduced the notion of a cache-oblivious algorithm: an algorithm that uses the cache to its advantage, regardless of the size or structure of said cache. Since then, various authors have designed cache-…
In recent years the Cache-Oblivious model of external memory computation has provided an attractive theoretical basis for the analysis of algorithms on massive datasets. Much progress has been made in discovering algorithms that are asymptotically optimal or near optimal. However, to date there are still relatively few successful experimental studies. In this paper we compare two different Cache-Oblivious priority queues based on the Funnel and Bucket Heap and apply them to the single source shortest path problem on graphs with positive edge weights. Our results show that when RAM is limited a…
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 alg…
In [SPAA2007], Bender et al. define a streaming B-tree (or index) as one that supports updates in amortized $o(1)$ IOs, and present a structure achieving amortized $O((\log N)/B)$ IOs and queries in $O(\log N)$ IOs. We extend their result to the partially-persistent case. For a version $v$, let $N_v$ be the number of keys accessible at $v$ and $N$ be the total number of updates. We give a data structure using space $O(N)$, supporting updates to a leaf version $v$ with $O((\log N_{v})/B)$ amortized IOs and answering range queries returning $Z$ elements with $O(\log N_{v} + Z/B)$ IOs on average …
In this paper we present randomized algorithms for sorting and convex hull that achieves optimal performance (for speed-up and cache misses) on the multicore model with private cache model. Our algorithms are cache oblivious and generalize the randomized divide and conquer strategy given by Reischuk and Reif and Sen. Although the approach yielded optimal speed-up in the PRAM model, we require additional techniques to optimize cache-misses in an oblivious setting. Under a mild assumption on input and number of processors our algorithm will have optimal time and cache misses with high probabilit…
We consider Oblivious Shuffling and K-Oblivious Shuffling, a refinement thereof. We provide efficient algorithms for both and discuss their application to the design of Oblivious RAM. The task of K-Oblivious Shuffling is to obliviously shuffle N encrypted blocks that have been randomly allocated on the server in such a way that an adversary learns nothing about the new allocation of blocks. The security guarantee should hold also with respect to an adversary that has learned the initial position of K touched blocks out of the N blocks. The classical notion of Oblivious Shuffling is obtained fo…
A dictionary (or map) is a key-value store that requires all keys be unique, and a multimap is a key-value store that allows for multiple values to be associated with the same key. We design hashing-based indexing schemes for dictionaries and multimaps that achieve worst-case optimal performance for lookups and updates, with a small or negligible probability the data structure will require a rehash operation, depending on whether we are working in the the external-memory (I/O) model or one of the well-known versions of the Random Access Machine (RAM) model. One of the main features of our cons…
We present a deterministic sorting algorithm, SPMS (Sample, Partition, and Merge Sort), that interleaves the partitioning of a sample sort with merging. Sequentially, it sorts $n$ elements in $O(n \log n)$ time cache-obliviously with an optimal number of cache misses. The parallel complexity (or critical path length) of the algorithm is $O(\log n \cdot \log\log n)$, which improves on previous bounds for optimal cache oblivious sorting. The algorithm also has low false sharing costs. When scheduled by a work-stealing scheduler in a multicore computing environment with a global shared memory and…
PACO: A Fully Cache-Oblivious Parallel FFT with One Global Redistribution
College of Computer Science and Artificial Intelligence, Fudan University（复旦大学计算机科学与人工智能学院）；School of Mathematical Sciences, School of Data Science, Fudan University（复旦大学数学科学学院、数据科学学部）；Shanghai Key Lab. of Intelligent Information Processing（上海市智能信息处理重点实验室）
We consider the design of efficient algorithms for a multicore computing environment with a global shared memory and p cores, each having a cache of size M, and with data organized in blocks of size B. We characterize the class of `Hierarchical Balanced Parallel (HBP)' multithreaded computations for multicores. HBP computations are similar to the hierarchical divide & conquer algorithms considered in recent work, but have some additional features that guarantee good performance even when accounting for the cache misses due to false sharing. Most of our HBP algorithms are derived from known cac…
We consider the problem of laying out a tree with fixed parent/child structure in hierarchical memory. The goal is to minimize the expected number of block transfers performed during a search along a root-to-leaf path, subject to a given probability distribution on the leaves. This problem was previously considered by Gil and Itai, who developed optimal but slow algorithms when the block-transfer size B is known. We present faster but approximate algorithms for the same problem; the fastest such algorithm runs in linear time and produces a solution that is within an additive constant of optima…
We consider the problem of constructing a single spanning tree for the single-source buy-at-bulk network design problem for doubling-dimension graphs. We compute a spanning tree to route a set of demands (or data) along a graph to or from a designated root node. The demands could be aggregated at (or symmetrically distributed to) intermediate nodes where the fusion-cost is specified by a non-negative concave function $f$. We describe a novel approach for developing an oblivious spanning tree in the sense that it is independent of the number of data sources (or demands) and cost function at int…
