Lazy Depth-First Search and Linear Graph Algorithms in Haskell

Depth-first 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 linear-time 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 calculational-style proof of a strongly-connected 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..