This paper introduces a new approach to optimising array algorithms in functional languages. We are specifically aiming at an efficient implementation of irregular array algorithms that are hard to implement in conventional array languages such as Fortran. We optimise the storage layout of arrays containing complex data structures and reduce the running time of functions operating on these arrays by meansofequationalprogramtransformations. Inparticular, this paper discusses a novel form of combinator loop fusion, whichbyremovingintermediatestructuresoptimisestheuse of the memory hierarchy. We identify a combinator named loopP that provides a general scheme for iterating over an array and that in conjunction with an array constructor replicateP is sufficient to express a wide range of array algorithms. On this basis, we define equational transformation rules that combine traversals of loopP and replicateP as well as sequences of applications of loopP into a single loopP traversal. Our approach naturally generalises to a parallel implementation and includes facilities for optimising load balancing and communication. A prototype implementation based on the rewrite rule pragma of the Glasgow Haskell Compiler is significantly faster than standard Haskell arrays and approaches the speed of hand coded C for simple examples. 1.

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 functional-style 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

Triangulations of a surface are of fundamental importance in computational geometry, computer graphics, and engineering and scientific simulations. Triangulations are ordinarily represented as mutable graph structures for which both adding and traversing edges take constant time per operation. These representations of triangulations make it di#cult to support persistence, including "multiple futures", the ability to use a data structure in several unrelated ways in a given computation; "time travel", the ability to move freely among versions of a data structure; or parallel computation, the ability to operate concurrently on a data structure without interference. We present a purely functional interface and representation of triangulated surfaces, and more generally of simplicial complexes in higher dimensions. In addition to being persistent in the strongest sense, the interface more closely matches the mathematical definition of triangulations (simplicial complexes) than do interfaces based on mutable representations. The representation, however, comes at the cost of requiring O(lg n) time for traversing or adding triangles (simplices), where n is the number of triangles in the surface. We show both analytically and experimentally that for certain important cases, this extra cost does not seriously a#ect end-to-end running time. Analytically, we present a new randomized algorithm for 3-dimensional Convex Hull based on our representations for which the running time matches the #(n lg n) lower-bound for the problem. This is achieved by using only O(n) traversals of the surface. Experimentally, we present results for both an implementation of the 3-dimensional Convex Hull and for a terrain modeling algorithm, which demonstrate that, although there is some cost to persis...

. Despite a lot of recent interest in purely functional data structures, for example [Ada93, Oka95, BO96, Oka96, OB97, Erw97], few have been benchmarked. Of these, even fewer have their performance qualified by how they are used. But how a data structure is used can significantly affect performance. This paper makes three original contributions. (1) We present an algorithm for generating a benchmark according to a given use of data structure. (2) We compare use of an automated tool based on this algorithm, with the traditional technique of hand-picked benchmarks, by benchmarking six implementations of random-access list using both methods. (3) We use the results of this benchmarking to present a decision tree for the choice of random-access list implementation, according to how the list will be used. 1 Motivation Recent years have seen renewed interest in purely functional data structures: sets [Ada93], random-access lists [Oka95], priority queues [BO96], arrays [OB97], gr...

When someone designs a new data structure, they want to know how well it performs. Previously, the only way to do this involves finding, coding and testing some applications to act as benchmarks. This can be tedious and time-consuming. Worse, how a benchmark uses a data structure may considerably affect the efficiency of the data structure. Thus, the choice of benchmarks may bias the results. For these reasons, new data structures developed for functional languages often pay little attention to empirical performance. We solve these problems by developing a benchmarking tool, Auburn, that can generate benchmarks across a fair distribution of uses. We precisely define "the use of a data structure", upon which we build the core algorithms of Auburn: how to generate a benchmark from a description of use, and how to extract a description of use from an application. We consider how best to use these algorithms to benchmark competing data structures. Finally, we test Auburn by benchmarking ...

One of the most serious shortcomings of multidimensional languages is their inability to collaborate with conventional programming languages and systems. Multidimensional languages are used in order to define (potentially infinite) streams, grids, cubes, and so on, concepts which resemble in nature to the familiar imperative arrays. The main difference is that the former entities are lazy while the latter are generally eager. This paper proposes the embedding of multidimensional languages into conventional ones as a form of definitional lazy arrays. The paper describes the details of an implementation of the proposed idea as well as the expressibility and the performance of the resulting system. The main advantage of the new approach is that multidimensional languages can now benefit from the advanced features that have been developed for conventional languages. Moreover, multidimensionality adds to conventional languages the idea of lazy arrays, which in many cases o#er significant advantages compared to the classical imperative arrays.

PHP is a popular language for server-side applications. In PHP, assignment to variables copies the assigned values, according to its so-called copy-on-assignment semantics. In contrast, a typical PHP implementation uses a copy-on-write scheme to reduce the copy overhead by delaying copies as much as possible. This leads us to ask if the semantics and implementation of PHP coincide, and actually this is not the case in the presence of sharings within values. In this paper, we describe the copy-on-assignment semantics with three possible strategies to copy values containing sharings. The current PHP implementation has inconsistencies with these semantics, caused by its naïve use of copy-on-write. We fix this problem by the novel mostly copy-on-write scheme, making the copy-onwrite implementations faithful to the semantics. We prove that our copy-on-write implementations are correct, using bisimulation with the copy-on-assignment semantics. D.3.0 [Programming Lan-