Derick Wood

A survey of adaptive sorting algorithms

The design and analysis of adaptive sorting algorithms has made important contributions to both theory and practice. The main contributions from the theoretical point of view are: the description of the complexity of a sorting algorithm not only in terms of the size of a problem instance but also in terms of the disorder of the given problem instance; the establishment of new relationships among measures of disorder; the introduction of new sorting algorithms that take advantage of the existing order in the input sequence; and, the proofs that several of the new sorting algorithms achieve maximal (optimal) adaptivity with respect to several measures of disorder. The main contributions from the practical point of view are: the demonstration that several algorithms currently in use are adaptive; and, the development of new algorithms, similar to currently used algorithms that perform competitively on random sequences and are significantly faster on nearly sorted sequences. In this survey, we present the basic notions and concepts of adaptive sorting and the state of the art of adaptive sorting algorithms.

Grail : a C++ library for automata and expressions

Abstract Grail is a package for symbolic manipulation of finite-state automata and regular expressions. It provides most standard operations on automata and expressions, including minimization, subset construction, conversion between automata and regular expressions, and language enumeration and testing. Grail s objects are parameterizable; users can provide their own classes to define the input alphabet of automata and expressions. Grail s operations are accessible either as individual programs or directly through a C++ class library.

Right invariant metrics and measures of presortedness

Abstract Right invariant metrics (ri-metrics) have several applications in the theory of rank correlation methods. For example, ranking models based on ri-metrics generalize Mallows ranking models. We explore the relationship between right invariant metrics and measures of presortedness (mops). The latter have been used to evaluate the behavior of sorting algorithms on nearly-sorted inputs. We give necessary and sufficient conditions for a measure of presortedness to be extended to a ri-metric; we characterize those ri-metrics that can be used as mops; and we show that those mops that are extendible to ri-metrics can be constructed from sets of sorting operations. Our results provide a paradigm for the construction of mops and ri-metrics.

Insertion reachability, skinny skeletons, and path length in red-black trees

Abstract The red-black trees are binary trees in which a longest path from every node v to a leaf is at most twice as long as a shortest path from v to a leaf. We examine sequences of simple insertions (insertions without promotions or any other restructuring operations) in red-black trees. We prove that the case of red-black trees is the same as the class of trees that is defined by an insertion algorithm for red-black trees that uses promotions to restore balance. The proof demonstrates that every red-black tree can be built from the empty tree by a sequence of simple insertions We also prove that every red-black tree contains a skinny red-black subtree of the same height (a red-black tree with the smallest number of nodes for its height) by giving a sequence of simple insertions that can be used to build the given red-black tree from the skinny red-black subtree. From this result we conclude that the skinny red-black trees have the minimum path length among all red-black trees of the same height, a result that seems intuitively obvious but whose proof is not.

Maximal path length of binary trees

Abstract We further refine the bounds on the path length of binary trees of a given size by considering not only their sizes, but also their heights and fringe thicknesses (the difference between the length of their shortest root-to-leaf paths and their heights). We characterize the maximum-path-length binary trees of a given height, size, and fringe thickness, and using this characterization, we give an algorithm to find the maximum-path-length binary trees of a given size and fringe thickness. The proof of the main result is based on two new tree transformations that preserve the height, size, and fringe thickness.

Staircase visibility and computation of kernels

AbstractLetn

Rediscovering Pushdown Machines

mathcal{O}

Structural Equivalence and ETOL grammars

n be some set of orientations, that is,n