Abstract
It has long been a fundamental open problem whether polylog time and {\em linear} processors are sufficient to find the strongly connected components of a directed graph and compute directed spanning trees for these components. This paper provides the first nontrivial partial solution to the tree problem: for a planar directed graph with $n$ vertices, if the graph is strongly connected, then a directed spanning tree rooted at a specified vertex can be built in $0(log^2~n)$ time using $0(n)$ processors. The algorithm is deterministic and runs on a parallel random access machine that allows concurrent reads and concurrent writes in its shared memory. The result complements an algorithm by Kao that computes the strongly connected components of a planar directed graph in $0(log^3~n)$ time and $0(n)$ processors.
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