Fig. 19. A user interface for controlling a stereo rendered automatically by Supple fora PDA (top) and a desktop computer (bottom). Fig. 20. Supple optimally uses the available space and robustly degrades the quality of the rendered interface if presented with a device with a smaller screen size. This figure shows three renderings of a classroom controller on three devices with progressively narrower screens. optimal interface. Conversely, given a very small amount of screen space, there will be very few or no legal renderings and the constraint propagation process will easily narrow down the solution space to a very small fraction of the original. The hardest problems are therefore somewhere in the middle, in the area where the problem transitions from being under- constrained to being over-constrained. When the existence and the location of these hardest problems are independent of the particular algorithm used, it is frequently referred to as the phase transition phenomenon [70,33,32]. For some problem spaces, the existence and the location of such phase transitions can be predicted analytically [89]. The space of user in-
