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Manufacturing Datatypes
, 1999
"... This paper describes a general framework for designing purely functional datatypes that automatically satisfy given size or structural constraints. Using the framework we develop implementations of different matrix types (eg square matrices) and implementations of several tree types (eg Braun trees, ..."
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Cited by 24 (3 self)
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This paper describes a general framework for designing purely functional datatypes that automatically satisfy given size or structural constraints. Using the framework we develop implementations of different matrix types (eg square matrices) and implementations of several tree types (eg Braun trees, 23 trees). Consider, for instance, representing square n \Theta n matrices. The usual representation using lists of lists fails to meet the structural constraints: there is no way to ensure that the outer list and the inner lists have the same length. The main idea of our approach is to solve in a first step a related, but simpler problem, namely to generate the multiset of all square numbers. In order to describe this multiset we employ recursion equations involving finite multisets, multiset union, addition and multiplication lifted to multisets. In a second step we mechanically derive datatype definitions from these recursion equations which enforce the `squareness' constraint. The tra...
A Simple Implementation Technique for Priority Search Queues
 Proceedings of the 2001 International Conference on Functional Programming
, 2001
"... This paper presents a new implementation technique for priority search queues. This abstract data type is an amazing blend of finite maps and priority queues. Our implementation supports logarithmic access to a binding with a given key and constant access to a binding with the minimum value. Priorit ..."
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Cited by 8 (2 self)
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This paper presents a new implementation technique for priority search queues. This abstract data type is an amazing blend of finite maps and priority queues. Our implementation supports logarithmic access to a binding with a given key and constant access to a binding with the minimum value. Priority search queues can be used, for instance, to give a simple, purely functional implementation of Dijkstra's singlesource shortestpaths algorithm.
An Algebra of Scans
 In Mathematics of Program Construction
, 2004
"... A parallel prefix circuit takes n inputs x1 , x2 , . . . , xn and produces the n outputs x1 , x1 x2 , . . . , x1 x2 xn , where `#' is an arbitrary associative binary operation. Parallel prefix circuits and their counterparts in software, parallel prefix computations or scans, have numerous app ..."
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Cited by 7 (0 self)
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A parallel prefix circuit takes n inputs x1 , x2 , . . . , xn and produces the n outputs x1 , x1 x2 , . . . , x1 x2 xn , where `#' is an arbitrary associative binary operation. Parallel prefix circuits and their counterparts in software, parallel prefix computations or scans, have numerous applications ranging from fast integer addition over parallel sorting to convex hull problems. A parallel prefix circuit can be implemented in a variety of ways taking into account constraints on size, depth, or fanout. Traditionally, implementations are either defined graphically or by enumerating the underlying graph. Both approaches have their pros and cons. A figure if well drawn conveys the possibly recursive structure of the scan but it is not amenable to formal manipulation. A description in form of a graph while rigorous obscures the structure of a scan and is equally hard to manipulate. In this paper we show that parallel prefix circuits enjoy a very pleasant algebra. Using only two basic building blocks and four combinators all standard designs can be described succinctly and rigorously. The rules of the algebra allow us to prove the circuits correct and to derive circuit designs in a systematic manner. lord darlington. . . . [Sees a fan lying on the table.] And what a wonderful fan! May I look at it? lady windermere. Do. Pretty, isn't it! It's got my name on it, and everything. I have only just seen it myself. It's my husband's birthday present to me. You know today is my birthday?  Oscar Wilde, Lady Windermere's Fan 1
Perfect Trees and Bitreversal Permutations
, 1999
"... A famous algorithm is the Fast Fourier Transform, or FFT. An efficient iterative version of the FFT algorithm performs as a first step a bitreversal permutation of the input list. The bitreversal permutation swaps elements whose indices have binary representations that are the reverse of each othe ..."
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Cited by 4 (2 self)
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A famous algorithm is the Fast Fourier Transform, or FFT. An efficient iterative version of the FFT algorithm performs as a first step a bitreversal permutation of the input list. The bitreversal permutation swaps elements whose indices have binary representations that are the reverse of each other. Using an amortized approach this operation can be made to run in linear time on a randomaccess machine. An intriguing question is whether a lineartime implementation is also feasible on a pointer machine, that is in a purely functional setting. We show that the answer to this question is in the affirmative. In deriving a solution we employ several advanced programming language concepts such as nested datatypes, associated fold and unfold operators, rank2 types, and polymorphic recursion. 1 Introduction A bitreversal permutation operates on lists whose length is n = 2 k for some natural number k and swaps elements whose indices have binary representations that are the reverse of eac...
Generic programming, now
 Generic Programming, Advanced Lectures, LNCS
, 2006
"... Abstract. Tired of writing boilerplate code? Tired of repeating essentially the same function definition for lots of different datatypes? Datatypegeneric programming promises to end these coding nightmares. In these lecture notes, we present the key abstractions of datatypegeneric programming, giv ..."
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Cited by 3 (1 self)
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Abstract. Tired of writing boilerplate code? Tired of repeating essentially the same function definition for lots of different datatypes? Datatypegeneric programming promises to end these coding nightmares. In these lecture notes, we present the key abstractions of datatypegeneric programming, give several applications, and provide an elegant embedding of generic programming into Haskell. The embedding builds on recent advances in type theory: generalised algebraic datatypes and open datatypes. We hope to convince you that generic programming is useful and that you can use generic programming techniques today! 1
Functional Pearls  Three Algorithms on Braun Trees
, 1997
"... Introduction Among the many flavors of balanced binary trees, Braun trees (Braun & Rem, 1983) are perhaps the most circumscribed. For any given node of a Braun tree, the left subtree is either exactly the same size as the right subtree, or one element larger. Braun trees always have minimum height, ..."
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Cited by 2 (0 self)
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Introduction Among the many flavors of balanced binary trees, Braun trees (Braun & Rem, 1983) are perhaps the most circumscribed. For any given node of a Braun tree, the left subtree is either exactly the same size as the right subtree, or one element larger. Braun trees always have minimum height, and the shape of each Braun tree is completely determined by its size. In return for this rigor, algorithms that manipulate Braun trees are often exceptionally simple and elegant, and need not maintain any explicit balance information. Braun trees have been used to implement both flexible arrays (Braun & Rem, 1983; Hoogerwoord, 1992; Paulson, 1996) and priority queues (Paulson, 1996; Bird, 1996). Most operations involving a single element (e.g. adding, removing, inspecting or updating an element) take O(logn) time since the trees are balanced. We consider three algorithmically interesting operations that manipulate entire trees. First, we give an<F24.