Showing papers on "Interval tree published in 1992"

A Simple Method for Estimating and Testing Minimum-Evolution Trees

Abstract: A simple method for estimating and testing phylogenetic trees under the principle of minimum evolution (ME) is presented. The basic procedure of this method is first to obtain the neighbor-joining (NJ) tree by Saitou and Nei’s method and then to search for a tree with the minimum value of the sum (S) of branch lengths by examining all trees that are closely related to the NJ tree. Once the ME tree is identified, a statistical test is conducted for the difference in S between this tree and other closely related trees. The mathematical method required for conducting this test is developed by using the least-squares approach. Computer simulation has shown that this method identifies the correct tree with a high probability, as long as the number of nucleotides examined is sufficiently large. It has also been shown that the topology of the NJ tree is almost always identical with that of the ME tree. A method for obtaining least-squares estimates (and their standard errors) of branch lengths for a given topology is also presented. This method can be used for testing the reliability of the branching pattern of the ME tree. However, the statistical test of S values is more powerful in rejecting incorrect trees than is the branch-length test or bootstrapping. Furthermore, both a mathematical method for computing the number of trees with a given value of topological difference from the NJ tree and a computer algorithm for identifying all the topologies are developed.

Fast K-dimensional tree algorithms for nearest neighbor search with application to vector quantization encoding

Abstract: Fast search algorithms are proposed and studied for vector quantization encoding using the K-dimensional (K-d) tree structure. Here, the emphasis is on the optimal design of the K-d tree for efficient nearest neighbor search in multidimensional space under a bucket-Voronoi intersection search framework. Efficient optimization criteria and procedures are proposed for designing the K-d tree, for the case when the test data distribution is available (as in vector quantization application in the form of training data) as well as for the case when the test data distribution is not available and only the Voronoi intersection information is to be used. The criteria and bucket-Voronoi intersection search procedure are studied in the context of vector quantization encoding of speech waveform. They are empirically observed to achieve constant search complexity for O(log N) tree depths and are found to be more efficient in reducing the search complexity. A geometric interpretation is given for the maximum product criterion, explaining reasons for its inefficiency with respect to the optimization criteria. >

Classification trees with neural network feature extraction

Abstract: The ideal use of small multilayer nets at the decision nodes of a binary classification tree to extract nonlinear features is proposed. The nets are trained and the tree is grown using a gradient-type learning algorithm in the multiclass case. The method improves on standard classification tree design methods in that it generally produces trees with lower error rates and fewer nodes. It also reduces the problems associated with training large unstructured nets and transfers the problem of selecting the size of the net to the simpler problem of finding a tree of the right size. An efficient tree pruning algorithm is proposed for this purpose. Trees constructed with the method and the CART method are compared on a waveform recognition problem and a handwritten character recognition problem. The approach demonstrates significant decrease in error rate and tree size. It also yields comparable error rates and shorter training times than a large multilayer net trained with backpropagation on the same problems. >

Hierarchical Steiner tree construction in uniform orientations

Abstract: A hierarchical approach to Steiner tree construction in lambda -geometry is proposed. The algorithm runs in time O(n log n) and the length of the constructed tree is at most ( sigma /cos( pi /2 lambda )) times (for lambda =2, 3/2 times) the length of the optimal Steiner tree where n is the cardinality of the point set and it was recently proved that sigma is (2/ square root 3). How to trade off between the running time of the algorithm and the length of the produced Steiner tree is shown. Given enough time, an optimal Steiner tree will be obtained. The algorithm is extended to construct a Steiner tree of a set of subtrees (i.e., partial trees) and runs in O( lambda N log N) time, where N is the total number of edges of the subtrees. >

On Spanning Trees with Low Crossing Numbers

Abstract: Every set S of n points in the plane has a spanning tree such that no line disjoint from S has more than O(√n) intersections with the tree (where the edges are embedded as straight line segments). We review the proof of this result (originally proved by Bernard Chazelle and the author in a more general setting), point at some methods for constructing such a tree, and describe some algorithmic and combinatorial applications.

