An implementation of range trees with fractional cascading in C++

Range trees are multidimensional binary trees which are used to perform d-dimensional orthogonal range searching. In this technical report we study the implementation issues of range trees with fractional cascading, named layered range trees. We also document our implementation of range trees with fractional cascading in C++ using STL and generic programming techniques.Comment: Technical repor

Efficient Authenticated Data Structures for Graph Connectivity and Geometric Search Problems

Authenticated data structures provide cryptographic proofs that their answers are as accurate as the author intended, even if the data structure is being controlled by a remote untrusted host. We present efficient techniques for authenticating data structures that represent graphs and collections of geometric objects. We introduce the path hash accumulator, a new primitive based on cryptographic hashing for efficiently authenticating various properties of structured data represented as paths, including any decomposable query over sequences of elements. We show how to employ our primitive to authenticate queries about properties of paths in graphs and search queries on multi-catalogs. This allows the design of new, efficient authenticated data structures for fundamental problems on networks, such as path and connectivity queries over graphs, and complex queries on two-dimensional geometric objects, such as intersection and containment queries.Comment: Full version of related paper appearing in CT-RSA 200

GPU LSM: A Dynamic Dictionary Data Structure for the GPU

We develop a dynamic dictionary data structure for the GPU, supporting fast insertions and deletions, based on the Log Structured Merge tree (LSM). Our implementation on an NVIDIA K40c GPU has an average update (insertion or deletion) rate of 225 M elements/s, 13.5x faster than merging items into a sorted array. The GPU LSM supports the retrieval operations of lookup, count, and range query operations with an average rate of 75 M, 32 M and 23 M queries/s respectively. The trade-off for the dynamic updates is that the sorted array is almost twice as fast on retrievals. We believe that our GPU LSM is the first dynamic general-purpose dictionary data structure for the GPU.Comment: 11 pages, accepted to appear on the Proceedings of IEEE International Parallel and Distributed Processing Symposium (IPDPS'18

Odds-On Trees

Let R^d -> A be a query problem over R^d for which there exists a data structure S that can compute P(q) in O(log n) time for any query point q in R^d. Let D be a probability measure over R^d representing a distribution of queries. We describe a data structure called the odds-on tree, of size O(n^\epsilon) that can be used as a filter that quickly computes P(q) for some query values q in R^d and relies on S for the remaining queries. With an odds-on tree, the expected query time for a point drawn according to D is O(H*+1), where H* is a lower-bound on the expected cost of any linear decision tree that solves P. Odds-on trees have a number of applications, including distribution-sensitive data structures for point location in 2-d, point-in-polytope testing in d dimensions, ray shooting in simple polygons, ray shooting in polytopes, nearest-neighbour queries in R^d, point-location in arrangements of hyperplanes in R^d, and many other geometric searching problems that can be solved in the linear-decision tree model. A standard lifting technique extends these results to algebraic decision trees of constant degree. A slightly different version of odds-on trees yields similar results for orthogonal searching problems that can be solved in the comparison tree model.Comment: 19 pages, 0 figure

LSM-based Storage Techniques: A Survey

Recently, the Log-Structured Merge-tree (LSM-tree) has been widely adopted for use in the storage layer of modern NoSQL systems. Because of this, there have been a large number of research efforts, from both the database community and the operating systems community, that try to improve various aspects of LSM-trees. In this paper, we provide a survey of recent research efforts on LSM-trees so that readers can learn the state-of-the-art in LSM-based storage techniques. We provide a general taxonomy to classify the literature of LSM-trees, survey the efforts in detail, and discuss their strengths and trade-offs. We further survey several representative LSM-based open-source NoSQL systems and discuss some potential future research directions resulting from the survey.Comment: This is a pre-print of an article published in VLDB Journal. The final authenticated version is available online at: https://doi.org/10.1007/s00778-019-00555-

Biased Range Trees

A data structure, called a biased range tree, is presented that preprocesses a set S of n points in R^2 and a query distribution D for 2-sided orthogonal range counting queries. The expected query time for this data structure, when queries are drawn according to D, matches, to within a constant factor, that of the optimal decision tree for S and D. The memory and preprocessing requirements of the data structure are O(n log n)

An Optimal Algorithm for Range Search on Multidimensional Points

This paper proposes an efficient and novel method to address range search on multidimensional points in $\theta(t)$ time, where $t$ is the number of points reported in $\Re^k$ space. This is accomplished by introducing a new data structure, called BITS $k$d-tree. This structure also supports fast updation that takes $\theta(1)$ time for insertion and $O(\log n)$ time for deletion. The earlier best known algorithm for this problem is $O(\log^k n+t)$ time in the pointer machine model

Hidden surface removal for rectangles

AbstractA simple but important special case of the hidden surface removal problem is one in which the scene consists of n rectangles with sides parallel to the x- and y-axes, with viewpoint at z=∞ (that is, an orthographic projection). This special case has application to overlapping windows in computer displays. An algorithm with running time O(n log n + k log n) is given for static scenes, where k is the number of line segments in the output. Algorithms are given for a dynamic setting (that is, rectangles may be inserted and deleted) that take time O(log2n log log n + k log2 n) per insert or delete, where k is now the number of visible line segments that change (appear or disappear). Algorithms for point location in the visible scene are also given

Discrepancy-Sensitive Dynamic Fractional Cascading, Dominated Maxima Searching, and 2-d Nearest Neighbors in Any Minkowski Metric

This paper studies a discrepancy-sensitive approach to dynamic fractional cascading. We provide an efficient data structure for dominated maxima searching in a dynamic set of points in the plane, which in turn leads to an efficient dynamic data structure that can answer queries for nearest neighbors using any Minkowski metric. We provide an efficient data structure for dominated maxima searching in a dynamic set of points in the plane, which in turn leads to an efficient dynamic data structure that can answer queries for nearest neighbors using any Minkowski metric.Comment: Expanded version of a paper that appeared in WADS 200

Stratified B-trees and versioning dictionaries

A classic versioned data structure in storage and computer science is the copy-on-write (CoW) B-tree -- it underlies many of today's file systems and databases, including WAFL, ZFS, Btrfs and more. Unfortunately, it doesn't inherit the B-tree's optimality properties; it has poor space utilization, cannot offer fast updates, and relies on random IO to scale. Yet, nothing better has been developed since. We describe the `stratified B-tree', which beats all known semi-external memory versioned B-trees, including the CoW B-tree. In particular, it is the first versioned dictionary to achieve optimal tradeoffs between space, query and update performance