Just Join for Parallel Ordered Sets

We address the vexing issue of deletions in balanced trees. Rebalancing after a deletion is generally more complicated than rebalancing after an insertion. Textbooks neglect deletion rebalancing, and many B-tree--based database systems do not do it. We describe a relaxation of AVL trees in which rebalancing is done after insertions but not after deletions, yet worst-case access time remains logarithmic in the number of insertions. For any application of balanced trees in which the number of updates is polynomial in the tree size, our structure offers performance competitive with that of classical balanced trees. With the addition of periodic rebuilding, the performance of our structure is theoretically superior to that of many, if not all, classic balanced tree structures. Our structure needs lg lg m + 1 bits of balance information per node, where m is the number of insertions and lg is the base-two logarithm, or lg lg n + O(1) with periodic rebuilding, where n is the number of nodes. An insertion takes up to two rotations and O(1) amortized time, not counting the time to find the insertion position. This is the same as in standard AVL trees. Using an analysis that relies on an exponential potential function, we show that rebalancing steps occur with a frequency that is exponentially small in the height of the affected node. Our techniques apply to other types of balanced trees, notably B-trees, as we show in a companion article, and particularly red-black trees, which can be viewed as a special case of B-trees.

We present a provably linearizable and lock-free relaxed AVL tree called the non-blocking ravl tree. At any time, the height of a non-blocking ravl tree is upper bounded by log_d (2m) + c, where d is the golden ratio, m is the total number of successful INSERT operations performed so far and c is the number of active concurrent processes that have inserted new keys and are still rebalancing the tree at this time. The most significant feature of the non-blocking ravl tree is that it does not rebalance itself after DELETE operations. Instead, it performs rebalancing only after INSERT operations. Thus, the non-blocking ravl tree is much simpler to implement than other self-balancing concurrent binary search trees (BSTs) which typically introduce a large number of rebalancing cases after DELETE operations, while still providing a provable non-trivial bound on its height. We further conduct experimental studies to compare our solution with other state-of-the-art concurrent BSTs using randomly generated data sequences under uniform distributions, and find that our solution achieves the best performance among concurrent self-balancing BSTs. As the keys in access sequences are likely to be partially sorted in system software, we also conduct experiments using data sequences with various degrees of presortedness to better simulate applications in practice. Our experimental results show that, when there are enough degrees of presortedness, our solution achieves the best performance among all the concurrent BSTs used in our studies, including those that perform self-balancing operations and those that do not, and thus is potentially the best candidate for many real-world applications.

A weight-balanced tree (WBT) is a binary search tree, whose balance is based on the sizes of the subtrees in each node. Although purely functional implementations on a variant WBT algorithm are widely used in functional programming languages, many existing implementations do not maintain balance after deletion in some cases. The difficulty lies in choosing a valid pair of rotation parameters: one for standard balance and the other for choosing single or double rotation. This paper identifies the exact valid range of the rotation parameters for insertion and deletion in the original WBT algorithm where one and only one integer solution exists. Soundness of the range is proved using a proof assistant Coq. Completeness is proved using effective algorithms generating counterexample trees. For two specific parameter pairs, we also proved in Coq that set operations also maintain balance. Since the difference between the original WBT and the variant WBT is small, it is easy to change the existing buggy implementations based on the variant WBT to the certified original WBT with a rational solution.

A class of binary trees is described for maintaining ordered sets of data. Random insertions, deletions, and retrievals of keys can be done in time proportional to log N where N is the cardinality of the data-set. Symmetric B-Trees are a modification of B-trees described previously by Bayer and McCreight. This class of trees properly contains the balanced trees.

A problem that arises in computational geometry, pattern matching, and other applications is the need to quickly determine which of a collection of intervals overlap a query point. The interval binary search tree (IBS-tree) has been proposed as a solution to this problem. A recently discovered randomized data structure called the skip list provides functionality and performance similar to balanced binary trees (e.g., AVL-trees), but is much simpler to implement than balanced trees. This paper introduces an extension of the skip list called the interval skip list, or IS-list, to support interval indexing. IS-lists remain dynamicly balanced, and show similar performance to IBS-trees, but can be implemented in about one-fourth as much high-level language code. Searching an IS-list containing n intervals to find intervals overlapping a point takes expected time O(log n+L) where L is the number of matching intervals. Inserting or deleting an mterval takes expected time O(log2n).

It has been shown that the cost W of a weight balanced binary tree satisfies the inequalities, H ≤ W ≤ H + 3, where H is the entropy of the set of the leaves. For a class of “smooth” distributions the inequalities, H ≤ W ≤ H + 2, are derived. These results imply that for sets with large entropy the search times provided by such trees cannot be substantially shortened when binary decisions are being used.

