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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
The Greedy Algorithms Class: Formalization, Synthesis and Generalization
, 1995
"... On the first hand, this report studies the class of Greedy Algorithms in order to find an as systematic as possible strategy that could be applied to the specification of some problems to lead to a correct program solving that problem. On the other hand, the standard formalisms underlying the G ..."
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On the first hand, this report studies the class of Greedy Algorithms in order to find an as systematic as possible strategy that could be applied to the specification of some problems to lead to a correct program solving that problem. On the other hand, the standard formalisms underlying the Greedy Algorithms (matroid, greedoid and matroid embedding) which are dependent on the particular type set are generalized to a formalism independent of any data type based on an algebraic specification setting.
Partitions Revisited
, 1993
"... Problems involving list partitions are found in many areas of computer science. This paper states theorems about programs that use strategies such as dynamic programming or greedy strategies to solve optimization problems, and applies the theorems to the solving of partition problems. The reasoning ..."
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Problems involving list partitions are found in many areas of computer science. This paper states theorems about programs that use strategies such as dynamic programming or greedy strategies to solve optimization problems, and applies the theorems to the solving of partition problems. The reasoning is in an equational style, using a calculus of relations and associated laws. Contents 1 Introduction 1 1.1 Partition Problems \Delta \Delta \Delta \Delta \Delta \Delta \Delta \Delta \Delta \Delta \Delta \Delta \Delta \Delta \Delta \Delta \Delta \Delta \Delta \Delta \Delta \Delta \Delta 2 2 A Calculus of Relations 4 2.1 Relations \Delta \Delta \Delta \Delta \Delta \Delta \Delta \Delta \Delta \Delta \Delta \Delta \Delta \Delta \Delta \Delta \Delta \Delta \Delta \Delta \Delta \Delta \Delta \Delta \Delta \Delta \Delta \Delta \Delta 4 2.2 Relators \Delta \Delta \Delta \Delta \Delta \Delta \Delta \Delta \Delta \Delta \Delta \Delta \Delta \Delta \Delta \Delta \Delta \Delta \Delta \Delta \Del...
Deriving the AhoCorasick Algorithms: A Case Study into the Synergy of Programming Methods
, 1993
"... Imperative programs can be derived using methods like stepwise refinement, but they do not lend themselves very well to transformational programming. The BirdMeertens approach offers a powerful transformation calculus for functional programs, but it pays little or no attention to imperative algorit ..."
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Imperative programs can be derived using methods like stepwise refinement, but they do not lend themselves very well to transformational programming. The BirdMeertens approach offers a powerful transformation calculus for functional programs, but it pays little or no attention to imperative algorithms. In this paper we will show how the two approaches can be combined by giving a transformational derivation of some efficient and practical imperative programs, viz. the AhoCorasick string pattern matching algorithms.
A Class of Graph Algorithms
, 1996
"... Several graph problems are shown to be instances of the abstract problem of selecting representative elements from a closuregenerated set. These are the singlesource minimum paths problem, the reachability set of a given vertex, and the construction of rooted trees. Algorithms to compute such ..."
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Several graph problems are shown to be instances of the abstract problem of selecting representative elements from a closuregenerated set. These are the singlesource minimum paths problem, the reachability set of a given vertex, and the construction of rooted trees. Algorithms to compute such representatives are derived from the specification of the general problem. Instances of the algorithms include Dijkstra's minimum paths algorithm, and depthfirst/breadthfirst graph traversals. A relational calculus for algorithms derivation is used as algebraic framework. This calculus, of a pointfree reasoning character, is extended with categorical points to allow a clearer phrasing and manipulation of some algorithmic expressions. Pointwise reasoning can thus be naturally carried out within the extended calculus.