Using the Minimum Set of Input Combinations to Minimize the Area of Local Routing Networks in Logic Clusters. FPGAs. Andy Ye Ryerson University
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1 Usng the Mnmum Set of Input Combnatons to Mnmze the Area of Local Routng Networks n Logc Clusters Contanng Logcally Equvalent I/Os n FPGAs Andy Ye Ryerson Unversty
2 Mnmum-area IIBs how to mnmze the area of a logc cluster wthout losng any functonalty?
3 FPGA Logc Clusters A Collecton of Look-Up Tables and Flp-Flops Share a Common Set of Inputs and Outputs Logcally Equvalent Logc Cluster Inputs Any nput sgnal can enter the cluster through any of the logc cluster nput pns Logcally Equvalent Logc Cluster Outputs Any output sgnal can ext the cluster through any of the logc cluster output pns
4 Beneft and Cost of Logc Equvalency Beneft: Reduces the Global Routng Area Increases the flexblty of global routng network Less routng tracks Less global routng area Cost: Increases Logc Cluster Area Needs specalzed local routng networks n every logc cluster Local routng networks cost area What s a mnmum local routng network desgn that can acheve full logc equvalency?
5 Beneft: Routng Track Reducton Consder a set of logc clusters each wth 2 output pns and 3 nput pns Logcally Non-Equvalent I/O Pns Logcally Equvalent I/O Pns
6 Cost: Local Routng Networks A Fully Connected Local Routng Network I 1 I 2 LUT O 1 I 3 I 4 I 5 LUT O 2 I 6
7 Less Than Full Connectvty? Yes! A Smple Example [Betz and Rose 98] Logc clusters contanng just one LUT each The number of nputs per cluster = the number of LUT nputs Can completely elmnate the local routng network Key Mechansm that Enables the Elmnaton of the Local Routng Network LUT reconfguraton reconfgure a LUT as the logc cluster nput assgnment changes
8 Logc Equvalency Through LUT Reconfguraton 1 LUT Cluster I 1 L 1 I 1 L 1 I 2 I 3 L 2 L 3 LUT O I 2 I 3 L 2 L 3 LUT O L 4 I 4 L 4 I 4 Logcal Equvalency Through Local Routng Network Logcal Equvalency Through LUT Reconfguraton
9 Logc Equvalency Through LUT Reconfguraton 1 LUT Cluster L1 L2 L3 L4 O f f f f f f f f f f f f f f f f15 LUT Confg. for Connecton to Cluster Input 1 L1 L2 L3 L4 O f f f f f f f f f f f f f f f f15 LUT Confg. for Connecton to Cluster Input 2 L1 L2 L3 L4O f f f f f f f f f f f f f f f f15 LUT Confg. for Connecton to Cluster Input 3 L1 L2 L3 L4 O f f f f f f f f f f f f f f f f15 LUT Confg. for Connecton to Cluster Input 4
10 Queston Addressed n Ths Research How about clusters wth more than 1 LUT? F 1 BLE Basc Logc Element F 2 I 1 LUT 1 O 1 I 2 I 3 I 4 I 5 LUT 2 O 2 I 6
11 Functons that can be Generated by a Local Routng Network A Logc Cluster wth k-nput LUT I logc cluster nputs N feedbacks Fx LUT Confguraton Reconfgure the Local Routng Network Maxmum Number of Functons that the LUT can Generate (I + N) k Many of these functons can be made redundant through LUT reconfguraton.
12 An Example k = 2 N = 2 I = 2: (2+2) 2 = 16 functons F 1 F 2 I 1 O 1 BLE 1 Local Routng Network O 2 BLE 2 I 2 f(f1f1) f(f1f2) f(f1i1) f(f1i2) f(f2f1) f(f2f2) f(f2i1) f(f2i2) f(i1f1) f(i1f2) f(i1i1) f(i1i2) f(i2f1) f(i2f2) f(i2i1) f(i2i2)
13 Local Routng Network 4:1 Muxes f(f1f1) f(f1f2) f(f1i1) f(f1i2) f(f2f1) f(f2f2) f(f2i1) f(f2i2) f(i1f1) f(i1f2) f(i1i1) f(i1i2) f(i2f1) f(i2f2) f(i2i1) f(i2i2) F 1 BLE 1 O 1 F 2 I 1 BLE 2 O 2 I 2
14 Commutatve Property of LUT Inputs Gven a k-input LUT confgured wth the functon: Connect the LUT to k ndependent Boolean nputs: ) ( 2 1 k a a a f k 2 1 There exsts another functon such that: = ) ( ' k y y y x x x f x y ' f ) ( k y y y x x x f y x k 1 2