We show that the internal path length of a red-black tree of size N is bounded above by 2N(log N−log log N)+O(N) and that this is, asymptotically, tight. This result further affirms a conjecture that whenever trees of size N, in a class of height-balanced binary trees, have height bounded from above by αlog2N, then the internal path length is bounded from above by α(log2N−log2 log2N)+O(N). We also prove that the asymptotic bound is tight for red-black trees. For this purpose, we introduce a class of red-black trees, the C(k, h) trees, which achieve the bound when k=h−2logh+e, where |e|≤1/2. This result solves the open problem of how bad red-black trees can be. In order to establish this result, we also prove that any red-black tree of a given height can be produced from a “skinniest” red-black tree of the same height by a series of simple insertions.

Stabbing queries in sets of intervals are usually answered using segment trees. A dynamic variant of segment trees has been presented by van Kreveld and Overmars, which uses red-black trees to do rebalancing operations. This paper presents zipping segment trees - dynamic segment trees based on zip trees, which were recently introduced by Tarjan et al. To facilitate zipping segment trees, we show how to uphold certain segment tree properties during the operations of a zip tree. We present an in-depth experimental evaluation and comparison of dynamic segment trees based on red-black trees, weight-balanced trees and several variants of the novel zipping segment trees. Our results indicate that zipping segment trees perform better than rotation-based alternatives.

Binary search trees have received a great deal of attention in recent years. As a result of this interest, several methods have been developed for balancing them; namely, random, height-balanced, bounded-balance, and weight-balanced trees. These methods which include weighted and non-weighted binary search trees are grouped into two classes: 1) dynamic balancing and 2) total restructuring. The rational and properties of the more significant methods are discussed and compared with other tree balancing algorithms. These comparisons provide insight about the conditions under which an algorithm is appropriate.

Three relevant types of suboptimal binary search trees are comparatively evaluated in this paper: two well-known representatives of height-balanced approaches (the AVL and red-black trees) and a popular self-adjusting splay tree. After a brief theoretical background, an evaluation method was described that employs a suitable synthetic workload method capable of producing diverse desired workload characteristics (different distributions and ranges of key values, varying input sequence lengths, etc.). Evaluation analysis was conducted for search, insert, and delete operations separately for each particular type and in appropriate combinations. Experimental results for an average operation cost as well as for tree maintenance cost are comparatively presented and carefully discussed. Finally, the suggested favorable conditions for application of each tree type are summarized.

Content authentication is a critical issue for online multimedia systems. A new content authentication scheme based on optimal binary search tree (OBST) was proposed and evaluated to reduce the authentication cost. A balanced tree based approach, however, can not achieve optimal authentication cost when each element is accessed with different probabilities, so it's not appropriate for a practical online multimedia system in which access of the contents is expected to obey Zipf's law. We prove that an OBST based scheme achieves minimized expected authentication cost; we also compare the authentication cost of our scheme to that of a binary search tree (BST) and a red-black tree.

We study an order-preserving general purpose data structure for binary data, the LPC-trie. The structure is a compressed trie, using both level and path compression. The memory usage is similar to that of a balanced binary search tree, but the expected average depth is smaller. The LPC-trie is well suited to modem language environments with efficient memory allocation and garbage collection. We present an implementation in the Java programming language and show that the structure compares favorably with a balanced binary search tree.

The quality-guide phase unwrapping is an important technique that is based on quality maps which guide the unwrapping process. The efficiency of this technique depends in the adjoin-list data structure implementation. There exists several proposals that improve the adjoin-list; Ming Zhao et. al. proposed an Indexed Interwoven Linked List (I2L2) that is based on dividing the quality values into intervals of equal size and inserting in a linked list those pixels with quality values within a certain interval. Ming Zhao and Qian Kemao proposed an improved I2L2 replacing each linked list in each interval by a heap data structure, which allows efficient procedures for insertion and deletion. In this paper, we propose an improved I2L2 which uses Red-Black trees (RBT) data structures for each interval. Our proposal has as main goal to avoid the unbalanced properties of the head and thus, reducing the time complexity of insertion. In order to maintain the same efficiency of the heap when deleting an element, we provide an efficient way to remove the pixel with the highest quality value in the RBT using a pointer to the rightmost element in the tree. We also provide a new partition strategy of the phase values that is based on a density criterion. Experimental results applied to phase shifting profilometry are shown for large images.

We introduce the relaxed k-tree, a search tree with relaxed balance and a height bound, when in balance, of (1 + Ɛ) log2 n + 1, for any Ɛ > 0. The rebalancing work is amortized O(1/Ɛ) per update. This is the first binary search tree with relaxed balance having a height bound better than c ċ log2 n for a fixed constant c. In all previous proposals, the constant is at least 1/ log2 Φ > 1.44, where Φ is the golden ratio. As a consequence, we can also define a standard (non-relaxed) k-tree with amortized constant rebalancing per update, which is an improvement over the original definition. Search engines based on main-memory databases with strongly fluctuating workloads are possible applications for this line of work.

Frontiers Events is a rapidly growing calendar management system dedicated to the scheduling of academic events. This includes announcements and invitations, participant listings and search functionality, abstract handling and publication, related events and post-event exchanges. Whether an organizer or participant, make your event a Frontiers Event!