Simultaneous Time-Space Upper Bounds for Certain Problems in Planar Graphs

In this paper, we show that given a weighted, directed planar graph G, and any > 0, there exists a polynomial time and O(n 1 2+ ) space algorithm that computes the shortest path between two fixed vertices in G. We also consider the RedBluePath problem, which states that given a graph G whose edges are colored either red or blue and two fixed vertices s and t in G, is there a path from s to t in G that alternates between red and blue edges. The RedBluePath problem in planar DAGs is NL-complete. We exhibit a polynomial time and O(n 1 2+ ) space algorithm (for any > 0) for the RedBluePath problem in planar DAG. In the last part of this paper, we consider the problem of deciding and constructing the perfect matching present in a planar bipartite graph and also a similar problem which is to find a Hall-obstacle in a planar bipartite graph. We show the time-space bound of these two problems are same as the bound of shortest path problem in a directed planar graph.