The Price of Routing in FPGAs
Florent de Dinechin
Projet Arenaire, LIPCNRSINRIA,
Ecole Normale Superieure de Lyon, France
Florent.de.Dinechin@enslyon.fr
Abstract: Studying the architectural evolution of mainstream
eld programmable gate arrays (FPGAs) leads to the following remark: in
these circuits, the proportion of silicon devoted to reconfigurable routing
is increasing, reducing the proportion of silicon available for computation
resources. A quantitative analysis shows that this trend, if pursued, will
lead to a widening gap between FPGA performance and VLSI performance. Some
prospective solutions to this problem are discussed.
Key Words: FPGA, reconfigurable computing, routing resources,
hardware complexity
Category: B.7
1 Introduction
1.1 The FPGA success story
Any hardware dataprocessing application requires some kind of dedicated
logic circuitry. As a minimum, when using only "offtheshelf"
components, the designer needs some amount of "glue logic" to
interface these components together, and to interface them with the rest
of the world (be it a computer backplane). But dedicated hardware may also
be needed for the bulk of the data processing itself, when no offthe shelf
component is available or when their performance is inadequate.
Until recently, to implement such dedicated logic, the designer had
to choose among the following options:
 using discrete components, which takes up a lot of physical space and
is very inefficient, and is therefore only suited for very small amounts
of glue logic,
 using some offtheshelf processor or microcontroller, which is very
exible due to programmability, but slow and poor in inputs/outputs,
 designing an applicationspecific integrated circuit (ASIC), which
guarantees the best results, but needs several months and is expensive
unless in the context of highvolume production.
Field programmable gate arrays (FPGAs) were therefore designed as offtheshelf
VLSI circuits able to emulate arbitrary logic, with performances close
to those of an ASIC, but with the exibility of software. Typically, a designer
evaluates the requirements of his dedicated circuitry (number of gates,
number of inputs and outputs, speed), buys an FPGA matching these requirements,
and programs it to get an ASIC replacement. The tools used to program FPGAs
are very similar to those used to design ASICs (for example they input
standard hardware description languages such as VHDL or Verilog), but the
designer may
test his circuit instantly, instead of having to go through the lengthly
and costly ASIC foundry process. Besides, some FPGAs are reconfigurable,
which allows for hardware debugging and upgrading. Being an offtheshelf
component produced in high volume, the cost of an FPGA in development time
is a fraction of the cost of an ASIC.
FPGAs have therefore been among the fastest growing sectors of the silicon
industry in the last years. They are mostly used as glorified glue logic,
replacing ASICs where time to market is critical, or where a small production
volume wouldn't justify the cost of an ASIC. Moreover, they have also spawned
new applications and research interests, from rapid prototyping of VLSI
[SKC + 95] to generalpurpose hardware acceleration
for numerical applications [Vui94, DeH94,
Bre98].
1.2 FPGA architectures
The FPGA architecture that allows to achieve these goals typically consists
of a large number of configurable logic blocs embedded in a network of
configurable interconnections [RM98]. We are not concerned
here by the exact nature of the logic blocs (which also contain some memory
elements), neither will we discuss the topology of the network, although
these two questions have been and still are the subject of abundant research
and discussion. We will not escape such discussion later in section 4,
but until then an abstracted view of typical FPGA architectures, such as
depicted on Fig. 1, will suffice.
Figure 1: A typical FPGA architecture. Grey boxes are functional units,
and lines are wires. To ensure that the interconnect network is programmable,
there must be some sort of programmable switch (not shown) wherever two
lines cross.
What Fig. 1 doesn't show is that the programmability of the FPGAs
has a significant hardware cost: there must be switches on the wires to
make the
interconnect network configurable, and there must be some kind of memory
holding this configuration along with the configuration of the logic blocks.
Finally there must be some dedicated logic and routing to allow for the
loading of this configuration.
All this consumes a significant proportion of the silicon of a chip,
with implications on both capacity and speed of the FPGA. Thus, at a given
time, an FPGA built using a stateoftheart VLSI process cannot hold the
biggest fullcustom circuits which can be fabricated in the same process.
Moreover, for those circuits which are small enough to t in the FPGA, this
FPGA implementation is slower than its fullcustom counterpart.
