On random cartesian trees

We destroy a finite tree of size n by cutting its edges one after the other and in uniform random order. Informally, the associated cut‐tree describes the genealogy of the connected components created by this destruction process. We provide a general criterion for the convergence of the rescaled cut‐tree in the Gromov‐Prohorov topology to an interval endowed with the Euclidean distance and a certain probability measure, when the underlying tree has branching points close to the root and height of order . In particular, we consider uniform random recursive trees, binary search trees, scale‐free random trees and a mixture of regular trees. This yields extensions of a result in Bertoin (Probab Stat 5 (2015), 478–488) for the cut‐tree of uniform random recursive trees and also allows us to generalize some results of Kuba and Panholzer (Online J Anal Combin (2014), 26) on the multiple isolation of vertices. The approach relies in the close relationship between the destruction process and Bernoulli bond percolation, which may be useful for studying the cut‐tree of other classes of trees. © 2017 Wiley Periodicals, Inc. Random Struct. Alg., 51, 404–427, 2017

We present a new linear-time algorithm for constructing multiway search trees with near-optimal search cost whose running time is independent of the size of the node in the tree. With the analysis of our construction method, we provide a new upper bound on the average search cost for multiway search trees that nearly matches the lower bound. In fact, it is tight for infinitely many probability distributions. This problem is well-studied in the literature for the case of binary search trees. Using our new construction method, we are able to provide the tightest upper bound on the average search cost for an optimal binary search tree.

On Robson's convergence and boundedness conjectures concerning the height of binary search trees

We study protected nodes in various classes of random rooted trees by putting them in the general context of fringe subtrees introduced by Aldous (1991). Several types of random trees are considered: simply generated trees (or conditioned Galton-Watson trees), which includes several cases treated separately by other authors, binary search trees and random recursive trees. This gives unified and simple proofs of several earlier results, as well as new results.

In this chapter, we will explain the following: The difference between a tree and a binary tree How to perform pre-order, in-order, and post-order traversals of a binary tree How to represent a binary tree in a computer program How to build a binary tree from given data What a binary search tree is and how to build one How to write a program to do a word-frequency count of words in a passage How to use an array as a binary tree representation How to write some recursive functions to obtain information about a binary tree How to delete a node from a binary search tree

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.

In a multiprocessor system, a lock-based scheme is used to ensure consistency and correctness during parallel processing. To manipulate a binary search tree in parallel, a process that modifies the current state of the data structure has to lock a certain portion of the tree. Lock-based schemes result in a longer wait time for the rest of the processes, particularly if a large portion of the tree has been locked. In a concurrent environment, the root of the binary search tree may therefore become a bottleneck, as it is the only entry point for all processes. To reduce the total wait time, it is possible to create and maintain a binary search tree with multiple roots, thereby allowing multiple processors to act upon different trees simultaneously. In this paper, we present static and dynamic methods to create and maintain a binary search tree with multiple roots. Without considerable overhead, a set of connected trees is created that resembles a forest, yet acts as a single binary search tree.

We study multi-finger binary search trees (BSTs), a far-reaching extension of the classical BST model, with connections to the well-studied k-server problem. Finger search is a popular technique for speeding up BST operations when a query sequence has locality of reference. BSTs with multiple fingers can exploit more general regularities in the input. In this paper we consider the cost of serving a sequence of queries in an optimal (offline) BST with k fingers, a powerful benchmark against which other algorithms can be measured. We show that the k-finger optimum can be matched by a standard dynamic BST (having a single root-finger) with an O(log{k}) factor overhead. This result is tight for all k, improving the O(k) factor implicit in earlier work. Furthermore, we describe new online BSTs that match this bound up to a (log{k})^{O(1)} factor. Previously only the "one-finger" special case was known to hold for an online BST (Iacono, Langerman, 2016; Cole et al., 2000). Splay trees, assuming their conjectured optimality (Sleator and Tarjan, 1983), would have to match our bounds for all k. Our online algorithms are randomized and combine techniques developed for the k-server problem with a multiplicative-weights scheme for learning tree metrics. To our knowledge, this is the first time when tools developed for the k-server problem are used in BSTs. As an application of our k-finger results, we show that BSTs can efficiently serve queries that are close to some recently accessed item. This is a (restricted) form of the unified property (Iacono, 2001) that was previously not known to hold for any BST algorithm, online or offline.