15 Commutatve Property Contnued L1 L2 L3 L4 O f f f f f f f f f f f f f f f f15 Before Exchangng L1 and L3 L3 L2 L1 L4O f f f f f f f f f f f f f f f f15 After Exchangng L1 and L3
16 Commutatve Property Contnued k = 2 N = 2 I = 2: 16 functons => 10 functons F 1 F 2 I 1 O 1 BLE 1 Local Routng Network O 2 BLE 2 I 2 f(f1f1) f(f1f2) f(f1i1) f(f1i2) f(f2f1) f(f2f2) f(f2i1) f(f2i2) f(i1f1) f(i1f2) f(i1i1) f(i1i2) f(i2f1) f(i2f2) f(i2i1) f(i2i2)
17 Local Routng Network 4:1 Muxes f(f1f1) f(f1f2) f(f1i1) f(f1i2) f(f2f1) f(f2f2) f(f2i1) f(f2i2) f(i1f1) f(i1f2) f(i1i1) f(i1i2) f(i2f1) f(i2f2) f(i2i1) f(i2i2) F 1 BLE 1 O 1 F 2 I 1 BLE 2 O 2 I 2
18 Duplcated-Constant Input Equvalence Gven a k-input LUT confgured wth the functon: Connect the LUT to k ndependent Boolean nputs: ) ( 2 1 k a a a f k 2 1 If x = y then here exsts another functon such that: = ' f ) ( k y y y x x x f y x k 1 2 ) ( ' k y y y x x x f x
19 Duplcated-Constant Input Equvalence Contnued L1 L2 L3 L4 O f f f f f f f f A Three Input Boolean Functon L1 L2 L3 L4 O f f f f x x x x x x x x f f f f7 Duplcated Input Implementaton L1 L2 L3 L4 O f f f f f f f f x x x x x x x x Constant Input ('0') Implementaton
20 Duplcated-Constant Input Equvalence Contnued k = 2 N = 2 I = 2: 16 functons => 10 functons F 1 F 2 I 1 O 1 BLE 1 Local Routng Network O 2 BLE 2 I 2 f(f1f1) f(f10) f(f1f2) f(f1i1) f(f1i2) f(f2f1) f(f2f2) f(f20) f(f2i1) f(f2i2) f(i1f1) f(i1f2) f(i1i1) f(i10) f(i1i2) f(i2f1) f(i2f2) f(i2i1) f(i2i2) f(i20)
21 Local Routng Network 4:1 Muxes f(f10) f(f1f2) f(f1i1) f(f1i2) f(f2f1) f(f20) f(f2i1) f(f2i2) f(i1f1) f(i1f2) f(i10) f(i1i2) f(i2f1) f(i2f2) f(i2i1) f(0i2) F BLE 1 O 1 F 2 I BLE 2 O 2 I 2
22 Constant-New Input Equvalence (Shannon Decomposton) Gven a k-input LUT confgured wth the functon: f a a a ) ( 1 2 k Connect the LUT to k ndependent Boolean nputs: 2 1 k If x = 0 then here exsts another functon f ' such that: = f '( 2 1 x z x+ 1 x+ 2 f ( 1 2 x 1 0 x+ 1 x+ 2 k ) where z s a new arbtrary nput 1 k )
23 Constant-New Input Equvalence Contnued L1 L2 L3 L4 O f f f f f f f f A Three Input Boolean Functon L1 L2 L3 L4 O f f f f x x x x x x x x f f f f7 Duplcated Input Implementaton L1 L2 L3 L4 O f f f f f f f f x x x x x x x x Constant Input ('0') Implementaton L1 L2 L3 L4 O f f f f f f f f f f f f f f f f7 New Input Implementaton
24 Constant-New Input Equvalence Contnued k = 2 N = 2 I = 2: 16 functons => 10 => 6 functons F 1 F 2 I 1 O 1 BLE 1 Local Routng Network O 2 BLE 2 I 2 f(f10) f(f1f2) f(f1i1) f(f1i2) f(f2f1) f(f20) f(f2i1) f(f2i2) f(i1f1) f(i1f2) f(i10) f(i1i2) f(i2f1) f(i2f2) f(i2i1) f(i20)
25 Local Routng Network 3:1 Muxes f(f10) f(f1f2) f(f1i1) f(f1i2) f(f2f1) f(f20) f(f2i1) f(f2i2) f(i1f1) f(i1f2) f(i10) f(i1i2) f(i2f1) f(i2f2) f(i2i1) f(i20) F 1 BLE 1 O 1 F 2 I 1 BLE 2 O 2 I 2
26 Summary Functon Reducton for Local Routng Network Wthout LUT Reconfguraton ( I + N ) Functons k n Commutatve Property Functons ( n = I + N ) j = 1 j k n Duplcated-Constant Input Equvalency Functons j = 1 j n Constant-New Input Equvalency Functons k What s the effect of functon reducton on the desgn of the local routng network? k
27 Summary Local Routng Network Desgn f(f1f1) f(f1f2) f(f1i1) f(f1i2) f(f2f1) f(f2f2) f(f2i1) f(f2i2) f(i1f1) f(i1f2) f(i1i1) f(i1i2) f(i2f1) f(i2f2) f(i2i1) f(i2i2) LUT Input 1 = {F1 F2 I1} LUT Input 2 = {F2 I1 I2} 3:1 Multplexers nstead of 4:1 Multplexers In General: From (I+N):1 Multplexers to (I+N-k+1):1 Multplexers
28 A More Complex Example k = 4 N = 2 I = 6: 4096 Functons => 70 Functons 8:1 Multplexers => 5:1 Multplexers Reducton n Fanout F 1 F 1 F 2 F 2 I 1 LUT 1 O O 1 I 1 LUT 1 O O 1 I 2 I 2 I 3 I 3 I 4 LUT 2 O 2 I 4 LUT 2 O 2 I 5 I 5 I 6 I 6
