Showing papers on "Binary tree published in 1992"

The complexity of reconstructing trees from qualitative characters and subtrees

Abstract: In taxonomy and other branches of classification it is useful to know when tree-like classifications on overlapping sets of labels can be consistently combined into a parent tree. This paper considers the computation complexity of this problem. Recognizing when a consistent parent tree exists is shown to be intractable (NP-complete) for sets of unrooted trees, even when each tree in the set classifies just four labels. Consequently determining the compatibility of qualitative characters and partial binary characters is, in general, also NP-complete. However for sets of rooted trees an algorithm is described which constructs the “strict consensus tree” of all consistent parent trees (when they exist) in polynomial time. The related question of recognizing when a set of subtrees uniquely defines a parent tree is also considered, and a simple necessary and sufficient condition is described for rooted trees.

Varieties of Increasing Trees

Abstract: An increasing tree is a labelled rooted tree in which labels along any branch from the root go in increasing order. Under various guises, such trees have surfaced as tree representations of permutations, as data structures in computer science, and as probabilistic models in diverse applications.

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. >

A note on optimal area algorithms for upward drawings of binary trees

Abstract: The goal of this paper is to investigate the area requirements for upward grid drawings of binary trees. First, we show that there is a family of binary trees with n vertices that require ω(n log n) area; this bound is tight to within a constant factor, i.e. any binary tree with n vertices can be drawn in O(n log n) area. Then we present an algorithm for constructing an upward drawing of a complete binary tree with n vertices in O(n) area, and, finally, we extend this result to the drawings of Fibonacci trees.

Efficient embeddings of trees in hypercubes

Abstract: THE BOOLEAN HYPERCUBE IS A PARTICULARLY VERSATILE NETWORK FOR PARALLEL COMPUTING. IT IS WELL KNOWN THAT MULTI-DIMENSIONAL GRID MACHINES CAN BE SIMULATED ON A HYPERCUBE WITH NO COMMUNICATIONS OVERHEAD. IN THIS PAPER WE SHOW THAT EVERY BOUNDED-DEGREE TREE CAN BE SIMULATED ON THE HYPERCUBE WITH CONSTANT COMMUNICATIONS OVERHEAD. OUR PROOF IN FACT SHOWS THAT EVERY BOUNDED-DEGREE GRAPH WITH AN 0(1)-SEPARATOR CAN BE EMBEDDED IN A HYPERCUBE OF THE SAME SIZE WITH DILATION AND CONGESTION BOTH 0(1). WE PROVE ALSO THAT NOT ALL BOUNDED-DEGREE GRAPHS CAN BE EFFICIENTLY EMBEDDED WITHIN THE HYPERCUBE.

Dynamic tree embeddings in butterflies and hypercubes

Abstract: This paper presents simple randomized algorithms for dynamically embedding M-node binary trees in either a butterfly or a hypercube network of N processors. These algorithms are dynamic in the sense that the tree to be embedded may start as one node and grow by dynamically spawning children. The nodes are incrementally embedded as they are spawned. Thus, the algorithm is especially suited for maintaining dynamic tree structures like those in divide-and-conquer and branch-and-bound algorithms.In the embeddings, the paper seeks to optimize the load on the processors of the network, the dilation of the tree edges, and the congestion on the network edges, in order to satisfy the demands of load balancing, process locality, and communication efficiency. The paper begins by presenting a simple level-by-level scheme for dynamically embedding trees in a butterfly network, and by successive modifications. The following results are obtained: 1. An embedding algorithm for the hypercube that achieves dilation 1 and, ...

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. >

On the deque conjecture for the splay algorithm

Abstract: Splay is a simple, efficient algorithm for searching binary search trees, devised by Sleator and Tarjan, that reorganizes the tree after each search by means of rotations. An open conjecture of Sleator and Tarjan states that Splay is, in essence, the fastest algorithm for processing any sequence of search operations on a binary search tree, using only rotations to reorganize the tree. Tarjan proved a basic special case of this conjecture, called theScanning Theorem, and conjectured a more general special case, called theDeque Conjecture. The Deque Conjecture states that Splay requires linear time to process sequences of deque operations on a binary tree. We prove the following results: 1. Almost tight lower and upper bounds on the maximum numbers of occurrences of various types of right rotations in a sequence of right rotations performed on a binary tree. In particular, the lower bound for right 2-turns refutes Sleator's Right Turn Conjecture. 2. A linear times inverse Ackerman upper bound for the Deque Conjecture. This bound is derived using the above upper bounds. 3. Two new proofs of the Scanning Theorem, one, a simple potential-based proof that solves Tarjan's problem of finding a potential-based proof for the theorem, the other, an inductive proof that generalizes the theorem.