Binary search trees built from the postorder traversal sequence of other binary search trees are characterized in terms of their binary tree structure. A connection is established between this structure and the Eulerian numbers. This yields considerable information concerning the “average” binary search tree with a given number of nodes. Periodicity is established for the process of repeatedly taking postorder sequences and building binary search trees. Finally, the average depth of a node in a postorder tree withn nodes is determined.

k-Protected vertices in binary search trees

Change-Point (CP) detection has attracted considerable attention in the fields of data mining and statistics; it is very meaningful to discuss how to quickly and efficiently detect abrupt change from large-scale bioelectric signals. Currently, most of the existing methods, like Kolmogorov-Smirnov (KS) statistic and so forth, are time-consuming, especially for large-scale datasets. In this paper, we propose a fast framework for abrupt change detection based on binary search trees (BSTs) and a modified KS statistic, named BSTKS (binary search trees and Kolmogorov statistic). In this method, first, two binary search trees, termed as BSTcA and BSTcD, are constructed by multilevel Haar Wavelet Transform (HWT); second, three search criteria are introduced in terms of the statistic and variance fluctuations in the diagnosed time series; last, an optimal search path is detected from the root to leaf nodes of two BSTs. The studies on both the synthetic time series samples and the real electroencephalograph (EEG) recordings indicate that the proposed BSTKS can detect abrupt change more quickly and efficiently than KS, t-statistic (t), and Singular-Spectrum Analyses (SSA) methods, with the shortest computation time, the highest hit rate, the smallest error, and the highest accuracy out of four methods. This study suggests that the proposed BSTKS is very helpful for useful information inspection on all kinds of bioelectric time series signals.

To quickly and efficiently recognize abnormal patterns from large-scale time series and pathological signals in epilepsy, this paper presents here a preliminary RSW&TST framework for Multiple Change-Points (MCPs) detection based on the Random Slide Window (RSW) and Trigeminal Search Tree (TST) methods. To avoid the remaining local optima, the proposed framework applies a random strategy for selecting the size of each slide window from a predefined collection, in terms of data feature and experimental knowledge. For each data segment to be diagnosed in a current slide window, an optimal path towards a potential change point is detected by TST methods from the top root to leaf nodes with O(log3(N)). Then, the resulting MCPs vector is assembled by means of TST-based single CP detection on data segments within each of the slide windows. In our experiments, the RSW&TST framework was tested by using large-scale synthetic time series, and then its performance was evaluated by comparing it with existing binary search tree (BST), Kolmogorov-Smirnov (KS)-statistics, and T-test under the fixed slide window (FSW) approach, as well as the integrated method of wild binary segmentation and CUSUM test (WBS&CUSUM). The simulation results indicate that our RSW&TST is both more efficient and effective, with a higher hit rate, shorter computing time, and lower missed, error and redundancy rates. When the proposed RSW&TST framework is executed for MCPs detection on pathological ECG (electrocardiogram)/EEG (electroencephalogram) recordings of people in epileptic states, the abnormal patterns are roughly recognized in terms of the number and position of the resultant MCPs. Furthermore, the severity of epilepsy is roughly analyzed based on the strength and period of signal fluctuations among multiple change points in the stage of a sudden epileptic attack. The purpose of our RSW&TST framework is to provide an encouraging platform for abnormal pattern recognition through MCPs detection on large-scale time series quickly and efficiently.

Binary Search Tree (BST) is an important data structure for managing ordered data. Many algorithms have been proposed for concurrent manipulation of a binary search tree in an asynchronous shared memory system that supports search, insert and delete operations based on both external and internal representations of a search tree. An important step in executing an operation on a tree is to traverse the tree from top-to-down in order to locate the operation's window. A process may need to perform this traversal several times to handle failures occurring due to other processes performing concurrent updates on the tree. Most concurrent algorithms that have been proposed so far use a naive approach and simply restart the traversal from the root of the tree. In this work, we present a new approach to recover from such failures more efficiently for certain types of concurrent binary search tree algorithms using local recovery by restarting the traversal from the middle of the tree in order to locate an operation's window (tree section of interest). Our approach is sufficiently general in the sense that it can be applied to most of the existing concurrent binary search trees using internal representation based on both blocking as well as non-blocking approaches. Using experimental evaluation, we demonstrate that our local recovery approach can significantly improve the performance of a concurrent internal BST by up to 124.7% when the contention is high.

The theory of records in sequences of independent identically distributed random variables leads to simple proofs of various properties of random trees, including among other things, the limit law for the depth of the last node of random ordered trees, random union-find trees, and random binary search trees.

This work is a case study of the formal verification and complexity analysis of some famous probabilistic algorithms and data structures in the proof assistant Isabelle/HOL. In particular, we consider the expected number of comparisons in randomised quicksort, the relationship between randomised quicksort and average-case deterministic quicksort, the expected shape of an unbalanced random Binary Search Tree, the randomised binary search trees described by Martínez and Roura, and the expected shape of a randomised treap. The last three have, to our knowledge, not been analysed using a theorem prover before and the last one is of particular interest because it involves continuous distributions.