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12
The UnderAppreciated Unfold
 In Proceedings of the Third ACM SIGPLAN International Conference on Functional Programming
, 1998
"... Folds are appreciated by functional programmers. Their dual, unfolds, are not new, but they are not nearly as well appreciated. We believe they deserve better. To illustrate, we present (indeed, we calculate) a number of algorithms for computing the breadthfirst traversal of a tree. We specify brea ..."
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Cited by 49 (10 self)
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Folds are appreciated by functional programmers. Their dual, unfolds, are not new, but they are not nearly as well appreciated. We believe they deserve better. To illustrate, we present (indeed, we calculate) a number of algorithms for computing the breadthfirst traversal of a tree. We specify breadthfirst traversal in terms of levelorder traversal, which we characterize first as a fold. The presentation as a fold is simple, but it is inefficient, and removing the inefficiency makes it no longer a fold. We calculate a characterization as an unfold from the characterization as a fold; this unfold is equally clear, but more efficient. We also calculate a characterization of breadthfirst traversal directly as an unfold; this turns out to be the `standard' queuebased algorithm.
Simple and efficient purely functional queues and deques
 JOURNAL OF FUNCTIONAL PROGRAMMING
, 1995
"... We present purely functional implementations of queues and doubleended queues (deques) requiring only O(1) time per operation in the worst case. Our algorithms are considerably simpler than previous designs with the same bounds. The inspiration for our approach is the incremental behavior of certai ..."
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Cited by 25 (6 self)
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We present purely functional implementations of queues and doubleended queues (deques) requiring only O(1) time per operation in the worst case. Our algorithms are considerably simpler than previous designs with the same bounds. The inspiration for our approach is the incremental behavior of certain functions on lazy lists.
BreadthFirst Numbering: Lessons from a Small Exercise in Algorithm Design
, 2000
"... Every programmer has blind spots. Breadthrst numbering is an interesting toy problem that exposes a blind spot common to manyperhaps mostfunctional programmers. Categories and Subject Descriptors D.1.1 [Programming Techniques]: Applicative (Functional) Programming General Terms Algorithms, De ..."
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Cited by 19 (0 self)
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Every programmer has blind spots. Breadthrst numbering is an interesting toy problem that exposes a blind spot common to manyperhaps mostfunctional programmers. Categories and Subject Descriptors D.1.1 [Programming Techniques]: Applicative (Functional) Programming General Terms Algorithms, Design Keywords Breadthrst numbering, breadthrst traversal, views 1. INTRODUCTION Breadthrst traversal of a tree is easy, but rebuilding the tree afterwards seems to be much harder, at least to functional programmers. At ICFP'98, John Launchbury challenged me with the following problem: Given a tree T , create a new tree of the same shape, but with the values at the nodes replaced by the numbers 1 : : : jT j in breadthrst order. For example, breadthrst numbering of the tree a b # c # # d # # Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for pr...
Inductive Graphs and Functional Graph Algorithms
, 2001
"... We propose a new style of writing graph algorithms in functional languages which is based on an alternative view of graphs as inductively defined data types. We show how this graph model can be implemented efficiently, and then we demonstrate how graph algorithms can be succinctly given by recursive ..."
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Cited by 16 (2 self)
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We propose a new style of writing graph algorithms in functional languages which is based on an alternative view of graphs as inductively defined data types. We show how this graph model can be implemented efficiently, and then we demonstrate how graph algorithms can be succinctly given by recursive function definitions based on the inductive graph view. We also regard this as a contribution to the teaching of algorithms and data structures in functional languages since we can use the functionalstyle graph algorithms instead of the imperative algorithms that are dominant today. Keywords: Graphs in Functional Languages, Recursive Graph Algorithms, Teaching Graph Algorithms in Functional Languages
An Overview of Edison
 Electronic Notes in Theoretical Computer Science
, 2000
"... Edison is a library of functional data structures implemented in Haskell. It supports three main families of abstractions: sequences, collections (e.g., sets and priority queues), and associative collections (e.g., nite maps). This paper summarizes the design of Edison, with particular attention to ..."
