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Functional Programming with Graphs
 2ND ACM SIGPLAN INT. CONF. ON FUNCTIONAL PROGRAMMING
, 1997
"... Graph algorithms expressed in functional languages often suffer from their inherited imperative, statebased style. In particular, this impedes formal program manipulation. We show how to model persistent graphs in functional languages by graph constructors. This provides a decompositional view of g ..."
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Cited by 29 (10 self)
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Graph algorithms expressed in functional languages often suffer from their inherited imperative, statebased style. In particular, this impedes formal program manipulation. We show how to model persistent graphs in functional languages by graph constructors. This provides a decompositional view of graphs which is very close to that of data types and leads to a "more functional" formulation of graph algorithms. Graph constructors enable the definition of general fold operations for graphs. We present a promotion theorem for one of these folds that allows program fusion and the elimination of intermediate results. Fusion is not restricted to the elimination of treelike structures, and we prove another theorem that facilitates the elimination of intermediate graphs. We describe an MLimplementation of persistent graphs which efficiently supports the presented fold operators. For example, depthfirstsearch expressed by a fold over a functional graph has the same complexity as the corresp...
Structuring DepthFirst Search Algorithms in Haskell
, 1995
"... Depthfirst search is the key to a wide variety of graph algorithms. In this paper we express depthfirst search in a lazy functional language, obtaining a lineartime implementation. Unlike traditional imperative presentations, we use the structuring methods of functional languages to construct alg ..."
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Cited by 26 (0 self)
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Depthfirst search is the key to a wide variety of graph algorithms. In this paper we express depthfirst search in a lazy functional language, obtaining a lineartime implementation. Unlike traditional imperative presentations, we use the structuring methods of functional languages to construct algorithms from individual reusable components. This style of algorithm construction turns out to be quite amenable to formal proof, which we exemplify through a calculationalstyle proof of a far from obvious stronglyconnected components algorithm. Classifications: Computing Paradigms (functional programming) ; Environments, Implementations, and Experience (programming, graph algorithms). 1 Introduction The importance of depthfirst search (DFS) for graph algorithms was established twenty years ago by Tarjan (1972) and Hopcroft and Tarjan (1973) in their seminal work. They demonstrated how depthfirst search could be used to construct a variety of efficient graph algorithms. In practice, this...
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
Graph Algorithms = Iteration + Data Structures?  The Structure of Graph Algorithms and a Corresponding Style of Programming
 of Programming, 18th Int. Workshop on GraphTheoretic Concepts in Computer Science, LNCS 657
, 1992
"... We encourage a specific view of graph algorithms, which can be paraphrased by "iterate over the graph elements in a specific order and perform computations in each step". Data structures will be used to define the processing order, and we will introduce recursive mapping/array definitions as a vehic ..."
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Cited by 9 (5 self)
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We encourage a specific view of graph algorithms, which can be paraphrased by "iterate over the graph elements in a specific order and perform computations in each step". Data structures will be used to define the processing order, and we will introduce recursive mapping/array definitions as a vehicle for specifying the desired computations. Being concerned with the problem of implementing graph algorithms, we outline the extension of a functional programming language by graph types and operations. In particular, we explicate an exploration operator that essentially embodies the proposed view of algorithms. Fortunately, the resulting specifications of algorithms, in addition to being compact and declarative, are expected to have an almost as efficient implementation as their imperative counterparts. 1 Introduction Looking at graph algorithms, we observe that many of them have a very similar structure, namely iterating over nodes or edges and thereby performing computations. Since grap...
Lazy DepthFirst Search and Linear Graph Algorithms in Haskell
 Glasgow Workshop on Functional Programming
, 1994
"... Depthfirst search is the key to a wide variety of graph algorithms. In this paper we explore the implementation of depth first search in a lazy functional language. For the first time in such languages we obtain a lineartime implementation. But we go further. Unlike traditional imperative presenta ..."
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Cited by 6 (0 self)
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Depthfirst search is the key to a wide variety of graph algorithms. In this paper we explore the implementation of depth first search in a lazy functional language. For the first time in such languages we obtain a lineartime implementation. But we go further. Unlike traditional imperative presentations, algorithms are constructed from individual components, which may be reused to create new algorithms. Furthermore, the style of program is quite amenable to formal proof, which we exemplify through a calculationalstyle proof of a stronglyconnected components algorithm. 1 Introduction Graph algorithms have long been a challenge to programmers of lazy functional languages. It has not been at all clear how to express such algorithms without using side effects to achieve efficiency. For example, many texts provide implementations of search algorithms which are quadratic in the size of the graph (see Paulson (1991), Holyer (1991), or Harrison (1993)), compared with the standard linear im...
An InitialAlgebra Approach to Directed Acyclic Graphs
, 1995
"... . The initialalgebra approach to modelling datatypes consists of giving constructors for building larger objects of that type from smaller ones, and laws identifying di#erentways of constructing the same object. The recursive decomposition of objects of the datatype leads directly to a recursiv ..."
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Cited by 5 (0 self)
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. The initialalgebra approach to modelling datatypes consists of giving constructors for building larger objects of that type from smaller ones, and laws identifying di#erentways of constructing the same object. The recursive decomposition of objects of the datatype leads directly to a recursive pattern of computation on those objects, whichisvery helpful for both functional and parallel programming. We showhow to model a particular kind of directed acyclic graph using this initialalgebra approach. Keywords. Graphs, data types, catamorphisms, initial algebras, BirdMeertens Formalism, program derivation. 1 Introduction It is now widely recognized that the traditional adhoc approaches to program construction do not yield reliable software; a more systematic and formal approach is required. One such approach consists of program veri#cationproving after the fact that a given program satis#es its formal speci#cation. This approach turns out to be di#cult to implement, not lea...
Lazy Least Fixed Points in ML
"... In this paper, we present an algorithm for computing the least solution of a system of monotone equations. This algorithm can be viewed as an effective form of the following wellknown fixed point theorem: Theorem Let V be a finite set of variables. Let (P, ≤, ⊥) be a partially ordered set ..."
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In this paper, we present an algorithm for computing the least solution of a system of monotone equations. This algorithm can be viewed as an effective form of the following wellknown fixed point theorem: Theorem Let V be a finite set of variables. Let (P, ≤, ⊥) be a partially ordered set
Purity, Impurity and Efficiency in Graph Algorithms
"... Introduction This chapter initially considers pure lazy functional languages: their philosophy, advantages and disadvantages. We then examine how to develop efficient lazy functional programs. One way to achieve efficiency is to introduce impurities. In the final section the two schools of lazy fun ..."
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Introduction This chapter initially considers pure lazy functional languages: their philosophy, advantages and disadvantages. We then examine how to develop efficient lazy functional programs. One way to achieve efficiency is to introduce impurities. In the final section the two schools of lazy functional programming, pure and impure, are assessed. The assessment centres around two partial implementations of the Hopcroft Tarjan graph planarity algorithm. Profiling tools are used to make an experimental comparison and optimisation of each program. 4.1 Lazy Functional Programming In his book [42] Reade suggests that the user of a traditional imperative language is required to do the following: 1. describe the result to be computed; 2. impose an order on the steps required in the computation; 3. create and destroy, as required, any data structures used by the computation. 74 The first item is concerned with the extensional prope