Showing papers on "Interval tree published in 1987"

Set operations on polyhedra using binary space partitioning trees

Abstract: We introduce a new representation for polyhedra by showing how Binary Space Partitioning Trees (BSP trees) can be used to represent regular sets. We then show how they may be used in evaluating set operations on polyhedra. The BSP tree is a binary tree representing a recursive partitioning of d-space by (sub-)hyperplanes, for any dimension d. Their previous application to computer graphics has been to organize an arbitrary set of polygons so that a fast solution to the visible surface problem could be obtained. We retain this property (in 3D) and show how BSP trees can also provide an exact representation of arbitrary polyhedra of any dimension. Conversion from a boundary representation (B-reps) of polyhedra to a BSP tree representation is described. This technique leads to a new method for evaluating arbitrary set theoretic (boolean) expressions on B-reps, represented as a CSG tree, producing a BSP tree as the result. Results from our language-driven implmentation of this CSG evaluator are discussed. Finally, we show how to modify a BSP tree to represent the result of a set operation between the BSP tree and a B-rep. We describe the embodiment of this approach in an interactive 3D object design program that allows incremental modification of an object with a tool. Placement of the tool, selection of views, and performance of the set operation are all performed at interactive speeds for modestly complex objects.

Computing on a free tree via complexity-preserving mappings

Abstract: The relationship between linear lists and free trees is studied. We examine a number of well-known data structures for computing functions on linear lists and show that they can be canonically transformed into data structures for computing the same functions defined over free trees. This is used to establish new upper bounds on the complexity of several query-answering problems.

On a Nonuniform Random Recursive Tree

Abstract: Publisher Summary This chapter discusses a nonuniform random recursive tree. A tree is a connected graph which has no cycles. Tree R with n vertices labelled 1, 2, …, n is a recursive tree if for each k such that 2≤ k ≤n the labels of vertices in the unique path from the first vertex to the k th vertex of a tree form an increasing subsequence of {1,2, . . ., n}. A random recursive tree with n vertices (n-RRT) is a tree picked at random from the family of all recursive trees with n vertices. The chapter deals with a random recursive tree such that the probability of joining a new vertex to the vertex i depends only on the degree of vertex i . In addition, the chapter considers two special cases of the random recursive tree.

Alternative methods for the reconstruction of trees from their traversals

Abstract: It is well-known that given the inorder traversal of a binary tree's nodes, along with either one of its preorder or postorder traversals, the original binary tree can be reconstructed using a recursive algorithm. In this short note we provide a short, elegent, iterative solution to this classical problem.

Pattern matching method for tree structured data

Abstract: In order to highly speed up the pattern matching of tree structured data in a logic programming language, the priority order is set when the data owned by the individual nodes of a tree structure are to be transversely sought, and the tree structured data are expressed in a vector type, in which they are arranged in that priority order, so that they are compared consecutively from the head for each element of the vector.

Binary search trees for vector quantisation

Abstract: This paper presents a data structure based on the k-d binary tree which substantially reduces the search complexity of a full search vector quantiser with negligible degradation in signal-to-noise ratio. The search complexity is k + O(\logN) rather than N for a codebook of dimension k and size N. Special features of the structure are (1) the use of a rotational transform prior to encoding and (2) the computational efficiency of the design algorithm due to the simple structure of the k-d tree.

Left distance binary tree representations

Abstract: We present a new scheme for representing binary trees. The scheme is based on rotations as a previous scheme of Zerling. In our method the items of a representation have a natural geometric interpretation, and the algorithms related to the method are simple. We give an algorithm for enumerating all the representations for trees onn nodes, and an algorithm for building the tree corresponding to a given representation.

Application of binary space partitioning trees to geometric modeling and ray-tracing

Abstract: The Binary Space Partitioning Tree (BSP Tree) is a binary tree used to represent an organization of continuous space by recursive subdivision. The BSP tree was initially introduced to organize a set of polygons so that visible surface renderings could be produced from an arbitrary viewing position. This thesis extends the use of the BSP tree in two ways. The first, constituting the bulk of the work, uses the BSP tree to model polyhedral solids. A number of algorithms relating to this representation of a polyhedral set are introduced: determining the boundary of the set, determining volume and center of mass, evaluating set operations (union, intersection, difference), classification of points, line segments, and polygons with respect to the set, constructing the BSP tree from the boundary of the set, and constructing a boundary representation of a set represented by a BSP tree. The BSP tree provides a unified approach to the basic problems encountered in constructing an interactive geometric modeling system. These are: representing polyhedra, rendering, geometric search, property calculation, and set operation evaluation. In previous work, these issues have been addressed with largely independent and unrelated algorithms. By solving these problems with a set of closely related algorithms based on the BSP tree, the task of constructing such a system is made easier. The other new use for the BSP tree discussed is its use in ray-tracing. Ray-tracing can produce very realistic images, but is plagued by the requirement of large amounts of computation. The problem of finding the closest intersection a ray with a collection of objects, the basic operation in ray-tracing, is essentially a searching problem. The BSP tree is used to organize the objects so that the search can be made efficient.

