Simple Linear - Time Algorithms

We present global and local algorithms for evaluating minimal xed points of dependency graphs, a general problem in xed-point computation and model checking. Our algorithms run in linear-time, matching the complexity of the best existing algorithms for similar problems, and are simple to understand. The main novelty of our global algorithm is that it does not use the counter and \reverse list" data structures commonly found in existing linear-time global algorithms. This distinction plays an essential role in allowing us to easily derive our local algorithm from our global one. Our local algorithm is distinguished from existing linear-time local algorithms by a combination of its simplicity and suitability for direct implementation. We also provide linear-time reductions from the problems of computing minimal and maximal xed points in Boolean graphs to the problem of minimal xed-point evaluation in dependency graphs. This establishes dependency graphs as a suitable framework in which to express and compute alternation-free xed points. Finally, we relate HORNSAT, the problem of Horn formula satissability, to the problem of minimal xed-point evaluation in dependency graphs. In particular, we present straightforward, linear-time reductions between these problems for both directions of reducibility. As a result, we derive a linear-time local algorithm for HORNSAT, the rst of its kind as far as we are aware.