Constant time BSR solutions to parenthesis matching, tree decoding,and tree reconstruction from its traversals

Recently Akl et al. introduced a new model of parallel computation, called BSR (broadcasting with selective reduction) and showed that it is more powerful than any CRCW PRAM and yet requires no more resources for implementation than even EREW PRAM. The model allows constant time solutions to sorting, parallel prefix and other problems. In this paper, we describe constant time solutions to the parenthesis matching, decoding binary trees in bitstring representation, generating next tree shape in B-order, and the reconstruction of binary trees from their traversals, using the BSR model. They are the first constant time solutions to mentioned problems on any model of computation. The number of processors used is equal to the input size, for each problem. A new algorithm for sorting integers is also presented     

The tree MVA algorithm

A new algorithm to solve product form queueing networks, especially those with large numbers of centers and chains, is presented. This algorithm is a Tree version of Mean Value Analysis (MVA). Tree MVA is analogous to the Tree version of Convolution developed by Lam and Lien. Like Tree Convolution, Tree MVA allows exact solution of large networks which are intractable with previous sequential algorithms. As with the sequential versions of Convolution and MVA, Tree MVA has better numerical properties than Tree Convolution. Further, Tree MVA avoids the computational complexity of sequential MVA in networks with several queue dependent centers. Thus, we consider Tree MVA to be the best algorithm for general product form networks.     

The hierarchical degree-of-visibility tree

In this paper, we present a novel structure called the hierarchical degree-of-visibility tree (HDoV-tree) for visibility query processing in visualization systems. The HDoV-tree builds on and extends the R-tree such that 1) the search space is pruned based on the degree of visibility of objects and 2) internal nodes store level-of-details (LoDs) that represent a collection of objects in a coarser form. We propose two tree traversal algorithms that balance performance and visual fidelity, explore three storage structures for the HDoV-tree, and develop novel caching techniques for disk-based HDoV-tree. We implemented the HDoV-tree in a prototype walkthrough system called VISUAL. Our experimental study shows that VISUAL can lead to high frame rates without compromising visual fidelity.     

The repeater tree construction problem

A tree-like substructure on a computer chip whose task is to carry a signal from a source circuit to possibly many sink circuits and which consists only of wires and so-called repeater circuits is called a repeater tree. We present a mathematical formulation of the optimization problems related to the construction of such repeater trees. Furthermore, we prove theoretical properties of a simple iterative procedure for these problems which was successfully applied in practice.     

The use of tree derivatives and a sample support parameter for inferring tree systems

Tree systems have been studied in the theoretical setting as an extension of finite automata. They have been found useful in the practical domain when applied to syntactic pattern recognition. The practical applications of tree systems have motivated the examination of inference techniques for tree grammars and tree automata. In this paper we present a tree automaton inference algorithm which incorporates three concepts-tree derivatives, grammatical expansion, and inferential strength. Tree derivatives are used for comparing tree forms. Grammatical expansion is a feedback mechanism which effectively enlarges the user sample. An inferential strength parameter is input by the user to indicate the amount of support required from the sample for inferences. The algorithm is also applied for inferring finite-state machines. Finally, we address an open problem posed by Joshi and Levy by demonstrating the use of our algorithm for the design of programming languages.     

The copying power of one-state tree transducers

One-state deterministic top-down tree transducers (or, tree homomorphisms) cannot handle “prime copying,” i.e., their class of output (string) languages is not closed under the operation L → {$(w$)^f(n) ∥ w?L, f(n) ? 1}, where f is any integer function whose range contains numbers with arbitrarily large prime factors (such as a polynomial). The exact amount of nonclosure under these copying operations is established for several classes of input (tree) languages. These results are relevant to the extended definable (or, restricted parallel level) languages, to the syntax-directed translation of context-free languages, and to the tree transducer hierarchy.     

The optimal strategy of a tree classifier

This paper deals with the decision rules of a tree classifier for performing the classification at each nonterminal node, under the assumption of complete probabilistic information. For given tree structure and feature subsets to be used, the optimal decision rules (strategy) are derived which minimize the overall probability of misclassification. The primary result is illustrated by an example.     

The nearest common ancestor in a dynamic tree

Summary We consider the problem of determining the nearest common ancestor of two given nodesx andy (denoted bynca(x,y)) in a dynamic arbitrary treeT. We present an implementation ofT by apointer machine which needslinear space, performsm arbitrary insertions and deletions in the initially empty tree T in timeO(m) and a query aboutnca(x,y) can be answeredon-line in timeO(log(min {depth(x), depth(y)}) + α(k,k)), where the second factor is amortized overk queries, α is a functional inverse of Ackermann’s function anddepth(x) the distance from nodex to the root ofT.     

The time complexity of typechecking tree-walking tree transducers

Tree-walking tree transducers can be typechecked in double exponential time. More generally, compositions of k tree-walking tree transducers can be typechecked in (k + 1)-fold exponential time. Consequently k-pebble tree transducers, which form a model of XML transformations and XML queries, can be typechecked in (k + 2)-fold exponential time. The results hold for both ranked and unranked trees.     

The nearest common ancestor in a dynamic tree

Summary We consider the problem of determining the nearest common ancestor of two given nodes x and y (denoted by nca(x, y)) in a dynamic arbitrary tree T. We present an implementation of T by a pointer machine which needs linear space, performs m arbitrary insertions and deletions in the initially empty tree T in time O(m) and a query about nca(x, y) can be answered on-line in time O(log(min{depth(x), depth(y))+(k,k))}, where the second factor is amortized over k queries, is a functional inverse of Ackermann's function and depth(x) the distance from node x to the root of T.     

The optimal binary search tree for Andersson's search algorithm

Andersson [1] presented a search algorithm for binary search trees that uses only two-way key comparisons by deferring equality comparisons until the leaves are reached. The use of a different search algorithm means that the optimal tree for the traditional search algorithm, which has been shown to be computable inO(n ²) time by Knuth [3], is not optimal with respect to the different search algorithm. This paper shows that the optimal binary search tree for Andersson's search algorithm can be computed inO(nlogn) time using existing algorithms for the special case of zero successful access frequencies, such as the Hu-Tucker algorithm [2].