## Between Dynamic Programming and Greedy: Data Compression (1995)

Venue: | Programming Research Group, 11 Keble Road, Oxford OX1 3QD |

Citations: | 5 - 0 self |

### BibTeX

@INPROCEEDINGS{Bird95betweendynamic,

author = {Richard S. Bird and Oege De Moor},

title = {Between Dynamic Programming and Greedy: Data Compression},

booktitle = {Programming Research Group, 11 Keble Road, Oxford OX1 3QD},

year = {1995}

}

### OpenURL

### Abstract

The derivation of certain algorithms can be seen as a hybrid form of dynamic programming and the greedy paradigm. We present a generic theorem about such algorithms, and show how it can be applied to the derivation of an algorithm for data compression. 1 Introduction Dynamic programming is a technique for solving optimisation problems. A typical dynamic programming algorithm proceeds by decomposing the input in all possible ways, recursively solving the subproblems, and combining optimal solutions to subproblems into an optimal solution for the whole problem. The greedy paradigm is also a technique for solving optimisation problems and differs from dynamic programming in that only one decomposition of the input is considered. Such a decomposition is usually chosen to maximise some objective function, and this explains the term `greedy'. In our earlier work, we have characterised the use of dynamic programming and the greedy paradigm, using the categorical calculus of relations to der...

### Citations

509 |
Programming from Specifications
- Morgan
(Show Context)
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449 |
Introduction to Higher Order Categorical Logic
- Lambek, Scott
- 1986
(Show Context)
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153 |
A calculus of refinements for program derivations
- Back
- 1988
(Show Context)
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106 |
Data compression via textual substitution
- Storer, Szymanski
- 1982
(Show Context)
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93 | Memo functions and machine learning - Michie - 1968 |

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- Crochemore
- 1986
(Show Context)
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40 |
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- Paulson
- 1987
(Show Context)
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27 |
Bicategories of spans and relations
- Carboni, Kasangian, et al.
- 1984
(Show Context)
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26 |
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- Ecker, Kupferschmid
- 1988
(Show Context)
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25 |
der Woude. A relational theory of datatypes
- Aarts, Backhouse, et al.
- 1992
(Show Context)
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- Cockett, Fukushima
- 1992
(Show Context)
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- Aarts
(Show Context)
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- Bird, Moor
- 1993
(Show Context)
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13 |
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- Carboni, Kelly, et al.
- 1991
(Show Context)
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- Bird, Moor
- 1992
(Show Context)
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- Moor
- 1992
(Show Context)
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- 1992
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- Helman
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- 1988
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- 1990
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- 1977
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- 1990
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