We develop novel algorithms to handle the single-edge updating and the batch edge updating. In this paper, we focus on dynamic algorithms for shortest point-to-point paths computation in directed graphs with positive edge weights. Work partially supported by NSF Grants CCR-8814977, CCR-9014605, by the ESPRIT II Basic Research Actions Program of the EC under contract No. This is the first sublinear-time algorithm known for this problem. Springer, Berlin, Heidelberg Shortest paths computation in graph is one of the most fundamental operation in many applications such as social network and sensor network.
A similar approach has been applied since the early 1980s to some polynomial time solvable optimization problems such as minimum spanning tree  and shortest path =-=[14, 31]-=- with the aim to maintain the optimum solution of the given problem under input modification (say elimination or insertion of an edge or update of an edge weight). This problem also subsume ..." The grammar problem, a generalization of the single-source shortest-path problem introduced by Knuth, is to compute the minimum-cost derivation of a terminal string from each non-terminal of a given context-free grammar, with the cost of a derivation being suitably defined. To address this problem, dynamic algorithm that computes the shortest-path in response to updates is in demand. When a large graph is updated with small changes, it is really expensive to recompute the new shortest path via the traditional static algorithms. Then a new instance I ′ is proposed, obtained by means of a slight perturbation of instance I. Then a new instance I ′ is proposed, obtained by means of a slight perturbation of instance ..." Abstract. We assume that an instance I of a computationally hard optimization problem has been solved and that we know the optimum solution of such instance. The data structure supports also insertions of new vertices and deletions of disconnected vertices in the same time bounds.