Suppose one has a line segment arrangement consisting entirely of vertical and horizontal segments, and one wants to find the shortest path from one point to another along these segments. Using known algorithms one can solve this in O(n2) time and in O(n2) space. We show that it is possible to find a shortest path in time O(n1.5 log n) and space O(n1.5). Furthermore, if only one path endpoint is known in advance, it is possible to preprocess the arrangement in the same time and space and then find shortest paths for query points in time O(log n).