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From Dynamic Programming to Greedy Algorithms
 Formal Program Development, volume 755 of Lecture Notes in Computer Science
, 1992
"... A calculus of relations is used to reason about specifications and algorithms for optimisation problems. It is shown how certain greedy algorithms can be seen as refinements of dynamic programming. Throughout, the maximum lateness problem is used as a motivating example. 1 Introduction An optimisat ..."
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Cited by 14 (3 self)
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A calculus of relations is used to reason about specifications and algorithms for optimisation problems. It is shown how certain greedy algorithms can be seen as refinements of dynamic programming. Throughout, the maximum lateness problem is used as a motivating example. 1 Introduction An optimisation problem can be solved by dynamic programming if an optimal solution is composed of optimal solutions to subproblems. This property, which is known as the principle of optimality, can be formalised as a monotonicity condition. If the principle of optimality is satisfied, one can compute a solution by decomposing the input in all possible ways, recursively solving the subproblems, and then combining optimal solutions to subproblems into an optimal solution for the whole problem. By contrast, a greedy algorithm considers only one decomposition of the argument. This decomposition is usually unbalanced, and greedy in the sense that at each step the algorithm reduces the input as much as poss...
A Generic Program for Sequential Decision Processes
 Programming Languages: Implementations, Logics, and Programs
, 1995
"... This paper is an attempt to persuade you of my viewpoint by presenting a novel generic program for a certain class of optimisation problems, named sequential decision processes. This class was originally identified by Richard Bellman in his pioneering work on dynamic programming [4]. It is a perfect ..."
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Cited by 13 (2 self)
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This paper is an attempt to persuade you of my viewpoint by presenting a novel generic program for a certain class of optimisation problems, named sequential decision processes. This class was originally identified by Richard Bellman in his pioneering work on dynamic programming [4]. It is a perfect example of a class of problems which are very much alike, but which has until now escaped solution by a single program. Those readers who have followed some of the work that Richard Bird and I have been doing over the last five years [6, 7] will recognise many individual examples: all of these have now been unified. The point of this observation is that even when you are on the lookout for generic programs, it can take a rather long time to discover them. The presentation below will follow that earlier work, by referring to the calculus of relations and the relational theory of data types. I shall however attempt to be light on the formalism, as I do not regard it as essential to the main thesis of this paper. Undoubtedly there are other (perhaps more convenient) notations in which the same ideas could be developed. This paper does assume some degree of familiarity with a lazy functional programming language such as Haskell, Hope, Miranda
Algorithms from Theorems
 Programming Concepts and Methods
, 1990
"... In this paper we show how algorithms are derived from their specification in the BirdMeertens form,dl,m. The BirdMeertens formalism is a programming methodology which provides a concise functional notation for algorithms and for every data structure a promotion theorem for proving equalities of ..."
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Cited by 11 (5 self)
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In this paper we show how algorithms are derived from their specification in the BirdMeertens form,dl,m. The BirdMeertens formalism is a programming methodology which provides a concise functional notation for algorithms and for every data structure a promotion theorem for proving equalities of functions.
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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Cited by 7 (0 self)
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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
Bridging the Algorithm Gap: A Lineartime Functional Program for Paragraph Formatting
 Science of Computer Programming
, 1997
"... In the constructive programming community it is commonplace to see formal developments of textbook algorithms. In the algorithm design community, on the other hand, it may be well known that the textbook solution to a problem is not the most efficient possible. However, in presenting the more eff ..."
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Cited by 3 (0 self)
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In the constructive programming community it is commonplace to see formal developments of textbook algorithms. In the algorithm design community, on the other hand, it may be well known that the textbook solution to a problem is not the most efficient possible. However, in presenting the more efficient solution, the algorithm designer will usually omit some of the implementation details, thus creating an algorithm gap between the abstract algorithm and its concrete implementation. This is in contrast to the formal development, which usually presents the complete concrete implementation of the less efficient solution. We claim that the algorithm designer is forced to omit some of the details by the relative expressive poverty of the Pascallike languages typically used to present the solution; the greater expressiveness provided by a functional language allows the whole story to be told in a reasonable amount of space. We therefore hope to bridge the algorithm gap between ab...
Using underspecification in the derivation of some optimal partition algorithms
, 1990
"... Indeterminacy is inherent in the specification of optimal partition (and many more) algorithms, even though the algorithms themselves may be fully determinate. Indeterminacy is a notoriously hard phenomenon to deal with in a purely functional setting. In the paper “A Calculus Of Functions for Progra ..."
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Cited by 1 (0 self)
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Indeterminacy is inherent in the specification of optimal partition (and many more) algorithms, even though the algorithms themselves may be fully determinate. Indeterminacy is a notoriously hard phenomenon to deal with in a purely functional setting. In the paper “A Calculus Of Functions for Program Derivation ” R. S. Bird tries to handle it by using underspecified functions. (Other authors have proposed to use ‘indeterminate ’ functions, and to use relations instead of functions.) In this paper we redo Bird’s derivation of the Leery and Greedy algorithm while being very precise about underspecification, and still staying in the functional framework. It turns out that Bird’s theorems are not exactly what one would like to have, and what one might understand from his wording of the theorems. We also give a derivation in the BirdMeertens style of a (linear time) optimal partition algorithm that was originally found by J. C. S. P. van der Woude. 1
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...