A certifying algorithm for a problem is an algorithm that provides a certificate with each answer that it produces. The certificate is a piece of evidence that proves that the answer has not been compromised by a bug in the implementation. We give linear-time certifying algorithms for recognition of interval graphs and permutation graphs, and for a few other related problems. Previous algorithms fail to provide supporting evidence when they claim that the input graph is not a member of the class. We show that our certificates of nonmembership can be authenticated in O(|V|) time.

) Abstract We consider the problem of maintaining the distances and the shortest paths from a single source in either a directed or an undirected graph with positive real edge weights, handling insertions, deletions and cost updates of edges. We propose fully dynamic algorithms with optimal space requirements and query time. The cost of update operations depends on the class of the considered graph and on the number of vertices that, due to an edge modification, either change their distance from the source or change their parent in the shortest path tree. In the case of graphs with bounded genus (including planar graphs), bounded degree graphs, bounded treewidth graphs and fi-near-planar graphs with bounded fi, the update procedures require O(log n) amortized time per vertex update, while for general graphs with n vertices and m edges they require O( p m log n) amortized time per vertex update. The solution is based on a dynamization of Dijkstra's algorithm [6] and requires simple ...

Recently two different linear time approximation algorithms for the weighted matching problem in graphs have been suggested [5][17]. Both these

Shape and location of objects in a spatial database are commonly represented by geometric data such as points, lines and regions. Numerous geometric operators and predicates have been proposed for spatial database systems. Existing work on their implementation concentrate on individual operators and predicates. This approach makes the realization of geometric operators and predicates in a spatial database system difficult since they are diverse and their implementation in general are complex. In this paper, we present a simple plane-sweep algorithm that can be easily modified to realize efficiently a set of frequently used line-region and region-region geometric operators and predicates. The design of this unified algorithm is based on the observation that the covering of elementary regions along the sweep line are updated locally and the implementation of these operators and predicates differ only with the output actions at an intersection point. Any geometric operator or predicate, t...

of maximal transversal algorithms