We solve the subgraph isomorphism problem in planar graphs in linear time, for any pattern of constant size. Our results are based on a technique of partitioning the planar graph into pieces of small tree-width, and applying dynamic programming within each piece. The same methods can be used to solve other planar graph problems including connectivity, diameter, girth, induced subgraph isomorphism, and shortest paths.

Heuristic search methods promise to find shortest paths for path-planning problems faster than uninformed search methods. Incremental search methods, on the other hand, promise to find shortest paths for series of similar path-planning problems faster than is possible by solving each path-planning problem from scratch. In this article, we develop Lifelong Planning A * (LPA*), an incremental version of A * that combines ideas from the artificial intelligence and the algorithms literature. It repeatedly finds shortest paths from a given start vertex to a given goal vertex while the edge costs of a graph change or vertices are added or deleted. Its first search is the same as that of a version of A * that breaks ties in favor of vertices with smaller g-values but many of the subsequent searches are potentially faster because it reuses those parts of the previous search tree that are identical to the new one. We present analytical results that demonstrate its similarity to A * and experimental results that demonstrate its potential advantage in two different domains if the path-planning problems change only slightly and the changes are close to the goal.

) 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 ...

Mobile robots often operate in domains that are only incompletely known, for example, when they have to move from given start coordinates to given goal coordinates in unknown terrain. In this case, they need to be able to replan quickly as their knowledge of the terrain changes. Stentz’ Focussed Dynamic A * (D*) is a heuristic search method that repeatedly determines a shortest path from the current robot coordinates to the goal coordinates while the robot moves along the path. It is able to replan faster than planning from scratch since it modifies its previous search results locally. Consequently, it has been extensively used in mobile robotics. In this article, we introduce an alternative to D * that determines the same paths and thus moves the robot in the same way but is algorithmically different. D * Lite is simple, can be rigorously analyzed, extendible in multiple ways, and is at least as efficient as D*. We believe that our results will make D*-like replanning methods even more popular and enable robotics researchers to adapt them to additional applications.

Incremental search reuses information from previous searches to find solutions to a series of similar search problems potentially faster than is possible by solving each search problem from scratch. This is important since many artificial intelligence systems have to adapt their plans continuously to changes in (their knowledge of) the world. In this article, we therefore give an overview of incremental search, focusing on Lifelong Planning A*, and outline some of its possible applications in artificial intelligence. Overview It is often important that searches be fast. Artificial intelligence has developed several ways of speeding up searches by trading off the search time and the cost of the resulting path. This includes using inadmissible heuristics (Pohl

The fully dynamic planarity testing problem consists of performing an arbitrary sequence of the following three kinds of operations on a planar graph G: (i) insert an edge if the resultant graph remains planar; (ii) delete an edge; and (iii) test whether an edge could be added to the graph without violating planarity. We show how to support each of the above operations in O(n2=3) time, where n is the number of vertices in the graph. The bound for tests and deletions is worst-case, while the bound for insertions is amortized. This is the first algorithm for this problem with sub-linear running time, and it affirmatively answers a question posed in [11]. The same data structure has further applications in maintaining the biconnected and triconnected components of a dynamic planar graph. The time bounds are the same: O(n2=3) worst-case time per edge deletion, O(n2=3) amortized time per edge insertion, and O(n2=3) worst-case time to check whether two vertices are either biconnected or triconnected.

We consider the problem of maintaining a large matching and a small vertex cover in a dynamically changing graph. Each update to the graph is either an edge deletion or an edge insertion. We give the first randomized data structure that simultaneously achieves a constant approximation factor and handles a sequence of K updates in K · polylog(n) time, where n is the number of vertices in the graph. Previous data structures require a polynomial amount of computation per update.

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