ELM-a fast addition algorithm discovered by a program

Abstract: A new addition algorithm, ELM, is presented. This algorithm makes use of a tree of simple processors and requires O(log n) time, where n is the number of bits in the augend and addend. The sum itself is computed in one pass through the tree. This algorithm was discovered by a VLSI CAD tool, FACTOR, developed for use in synthesizing CMOS VLSI circuits. >

The SB -tree: an index-sequential structure for high-performance sequential access

Abstract: A variant of aB-tree known as anSB-tree is introduced, with the object of offering high-performance sequential disk access for long range retrievals. The key to this efficiency is a structure that supports multi-page reads (or writes) during sequential access to any node level below the root, even following significant node splitting. In addition, theSB-tree will support a policy to ‘stripe’ successive multi-page blocks on multiple disks to achieve maximum parallelism. Compared to traditionalB-tree structures,SB-tree performance characteristics are less subject to degradation resulting from modifications entailed in growing and shrinking;SB-trees are therefore more appropriate for use in situations where frequent reorganization is not possible. A performance analysis reveals the strengths of theSB-tree by comparing its performance under various circumstances to theB+-tree and the bounded disorder (BD) file of [11]. The performance analysis formulates a new useful concept, the ‘effective depth’ of anSB- orB+-tree, defined as the expected number of pages read from disk to perform a random retrieval search given standard buffering behavior. A graph of effective depth against tree size is shown to have a scalloped appearance, reflecting the changing effectiveness of incremental additions to buffer space.

A fast recognition-complete processor allocation strategy for hypercube computers

Abstract: Fully recognizing various subcubes in a hypercube computer efficiently is addressed. A method with much less complexity than the multiple-GC strategy in generating the search space, while achieving complete subcube recognition, is proposed. This method is referred to as a dynamic processor allocation scheme because the search space generated is dependent on the dimension of the requested subcube dynamically. The basic idea lies in collapsing the binary tree representations of a hypercube successively so that the nodes which form a subcube but are distant are brought close to each other for recognition. The strategy can be implemented efficiently by using right rotating operations on the notations of the sets of subcubes corresponding to the nodes at a certain level of binary tree representations. Results of extensive simulation runs carried out to collect performance measures for different allocation strategies are discussed. It is shown that this strategy compares favorably in most situations with other known allocation schemes capable of achieving complete subcube recognition. >

A Two-Stage Predictive Splitting Algorithm in Binary Segmentation

Abstract: In the framework of binary segmentation, we propose a two-stage splitting algorithm which optimizes a defined predictability function. The idea is to find a binary tree whose nodes are internally most homogeneous and externally most heterogeneous with respect to the predictability of their cases. The main steps of the algorithm will be described. Some relations with the CART splitting procedure will be discussed and an example will be shown.

Paraunitary filter banks and wavelet packets

Abstract: Binary tree-structured filter banks have been employed in the past to generate wavelet bases. The equivalence between binary paraunitary tree-structures and orthonormal wavelet bases is proven. It is known that a binary tree with paraunitary filters on each level generates a discrete time orthonormal wavelet basis. It is shown that every discrete-time orthonormal wavelet basis can be generated using paraunitary binary trees. Next, this analysis is extended to arbitrary tree structures such as those used for generating wavelet packets. It is shown that an arbitrary tree with paraunitary filers gives an orthonormal basis of wavelet packets. The strict converse is not true, but a weaker result is presented. >

Clock Tree Synthesis Based On Rc Delay Balancing

Abstract: This paper presents a novel clock tree synthesis method based on top down binary tree constructicm, optimal buffer insertion, and bottom up wiring with precise RC delay balancing. The proposed method has achieved near-zero clock skews with minimal clock delays while achileving similar total wire lengths in comparison with previous heuristics.

Efficient generation of k -ary trees in natural order

Abstract: A k-ary tree with n nodes can be encoded as a k-inorder-preorder sequence of length (k-1)n by labelling the tree from 1 to (k-1)n in generalised inorder and then reading off the labelled integers in generalised preorder. The natural ordering of a set of k-ary trees with n nodes is shown to be preserved in the lexicographic ordering of a set of k-inorder-preorder sequences of length (k-1)n. An efficient algorithm for generating a set of k-inorder-preorder sequences in lexicographic order is presented

Reconfiguration strategies for VLSI processor arrays and trees using a modified Diogenes approach

Abstract: The authors deal with reconfiguration of a rectangular array of processors arranged as an N*N mesh, and a complete binary tree of N processors. They present new reconfiguration techniques that are modifications of the Diogenes approach proposed earlier by A.L. Rosenberg et al. (1983). These techniques reduce the overheads incurred in the earlier Diogenes schemes. Some of the previous approaches to the problem are summarized. Two schemes are presented for reconfiguring rectangular arrays and a scheme for reconfiguring trees. For the analysis of the different schemes presented, it is assumed that a processor has a square layout. These schemes are analyzed and their performance results are presented. >

DCT-based codebook design for vector quantization of images

Abstract: A codebook design algorithm based on a two-dimensional discrete cosine transform (2-D DCT) is presented for vector quantization (VQ) of images. The significant features of training images are extracted by using the 2-D DCT. A codebook is generated by partitioning the training set into a binary tree. Each training vector at a nonterminal node of the binary tree is directed to one of the two descendants by comparing a single feature associated with that node to a threshold. Compared with the pairwise nearest neighbor (PNN) algorithm, the algorithm results in a considerable reduction in computation time and shows better picture quality. >