Abstract: We present a simple parallel algorithm for the single-source shortest path
problem in planardigraphs with nonnegative real edge weights. The algorithm runs
on the EREW PRAM model of parallel computation in O((n2=+n1&=) log n)
time, performing O(n1+= log n) work for any 0<{\aa}<1/2. The strength of the
algorithm is its simplicity, making it easy to implement and presumable quite
efficient in practice. The algorithm improves upon the work of all previous
parallel algorithms. Our algorithm is based on a region decomposition of the
input graph and uses a well-known parallel implementation of Dijkstra's
algorithm. The logarithmic factor in both the work and the time can be
eliminated by plugging in a less practical, sequential planar shortest path
algorithm together with an improved parallel implementation of Dijkstra's
algorithm.

Abstract: We describe algorithms for finding shortest paths and distances in outerplanar and planardigraphs
that exploit the particular topology of the input graph. An important feature of our algorithms is that they can
work in a dynamic environment, where the cost of any edge can be changed or the edge can be deleted. In the
case of outerplanardigraphs, our data structures can be updated after any such change in only logarithmic time.
A distance query is also answered in logarithmic time. In the case of planardigraphs, we give an interesting
tradeoff between preprocessing, query, and update times depending on the value of a certain topological
parameter of the graph. Our results can be extended to n-vertex digraphs of genus O.n1¡"/ for any " > 0.

Abstract: We consider the problem of preprocessing an n-vertex digraph with real edge weights so that subsequent queries for the shortest path or distance between any two vertices can be efficiently answered. We give parallel algorithms for the EREW PRAM model of computation that depend on the treewidth of the input graph. When the treewidth is a constant, our algorithms can answer distance queries in O({\'a}(n)) time using a single processor, after a preprocessing of O(log2n) time and O(n) work, where {\'a}(n) is the inverse of Ackermann's function. The class of constant treewidth graphs contains outerplanar graphs and series-parallel graphs, among others. To the best of our knowledge, these are the first parallel algorithms which achieve these bounds for any class of graphs except trees. We also give a dynamic algorithm which, after a change in an edge weight, updates our data structures in O(log n) time using O(n{\^a}) work, for any constant 0 < {\^a} < 1. Moreover, we give an algorithm of independent interest: computing a shortest path tree, or finding a negative cycle in O(log2n) time using O(n) work.