Motivated by privacy preservation for outsourced data, data-oblivious external memory is a computational framework where a client performs computations on data stored at a semi-trusted server in a way that does not reveal her data to the server. This approach facilitates collaboration and reliability over traditional frameworks, and it provides privacy protection, even though the server has full access to the data and he can monitor how it is accessed by the client. The challenge is that even if data is encrypted, the server can learn information based on the client data access pattern; hence,…
Stony Brook University（石溪大学）；Tokutek, Inc.（Tokutek公司）；Computer Science and Artificial Intelligence Laboratory, Massachusetts Institute of Technology（麻省理工学院计算机科学与人工智能实验室）；University of Southern California（南加州大学）；Akamai Technologies, Inc.（Akamai科技公司）；Boston University（波士顿大学）
As secure processors such as Intel SGX (with hyperthreading) become widely adopted, there is a growing appetite for private analytics on big data. Most prior works on data-oblivious algorithms adopt the classical PRAM model to capture parallelism. However, it is widely understood that PRAM does not best capture realistic multicore processors, nor does it reflect parallel programming models adopted in practice. In this paper, we initiate the study of parallel data oblivious algorithms on realistic multicores, best captured by the binary fork-join model of computation. We first show that data-ob…
The next-generation non-volatile memory (NVM) is striding into computer systems as a new tier as it incorporates both DRAM's byte-addressability and disk's persistency. Researchers and practitioners have considered building persistent memory by placing NVM on the memory bus for CPU to directly load and store data. As a result, cache-friendly data structures have been developed for NVM. One of them is the prevalent B+-tree. State-of-the-art in-NVM B+-trees mainly focus on the optimization of write operations (insertion and deletion). However, search is of vital importance for B+-tree. Not only …
Emerging memory technologies have a significant gap between the cost, both in time and in energy, of writing to memory versus reading from memory. In this paper we present models and algorithms that account for this difference, with a focus on write-efficient sorting algorithms. First, we consider the PRAM model with asymmetric write cost, and show that sorting can be performed in $O\left(n\right)$ writes, $O\left(n \log n\right)$ reads, and logarithmic depth (parallel time). Next, we consider a variant of the External Memory (EM) model that charges $ω> 1$ for writing a block of size $B$ to th…
An oblivious data structure is a data structure where the memory access patterns reveals no information about the operations performed on it. Such data structures were introduced by Wang et al. [ACM SIGSAC'14] and are intended for situations where one wishes to store the data structure at an untrusted server. One way to obtain an oblivious data structure is simply to run a classic data structure on an oblivious RAM (ORAM). Until very recently, this resulted in an overhead of $ω(\lg n)$ for the most natural setting of parameters. Moreover, a recent lower bound for ORAMs by Larsen and Nielsen [C…
We present a new oblivious RAM that supports variable-sized storage blocks (vORAM), which is the first ORAM to allow varying block sizes without trivial padding. We also present a new history-independent data structure (a HIRB tree) that can be stored within a vORAM. Together, this construction provides an efficient and practical oblivious data structure (ODS) for a key/value map, and goes further to provide an additional privacy guarantee as compared to prior ODS maps: even upon client compromise, deleted data and the history of old operations remain hidden to the attacker. We implement and m…
Hash tables are a ubiquitous class of dictionary data structures. However, standard hash table implementations do not translate well into the external memory model, because they do not incorporate locality for insertions. Iacono and Patracsu established an update/query tradeoff curve for external hash tables: a hash table that performs insertions in $O(λ/B)$ amortized IOs requires $Ω(\log_λN)$ expected IOs for queries, where $N$ is the number of items that can be stored in the data structure, $B$ is the size of a memory transfer, $M$ is the size of memory, and $λ$ is a tuning parameter. They p…
We consider the single-source (or single-sink) buy-at-bulk problem with an unknown concave cost function. We want to route a set of demands along a graph to or from a designated root node, and the cost of routing x units of flow along an edge is proportional to some concave, non-decreasing function f such that f(0) = 0. We present a polynomial time algorithm that finds a distribution over trees such that the expected cost of a tree for any f is within an O(1)-factor of the optimum cost for that f. The previous best simultaneous approximation for this problem, even ignoring computation time, wa…
We consider the top tree compression scheme introduced by Bille et al. [ICALP 2013] and construct an infinite family of trees on $n$ nodes labeled from an alphabet of size $σ$, for which the size of the top DAG is $Θ(\frac{n}{\log_σn}\log\log_σn)$. Our construction matches a previously known upper bound and exhibits a weakness of this scheme, as the information-theoretic lower bound is $Ω(\frac{n}{\log_σn})$. This settles an open problem stated by Lohrey et al. [arXiv 2017], who designed a more involved version achieving the lower bound. We show that this can be also guaranteed by a very minor…
LSM-tree has been widely used in cloud computing systems by Google, Facebook, and Amazon, to achieve high performance for write-intensive workloads. However, in LSM-tree, random key-value queries can experience long latency and low throughput due to the interference from the compaction, a basic operation in the algorithm, to caching. LSM-tree relies on frequent compaction operations to merge data into a sorted structure. After a compaction, the original data are reorganized and written to other locations on the disk. As a result, the cached data are invalidated since their referencing addresse…
