AN OPTIMAL ALGORITHM FOR SHORTEST PATHS ON WEIGHTED INTERVAL AND CIRCULAR-ARC GRAPHS, WITH APPLICATIONS

We give the first linear-time algorithm for computing single-source shortest paths in a weighted interval or circular-are graph, when we are given the model of that graph, i.e., the actual weighted intervals or circular-arcs and the sorted list of the interval endpoints. Our algorithm solves this problem optimally in O(n) time, where n is the number of intervals or circular-arcs in a graph. An immediate consequence of our result is an O(qn + n log n)-time algorithm for the minimum-weight circle-cover problem, where q is the minimum number of arcs crossing any point on the circle; the n log n term in this time complexity is from a preprocessing sorting step when the sorted list of endpoints is not given as part of the input. The previously best time bounds were O(n log n) for this shortest paths problem, and O(qn log n) for the minimum-weight circle-cover problem. Thus we improve the bounds of both problems. More importantly, the techniques we give hold the promise of achieving similar (log n)-factor improvements in other problems on such graphs.