Optimal Binary Search Tree Research Articles

Four-splitting based coarse-grained multicomputer parallel algorithm for the optimal binary search tree problem

ABSTRACT This paper presents a parallel solution based on the coarse-grained multicomputer (CGM) model using the four-splitting technique to solve the optimal binary search tree problem. The well-known sequential algorithm of Knuth solves this problem in time and space, where n is the number of keys used to build the optimal binary search tree. To parallelise this algorithm on the CGM model, the irregular partitioning technique, consisting in subdividing the dependency graph into subgraphs (or blocks) of variable size, has been proposed to tackle the trade-off of minimising the number of communication rounds and balancing the load of processors. This technique, however, induces a high latency time of processors (which accounts for most of the global communication time) because varying the blocks' sizes does not enable them to start evaluating some blocks as soon as the data they need are available. The four-splitting technique proposed in this paper solves this shortcoming by evaluating a block as a sequence of computation and communication steps of four subblocks. This CGM-based parallel solution requires execution time with communication rounds, where p is the number of processors and k is the number of times the size of blocks is subdivided. An experimental study conducted to evaluate the performance of this CGM-based parallel solution showed that compared to the solution based on the irregular partitioning technique where the speedup factor is up to ×10.39 on 128 processors with 40,960 keys when k = 2, the speedup factor of this solution is up to ×13.12 and rises up to ×14.93 when k = 5.

Endogenous efficiency of the dynamic profit maximization in the intertemporal production models of venture behavior

How do ventures manage adjustment costs, input elasticities, and productivity growth? We draw on the intertemporal production decisions of ventures that are quasi-fixed, costly to adjust, and endogenous. Using a modified version of the Bayesian Exponentially Tilted Empirical Likelihood (BETEL) method adjusted for the presence of dynamic latent variables, the proposed moment-based multiple-equation estimation system incorporates dynamic and static optimality conditions derived from a firm's expected intertemporal profit maximization. Using reliable tax information data from a sample of 72,035 Portuguese ventures founded between 2010 and 2017, we find that the most important inputs are labor, equity, and inventories. However, the technical change is small, as is productivity growth. Ventures have a high degree of labor input elasticity but much lower elasticities for equity, inventories, capital, and advertising. The findings provide an understanding of the intertemporal behavior of ventures in managing adjustment costs, input elasticities, and productivity growth where adjustments to efficiency are “cheaper” than adjustments to capital. Labor elasticity is the highest, and productivity growth is very small.

Dynamic Optimal Mean-Variance Investment with Mispricing in the Family of 4/2 Stochastic Volatility Models

This paper considers an optimal investment problem with mispricing in the family of 4/2 stochastic volatility models under mean–variance criterion. The financial market consists of a risk-free asset, a market index and a pair of mispriced stocks. By applying the linear–quadratic stochastic control theory and solving the corresponding Hamilton–Jacobi–Bellman equation, explicit expressions for the statically optimal (pre-commitment) strategy and the corresponding optimal value function are derived. Moreover, a necessary verification theorem was provided based on an assumption of the model parameters with the investment horizon. Due to the time-inconsistency under mean–variance criterion, we give a dynamic formulation of the problem and obtain the closed-form expression of the dynamically optimal (time-consistent) strategy. This strategy is shown to keep the wealth process strictly below the target (expected terminal wealth) before the terminal time. Results on the special case without mispricing are included. Finally, some numerical examples are given to illustrate the effects of model parameters on the efficient frontier and the difference between static and dynamic optimality.

