  In this paper, we show that given a weighted, directed planar graph $G$, and
any $\epsilon >0$, there exists a polynomial time and
$O(n^{\frac{1}{2}+\epsilon})$ space algorithm that computes the shortest path
between two fixed vertices in $G$.
  We also consider the {\RB} 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 {\RB} problem in planar DAGs is {\NL}-complete. We exhibit a
polynomial time and $O(n^{\frac{1}{2}+\epsilon})$ space algorithm (for any
$\epsilon >0$) for the {\RB} 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.
