We propose a format of a binary phrase structure grammar with composite labels. The grammar adopts binary rules so that the dependency between two sub-trees can be represented in the label of the tree. The label of a tree is composed of two attributes, each of which is extracted from each sub-tree, so that it can represent the compositional information of the tree. The composite label is genera...

We study losses for binary classification and class probability estimation and extend the understanding of them from margin losses to general composite losses which are the composition of a proper loss with a link function. We characterise when margin losses can be proper composite losses, explicitly show how to determine a symmetric loss in full from half of one of its partial losses, introduc...

An amortized analysis of the insertion and deletion algorithms of k-dimensional balanced binary trees (kBB-trees) is performed. It is shown that the total rebalancing time for a mixed sequence of m insertions and deletions in a kBB-tree of size n is O(k(m+n)). Based on 2BB-trees, a self-organizing tree, called a self-organizing balanced binary tree, is defined. It is shown that the average acce...

Deep neural networks have been proved a success in multiple fields. However, researchers still favor traditional approaches to obtain more interpretable models, such as Bayesian methods and decision trees when processing heterogeneous tabular data. Such models are hard differentiate, thus inconvenient be integrated into end-to-end settings. On the other hand, differentiable but perform poorly o...

Introduced a new binary search tree algorithm, cache-splay, that was the first to achieve the Unified Bound of Iacono since this problem was posed in 2001, and showed that the lower bound framework (see below) applied to a more general model of computation than binary search. Additionally, my thesis summarized my previous work on developing both lower bounds and upper bounds on the cost of sequ...

Trees are a common structure to represent the inter-task communication pattern of a parallel algorithm. In this paper, we consider the embedding a complete binary tree in a star graph with the objective of minimizing congestion and dilation. We develop two embeddings: i) a congestion-free, dilation-2, load-1 embedding of a level-p binary tree, and ii) a congestion-free, dilation-2, load-2 k emb...

In this paper we are concerned with the analysis of special recursive algorithms for traversing the nodes of a planted plane tree (ordered tree ; planar tree) . Some by now classical results in this area are due to (de Bruijn, Knuth, and Rice, 1972 ; Knuth, 1973 ; Flajolet, 1981 ; Flajolet and Odlyzko, 1982 ; Flajolet, Raoult, and Vuillemin, 1979 ; Kemp, 1979) and others and are summarized in t...

This paper proposes a new coding scheme and an algorithm to determine ancestordescendant relationship among nodes in a tree without tree traversal. Each node in a tree is assigned a unique binary code. The algorithm and the binary coding approach were tested with trees of different heights and widths. The algorithm is of O(1) complexity versus O(d) for tree traversal, where d is the depth of th...

This article is a continuation of an article on deening functions on trees (see 7]). In this article we develop terminology specialized for binary trees, rst deening binary trees and binary grammars. We recast the induction principle for the set of parse trees of binary grammars and the scheme of deening functions inductively with the set as domain. We conclude with deening the scheme of deenin...

The rank/activity restriction on binary resolution is introduced. It accepts only a single derivation tree from a large equivalence class of such trees. The equivalence classes capture all trees that are the same size and differ only by reordering the resolution steps. A proof procedure that combines this restriction with the authors’ minimal restriction of binary resolution computes each minim...