Finding Many Optimal Paths Without Growing Any Optimal Path Trees

Abstract

Many algorithms seek to compute actual optimal paths in weighted directed graphs. The standard approach for reporting an actual optimal path is based on building a single-source optimal path tree. A technique was given in [1] for a class of problems such that a single actual optimal path can be reported without maintaining any single-source optimal path tree, thus significantly reducing the space bound of those problems with no or little increase in their running time. In this paper, we extend the technique in [1] to the generalized problem of reporting many actual optimal paths with different starting and ending vertices in certain directed graphs. We show how this new technique yields improved results on several application problems, such as reconstructing a 3-D surface band bounded by two simple closed curves, finding various constrained segmentation of 2-D medical images, and circular string-to-string correction. Although the generalized many-path problem seems more difficult, our algorithms have nearly the same space and time bounds as those of the single-path cases. Our technique is likely to help improve other optimal paths or dynamic programming algorithms. We also correct an error in the time/space complexity for the circular string-to-string correction algorithm in [7] and give improved results for it.