Optimal Binary Search Trees with Costs Depending on the Access Paths

Optimal binary search trees with costs depending on the access paths

We design an efficient sublinear time parallel construction of optimal binary search trees. The efficiency of the parallel algorithm corresponds to its total work (the product time × processors). Our algorithm works in O(n1−ɛ log n) time with the total work O(n2−2ɛ), for an arbitrarily small constant 0 < e ≤ 1/2. This is optimal within a factor n2ɛ with respect to the best known sequential algorithm given by Knuth, which needs only O(n2) time due to a monotonicity property of optimal binary search trees, see [6]). It is unknown how to explore this property in an efficient NC construction of binary search trees. Here we show that it can be effectively used in sublinear time parallel computation. Our improvement also relies on the use (in independently processed small subcomputations) of the parallelism present in Knuth's algorithm. The best known sublinear time algorithms for the construction of binary search trees (as an instance of a more general problem) have O(n3) work for time larger than n3/4, see [3] and [7]. For time √n these algorithms need n4 work, while our algorithm needs for this time only n3 work, thus improving the known algorithms by a linear factor. Also if time is O(n1−ɛ) and e is very small our improvement is close to O(n). Such improvement is similar to the one implied by the monotonicity property in sequential computations (from n3 sequential time for a more general dynamic programming problem to n2 time for the special case of optimal binary search trees).

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 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.

Growing nearly optimal binary search trees

A sublinear parallel algorithm for some dynamic programming problems

OBST-based segmentation approach to financial time series

Following work of Stroud and Saeger (Proceedings of ISI, Springer Verlag, New York, 2006) and Anand et al. (Proceedings of Computer, Communication and Control Technologies, 2003), we formulate a port of entry inspection sequencing task as a problem of finding an optimal binary decision tree for an appropriate Boolean decision function. We report on new algorithms for finding such optimal trees that are more efficient computationally than those presented by Stroud and Saeger and Anand et al. We achieve these efficiencies through a combination of specific numerical methods for finding optimal thresholds for sensor functions and two novel binary decision tree search algorithms that operate on a space of potentially acceptable binary decision trees. The improvements enable us to analyze substantially larger applications than was previously possible. © 2011 Wiley Periodicals, Inc. Naval Research Logistics, 2011

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.

Optimum binary search trees

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

In the past ten years, there have been a number of data structures that, given a distribution of planar point location queries, produce a planar point location data structure that is tuned for the provided distribution. These structures all suffer from the requirement that the query distribution be provided in advance. For the problem of point location in a triangulation, a data structure is presented that performs asymptotically as well as these structures, but does not require the distribution to be provided in advance. This result is the 2-d analogue of the jump from the optimum binary search trees of Knuth in 1971 which required that the distribution be provided, to the splay trees of Sleator and Tarjan in 1985 where in the static optimality theorem it was proven that splay trees had the same asymptotic performance of optimum search trees without being provided the probability distribution.

Optimal classification trees with leaf-branch and binary constraints

In this paper, the problem of traveling disjoint segment sequences in the plane will be studied. The goal is to find the shortest path from the start point, then visiting each segment in the given order, and finally to the target point. By adopting the techniques of division of convex chain, combination optimization and binary search tree, we design a fast algorithm with the O(nlog2n) time complexity to solve it, denoted by BST algorithm, where n is the total number of all segments, and we introduce the main techniques used in this paper in detail. Furthermore, we generate a large amount of test data to test BST algorithm, and compare the efficiency of BST algorithm and Rubber-band algorithm, which is the better solution to this problem. The results show that BST algorithm is superior to Rubber-band algorithm, and it is the optimal algorithm for visiting the disjoint segment sequences so far.

Indexing technology is hardly studied when data access is skew in real time system. On one hand indexing considers the probability of data accessed, on the other hand indexing needs to satisfy data real-time constraints, and meanwhile, the time complexity of arithmetic computing for indexing must be low. The nearly optimal binary index tree NOBIT is based on the idea of static optimal search tree. We introduce real-time weight when it deals with nodes probabilities weight so as to quickly construct a binary index tree with nearly optimal performance. The search process of NOBIT is similar to binary search, and its average time complexity is O(logN). The NOBIT considers the probability of accessed data and data which are accessed frequently are put into the front part of broadcast sequence so that average tuning time is shortened. Meanwhile, the NOBIT takes the real-time constraints of accessed data into account and data with rigid time constraints also are put into the front part of broadcast sequence so that success rate of data requested is improved.