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18
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.
Optimal Purely Functional Priority Queues
 JOURNAL OF FUNCTIONAL PROGRAMMING
, 1996
"... Brodal recently introduced the first implementation of imperative priority queues to support findMin, insert, and meld in O(1) worstcase time, and deleteMin in O(log n) worstcase time. These bounds are asymptotically optimal among all comparisonbased priority queues. In this paper, we adapt B ..."
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Cited by 18 (1 self)
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Brodal recently introduced the first implementation of imperative priority queues to support findMin, insert, and meld in O(1) worstcase time, and deleteMin in O(log n) worstcase time. These bounds are asymptotically optimal among all comparisonbased priority queues. In this paper, we adapt Brodal's data structure to a purely functional setting. In doing so, we both simplify the data structure and clarify its relationship to the binomial queues of Vuillemin, which support all four operations in O(log n) time. Specifically, we derive our implementation from binomial queues in three steps: first, we reduce the running time of insert to O(1) by eliminating the possibility of cascading links; second, we reduce the running time of findMin to O(1) by adding a global root to hold the minimum element; and finally, we reduce the running time of meld to O(1) by allowing priority queues to contain other priority queues. Each of these steps is expressed using MLstyle functors. The last transformation, known as datastructural bootstrapping, is an interesting application of higherorder functors and recursive structures.
Purely Functional RandomAccess Lists
 In Functional Programming Languages and Computer Architecture
, 1995
"... We present a new data structure, called a randomaccess list, that supports array lookup and update operations in O(log n) time, while simultaneously providing O(1) time list operations (cons, head, tail). A closer analysis of the array operations improves the bound to O(minfi; log ng) in the wor ..."
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Cited by 17 (2 self)
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We present a new data structure, called a randomaccess list, that supports array lookup and update operations in O(log n) time, while simultaneously providing O(1) time list operations (cons, head, tail). A closer analysis of the array operations improves the bound to O(minfi; log ng) in the worst case and O(log i) in the expected case, where i is the index of the desired element. Empirical evidence suggests that this data structure should be quite efficient in practice. 1 Introduction Lists are the primary data structure in every functional programmer 's toolbox. They are simple, convenient, and usually quite efficient. The main drawback of lists is that accessing the ith element requires O(i) time. In such situations, functional programmers often find themselves longing for the efficient random access of arrays. Unfortunately, arrays can be quite awkward to implement in a functional setting, where previous versions of the array must be available even after an update. Since arra...
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
Fusion of Recursive Programs with Computational Effects
 Theor. Comp. Sci
, 2000
"... Fusion laws permit to eliminate various of the intermediate data structures that are created in function compositions. The fusion laws associated with the traditional recursive operators on datatypes cannot in general be used to transform recursive programs with effects. Motivated by this fact, t ..."
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Cited by 14 (4 self)
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Fusion laws permit to eliminate various of the intermediate data structures that are created in function compositions. The fusion laws associated with the traditional recursive operators on datatypes cannot in general be used to transform recursive programs with effects. Motivated by this fact, this paper addresses the definition of two recursive operators on datatypes that capture functional programs with effects. Effects are assumed to be modeled by monads. The main goal is thus the derivation of fusion laws for the new operators. One of the new operators is called monadic unfold. It captures programs (with effects) that generate a data structure in a standard way. The other operator is called monadic hylomorphism, and corresponds to programs formed by the composition of a monadic unfold followed by a function defined by structural induction on the data structure that the monadic unfold generates. 1 Introduction A common approach to program design in functional programmin...
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.
Simple Confluently Persistent Catenable Lists
 SIAM JOURNAL ON COMPUTING
, 1998
"... We consider the problem of maintaining persistent lists subject to concatenation and to insertions and deletions at both ends. Updates to a persistent data structure are nondestructive  each operation produces a new list incorporating the change, while keeping intact the list or lists to which it a ..."
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Cited by 11 (2 self)
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We consider the problem of maintaining persistent lists subject to concatenation and to insertions and deletions at both ends. Updates to a persistent data structure are nondestructive  each operation produces a new list incorporating the change, while keeping intact the list or lists to which it applies. Although general techniques exist for making data structures persistent, these techniques fail for structures that are subject to operations, such as catenation, that combine two or more versions. In this paper we develop a simple implementation of persistent doubleended queues with catenation that supports all deque operations in constant amortized time. Our implementation is functional if we allow memoization.
Pattern Guards and Transformational Patterns
, 2000
"... We propose three extensions to patterns and pattern matching in Haskell. The first, pattern guards, allows the guards of a guarded equation to match patterns and bind variables, as well as to test boolean condition. For this we introduce a natural generalisation of guard expressions to guard quali ..."
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Cited by 7 (0 self)
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We propose three extensions to patterns and pattern matching in Haskell. The first, pattern guards, allows the guards of a guarded equation to match patterns and bind variables, as well as to test boolean condition. For this we introduce a natural generalisation of guard expressions to guard qualifiers. A frequentlyoccurring special case is that a function should be applied to a matched value, and the result of this is to be matched against another pattern. For this we introduce a syntactic abbreviation, transformational patterns, that is particularly useful when dealing with views. These proposals can be implemented with very modest syntactic and implementation cost. They are upward compatible with Haskell; all existing programs will continue to work. We also offer a third, much more speculative proposal, which provides the transformationalpattern construct with additional power to explicitly catch pattern match failure. We demonstrate the usefulness of the proposed extension by several examples, in particular, we compare our proposal with views, and we also discuss the use of the new patterns in combination with equational reasoning.
Amortization, Lazy Evaluation, and Persistence: Lists with Catenation via Lazy Linking
 Pages 646654 of: IEEE Symposium on Foundations of Computer Science
, 1995
"... Amortization has been underutilized in the design of persistent data structures, largely because traditional accounting schemes break down in a persistent setting. Such schemes depend on saving "credits" for future use, but a persistent data structure may have multiple "futures", each competing for ..."
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Cited by 7 (1 self)
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Amortization has been underutilized in the design of persistent data structures, largely because traditional accounting schemes break down in a persistent setting. Such schemes depend on saving "credits" for future use, but a persistent data structure may have multiple "futures", each competing for the same credits. We describe how lazy evaluation can often remedy this problem, yielding persistent data structures with good amortized efficiency. In fact, such data structures can be implemented purely functionally in any functional language supporting lazy evaluation. As an example of this technique, we present a purely functional (and therefore persistent) implementation of lists that simultaneously support catenation and all other usual list primitives in constant amortized time. This data structure is much simpler than the only existing data structure with comparable bounds, the recently discovered catenable lists of Kaplan and Tarjan, which support all operations in constant worstca...