Showing papers on "Interval tree published in 1986"

Optimal simulations of tree machines

Abstract: Universal networks offer the advantage that they can execute programs written for simpler architectures without significant run-time overhead. In this paper we investigate simulations of tree machines; the fact that divide-and-conquer algorithms are programmed naturally on trees motivates our investigation. Among various proposals for parallel computing the boolean hypercube has emerged as a particularly versatile network. It is well known that programs for multidimensional grid machines, for example, can be executed on a hypercube with no communications overhead by embedding the grid as a subgraph of the hypercube. Our first result is that a program for any tree machine can be executed on the hypercube with constant overhead. More precisely, every cycle of a synchronous binary tree can be simulated in O(1) cycles on a hypercube, independent of the shape of the tree. The algorithm to embed the tree within the hypercube runs in polynomial time. We also give efficient simulations of arbitrary binary trees on the complete binary tree, the FFT and shuffle-exchange networks. It is natural to ask if any sparse network can simulate every binary tree efficiently. Somewhat surprisingly, we construct a universal bounded-degree network on N nodes for which every N node binary tree is a spanning tree. In other words, every binary tree can be simulated on our universal network with no overhead. This improves previous bounds on the sizes of universal graphs for trees.

Estimating the reliability of evolutionary trees.

Abstract: Six protein sequences from the same 11 mammalian taxa were used to estimate the accuracy and reliability of phylogenetic trees using real, rather than simulated, data. A tree comparison metric was used to measure the increase in similarity of minimal trees as larger, randomly selected subsets of nucleotide positions were taken. The ratio of the observed to the expected number of incompatibilities for each nucleotide position (character) is a good predictor of the number of changes required at that position on the minimal (most-parsimonious) tree. This allows a higher weighting of nucleotide positions that have changed more slowly and should result in the minimal length tree converging to the correct tree as more sequences are obtained. An estimate was made of the smallest subset of trees that need to be considered to include the actual historical tree for a given set of data. It was concluded that it is possible to give a reasonable estimate of the reliability of the final tree, at least when several sequences are combined. With the present data, resolving the rodent-primate-lagomorph (rabbit) trichotomy is the least certain aspect of the final tree, followed then by establishing the position of dog. In our opinion, it is unreasonable to publish an evolutionary tree derived from sequence data without giving an idea of the reliability of the tree.

Optimal variable weighting for ultrametric and additive tree clustering

Abstract: A method is developed which for a given objects by variables data matrix estimates weighted inter-object distances that are optimally suited for either an ultrametric or an additive tree representation The effectiveness of the method is demonstrated on two synthetic data sets having a known tree structure and on one real data set In the final section, some possible extensions of the present method are discussed

Tree rebalancing in optimal time and space

Abstract: A simple algorithm is given which takes an arbitrary binary search tree and rebalances it to form another of optimal shape, using time linear in the number of nodes and only a constant amount of space (beyond that used to store the initial tree). This algorithm is therefore optimal in its use of both time and space. Previous algorithms were optimal in at most one of these two measures, or were not applicable to all binary search trees. When the nodes of the tree are stored in an array, a simple addition to this algorithm results in the nodes being stored in sorted order in the initial portion of the array, again using linear time and constant space.

Lower bounds for accessing binary search trees with rotations

Abstract: We describe two methods for obtaining lower bounds on the cost of accessing a sequence of nodes of a symmetrically ordered binary search tree, where rotations can be done on the tree. The bounds apply to offline as well as online algorithms.

Editorial method in display of tree

Abstract: PURPOSE: To simplify a longitudinal search for tree structure data by using a stack storing the name of node and the number of branches. CONSTITUTION: A window 201 is set inside a display screen 200 and inside the window 201, a tree structure 202 consisting of the nodes and branches is displayed. This example shows that the structure consists of a root 203 which is the node in the highest order and next the head 204, the text 205 and the supplement 206 of a document which are the nodes under the root 203. At this time, the root 203 in the highest order and the head 204, for instance, are the nodes which shows a parent-child relation and the root 203 and the head 204 are connected with the branch 217 shown with a segment. A title 207 and an author name 204 are the nodes which are under the head 204 in this example. The display co-ordinate for the tree structure is obtained by the longitudinal search based on the stack. COPYRIGHT: (C)1988,JPO&Japio

Cyclic tree traversal

Abstract: Programs which process tree structures usually cannot handle cyclic trees. This paper describes some new, very simple, and efficient algorithms for detecting and traversing cyclic trees. Traversed structures do not have to be modified. Tail recursion optimisation can be used, which reduces stack requirements greatly. The over head for non-cyclic structures is very small.

Tree information: a split screen system

Abstract: A split-screen, main frame computerized tree infor- mation system has been developed in Windsor, Ontario, to in- tegrate a manual tree inventory and work record system with a manual current work order system. The computer system pro- vides tree information such as diameter, height, tree condition, location, origin year and survey year, for individual municipal addresses. This information is shown on a tree inventory file which is linked to the current work order and work history file. Simplified parameters were utilized for easy updating and infor- mation retrieval by municipal staff. A tree code was developed using a genus/species four-letter code from common tree name abbreviations. Tree locations were indicated by a floating tree ordering system in relation to the curb and posi- tion on the property, rather than using coordinates for each tree. The parameters provided on the tree inventory file allows the sorting of data to produce tree profiles by planning districts. Management of the tree population is thusly up- graded, enhanced, and systematized.

Tree resolution and generalized semantic tree

Abstract: A resolution proof or a derivation of the empty clause from a set of clauses S = {C1, C2, …, Ck} is called a tree resolution if no clause Ci is used in more than one resolvent. We show that an unsatisfiable set of clauses S has a tree resolution proof if and only if there is a general semantic tree for S in which no clause appears in more than one terminal node. As an important application of this result, we derive a simple algorithm for obtaining a tree resolution proof, if one exists. The tree resolution proofs are important because they allow us to obtain the shortest “explanation”.

Time and Space

Abstract: A simple algorithm is given which takes an arbitrary binary search tree and rebalances it to form another of optimal shape, using time linear in the number of nodes and only a constant amount of space (beyond that used to store the initial tree). This algorithm is therefore optimal in its use of both time and space. Previous algorithms were optimal in at most one of these two measures, or were not applicable to all binary search trees. When the nodes of the tree are stored in an array, a simple addition to this algorithm results in the nodes being stored in sorted order in the initial portion of the array, again using linear time and constant space.