G Gerard Zwaan

On computing a longest path in a tree

The primary purpose of this note is to present an exercise in proof design. For us, such a design consists in isolating the relevant concepts for the problem at hand, introducing special-purpose notation for them that is geared to manipulation and to crisp formal specification, and then solving the problem in a demand-driven way, while onthe-fly extracting from the calculation additional theory useful for solving the problem proper. The problem chosen is demonstrating the correctness of an algorithm for computing the longest path in a tree. Given a finite tree with all edges having positive length, we wish to compute a longest path. This can be done using a procedure invented by Edsger W. Dijkstra around 1960, which is as follows. Build a physical model of the tree by connecting each pair of adjacent nodes by a piece of string of the given edge length. Now pick up the physical tree at an arbitrary node U , l t the contraption hang down, and determine a deepest node X. Then pick up the tree at X and determine a deepest node Y . The claim is that the path between X andY is a longest path in the tree. We have never seen a formal proof of this claim, and the purpose of this note is to provide one.

A taxonomy of sublinear multiple keyword pattern matching algorithms

This article presents a taxonomy of sublinear keyword pattern matching algorithms related to the Boyer-Moore algorithm [3] and the Commentz-Walter algorithm [5, 6]. The taxonomy includes, amongst others, the multiple keyword generalization of the single keyword Boyer-Moore algorithm and an algorithm by Fan and Su [9, 10]. The corresponding precomputation algorithms are presented as well. The taxonomy is based on the idea of ordering algorithms according to their essential problem and algorithm details, and deriving all algorithms from a common starting point by successively adding these details in a correctness preserving way. This way of presentation not only provides a complete correctness argument of each algorithm, but also makes very clear what algorithms have in common (the details of their nearest common ancestor) and where they differ (the details added after their nearest common ancestor). Introduction of the notion of safe shift distances proves to be essential in the derivation and classification of the algorithms. Moreover, the article provides a common derivation for and a uniform presentation of the precomputation algorithms, not yet found in the literature.

A new taxonomy of sublinear right-to-left scanning keyword pattern matching algorithms

A new taxonomy of sublinear (multiple) keyword pattern matching algorithms is presented. Based on an earlier taxonomy by the second and third authors, this new taxonomy includes not only suffix-based algorithms, but also factor- and factor-oracle-based algorithms. In particular, we show how suffix-based (Commentz-Walter like), factor- and factor-oracle-based sublinear keyword pattern matching algorithms can be seen as instantiations of a general sublinear algorithm skeleton. During processing, such algorithms shift or jump through the text in a forward or left-to-right direction, and read backward or right-to-left starting from positions in the text, i.e. they read suffixes of certain prefixes of the text. They use finite automata for efficient computation of string membership in a certain language. In addition, we show shift functions defined for the suffix-based algorithms to be reusable for factor- and factor-oracle-based algorithms. The taxonomy is based on deriving the algorithms from a common starting point by adding algorithm and problem details, to arrive at efficient or well-known algorithms. Such a presentation provides correctness arguments for the algorithms as well as clarity on how the algorithms are related to one another. In addition, it is helpful in the construction of a toolkit of the algorithms.

Automaton-based sublinear keyword pattern matching

We show how automaton-based sublinear keyword pattern matching (skpm) algorithms appearing in the literature can be seen as different instantiations of a general automaton-based skpm algorithm skeleton. Such algorithms use finite automata (FA) for efficient computation of string membership in a certain language. The algorithms were formally derived as part of a new skpm algorithm taxonomy, based on an earlier suffix-based skpm algorithm taxonomy [1]. Such a taxonomy is based on deriving the algorithms from a common starting point by successively adding algorithm and problem details and has a number of advantages. It provides correctness arguments, clarifies the working of the algorithms and their interrelationships, helps in implementing the algorithms, and may lead to new algorithms being discovered by finding gaps in the taxonomy. We show how to arrive at the general algorithm skeleton and derive some instantiations, leading to well-known factor- and factor oracle-based algorithms. In doing so, we show the shift functions used for them can be (strengthenings of) shift functions used for suffix-based algorithms. This also results in a number of previously undescribed factor-based skpm algorithm variants, whose performance remains to be investigated.

A systolic design for acceptors of regular languages

In this paper we present a derivation method for networks of systolic cells. The method is calculational and can be applied for specifications that are data-independent, i.e. the order of communications does not depend on the values communicated. The techniques used in this style of parallel programming are comparable with techniques used in sequential programming. The presentation is done by means of an example: the derivation of a set of building blocks for the construction of acceptors for regular languages.