This cost may be expressed as a time lag between FPGA and
fullcustom. For example, in 1999, one may observe that topoftherange
FPGAs from various vendors claim to contain the equivalent of one to two
million programmable gates, with systemlevel operating frequencies above
100MHz. These numbers are comparable with the transistor count (3.1 million,
meaning roughly 1 million gates) and the operating frequency (60 to 120
MHz) of Pentium processors built in 19931994 : here we have a time lag
of five or six years.
One may think that this time lag is constant, which would mean that
we only need to wait for another ve years to get the current fullcustom
performance out of an FPGA. The purpose of this paper is to question this
assumption: we will show that the current trend in FPGA technology, if
pursued, will entail an increase in this time lag. The reason for it is
that an ever increasing proportion of FPGA silicon is devoted to
the routing architecture, and therefore wasted for computing itself. Therefore
the curve of FPGA performance doesn't follow the curve of VLSI integration.
1.3 Rent's rule and FPGAs
Table 1 gives a summary of the evolution of routing resources per lookup
table (or LUT, the computing unit) in the three most recent FPGA families
from Xilinx [Xil97, Xil98]. Figure
2 shows a logical view of an actual recent FPGA, the Xilinx Virtex [Xil98].
This table shows that the amount of routing resources per LUT increases
almost proportionally to the number of LUTs per line or column. This general
trend [TMM + 98] may be explained by the FPGA version
of Rent's experimental law [SKC + 95, Ull84]
which states that, as a logic circuit is partitioned, the number of signals
crossing the boundary of the partition is proportional to the number of
gates on each side, raised to a power r which ranges from 0.5 to
0.8, depending on the class of application.
Let us transpose this rule to our FPGA (an extensive review on this
kind of problems in the context of FPGAs may be found in section 7.6 of
DeHon's thesis [DeH96]). Let us consider a vertical
line splitting an FPGA in two halves. If N is the number of LUTs
in a row/column, then there are O(N^{2}) gates on
each side of the partition. Rent's rule tells that the FPGA needs O(N^{2r})
wires across the vertical line. This line crosses N channels, so
we need O(N^{2r1}) wires per channel. For r
between 0.5 and 0.8, the exponent 2r1 ranges from 0 (constant channel
width) to 0.6.
However, the value of 2r1 extracted from Table 1 seems to be
quite close
Table 1: Xilinx FPGAs
to 1. This can be explained in several ways:
 FPGAs are still in their infancy, and extrapolating asymptotic laws
from Table 1 is obviously hazardous.
 An FPGA has to accomodate any application, and thus implements a very
conservative, worst case Rent's exponent
 In the FPGA market, the speed of the design cycle is an increasingly
important selling point. The longest step of this design cycle is the (NPcomplete)
placeandroute phase, and with current methodologies and heuristics, this
step greatly benefits from an excess in routing resources. The lost silicon
area is then more than compensated by the improvement in FPGA design time
[RH97].
Whatever the exact growth of routing resources, one obvious consequence
is that the proportion of silicon devoted to routing increases with
integration. This means in turn that routing has an increasing impact on
FPGA performances. This is all the more true as wires, in FPGA, also carry
logic: a signal from one gate to another one has to run through the various
switches which ensure the programmability of the routing network.
This is one major difference when comparing FPGA and fullcustom solutions.
The purpose of this paper is to study the longterm implications of
this evolution. Section 2 will attempt to build a very
rough but realistic model of the trend in FPGA technology demonstrated
by Table 1. Section 3 then shows
that this trend would lead to an increasing performance gap between FPGA
and fullcustom VLSI. The third section concludes by questioning the model,
considering several possible evolutions of FPGA technology which would
address the problems exposed here.
^{1}The 4000 series contains
2 LUTs per configurable logic bloc (CLB). The Virtex series contains 4
LUTs per CLB. The LUT matrix is therefore given as (CLBs per line) x (CLBs
per column) x (LUT per CLB).
Figure 2: A closeup on Xilinx Virtex chip routing resources in the
EPIC tool. The black boxes are computing units, the rest is wiring. Here
again, routing switches are not shown, but the white boxes around the computing
units are their input/output crossbars.
2 Definitions and hypotheses
2.1 Very large scale integration
The object of our study will be the evolution of FPGA computing power
with respect to VLSI integration. To quantify this integration, we will
use a universal measure, the typical length
of a given VLSI process. Currently between 0.5
and 0.1, this
length is related to the size of the smallest possible transistor. We shall
express the various quantities we study as a function of .
