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A Relational Approach To Optimization Problems
, 1996
"... The main contribution of this thesis is a study of the dynamic programming and greedy strategies for solving combinatorial optimization problems. The study is carried out in the context of a calculus of relations, and generalises previous work by using a loop operator in the imperative programming s ..."
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The main contribution of this thesis is a study of the dynamic programming and greedy strategies for solving combinatorial optimization problems. The study is carried out in the context of a calculus of relations, and generalises previous work by using a loop operator in the imperative programming style for generating feasible solutions, rather than the fold and unfold operators of the functional programming style. The relationship between fold operators and loop operators is explored, and it is shown how to convert from the former to the latter. This fresh approach provides additional insights into the relationship between dynamic programming and greedy algorithms, and helps to unify previously distinct approaches to solving combinatorial optimization problems. Some of the solutions discovered are new and solve problems which had previously proved difficult. The material is illustrated with a selection of problems and solutions that is a mixture of old and new. Another contribution is the invention of a new calculus, called the graph calculus, which is a useful tool for reasoning in the relational calculus and other nonrelational calculi. The graph
An InitialAlgebra Approach to Directed Acyclic Graphs
, 1995
"... . The initialalgebra approach to modelling datatypes consists of giving constructors for building larger objects of that type from smaller ones, and laws identifying di#erentways of constructing the same object. The recursive decomposition of objects of the datatype leads directly to a recursiv ..."
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. The initialalgebra approach to modelling datatypes consists of giving constructors for building larger objects of that type from smaller ones, and laws identifying di#erentways of constructing the same object. The recursive decomposition of objects of the datatype leads directly to a recursive pattern of computation on those objects, whichisvery helpful for both functional and parallel programming. We showhow to model a particular kind of directed acyclic graph using this initialalgebra approach. Keywords. Graphs, data types, catamorphisms, initial algebras, BirdMeertens Formalism, program derivation. 1 Introduction It is now widely recognized that the traditional adhoc approaches to program construction do not yield reliable software; a more systematic and formal approach is required. One such approach consists of program veri#cationproving after the fact that a given program satis#es its formal speci#cation. This approach turns out to be di#cult to implement, not lea...
Ideal stream algebra
 Lecture Notes in Computer Science 1546
, 1998
"... We provide some mathematical properties of behaviours of systems, where the individual elements of a behaviour are modeled by ideals of a suitable partial order. It is wellknown that the associated ideal completion provides a simple way of constructing algebraic cpos. An ideal can be viewed as a ..."
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We provide some mathematical properties of behaviours of systems, where the individual elements of a behaviour are modeled by ideals of a suitable partial order. It is wellknown that the associated ideal completion provides a simple way of constructing algebraic cpos. An ideal can be viewed as a set of consistent finite or compact approximations of an object which itself may even be infinite. A special case is the domain of streams where the finite approximations are the finite prefixes of a stream. We introduce a special way of characterising behaviours through sets of relevant approximations. This is a generalisation of the technique used earlier for the case of streams. Given a set P ` M of a partial order (M;), we define ide P: = fQ: Q ` P directedg; where Q
Layered Graph Traversals and Hamiltonian Path Problems  An Algebraic Approach
 MATHEMATICS OF PROGRAM CONSTRUCTION. LECTURE NOTES IN COMPUTER SCIENCE 1422
, 1997
"... Using an algebra of paths we present abstract algebraic derivations for two problem classes concerning graphs, viz. layer oriented traversal and computing sets of Hamiltonian paths. In the first case, we are even able to abstract to the very general setting of Kleene algebras. Applications include r ..."
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Using an algebra of paths we present abstract algebraic derivations for two problem classes concerning graphs, viz. layer oriented traversal and computing sets of Hamiltonian paths. In the first case, we are even able to abstract to the very general setting of Kleene algebras. Applications include reachability and a shortest path problem as well as topological sorting, cycle detection and finding maximum cardinality matchings.