29 Local Routng Network Area Reducton k = 4 70% Mn. Mem. Two Level Mn. Level 60% 50% % Reducto on 40% 30% 20% 10% 0% N
30 Local Routng Network Area Reducton k = 7 80% Mn. Mem. Two Level Mn. Level 70% 60% % Reducto on 50% 40% 30% 20% 10% 0% N
31 Logc Cluster Area Reducton k = 4 % Reducto on 24% 22% 20% 18% 16% 14% 12% 10% 8% 6% 4% 2% Mn. Mem. Two Level Mn. Level N
32 Logc Cluster Area Reducton k = 7 % Reducto on 24% 22% 20% 18% 16% 14% 12% 10% 8% 6% 4% 2% Mn. Mem. Two Level Mn. Level N
33 Fanout Reducton 80% k=4 k=5 k=6 k=7 % Reducto on 70% 60% 50% 40% 30% 20% 10% 0% N
34 Fanout Adjusted Logc Cluster Area Reducton N = 6 k I Trad. Mn. Mem Mn. Level Two Level New % % % Trad. New Trad. New Reduc. Reduc. Reduc % % % % % % % % % % % %
35 Conclusons Examned the relatonshp between logc equvalency and LUT reconfguraton. LUT reconfguraton can reduce mux sze from (I+N):1 to (I+N-k+1):1. (I+N-k+1):1 s the mnmum sze requred to retan logc equvalency (proved n paper TVLSI Jan 2010). Local Routng Network Area Reducton 3.7%-72% Logc Cluster Area Reducton 2.9%-25% Fanout Reducton 5%-75% Fanout Adjust Logc Cluster Area Reducton (N=6) 6.2% (k=7 Mn. Mem) 12.4% (k=6 Mn. Level)
36 Questons?
37 Future Work 1: Based on [Lemeux01] Proposed a sparse local routng network N=6 (k=4-7) => over 50% reducton n local routng network area compare to the baselne (full connectvty) Ths Work => baselne can be mproved by 8.5%-13% Ddn t study smaller values of N n detal (e.g. N=4) Ddn t study smaller values of N n detal (e.g. N=4) Ths Work => N=4: baselne can be mproved by 24% (k=6 and k=7) Is sparse local routng network stll more area effcent than the mproved baselne archtecture at N=4? Overall s the sparse archtecture stll more area effcent than the mproved baselne archtecture when all values of N are consdered?
38 Future Work 2: Based on [Feng08] Actel Logc Cluster => 8 4-nput LUTs 32 logc cluster nputs 8:1 mux per LUT nput. Sacrfce logc equvalency and local feedbacks for larger logc cluster sze Justfcaton: 8:1 mux can only be used to construct VPR type logc clusters wth 2 4-nput LUTs only; 2 4-nput LUTs are not effcent Ths work: 8:1 mux can be used to construct logc clusters wth 3 4-nput LUTs wth full feedbacks or 4 4- nput LUTs wthout feedbacks => VPR style clusters become compettve agan Need further expermental studes
39 Impact on Prevous Work [Lemeux01] N = 6 (k=4-7) => 50% reducton n local routng network area compare to full connectvty as baselne Our Work => baselne can be mproved by 8.5%-13% Ddn t study smaller values of N n detal (e.g. N=4) Our Work => N=4: baselne can be mproved by 24% (k=6 and k=7) How much sparse local routng network can mprove for N = 4 (probably much less than 50% as observed for N = 6)? tghtly connected LUTs as an alternatve? Fgure 4 needs update N=2 k=7 => 35.8% reducton n local routng network area for baselne
40 Future Work: Impact on Prevous Work [Lemeux01] Proposed a sparse local routng network N=6 (k=4-7) => over 50% reducton n local routng network area compare to the baselne (full connectvty) Our Work => baselne can be mproved by 8.5%-13% Ddn t study smaller values of N n detal (e.g. N=4) Ddn t study smaller values of N n detal (e.g. N=4) Our Work => N=4: baselne can be mproved by 24% (k=6 and k=7) Is sparse local routng network stll more area effcent than the mproved baselne archtecture at N=4? Overall s the sparse archtecture more area effcent than the baselne archtecture when all values of N are consdered? N=6 => tghtly connected LUTs as an alternatve?
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