Generalized Digital Trees and Their Difference—Differential Equations

Abstract: Consider a tree partitioning process in which n elements are split into b at the root of a tree (b a design parameter), the rest going recursively into two subtrees with a binomial probability distribution. This extends some familiar tree data structures of computer science like the digital trie and the digital search tree. The exponential generating function for the expected size of the tree satisfies a difference-differential equation of order b, The solution involves going to ordinary (rather than exponential) generating functions, analyzing singularities by means of Mellin transforms and contour integration. The method is of some general interest since a large number of related problems on digital structures can be treated in this way via singularity analysis of ordinary generating functions. 0 1992 John Wiley & Sons, Inc.

Classification trees with neural network feature extraction

Abstract: The use of small multilayer nets at the decision nodes of a binary classification tree to extract nonlinear features is proposed. This approach exploits the power of tree classifiers to use appropriate local features at the different levels and nodes of the tree. The nets are trained and the tree is grown using a gradient-type learning algorithm in conjunction with a heuristic class aggregation algorithm. The method improves on standard classification tree design methods in that it generally produces trees with lower error rates and fewer nodes. It also provides a structured approach to neural network classifier design which reduces the problem associated with training large unstructured nets, and transfers the problem of selecting the size of the net to the simpler problem of finding the right size tree. Example concern waveform and handwritten character recognition. >

Locating information in an unsorted database utilizing a B-tree

Abstract: A binary search tree is created having a plurality of linked tree nodes, each of which store a key generated using a predetermined hashing function for each record in a database based upon a predetermined set of fields of each record. Tree nodes consist of such keys and the address of the corresponding record. The keys are arranged on the tree relative to the numerical value of keys previously entered in the tree. To locate a particular record, the key is generated for the record and the tree is traversed to locate a tree node having an identical key value. The address stored with the matching node in the tree is utilized to access the record.

On optimal interconnections

Abstract: Many applications require algorithms for determining optimal interconnections. This dissertation centers on new geometric formulations and approximation algorithms for optimizing interconnection objectives which are of particular interest in the design of high-performance VLSI systems. These formulations include Steiner trees, pathlength-balanced trees, bounded-radius trees, and prescribed-width paths; we also address the closely related question of efficiently testing physical interconnections. For most cases, we have new, best-known results, and in all cases we have empirically demonstrated significant improvements over the best previous methods. We give the best-performing rectilinear Steiner tree heuristic to date: the algorithm has worst-case performance ratio strictly less than 3/2 times optimal, settling a long-standing open problem. We also give a class of instances which are pathological for virtually all existing Steiner tree heuristics in the literature, thus disproving several conjectures and claimed performance bounds. We propose a matching-based method for pathlength-balanced trees: the construction yields near-zero average pathlength skew while maintaining small total tree cost. To address a separate objective, we also offer the first general formulation of performance-driven routing, allowing a smooth tradeoff of tree cost for tree radius. Our algorithm melds the two classical constructions of the minimum spanning tree and the shortest paths tree, and has worst-case performance bounded by a constant times optimal with respect to both tree cost and tree radius. Motivated by recent circuit testing applications, we formulate connectivity testing as a problem of tree verification via k-probes. We present linear-time algorithms which compute a minimum probe set achieving complete coverage of both edge and node fault classes. Actual testing demands the efficient scheduling of probe operations: we show that this entails a special type of metric traveling salesman optimization, and we give provably good heuristics. Finally, we address a fundamental problem in routing and path planning: determining a minimum-cost path of prescribed width which connects a given source-destination pair in an arbitrarily costed region. We give the first known polynomial-time algorithm for this problem, and extend our approach to solve a discrete version of the classical Plateau problem on minimal surfaces.