As we study integration, we will consider circuits of xed size,
say square circuits of unit size^{2}.
As measures
the size of a transistor, the number of transistors on this unit
size chip grows as O(^{2}).
Besides, integration increases the frequency of the chip. As a very
rough estimate we may assume that the switching time of a transistor is
proportional to its width, thus to .
Therefore the typical frequency
^{2}It should be noted that
the typical commercial size of a VLSI chip has been almost stable over
the decade, at a few square centimeters. For example, the die size of the
Intel Pentium II is roughly that of the 80286, even though intermediate
processors of the same company have been bigger. In any case, the growth
of die size doesn't compare to the growth of 1/.
will grow as O(^{1}),
and the maximum raw power of the unit size chip grows as O(^{3}).
This is only an upper bound on computing power: it assumes that all
the silicon is used for transistors, whereas in any circuit, you have of
course to dedicate some silicon to wires as well.
These hypotheses are linked to Noyce's thesis (as cited by Vuillemin
[Vui94]) according to which
is reduced by a constant factor
1.25 every year. This integration leads to an increase of theoretical computing
power of 2 every
year. Vuillemin concludes from this that the peak FPGA performance also
doubles every year, which we question in the following.
2.2 Fieldprogrammable gate arrays
Let us now observe a family of theoretical FPGAs which closely follows
VLSI integration: we will note FPGA ()
the FPGA built in process .
More precisely, FPGA ()
has the following characteristics.
 We will assume that the architecture of the computing unit doesn't
depend on integration. This is a very strong assumption, to be discussed
later in section 4. We motivate it by the following observation: the base
block of this computing unit, a 4input lookup table, is the main common
characteristic of the mainstream FPGA architectures from the two major
FPGA vendors, Xilinx and Altera, and has been for almost a decade now.
 Let us assume this computing unit is a square of size X_{l}.
 Wires also have a width which, expressed in ,
is a constant, say X_{w}.
In other terms you can have X_{w}/X_{l} wires in
the width of a logic bloc, and this value doesn't depend on .
The last assumption may be discussed, considering that technology allows
for more and more metal layers to put wires in (current FPGAs use 5 metal
layers [Xil98]). The point is, however, that any FPGA
wire must have some logic on it: in FPGAs, wires begin and end with switches
which ensure the programmability of the routing, and these switches are
built out of transistors, and thus consume transistor space.
We now need to quantify the number of wires we have in FPGA ().
2.3 A simple model of routing
It is impossible to give a model of all existing and possible routing
architectures, therefore we will restrict the study to quantitative aspects.
Let N () be
the number of LUTs per row or column of FPGA ().
In [Vui94], Vuillemin neglects the space used by routing
resources, and therefore assumes that N()
grows as O(^{1}).
This assumption has been proven wrong (see table 1): the number of wires
per LUT tends to increase with N().
As a consequence, N()
is no longer proportional to ^{1}.
Now we have to quantify this increase of routing. Table 1 shows that
routing per channel grows almost linearly with N. We will therefore
consider a number of wires per LUT (in other words a number of wires per
routing channel) which is proportional to N()
(let k be the proportionality factor). This corresponds to
a Rent's exponent of 1. However the same results hold for values
of the exponent between 0.5 and 1, as the interested reader may nd in appendix.
We choose here to keep the model simple.
3 The price of programmability
3.1 Spatial cost
Let us first estimate N().
The width of an FPGA tile is now (X _{l} +k.N().X_{w}
). As N()
increases we soon have X_{l} k.N().X_{w}.
Note that this is already the case in current FPGAs as shown by Fig. 2.
In this figure, the two black boxes are two fourinput lookup tables each,
and the rest consists of routing and routing switches. Beware that this
picture is a logical view from the Epic tool, and not an actual photograph
of the layout: as such it might not perfectly re ect the scale of the various
components, however it makes the point.
We now have an idea of the evolution of N()
with respect to :
hence
Lemma 1. The number N()
of LUTs in a row of FPGA grows as O(
^{1/2} ).
In other words, in order to double the number of LUTs on a row, we have
to wait until
has been divided by 4. Or: an FPGA will have twice the number of LUTs per
surface unit only when the technology allows four times as many transistors.
3.2 Time cost
The previous analysis also has implications on the maximal operating
frequency of FPGA().
Its inverse, the period, is the sum of two terms:
 The switching time for a logic bloc, which in our approximation is
proportional to .