Dynamic Relocations in Car-Sharing Networks

We propose a novel dynamic car relocation policy for a car-sharing network with centralized control and uncertain, unbalanced demand. The policy is derived from a reformulation of the linear programming fluid model approximation of the dynamic problem. We project the full-dimensional fluid approximation onto the lower-dimensional space of relocation decisions only. This projection results in a characterization of the problem as n+1 linear programs, where n is the number of nodes in the network. The reformulation uncovers structural properties that are interpretable using absorbing Markov chain concepts and allows us to write the gradient with respect to the relocation decisions in closed form. Our policy exploits these gradients to make dynamic car relocation decisions. We provide extensive numerical results where our dynamic car relocation policy consistently outperforms the standard static policy in realistic instances from the literature. In fact, it reduces the static policy's optimality gap by more than 13% in all of these instances and up to 35% in some.

3D parallel tiled code implementing a modified Knuth's optimal binary search tree algorithm

This paper presents an approach to generation of 3D parallel tiled code implementing an optimal binary search tree (OBST) algorithm. We demonstrate that the features of data dependences available in the code implementing Knuth's OBST algorithm allow us to generate only 2D tiled code. We suggest a way of transformation of Knuth's OBST algorithm to a modified one exposing dependences allowing us to generate 3D parallel tiled code. The polyhedral model and the corresponding tools supporting that model are used by us to generate 3D target tiled code on the basis of the modified Knuth's OBST algorithm. Program parallelism is based on the wavefront technique and it is presented in the OpenMP C/C++ standard. Experiments carried out by us with obtained 3D tiled code demonstrate that this code considerably outperforms 2D tiled code generated on the basis of serial code implementing classic Knuth's OBST algorithm. Increased code performance is achieved due to much larger locality of 3D tiled code in comparison with that of 2D one.

Endogenous dynamic efficiency in the intertemporal optimization models of firm behavior

Existing methods for the measurement of technical efficiency in the dynamic production models obtain it from the implied distance functions without making use of the information about intertemporal economic behavior in the estimation beyond an indirect appeal to duality. The main limitation of such an estimation approach is that it does not allow for the dynamic evolution of efficiency that is explicitly optimized by the firm. This paper introduces a new conceptualization of efficiency that directly enters the firm’s intertemporal production decisions and is both explicitly costly and endogenously determined. We build a moment-based multiple-equation system estimation procedure that incorporates both the dynamic and static optimality conditions derived from the firm’s intertemporal expected cost minimization. We operationalize our methodology using a modified version of a Bayesian Exponentially Tilted Empirical Likelihood adjusted for the presence of dynamic latent variables in the model, which we showcase using the 1960–2004 U.S. agricultural farm production data. We find that allowing for potential endogenous adjustments in efficiency over time produces significantly higher estimates of technical efficiency, which is likely due to inherent inability of the more standard exogenous-efficiency model to properly credit firms for incurring efficiency-improvement adjustment costs. Our results also suggest material improvements in efficiency over time at an about 2.6% average annual rate, which contrasts with near-zero estimates of the exogenous efficiency change.

Efficient Aspect Finding in Sentiment Analysis using Optimal Binary Search Tree

Sentiment Analysis is an interesting research area. It useful for analyze the user comments with the help of natural language processing. While consider a huge data and improper data, the sentiment classification with better accuracy is a very big challenge in sentiment analysis. The social media records relevant to Sentiment analysis is a difficult one to identify with a people’s situation, thoughts, and view in the direction of a certain event, which has a number of applications such as election guess and product assessment. This paper focuses on the basic idea of optimal binary search Tree Technique to find the optimal solution (aspect) from a sentence among the several aspects; specifically, we perform the classification process appropriate to various features, which is used to identify the best one aspect among the several aspect terms in a sentence. In the proposed system, We have introduced optimal binary search Tree aspect level sentiment analysis technique. Through this we are getting best and efficient aspect level solution compare to General Tree. The main novelty of the paper is analyzing how aspect sentiment extracted from each word in a sentence through Dictionary based approach with help of the syntactic Patterns. It provides a well understanding of the appropriate role of every words to obtain optimal Aspect in an efficiency way.