Hybrid backtrack/lookahead search technique for constraint-satisfaction problems

Abstract: A method of solving a constraint-satisfaction problem with a data processor includes the steps of (a) providing a search tree structure (10) representing a plurality (N) of variables (X), the search tree structure having a plurality of levels; (b) searching (L) shallow levels of the search tree structure by employing a backtrack search method wherein (L) is less than or equal to a specified value H; and (c) searching (M) remaining, deeper, levels of the search tree structure by employing a lookahead search method. The step of searching (L) shallow levels of the search tree structure includes a step of binding a set of X 1 through X H variables each to an element from its domain such that no constraints are violated. The step of searching (M) remaining, deeper, levels of the search tree structure includes the steps of, given the bindings for the set of variables X 1 through X H , determining for each variable X i , H

A spanning tree transitive closure algorithm

Abstract: The authors present a transitive closure algorithm that maintains a spanning tree of successors for each node rather than a simple successor list. This spanning tree structure promotes sharing of information across multiple nodes and leads to more efficient algorithms. An effective relational implementation of the spanning tree storage structure is suggested, and it is shown how blocking can be applied to reduce the input/output cost of the algorithm. The algorithm can handle path problems also. Analytical and experimental evidence is presented that demonstrates the utility of the algorithm, especially in a graph with many alternate paths between the nodes. The spanning tree storage structure can be compressed and updated incrementally in response to changes in the underlying graph. >

Optimal pruning for tree-structured vector quantization

Abstract: Tree-structured vector quantization (VQ) is a technique designed to represent a codebook that simplifies encoding as well as vector quantizer design. Most design algorithms for tree-structured VQ used in the past are based on heuristics that successively partition the input space. Recently, Chou, Lookabaugh and Gray proposed a tree-pruning heuristic in which a given initial tree is pruned backwards according to certain optimization criterion. We define the notion of an optimal pruned tree subject to a cost constraint and study the computational complexity of finding such an optimal tree for various cost functions. Under the assumption that all trees are equally probable, we show that, on the average, the number of pruned trees in a given tree is exponential in the number of leaves. Furthermore, we prove that finding an optimal pruned tree subject to constraints such as entropy or the expected-depth is NP-hard. However, we show that when the constraint is the number of leaves, the problem can be solved in polynomial time. We develop an algorithm to find the optimal pruned tree in O(nk) time, where n is the size of the initial tree and kis the constraint size.

Dense edge-disjoint embedding of binary trees in the mesh

Abstract: We present an embedding of the complete binary tree with n leaves in the Vn x Vn mesh, for any n = 2exp(2m) where m is a positive integer. The embedding has the following properties: at most two tree nodes (one of which is a leaf) are mapped onto each mesh node, paths of the tree are mapped onto edge-disjoint paths in the mesh (each mesh edge considered as two anti-parallel directed edges) and the maximum distance from a leaf to the root of the tree is Vn + O (log n) mesh steps. This embedding facilitates efficient implementation of many P-RAM algorithms on the mesh, particularly those using the balanced binary tree technique. Such an embedding offers greater flexibility of use and improves the time complexity of these implementations by a constant factor compared with previously described embeddings.

Optimal parallel encoding and decoding algorithms for trees

Abstract: Encoding the shape of a binary tree is a basic step in a number of algorithms in integrated circuit design, automated theorem proving, and game playing. We propose cost-optimal parallel algorithms to solve the binary tree encoding/decoding problem. Specifically, we encode the relevant shape information of an n-node binary tree in a 2n bitstring. Conversely, given an arbitrary 2n bitstring we reconstruct the shape of the corresponding binary tree, if such a tree exists. All our algorithms run in O(log n) time using O(n/log n) processors in the EREW-PRAM model of computation.