 The time to go through routing, which in turn is the sum of two terms:
 A term which is at least proportional to the distance between the LUTs
connected. This value is difficult to estimate, as it depends widely on
the application. However this term is victim of the evolution of the distance
between two LUTs, which is 1/N(),
and as such evolves at best as ^{1/2}
 A term which measures the time needed to go through routing switches.
This term is the traversal time for one switch (proportional to ,
still in the same approximation), multiplied by the the number of switches
on a wire. Note that these switches don't actually switch, except at configuration
time, so their contribution is not dependent on the switching time of a
transistor. However, transistors have a higher resistance than the wires
between them, and we may consider that their traversal time evolves as
Finally, as
decreases, the most significant term evolves in ^{1/2}:
Lemma 2. The frequency F()
of FPGA() grows
as O(^{1/2}).
3.3 Peak computing power
And finally the peak computing power grows as N()^{2}
x F(),
that is to say as O(^{3/2}).
If for example
is divided by 1.25 every year (following Noyce's thesis), the theoretical
computing power of our family of FPGAs will double every other year, whereas
that of pure VLSI will double every year. This means that the performance
lag between FPGA and VLSI will increase (doubling every other year).
A sensible objection to our study is that we don't compare like for
like, as we compare an FPGA with routing to VLSI without routing. This
is certainly true: we are comparing peak, or theoretical, computing power.
The point is that there exist very efficient architectures with limited
or localized routing which, implemented in FPGA, will not benefit from
this economy in routing when compared to a VLSI implementation.
Consider for example a microprocessor, and suppose that the critical
component of its arithmetic unit is, say, a multiplier. This component
will therefore be heavily optimized for performance. The designers of this
multiplier won't for example allow any external routing channel through
it: external routing will ow around the multiplier (or above if there are
metal layers available). Now imagine the same processor, with the same
multiplier, implemented in FPGA():
these optimizations are no longer possible, because routing channels, most
of which are used by external routing, cut through the multiplier.
This is, once again, probably the main difference between FPGA and fullcustom
VLSI: in fullcustom one pays the performance cost of the routing one gets,
whereas in FPGA one pays the cost of all the potential routing even if
it isn't being used.
4 Discussion
This study is not meant to be pessimistic: we trust FPGA vendors for
contradicting this dooming view in the forthcoming years. To do so, they
just have to contradict one of our hypotheses. Note that, as shown in appendix,
taking into consideration a Rent's exponent smaller than 1 doesn't change
our conclusion (as long as r > 0.5, which is a prerequisite for
an FPGA).
4.1 Constant granularity
We have supposed that the size of the basic building block will remain
constant, while their number grows. However it is to be expected that the
size of the
computing unit will grow as well, in order to keep a balance in the
routing resources. Note that it is the case in the evolution form the 4000
series to the Virtex series, as the size of the configurable logic bloc
has increased from 2 to 4 LUTs. Another example of this trend is Chess
[MSK + 99], a reconfigurable array aimed at multimedia
applications, where the unit of both computing and routing resources is
4 bits. This architecture is closer in many respects to the MasPar massively
parallel computers [HPK95] than to mainstream FPGAs.
It is difficult here to resist comparison with the history of parallel
computing: the number of processors in parallel computers doesn't grow
anymore. Today's biggest parallel computer have no more than a few hundreds
of processors, much less than their predecessors of the previous decade
(MasPar and TMC computers had tens of thousands). The growth is now in
the performance of each processor. We should expect the same kind of evolution
in the FPGA world.
Exploring this parallel further, let us notice that increasing emphasis,
in the parallel computing community, is set on architectural and system
techniques designed to hide both the complexity and inefficiency of the
communications (e.g. distributed shared memory and caches). Could this
trend not lter down to the FPGA world? This is obviously not only a question
of architecture.
4.2 Regular architecture
We also expect more and more hierarchy in the FPGA structure [RH97]:
nearlinear routing is probably not necessary at any scale.
Attempts to hierarchical architectures have already been made, for example
in the Teramac system [SKC + 95], or in the Xilinx
Series 6000 which had a logarithmic routing architecture. The newest Actel
FPGAs [KBK + 99] claim to be "semihierarchical",
with three levels of hierarchy, each level being the usual mesh.
