Philip Brisk2

Paolo Ienne2

Hadi Parandeh-Afshar1,2

1: University of Tehran, ECE Department

2: EPFL, School of Computer and Communication Sciences

Efficient Synthesis of Compressor Trees on FPGAs

January 22, 2008 2

Outline

State of the Art: FPGAs

Motivation

Generalized Parallel Counters

Mapping Heuristic

Experimental Results

Conclusion

January 22, 2008 3

Outline

State of the Art: FPGAs

Motivation

Generalized Parallel Counters

Mapping Heuristic

Experimental Results

Conclusion

January 22, 2008 4

FPGA vs. ASIC

Performance

Area Utilization

Power Consumption

Flexibility

Time-to-Market

ASIC FPGA

√

√

√

√

√

January 22, 2008 5

FPGA Arithmetic Features

Poor Performance for Arithmetic Operations Compared to ASIC IP Cores

High Routing Costs Limited Flexibility; 18-bit Adder/Multiplier

Full Adder Implemented in CLB Structure Fast Carry-Chain (Xilinx and Altera)

Reduces Routing Delay

Cannot Use Compressor Trees to Add k>2 Values Wallace/Dadda/3-Greedy

January 22, 2008 6

Outline

State of the Art: FPGAs

Motivation

Generalized Parallel Counters

Mapping Heuristic

Experimental Results

Conclusion

January 22, 2008 7

Motivation: Compressor Trees

Partial product reduction in parallel multiplication Wallace and Dadda in the 1960s

Multi-input addition occurs in many multimedia and signal processing H.264/AVC Variable Block Size Motion Estimation FIR Filters 3G Wireless Base Station Channel Cards

Flow graph transformations expose opportunities to use compresor trees in high-level synthesis [Verma and Ienne, ICCAD 04]

January 22, 2008 8

Flow Graph Transformationstep 3

>>

&

delta

7

&4

SEL =

0+

SEL

+

step 1

>>

&

2

=

0

SEL

+

step 2

>>

&

1

=

0

vpdiff

step 3

>>

=

delta

1

&0

step 2

>>

SEL

0

=

delta

2

&0

step 1

>>

SEL

0

=

delta

4

&0

step 0

>>

SEL

0

vpdiff

∑

+Compressor

TreeADPCM

January 22, 2008 9

Outline

State of the Art: FPGAs

Motivation

Generalized Parallel Counters

Mapping Heuristic

Experimental Results

Conclusion

January 22, 2008 10

Counters

m

n

m:n counter

n = log2(m+1)

Count #of Input Bits Set to 1

Output # as a Binary Value

Counters You Know

2:2 – Half Adder

3:2 – Full Adder(Carry-Save Adder)

The correct building block for computing sums of k>2 numbers

Counters do not map well onto LUTs or carry chains

January 22, 2008 11

Generalized Parallel Counters (GPCs) Sum bits having different ranks

m:n counter: all bits have rank 0, i.e.: 20 = 1 Representation:

(Kn-2, Kn-1, …, K0; S) Ki – number of input bits of rank i S – number of output bits

(0, 4; 3) – typical 4:3 counter (2, 3; 3) – maximum value: 2*21 + 4*20

= 12 Range [0, 12] requires S = 4 output bits

Examples usingdot notation (3, 3; 4) GPC (5, 5; 4) GPC

January 22, 2008 12

GPC Implementation

For ASICs Basic gates, e.g. AND, XOR Built from m:n counters, e.g., just like a

compressor tree FPGA Implementation

K-input GPC maps nicely onto K-LUTs One logic level required

K = 6 for Xilinx Virtex-5 and Altera Stratix II and III Three 6-LUTs for 6-input, 3-output GPC Four 6-LUTs for 6-input, 4-output GPC

January 22, 2008 13

Outline

State of the Art: FPGAs

Motivation

Generalized Parallel Counters

Mapping Heuristic

Experimental Results

Conclusion

January 22, 2008 14

Definitions

Primitive GPCs: Satisfies given I/O Constraints 12-primitive GPCs for 6 inputs, 3 outputs