Location Of A Tree Shaped Facility In A Network

Abstract: We consider the problem of locating a tree shaped facility in an undirected network. If the facility is of obnoxious type the problem is one of finding a tree whose length is at least a stated value such that the minimum of the shortest distances from the nodes on the tree to nodes that are not on the tree is maximum. Analogously, in the case of a friendly facility we wish to find a tree whose length is at most a stated value such that the maximum of the above shortest distances is minimum. For the first case we provide a polynomial algorithm but we show that the second case belongs to the class of NP-hard problems. Two analogous problems, each involving a path shaped facility, are also shown to be NP-hard.

A synchronous algorithm for shortest paths on a tree machine

Abstract: This paper presents a synchronous (SIMD) algorithm for solving the single source problem for finding shortest paths in a network on a tree machine model. The algorithm requires O(N log2 N) complexity time using a tree machine with N leaf processing elements.

Searching tree structures on a mesh of processors

Abstract: In this paper, we present efficient mesh algorithms for performing n search processes on data structures that can be modeled as a tree T of n constant degree nodes. Each of the n search process consists of tracing a search path, i.e. a sequence ofadjacent nodes, in Tonline. Letkbe the maximum length of the traced search paths (note that k is not known ahead of time and must be determined online). We achieve the following time bounds on a @ x @ mesh of processors: (i) when each search path forms a simple path in T, we achieve the optimal 0(++ k) time bound, and (ii) when each search path forms an Euler traversal of a subtree of T, we achieve the 0(/7i(nc+ ~l_Lk)C )) time bound, for any fixed integer in> 2;, %: a constant e, o < c < J’_l. The best previous bound is —]) where a search path can be an ar%mlo;n bitrary sequence of adjacent nodes. As a consequence, we prove that a search tree of height O(W) performs as well as the conventional one of height O (log n) on a mesh of n processors, provided that each search path is a simple path. Our algorithms are based on tree decompositions which are of independent interest.

Improving search for tree-structured vector quantization

Abstract: The authors analyze the approximate performance of tree search and provide tight upper bounds on the amount of error resulting from tree search and for a single input vector. These bounds are not encouraging but fortunately, the performance of tree-structured VQ in practice does not seem to be as bad. From the analysis, they derive a simple heuristic to improve the approximation of tree search. The strategy is to identify for each code vector some of its closest neighboring code vectors determined by the partition. After a code vector is found for an input vector by tree search, the closest neighboring code vectors are then searched for the best match. Unfortunately, the average number of neighboring code vectors of a given code vector can be as many as the total number of code vectors. Thus, the performance improvement of the strategy depends on the number of code vectors that are searched. Experimental results show that a number logarithmic in the size of the codebook provides significant performance gain while preserving the asymptotic search time complexity. >

Degenerate Gilbert—Steiner trees

Abstract: The minimal Gilbert network problem is a generalization of the Steiner minimal tree problem derived by adding flow-dependent weights to the edges. The minimal Gilbert network is referred to as a minimal Gilbert—Steiner tree if it has a Steiner topology. All trees with Steiner topologies, obtained by sequentially contracting edges connecting given points and their incident Steiner points, form a degenerate tree family. Similarly to the Steiner problem, a relatively minimal tree satisfying some angle conditions is referred to as a Gilbert–Steiner tree. This work shows that, unlike the Steiner problem, there may exist more than one Gilbert–Steiner tree in a degenerate tree family. It is proved that the maximum number of Gilbert–Steiner trees in a degenerate tree family is 2[(n−2)/2]. A necessary condition for the minimal Gilbert–Steiner tree is also given.