However the Xilinx 6000 series was a commercial failure. To say that
it was due to its hierarchical routing would be exaggerating: its main
weakness, in our sense, was in its vendor placeandroute tools. Altough
they were very good at allowing the designer to organize a hierarchy of
subcircuits, they were very bad at doing anything automatically.
What this example makes clear is that hierarchical routing architectures
need sounder bases on the software side to really succeed. This is only
one aspect of the evolutions in FPGA support software needed to address
the problems shown in this paper.
4.3 FPGA synthesis tools
Current FPGA synthesis tools discard the logical hierarchy of a circuit
to perform a global optimization of the place and route problem. In the
VLSI world, this approach has long been replaced with a more hierarchical
one, where a big circuit is decomposed in smaller functional blocks which
are then optimized locally (or found in libraries).
This doesn't only ease the task of the designer, but also leads to efficient
architectures by breaking up global optimization problems into tractable,
local ones.
This point is obviously linked to the previous one: for the hierarchical
approach to have the same performance benefits in FPGAs, there must be
some form of hardware hierarchy. This was the case in the Xilinx 6000 series.
4.4 General purpose FPGAs, or not
Some classes of applications don't need general purpose routing, and
would be contented with more scalable architectures. An example is that
of datapathoriented FPGAs, as studied in the Garp project [HW97]
among other. Chess [MSK + 99] is dedicated to multimedia
processing, and this loss of generality allows for a more efficient architecture
in its application eld. An even more radical example is the class of systolic
arrays, as obtained using automatic parallelization techniques [Qui84,
dD97]: these techniques yield designs with only clock
and local routing, which scale well. Could a FPGA family with constant,
local routing dedicated to systolic applications be designed? This class
of applications, unfortunately, is probably too restricted to make a such
a systolic FPGA commercially viable.
5 Conclusion
The increase of road traffic in expanding cities is a fatality for complexity
reasons very similar to those presented here. Most big cities have tried
to address this traffic increase by widening the roads and increasing the
number of lanes. Since the eighties, however, this approach to city traffic
is generally considered a failure. One reason is specific to the city metaphor:
city planners, unlike FPGA designers, can't move blocks further apart when
they want more lanes between them. But even if they could, they would face
the citizens' critics: the useful part of the city is the blocks, that
is where people live and interact. Cityresidents are increasingly reluctant
to sacrifice this living space to traffic lanes, all the more as they are
conscious of the short lifespan of any traffic improvement.
Similarly, this study showed that the current evolution of FPGAs gives
an increasing proportion of the silicon resources to routing, which is
computationnaly useless. It establishes that this trend, if pursued, would
lead to an increase of the performance gap between FPGA and fullcustom
VLSI. The hope is to push longterm researchers to explore alternative
possibilities.
For example, some cities manage their trafic increase quite well by
promoting public transportation (the metaphor of a bigger granularity),
local shopping and working (the metaphor for architectural solutions placing
the emphasis on local routing), and cycling.
Alas, as far as cycling is concerned, we couldn't find an FPGA metaphor.
Acknowledgements
Special thanks should go to Wayne Luk, Nabeel Shirazi and the ALA team
at Imperial College, London, and to Dominique Lavenier of IRISA, Rennes,
for many interesting discussions and comments. This work was partly supported
by an INRIA postdoctoral fellowship at Imperial College, London, UK.
References
[Bre98] Gordon Brebner. Fieldprogrammable logic:
Catalyst for new computing paradigms. In International Workshop on Field
Programmable Logic and Applications, Tallin, Estonia, September 1998.
[dD97] Florent de Dinechin. Libraries of schedulefree
operators in Alpha. In Application Specific Array Processors. IEEE
Computer Society Press, July 1997.
[DeH94] Andre DeHon. DPGAcoupled microprocessors:
Commodity ICs for the early 21st century. In IEEE Workshop on FPGAs for
Custom Computing Machines, 1994.
[DeH96] Andre DeHon. Reconfigurable Architectures
for GeneralPurpose Computing. PhD thesis, MIT, August 1996.
[HPK95] M. Hamdi, Y. Pan, and W. T. Kwong. Effcient
image processing applications on the maspar massively parallel computers.
International Journal of High Speed Computing, 7(4):489514, 1995.
[HW97] John R. Hauser and John Wawrzynek. Garp: a
MIPS processor with a reconfigurable coprocessor. In IEEE Symposium
on FPGAs for Custom Computing Machines, pages 1221, Napa Valley, CA,
April 1997.