Endogenous Dynamic Efficiency in the Intertemporal Optimization Models of Firm Behavior

Existing methods for the measurement of technical efficiency in the dynamic production models obtain it from the implied distance functions without making use of the information about intertemporal economic behavior in the estimation beyond an indirect appeal to duality. The main limitation of such an estimation approach is that it does not allow for the dynamic evolution of efficiency that is explicitly optimized by the firm. This paper introduces a new conceptualization of efficiency that directly enters the firm's intertemporal production decisions and is both explicitly costly and endogenously determined. We build a moment-based multiple-equation system estimation procedure that incorporates both the dynamic and static optimality conditions derived from the firm's intertemporal expected cost minimization. We operationalize our methodology using a modified version of a Bayesian Exponentially Tilted Empirical Likelihood adjusted for the presence of dynamic latent variables in the model, which we showcase using the 1960-2004 U.S. agricultural farm production data. We find that allowing for potential endogenous adjustments in efficiency over time produces significantly higher estimates of technical efficiency, which is likely due to inherent inability of the more standard exogenous-efficiency model to properly credit firms for incurring efficiency-improvement adjustment costs. Our results also suggest material improvements in efficiency over time at an about 2.6% average annual rate, which contrasts with near-zero estimates of the exogenous efficiency change.

Efficient and secure conditional access system for pay-TV systems

In Digital TV service, subscribers get the access of channels of their interest. In Conditional Access Systems (CAS), multimedia content is delivered securely using the Control Word (CW). To frequent update of the CW, a large number of messages are exchanged in the conventional key distribution systems. In this paper, a centralized key distribution scheme based on dynamicity of the groups and with freedom of channel package composition is proposed. Scheme has also efficient mechanism of load balancing at Group Controller (GC) side. To speed up the search of the channel package in the existing database, Optimal Binary Search Tree (OBST) data structure and Finite State Machine (FSM) are used. Scheme also provides the efficient leaving and joining mechanism for single user as well as batch users.

Better analysis of binary search tree on decomposable sequences

In their seminal paper Sleator and Tarjan (1985) [15], the authors conjectured that the splay tree is a dynamically optimal binary search tree (BST). In spite of decades of intensive research, the problem remains open. Perhaps a more basic question, which has also attracted much attention, is if there exists any dynamically optimal BST algorithm. One such candidate is ▪ which is a simple and intuitive BST algorithm Lucas (1988) [11]; Munro (2000) [12]; Demaine et al. (2009) [6]. Demaine et al. (2009) [6] showed a novel connection between a geometric problem and the binary search tree problem related to the above conjecture. However, there has been little progress in solving this geometric problem too.Since dynamic optimality conjecture in its most general form remains elusive despite much effort, researchers have studied this problem on special sequences. Recently, Chalermsook et al. (2015) [2] studied a type of sequences known as k-decomposable sequences in this context, where k parametrizes easiness of the sequence. Using tools from forbidden submatrix theory, they showed that ▪ takes n2O(k2) time on this sequence and explicitly raised the question of improving this bound.In this paper, we show that ▪ takes O(nlog⁡k) time on k-decomposable sequences. In contrast to the previous approach, ours is based on first principles. One of the main ingredients of our result is a new construction of a lower bound certificate on the performance of any algorithm. This certificate is constructed using the execution of ▪, and is more nuanced and possibly more flexible than the previous independent set certificate of Demaine et al. This result, which is applicable to all sequences, may be of independent interest and may lead to further progress in analyzing ▪ on k-decomposable as well as general sequences.

Statically Optimal Binary Search Tree Computation Using Non-Serial Polyadic Dynamic Programming on GPU's

Modern GPUs perform computation at a very high rate when compared to CPUs; as a result, they are increasingly used for general purpose parallel computation. Determining if a statically optimal binary search tree is an optimization problem to find the optimal arrangement of nodes in a binary search tree so that average search time is minimized. Knuth's modification to the dynamic programming algorithm improves the time complexity to O(n2). We develop a multiple GPU-based implementation of this algorithm using different approaches. Using suitable GPU implementation for a given workload provides a speedup of up to four times over other GPU based implementations. We are able to achieve a speedup factor of 409 on older GTX 570 and a speedup factor of 745 is achieved on a more modern GTX 1060 when compared to a conventional single threaded CPU based implementation.

