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where Pc and Pm are respectively the conventional and minimal projections as in De nition 6.3.7 and De nition 6.3.8. In other words, Vab contains both a and b and is parallel to the l3 axis. The degenerate case where ab is parallel to the l3 axis is simple to handle and is not considered. De nition 6.3.15. Let Ls and Lt be the given start and target con gurations of the arm, and let S = {j J|L(j ) = Ls } J and T = {j J|L(j ) = Lt } J, respectively, be the sets of points corresponding to Ls and Lt . Let f : J C be the projection as in De nition 6.3.12. Then the vertical surface V C is de ned as V = {f (j ) C|j Vst for all s S and t T}

The problem that arises with register ordering is that it may prevent a group of registers that have added pipelining for route balancing to be grouped together. This will prevent the additional registers from dividing the routing delay as originally intended. Consider the delay balancing register used above when packing registers into I/O buffers. As can be seen from Figure 16.12, the bene t of the pipelined register has been eliminated due to the fact that it has been placed adjacent to its driver.

For the RRP arm, which is the most general case among XXP arms, V consists of four components Vi , i = 1, 2, 3, 4. Each Vi represents a class of vertical planes in J and can be determined by the rst two coordinates of a pair of points drawn s s s t t t respectively from S and T. If js = ( 1 , 2 , l3 ) and jt = ( 1 , 2 , l3 ) are the robot s start and target con gurations, the four components of the vertical surface V can be represented as follows:

Craig had tipped me off to the idea in the first place, and Harvey had confirmed it Could I repeat this I opened up the daily chart for the Euro and went to work.

Designers typically do not like the concept that there may be a certain degree of randomness in their design ow or implementation tools. However, it is important to note that a placement for a given design with a set of constraints is not entirely deterministic. In other words, there is no single optimal placement for any given design; at least, not one that can be easily determined by today s technology prior to the placement process itself. In fact, as shown in Figure 16.13 this is not an obvious problem at all, even for an automated placer, to determine the optimal starting point for the placement process. Thus, the placement tool needs to be seeded similar to a process for random number generation. The exact de nition of a seed is a bit abstracted from the standpoint of the designer, but for useful purposes, different seeds essentially provide the placer with slightly different starting points by which the rest of the placement process branches. This is illustrated in Figure 16.14. Some vendors extend this beyond simply the initialization to other random parameters of the placement process such as relative weighting of the various constraints, desirable placement distances, and estimates for routing congestion. Xilinx calls this the placement cost table, which affects the various placement parameters, but abstracts this to an integer (1, 2, 3, etc.). Each integer corresponds with a different set of initial conditions and placement parameters, but because of the low level complexity, it is abstracted to a single number and the actual functionality is hidden from the parameter table.

t t s s V1 : ( 1 , 2 ) ( 1 , 2 ) t t t s s s V2 : ( 1 , 2 ) ( 1 , 2 2 sign( 2 2 ))

(6.7)

The bene t to adjusting the placement seed is that the user can take another shot at meeting the timing constraints without actually changing anything in the design. This can be automated with a process called multipass place and route. When this option is selected, the place and route tool will run the implementation multiple times and automatically vary the seed on every run. This will of course require a large amount of time to produce effective results, so this is typically used on overnight runs where the target design is not already meeting the timing constraints in a repeatable manner. It is important to note here that the place and route algorithms are actually very good regardless of the seed, and any changes to the various parameters will have a very small effect. The danger some newer designers run into is to spend

t s 2 sign( 2 2 ))