Including (1, 3; 3), (2, 3; 3)

Covering GPCs Functionality cannot be implemented by other

GPCs, given the I/O constraints e.g., (2, 3; 3) GPC can implement a (1, 3; 3) GPC

Set a rank-1 input bit to 0

January 22, 2008 15

Definitions

Unreasonable GPCs: Single bit in rank-0 column

(3, 1; 3) GPC rank-0 output bit = rank-0 input bit

No reduction in bits (1, 2; 3) GPC 3 input bits: Output value in range [0, 4] 3 output bits

January 22, 2008 16

Definitions

Compression Ratio (CR): # Input Bits / # Output Bits (3, 3; 4) GPC

CR = 6/4 = 1.5 (2, 3; 3) GPC

CR = 5/3 = 1.67

Using GPCs with large CR tends to reduce the number of bits to sum at the next logic level # logic levels = # LUTs on critical path in an FPGA

January 22, 2008 17

Input: Columns of bits to sum

Example: 3-tap FIR filter Each FIR filter is different, depending on

constants used

0rank

January 22, 2008 18

Mapping Heuristicmap_algorithm(Integer : M, Integer : N ,Array of Integers : columns)

{step1: find_covering_GPCs ;) (step2: find_primitive_GPCs;) (step3: order_primitive_GPCs ;) (

Repeat{ step4: Repeat{

col_indx = find_highest_column ;) ( find_next_GPC (col_indx);

remove_covered_dots;) ( } until all dots are covered or no reasonable GPC is found

step5: connect_GPCs_IOs;) (step6: generate_next_stage_dots;) (

} until three rows of dots remains;}step7: generate_final_cpa( columns )

Virtex-5 and Stratix II & III support ternary addition

Attack the tallest column first (greedy approach)

January 22, 2008 19

4

3

1

Example

2

Map to ternary adder

January 22, 2008 20

Outline

State of the Art: FPGAs

Motivation

Generalized Parallel Counters

Mapping Heuristic

Experimental Results

Conclusion

January 22, 2008 21

Experimental Methodology Altera Stratix-II

90nm CMOS Technology Implementations of multi-input addition

ADD – Ternary adder tree State of the art for FPGAs

3GD – 3-greedy algorithm (3:2 and 2:2 counters) [Stelling et al., TCOMP 98] 2 and 3-input counters do not map well onto 6-LUTs!

GPCs – Heuristic described here

January 22, 2008 22

Experimental Results (Delay)

27% on average GPC is faster than ADD

Delay (ns)

0123456789

adpc

m

add2

Q fir3

G72x_

2m

ac

Mot

ionEst.

m12

x12

m16

x16

RQGQBQ

RYGYBY

sam

ul

Avera

ge

ADD 3GD GPC

January 22, 2008 23

Experimental results (Area)

5% increase in ALMs usage for GPC compared to ADD

Area (ALM)

0

100

200

300

400

500

600

adpc

m

add2

Q fir3

G72x_

2m

ac

Mot

ionEst.

m12

x12

m16

x16

RQGQBQ

RYGYBY

sam

ul

Avera

ge

ADD 3GD GPC

January 22, 2008 24

Are DSP/MAC Blocks Useful?

Resource Utilization for Adder Trees Using DSPs

0

10

20

30

40

50

60

70

80

DSPs ALMs

No! On average, delay using DSP/MAC blocks was more than 2x worse than 3GD

January 22, 2008 25

Outline

State of the Art: FPGAs

Motivation

Generalized Parallel Counters

Mapping Heuristic

Experimental Results

Conclusion

January 22, 2008 26

Conclusion Conventional wisdom has held that adder trees

outperform compressor trees on FPGAs Ternary adder trees were a major selling point of

the Altera Stratix II architecture This led to their inclusion in Xilinx Virtex-5 devices

Conventional wisdom is wrong! GPCs map nicely onto LUTs

Compressor trees on FPGAs, are faster than adder trees when built from GPCs