A Novel Time Series Pattern Matching Model Combined with Ant Colony Optimization and Optimal Binary Search Trees Based Segmentation Approach

A Novel Time Series Pattern Matching Model Combined with Ant Colony Optimization and Optimal Binary Search Trees Based Segmentation Approach

Arboral satisfaction: Recognition and LP approximation

A point set P is arborally satisfied if, for any pair of points with no shared coordinates, the box they span contains another point in P. At SODA 2009, Demaine, Harmon, Iacono, Kane, and Pǎtrascu proved a connection between the longstanding dynamic optimality conjecture about binary search trees and the problem of finding the minimum-size arborally satisfied superset of a given 2D point set [1].We study two basic problems about arboral satisfaction. First, we develop two nontrivial algorithms to test whether a given point set is arborally satisfied. In 2D, both of our algorithms run in O(nlog⁡n) time, and one of them achieves O(n) runtime if the points are presorted; we also show a matching Ω(nlog⁡n) lower bound in the algebraic decision tree model. In d dimensions, our algorithm runs in O(dnlog⁡n+nlogd−1⁡n) time. Second, we study a natural integer linear programming formulation of finding the minimum-size arborally satisfied superset of a given 2D point set, which is equivalent to finding offline dynamically optimal binary search trees. Unfortunately, we conclude that the linear programming relaxation has large integrality gap, making it unlikely to find an approximation algorithm via this approach.

Property Testing on Product Distributions

The primary problem in property testing is to decide whether a given function satisfies a certain property or is far from any function satisfying it. This crucially requires a notion of distance between functions. The most prevalent notion is the Hamming distance over the uniform distribution on the domain. This restriction to uniformity is rather limiting, and it is important to investigate distances induced by more general distributions. In this article, we provide simple and optimal testers for bounded derivative properties over arbitrary product distributions . Bounded derivative properties include fundamental properties, such as monotonicity and Lipschitz continuity. Our results subsume almost all known results (upper and lower bounds) on monotonicity and Lipschitz testing over arbitrary ranges. We prove an intimate connection between bounded derivative property testing and binary search trees (BSTs). We exhibit a tester whose query complexity is the sum of expected depths of optimal BSTs for each marginal. Furthermore, we show that this sum-of-depths is also a lower bound. A technical contribution of our work is an optimal dimension reduction theorem for all bounded derivative properties that relates the distance of a function from the property to the distance of restrictions of the function to random lines. Such a theorem has been elusive even for monotonicity, and our theorem is an exponential improvement to the previous best-known result.

High Performance CGM-based Parallel Algorithms for the Optimal Binary Search Tree Problem

In this paper, the authors highlight the existence of close relations between the execution time, efficiency and number of communication rounds in a family of CGM-based parallel algorithms for the optimal binary search tree problem (OBST). In this case, these three parameters cannot be simultaneously improved. The family of CGM (Coarse Grained Multicomputer) algorithms they derive is based on Knuth's sequential solution running in time and space, where n is the size of the problem. These CGM algorithms use p processors, each with local memory. In general, the authors show that each algorithms runs in with communications rounds. is the granularity of their model, and is a parameter that depends on and . The special case of yields a load-balanced CGM-based parallel algorithm with communication rounds and execution steps. Alternately, if , they obtain another algorithm with better execution time, say , the absence of any load-balancing and communication rounds, i.e., not better than the first algorithm. The authors show that the granularity has a crucial role in the different techniques they use to partition the problem to solve and study the impact of each scheduling algorithm. To the best of their knowledge, this is the first unified method to derive a set of parameter-dependent CGM-based parallel algorithms for the OBST problem.