Random Tree Models in the Analysis of Algorithms

Abstract: We present three classes of random tree models tbat occur in the average case analysis of a variety of computer algorithms including symbolic manipulation algorithms, rompilling, comparison based searching and sorting, digital retrieval techniques, ble systems and communication protocols. Each model carries a coherent set of algebraic and analytic techniques, whicb we illustrate by reviewing a few characteristic examples.

Binary search on a tape.

Abstract: Given n records stored alphabetically on a tape, any comparison search procedure can be characterized by a binary tree. The complete binary tree (binary search) uses the minimum number of comparisons but not the minimum number of movements. The linear binary tree (sequential search) uses the minimum number of movements but not the minimum number of comparisons. A tape-optimal tree is a tree which minimizes the total cost of comparisons and movements. The tape-optimal tree is a “hybrid” of the linear tree and the complete binary tree, and is characterized for arbitrary n.

Decomposing an N-ary relation into a tree of binary relations

Abstract: We present an efficient algorithm for decomposing an n-ary relation into a tree of binary relations, and provide an efficient test for checking whether or not the tree formed represents the relation. If there exists a tree-decomposition, the algorithm is guaranteed to find one, otherwise, the tree generated will fail the test, then indicating that no tree decomposition exist. The unique features of the algorithm presented in this paper, is that it does not apriori assume any dependencies in the initial relation, rather it derives such dependencies from the bare relation instance.

A Parallel Tree Contraction Algorithm

Abstract: A simple reduction from the tree contraction problem to the list ranking problem is presented. The reduction takes O$(\log n)$ time for a tree with $n$ nodes, using O$(n / \log n)$ EREW processors. Thus tree contraction can be done as efficiently as list ranking. A broad class of parallel tree computations to which the tree contraction techniques apply is described. This subsumes earlier characterizations. Applications to the computation of certain properties of cographs are presented in some detail.

On enumerating tree permutations in natural order

Abstract: An efficient algorithm for enumerating all tree permutations of n integers in the natural order is presented. The enumeration problem is solved by considering that a tree permutation hLR is simply a linearized representation of the corresponding binary tree, such that h is the node value and L and R are its left and right subtrees, respectively. The best-case, average-case and worst-case time-complexities of the enumeration algorithm are 0(1)0 (3) and 0(n) respectively, whereas its space-complexity is 0(n). Furthermore, it is shown that the conversion from a tree permutation to the conventional representation of a binary tree using records (nodes) and pointers can be accomplished in 0(n) units of time.

Structural processing of waveforms as trees

Abstract: Waveforms may be represented symbolically such that their underlying, global structural composition is emphasized. One such symbolic representation is the relational tree. The relational tree is a computer data structure that describes the relative size and placement of peaks and valleys in a waveform. Researchers have developed various distance measures which serve as tree metrics. A tree metric defines a tree space. We are able to cluster groups of trees by their proximity in a tree space. Linear discriminants are used to reduce vector space dimensionality and to improve cluster performance. A tree transformation operating on a regular tree language accomplishes this same goal in a tree space. Under certain restrictions, relational trees form a regular tree language. Combining these concepts yields a waveform recognition system. This system recognizes waveforms even when they have undergone a monotonic transformation of the time axis. The system performs well with high signal to noise ratios, but further refinements are necessary for a working waveform interpretation system.

Optimal attribute ranking in multiple attribute tree

Abstract: In a Multiple Attribute Tree (MAT) based data organization, the average case response to a specific range query depends on the structural properties of MAT. These structural properties depend very much on the interrelationships among the data elements. Efficiency in searching can be achieved by exploiting the data properties in the construction of MAT. The order or ranking of attributes is a key factor in deciding the profile of the MAT for given data. In this paper, we estimate the average cost of a range query in MAT based data organization. We then prove that the average performance can be improved by ranking the attributes in such a way that the average size of the filial sets decreases towards the lower levels of the tree structure.