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Cited by 15 (0 self)
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Edison is a library of functional data structures implemented in Haskell. It supports three main families of abstractions: sequences, collections (e.g., sets and priority queues), and associative collections (e.g., nite maps). This paper summarizes the design of Edison, with particular attention to how that design is inuenced by details of Haskell. 1 Introduction There is a growing recognition that a useful set of libraries is at least as important to the acceptance of a programming language as the design of the language itself. A library of fundamental data structures such as queues, sets, and nite maps is particularly important in this regard. However, highquality examples of such libraries, such as the STL [14] in C++ or the the collection classes [3] in Smalltalk, are rare. Edison is a library of ecient data structures suitable for implementation and use in functional programming languages. It is named after Thomas Alva Edison and for the mnemonic value of EDiSon (Ecient Data ...
The Role of Lazy Evaluation in Amortized Data Structures
 In Proc. of the International Conference on Functional Programming
, 1996
"... Traditional techniques for designing and analyzing amortized data structures in an imperative setting are of limited use in a functional setting because they apply only to singlethreaded data structures, yet functional data structures can be nonsinglethreaded. In earlier work, we showed how lazy e ..."
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Cited by 13 (2 self)
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Traditional techniques for designing and analyzing amortized data structures in an imperative setting are of limited use in a functional setting because they apply only to singlethreaded data structures, yet functional data structures can be nonsinglethreaded. In earlier work, we showed how lazy evaluation supports functional amortized data structures and described a technique (the banker's method) for analyzing such data structures. In this paper, we present a new analysis technique (the physicist's method) and show how one can sometimes derive a worstcase data structure from an amortized data structure by appropriately scheduling the premature execution of delayed components. We use these techniques to develop new implementations of FIFO queues and binomial queues. 1 Introduction Functional programmers have long debated the relative merits of strict versus lazy evaluation. Although lazy evaluation has many benefits [11], strict evaluation is clearly superior in at least one area:...
Purely Functional, RealTime Deques with Catenation
 Journal of the ACM
, 1999
"... We describe an efficient, purely functional implementation of deques with catenation. In addition to being an intriguing problem in its own right, finding a purely functional implementation of catenable deques is required to add certain sophisticated programming constructs to functional programming ..."
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Cited by 13 (2 self)
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We describe an efficient, purely functional implementation of deques with catenation. In addition to being an intriguing problem in its own right, finding a purely functional implementation of catenable deques is required to add certain sophisticated programming constructs to functional programming languages. Our solution has a worstcase running time of O(1) for each push, pop, inject, eject and catenation. The best previously known solution has an O(log k) time bound for the k deque operation. Our solution is not only faster but simpler. A key idea used in our result is an algorithmic technique related to the redundant digital representations used to avoid carry propagation in binary counting.
Active Patterns
 In 8th Int. Workshop on Implementation of Functional Languages, LNCS 1268
, 1996
"... . Active patterns apply preprocessing functions to data type values before they are matched. This is of use for unfree data types where more than one representation exists for an abstract value: in many cases there is a specific representation for which function definitions become very simple, and a ..."
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Cited by 12 (3 self)
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. Active patterns apply preprocessing functions to data type values before they are matched. This is of use for unfree data types where more than one representation exists for an abstract value: in many cases there is a specific representation for which function definitions become very simple, and active patterns just allow to assume this specific representation in function definitions. We define the semantics of active patterns and describe their implementation. 1 Introduction Pattern matching is a wellappreciated concept of (functional) programming. It contributes to concise function definitions by implicitly decomposing data type values. In many cases, a large part of a function's definition is only needed to prepare for recursive application and to finally lead to a base case for which the definition itself is rather simple. Such function definitions could be simplified considerably if these preparatory computations could be factorized and given in a separate place. Recognizing t...
Programming with Variable Functions
 In Proceedings of the 1998 ACM SIGPLAN International Conference on Functional Programming
, 1998
"... What is a good method to specify and derive imperative programs? This paper argues that a new form of functional programming fits the bill, where variable functions can be updated at specified points in their domain. Traditional algebraic specification and functional programming are a powerful pair ..."
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Cited by 8 (0 self)
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What is a good method to specify and derive imperative programs? This paper argues that a new form of functional programming fits the bill, where variable functions can be updated at specified points in their domain. Traditional algebraic specification and functional programming are a powerful pair of tools for specifying and implementing domains of discourse and operations on them. Recent work on evolving algebras has introduced the function update in algebraic specifications, and has applied it with good success in the modelling of reactive systems. We show that similar concepts allow one to derive efficient programs in a systematic way from functional specifications. The final outcome of such a derivation can be made as efficient as a traditional imperative program with pointers, but can still be reasoned about at a high level. Variable functions can also play an important role in the structuring of large systems. They can subsume objectoriented programming languages, without incu...