Network design problems aim to compute low-cost structures such as routes, trees and subgraphs. Often, it is natural and desirable to require that these structures have small hop length or hop diameter. Unfortunately, optimization problems with hop constraints are much harder and less well understood than their hop-unconstrained counterparts. A significant algorithmic barrier in this setting is the fact that hop-constrained distances in graphs are very far from being a metric. We show that, nonetheless, hop-constrained distances can be approximated by distributions over "partial tree metrics."…
Over the years, many different indexing techniques and search algorithms have been proposed, including CSS-trees, CSB+ trees, k-ary binary search, and fast architecture sensitive tree search. There have also been papers on how best to set the many different parameters of these index structures, such as the node size of CSB+ trees. These indices have been proposed because CPU speeds have been increasing at a dramatically higher rate than memory speeds, giving rise to the Von Neumann CPU--Memory bottleneck. To hide the long latencies caused by memory access, it has become very important to well-…
We study the problem of explainable clustering in the setting first formalized by Dasgupta, Frost, Moshkovitz, and Rashtchian (ICML 2020). A $k$-clustering is said to be explainable if it is given by a decision tree where each internal node splits data points with a threshold cut in a single dimension (feature), and each of the $k$ leaves corresponds to a cluster. We give an algorithm that outputs an explainable clustering that loses at most a factor of $O(\log^2 k)$ compared to an optimal (not necessarily explainable) clustering for the $k$-medians objective, and a factor of $O(k \log^2 k)$ f…
It has been shown in the indexing literature that there is an essential difference between prefix/range searches on the one hand, and predecessor/rank searches on the other hand, in that the former provably allows faster query resolution. Traditionally, prefix search is solved by data structures that are also dictionaries---they actually contain the strings in $S$. For very large collections stored in slow-access memory, we propose much more compact data structures that support \emph{weak} prefix searches---they return the ranks of matching strings provided that \emph{some} string in $S$ start…
We attempt to determine the best order and search algorithm to store $n$ comparable data items in an array, $A$, of length $n$ so that we can, for any query value, $x$, quickly find the smallest value in $A$ that is greater than or equal to $x$. In particular, we consider the important case where there are many such queries to the same array, $A$, which resides entirely in RAM. In addition to the obvious sorted order/binary search combination we consider the Eytzinger (BFS) layout normally used for heaps, an implicit B-tree layout that generalizes the Eytzinger layout, and the van Emde Boas la…
The Fenwick tree is a classical implicit data structure that stores an array in such a way that modifying an element, accessing an element, computing a prefix sum and performing a predecessor search on prefix sums all take logarithmic time. We introduce a number of variants which improve the classical implementation of the tree: in particular, we can reduce its size when an upper bound on the array element is known, and we can perform much faster predecessor searches. Our aim is to use our variants to implement an efficient dynamic bit vector: our structure is able to perform updates, ranking …
Applications making excessive use of single-object based data structures (such as linked lists, trees, etc...) can see a drop in efficiency over a period of time due to the randomization of nodes in memory. This slow down is due to the ineffective use of the CPU's L1/L2 cache. We present a novel approach for mitigating this by presenting the design of a single-object memory allocator that preserves memory locality across randomly ordered memory allocations and deallocations.
We present a technique that allows for improving on some relative greedy procedures by well-chosen (non-oblivious) local search algorithms. Relative greedy procedures are a particular type of greedy algorithm that start with a simple, though weak, solution, and iteratively replace parts of this starting solution by stronger components. Some well-known applications of relative greedy algorithms include approximation algorithms for Steiner Tree and, more recently, for connectivity augmentation problems. The main application of our technique leads to a $(1.5+ε)$-approximation for Weighted Tree Au…
Mining large graphs for information is becoming an increasingly important workload due to the plethora of graph structured data becoming available. An aspect of graph algorithms that has hitherto not received much interest is the effect of memory hierarchy on accesses. A typical system today has multiple levels in the memory hierarchy with differing units of locality; ranging across cache lines, TLB entries and DRAM pages. We postulate that it is possible to allocate graph structured data in main memory in a way as to improve the spatial locality of the data. Previous approaches to improving c…
This paper is about metric data structures in high-dimensional or non-Euclidean space that permit cached sufficient statistics accelerations of learning algorithms. It has recently been shown that for less than about 10 dimensions, decorating kd-trees with additional "cached sufficient statistics" such as first and second moments and contingency tables can provide satisfying acceleration for a very wide range of statistical learning tasks such as kernel regression, locally weighted regression, k-means clustering, mixture modeling and Bayes Net learning. In this paper, we begin by defining the …
$k$d-trees are widely used in parallel databases to support efficient neighborhood/similarity queries. Supporting parallel updates to $k$d-trees is therefore an important operation. In this paper, we present BDL-tree, a parallel, batch-dynamic implementation of a $k$d-tree that allows for efficient parallel $k$-NN queries over dynamically changing point sets. BDL-trees consist of a log-structured set of $k$d-trees which can be used to efficiently insert or delete batches of points in parallel with polylogarithmic depth. Specifically, given a BDL-tree with $n$ points, each batch of $B$ updates …