Tối ưu hoá cây nhị phân một chiều với thông tin chứa ở các đỉnh trong

The notion of a search tree plays an important role in computer science, especially in the theory of data structures. For that reason we can find many papers concerned with the theory of search trees in literature. After having read these papers we noticed that, above all, questions of the optimal construction and inductive generation of search trees are studied, where equivalent transformation of search trees are often used. In this paper we intend to further develop the conception of search trees in [4]. A prove of a theorem which shows that each binary search tree can be uniquely transformed into optimal binary search tree by using axiom schemes and the rules are introduced in this paper.

Parallel dynamic programming for solving the optimal search binary tree problem on CGM

This paper presents an efficient coarse grain multicomputer CGM parallel algorithm for the cost of the optimal binary search tree problem OBST problem. In the previous best CGM algorithm for this problem, the size of the local memory of each processor is not bounded. And even, in the worst case, its running time reduces to the one of Knuth's sequential algorithm for the same problem, bringing no gain by parallelising OBST. Our CGM algorithm uses Knuth's sequential algorithm for local computations to perform in time as well as the best previous CGM algorithm. Moreover, each processor can process at most two blocks and thus avoid the severe drawback raised above of the previous work.

OBST-based segmentation approach to financial time series

Financial time series data are large in size and dynamic and non-linear in nature. Segmentation is often performed as a pre-processing step for locating technical patterns in financial time series. In this paper, we propose a segmentation method based on Turning Points (TPs). The proposed method selects TPs from the financial time series in question based on their degree of importance. A TP's degree of importance is calculated on the basis of its contribution to the preservation of the trends and shape of the time series. Algorithms are also devised to store the selected TPs in an Optimal Binary Search Tree (OBST) and to reconstruct the reduced sample time series. Comparison with existing approaches show that the time series reconstructed by the proposed method is able to maintain the shape of the original time series very well and preserve more trends. Our approach also ensures that the average retrieval cost is kept at a minimum.

A STATIC OPTIMALITY TRANSFORMATION WITH APPLICATIONS TO PLANAR POINT LOCATION

Over the last decade, there have been several data structures that, given a planar subdivision and a probability distribution over the plane, provide a way for answering point location queries that is fine-tuned for the distribution. All these methods suffer from the requirement that the query distribution must be known in advance. We present a new data structure for point location queries in planar triangulations. Our structure is asymptotically as fast as the optimal structures, but it requires no prior information about the queries. This is a 2-D analogue of the jump from Knuth's optimum binary search trees (discovered in 1971) to the splay trees of Sleator and Tarjan in 1985. While the former need to know the query distribution, the latter are statically optimal. This means that we can adapt to the query sequence and achieve the same asymptotic performance as an optimum static structure, without needing any additional information.

The Knuth-Yao quadrangle-inequality speedup is a consequence of total monotonicity

There exist several general techniques in the literature for speeding up naive implementations of dynamic programming. Two of the best known are the Knuth-Yao quadrangle inequality speedup and the SMAWK algorithm for finding the row-minima of totally monotone matrices. Although both of these techniques use a quadrangle inequality and seem similar, they are actually quite different and have been used differently in the literature. In this article we show that the Knuth-Yao technique is actually a direct consequence of total monotonicity. As well as providing new derivations of the Knuth-Yao result, this also permits to solve the Knuth-Yao problem directly using the SMAWK algorithm. Another consequence of this approach is a method for solving online versions of problems with the Knuth-Yao property. The online algorithms given here are asymptotically as fast as the best previously known static ones. For example, the Knuth-Yao technique speeds up the standard dynamic program for finding the optimal binary search tree of n elements from Θ( n 3 ) down to O ( n 2 ), and the results in this article allow construction of an optimal binary search tree in an online fashion (adding a node to the left or the right of the current nodes at each step) in O ( n ) time per step.