[KBK + 99] S. Kaptanoglu, G. Bakker, A. Kundu, I.
Corneillet, and B. Ting. A new high density and very low cost reprogrammable
FPGA architecture. In FPGA'99, ACM/SIGDA International Symposium on
FPGAs, pages 312, Monterey, CA, February 1999.
[MSK + 99] A. Marshall, T. Stans eld, I. Kostarnov,
J. Vuillemin, and B. Hutchings. A reconfigurable arithmetic array for multimedia
applications. In FPGA'99, ACM/SIGDA International Symposium on FPGAs,
pages 135143, Monterey, CA, February 1999.
[Qui84] P. Quinton. Automatic synthesis of systolic
arrays from recurrent uniform equations. In 11th Annual Int. Symp. Computer
Arch., Ann Arbor, pages 20814, June 1984.
[RH97] J. Rose and D. Hill. Architectural and physical
design challenges for onemillion gate FPGAs and beyond. In FPGA'97,
ACM International Symposium on FPGAs, pages 129132, Monterey, CA,
February 1997.
[RM98] B. Radunovic and V. Milutinovic. A survey
of reconfigurable computing architectures. In International Workshop
on Field Programmable Logic and Applications, Tallin, Estonia, September
1998.
[SKC + 95] G. Snider, P. Kuekes, W. B. Culbertson,
R. J. Carter, A. S. Berger, and R. Amerson. The Teramac configurable computer
engine. In Field Programmable Logic and Applications, pages 4453.
LNCS 975, September 1995.
[TMM + 98] A. Takahara, T. Miyazaki, T. Murooka,
M. Katayama, K. Hayashi, A. Tsutsui, T. Ichimori, and K. Fukami. More wires
and fewer LUTs: a design methodology for FPGAs. In FPGA'98, ACM/SIGDA
International Symposium on FPGAs, pages 1219, Monterey, CA, February
1998.
[Ull84] Jeffrey D. Ullman. Computational Aspects
of VLSI. Principles of Computer Science. Computer Science Press, 1984.
[Vui94] Jean Vuillemin. On computing power. In Programming
Languages and System Architectures, LNCS 782, pages 6986, Zurich,
Switzerland, June 1994.
[Xil97] Xilinx Corporation. XC4000E and XC4000X
Series Field Programmable Gate Arrays, November 1997.
[Xil98] Xilinx Corporation. Virtex 2.5V Field
Programmable Gate Arrays, October 1998.
A When Rent's exponent is smaller than 1
This section follows the computation of section 3 in the case of a Rent's
exponent between 0.5 (constant width channels) and 1.
A.1 Spatial cost
As exposed in the introduction, a Rent's exponent r corresponds
to a channel width of 2r  1. The width of an FPGA tile is thus
X_{ l }+ k.N()
^{2r1} .X_{w}, and we still have X_{l}
k.N().X_{w}
, leading to
hence
Lemma 3. The number N()
of LUTs in a row of FPGA grows as O.
A.2 Time cost
The period is the sum of:
 The switching time for a logic bloc, proportional to .
 The time to go through routing, which in turn is the sum of two terms:
 A term expressing the time to go through the wires themselves (i.e.
to load the corresponding capacitor), at least proportional to the distance
between the connected LUTs; two LUTs are at least separated by one routing
channel, thus the minimal distance between two LUTs is 1/N().
This term evolves therefore at best as ;
 A term which measures the time needed to go through routing switches.
This term is the traversal time for one switch (proportional to ),
multiplied by the the number of switches on a wire. The number of switches
on a net, assuming a sensible routing algorithm, is at least log_{2}
N(), in
which case the corresponding term in the period is .
log_{2} N()
and can be neglected, and at most N(),
in which case this term evolves as .
As 1/2 < r < 1, we have 0 < 1 
< . As
decreases, the most significant term in the period is
in this case. However this term also assumes a very bad routing architecture,
in which the number of switches on a typical net is proportional to the
size of the chip. This is what you would get on an FPGA with only local
routing, for instance.
Let us be optimistic and assume that the routing is good enough for
the period to be limited by the second term in .
We get the following optimistic asymptotic frequency:
Lemma 4. The frequency F ()
grows as O .
A.3 Peak computing power
And finally the peak computing power grows as N()
^{2} x F(),
that is to say as O().
This result yields the same conclusions with respect to an increase
of the time lag between fullcustom VLSI and FPGA peak computing power.
