A linear-time algorithm for routing in a convex grid

An algorithm for the problem of routing two-terminal nets in a convex grid is presented. A convex grid is a subset R of the planar rectangular grid without any nontrivial holes, i.e. every finite face has exactly four incident vertices, so that every vertical and horizontal line crosses the boundary of the grid at most twice. A net is a pair of vertices of nonmaximal degree on the boundary of A. A solution of the problem is a set of edge-disjoint paths, one for each net. The vertices of a net are called its terminals. The algorithm is based on a theorem of H. Okamura and P.D. Seymour (J. Combinatorial Theory, vol.31, series B., p.75-81, 1981) on multicommodity flows in a planar graphs. The algorithm is very simple, uses only one simple data structure and works in time O(n) on a more general routing region