Suppose you're a consultant for the Ergonomic Architecture Commission, and they come t...

Suppose you're a consultant for the Ergonomic Architecture Commission, and they come to you with the following problem.

They're really concerned about designing houses that are "user-friendly," and they've been having a lot of trouble with the setup of light fixtures and switches in newly designed houses. Consider, for example, a one-floor house with n light fixtures and n locations for light switches mounted in the wall. You'd like to be able to wire up one switch to control each light fixture, in such a way that a person at the switch can see the light fixture being controlled.

Sometimes this is possible and sometimes it isn't. Consider the two simple floor plans for houses in Figure 1 7.28. There are three light fixtures (labeled a, b, c) and three switches (labeled 1, 2, 3). It is possible to wire switches to fixtures in Figure 1 (a) so that every switch has a line of sight to the fixture, but this is not possible in Figure 1 (b).

Let's call a floor plan, together with n light fixture locations and n switch locations, ergonomic if it's possible to wire one switch to each fixture so that every fixture is visible from the switch that controls it. A floor plan will be represented by a set of m horizontal or vertical line segments in the plane (the walls), where the ith wall has endpoints . Each of the n switches and each of the n fixtures is given by its coordinates in the plane. A fixture is visible from a switch if the line segment joining them does not cross any of the walls.

Give an algorithm to decide if a given floor plan is ergonomic. The running time should be polynomial in m and n. You may assume that you have a subroutine with O(1) running time that takes two line segments as input and decides whether or not they cross in the plane.

Figure 1 The floor plan in (a) is ergonomic, because we can wire switches to fixtures in such a way that each fixture is visible from the switch that controls it. (This can be done by wiring switch 1 to a, switch 2 to b, and switch 3 to c.) The floor plan in (b) is not ergonomic, because no such wiring is possible.