Optimal placement of identical resources in a tree

Abstract: The problem of placing a number t of identical resources at nodes of a tree so as to minimize the total expected cost of servicing a set of t requests arriving randomly at nodes is considered. The cost of servicing a particular set of requests is the total distance in the tree between each request and its assigned resource. Distance is measured by the number of edges along the unique path from the request to the resource. Optimal placements can be found in time O(mt), where m is the number of edges in the tree. Allowing resources to be split into fractional-sized pieces which can be placed separately neither reduces the cost of an optimal placement nor provides an obvious way to find optimal placements significantly faster. Simple, natural “fair” placements whose cost differs from optimality by at most the number of edges in the tree are described. For any fixed tree T, the cost of these placements grows as O( t ) , where the constant implicit in the “O” notation depends on the size and shape of T. In the case of balanced trees with k leaves, that constant is at most 2k φ . The placement problem becomes somewhat simpler for a complete (rooted) d-ary tree with a symmetric probability density function for request arrivals, and in that case slightly stronger results are possible. For example, an optimal placement can be found in time O(min{l, logdt} + t), where l is the height of the tree, and the placement is symmetric and fair.

Parallel Recognition of Multidimensional Images Using Regular Tree Grammars

Abstract: In this paper, we use tree grammars, and tree automata for representing a set of Multidimensional images. We show that the set of all full 2d-trees (Quadtrees, Octrees,...etc) is not a regular set. But every finite set of full 2d-trees can be represented by a regular tree grammar. We give optimal algorithms for solving the S-equivalence problem of two 2d-trees and the reduction (i.e. compression) problem of a full 2d-trees. We give some characterizations of regular trees sets. We present a parallel algorithm for recognizing a multidimensional image of size N in O(log(N)) time with O(N) = N/d processors on EREW-PRAM model.

Adaptive representation of histogram using interval tree

Abstract: Proposes a new data structure, called histogram interval tree, to represent a distribution of gray levels in terms of intervals over a given gray-level range. The proposed structure will allow a user to refine the intervals adaptively as needed in each individual application. The author demonstrates that by using this new representation, the interactive specification of desired histogram becomes particularly easy and effective. Results from the proposed technique is compared with that from existing techniques for image enhancement using histogram transformation. The applications of the histogram interval tree can be extended when certain properties associated with histograms are attached to the tree. The author illustrates an example of such extension by building a threshold hierarchy for image segmentation. >

Parallel techniques for construction of trees and related problems

Abstract: The concept of a tree has been used in various areas of mathematics for over a century. In particular, trees appear to be one of the most fundamental notions in computer science. Sequential algorithms for trees are generally well studied. Unfortunately many of these sequential algorithms use methods which seem to be inherently sequential. One of the contributions of this thesis is the introduction of several parallel techniques for the construction of various types of trees and the presentation of new parallel tree construction algorithms using these methods. Along with the parallel tree construction techniques presented here, we develop techniques which have broader applications. We use the Parallel Random Access Machine as our model of computation. We consider two basic methods of constructing trees: tree expansion and tree synthesis. In the tree expansion method, we start with a single vertex and construct a tree by adding nodes of degree one and/or by subdividing edges. We use the parallel tree expansion technique to construct the tree representation for graphs in the family of graphs known as cographs. In the tree synthesis method, we start with a forest of single node subtrees and construct a tree by adding edges or (for rooted trees) by creating parent nodes for some roots of the trees in the forest. We present a family of parallel and sequential algorithms to construct various approximations to the Huffman tree. All these algorithms apply the tree synthesis method by constructing a tree in a level-by-level fashion. To support one of the algorithms in the family we develop a technique which we call the cascading sampling technique. One might suspect that the parallel tree synthesis method can be applied only to trees of polylogarithmic height, but this is not the case.We present a technique which we call the valley filling technique and develop its accelerated version called the accelerated valley filling technique. We present an application of this technique to an optimal parallel algorithm for construction of minimax trees.

A new parallel algorithm for breadth-first search on interval graphs

Abstract: The authors design an efficient parallel algorithm for constructing a breadth-first spanning tree of an interval graph. Their novel approach is based on elegantly capturing the structure of a given collection of intervals. This structure reveals important properties of the corresponding interval graph, and is found to be instrumental in solving many other problems including the computation of a breadth-depth spanning tree, which they report for the first time. The algorithm requires O(logn) time employing O(n) processors on the EREW PRAM model. >