A Faster Approximation Scheme for Timing A Faster Approximation - - PowerPoint PPT Presentation
A Faster Approximation Scheme for Timing A Faster Approximation Scheme for Timing Driven Minimum Cost Layer Assignment Driven Minimum Cost Layer Assignment Shiyan Hu* , Shiyan Hu * , Zhuo Zhuo Li* * , and Charles J. Alpert* * Li* * , and
Shiyan Hu Shiyan Hu* , * , Zhuo Zhuo Li* * , and Charles J. Alpert* * Li* * , and Charles J. Alpert* * * Dept of ECE, Michigan Technological University * Dept of ECE, Michigan Technological University * * IBM Austin Research Lab * * IBM Austin Research Lab
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1 1X X 2 2X X 4 4X X
In 45nm technology, layer assignment is critical for timing and buffer area optimization timing and buffer area optimization
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10 20 30 40 50 60 70 80 90 0.1 0.2 0.3 0.4 0.5 0.6 0.7 wire length w ire d e la y M2 M4 M6
Wire in higher Wire in higher layer has much layer has much smaller delay smaller delay
0.00E+ 00 5.00E-01 1.00E+ 00 1.50E+ 00 2.00E+ 00 Resistance M2 M4 M6 0.00E+ 00 5.00E+ 01 1.00E+ 02 1.50E+ 02 2.00E+ 02 2.50E+ 02 Capacitance M2 M4 M6
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A buffer can drive longer drive longer distance in distance in higher layer higher layer
Timing is improved improved
Fewer buffers are needed are needed
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IP IP IP IP
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Find a minimal cost minimal cost layer assignment such that the layer assignment such that the timing constraint is satisfied. timing constraint is satisfied.
Same Layer Same Layer Can be different Can be different layers layers
– – A buffered Steiner tree A buffered Steiner tree with n wire segments with n wire segments – – Timing constraint Timing constraint – – m wire layers with RC m wire layers with RC parameters and cost parameters and cost
A layer refers to a pair of horizontal and vertical layers with similar RC characteristics layers with similar RC characteristics
Between any buffers, one layer is used
In early design stage, when buffering effect is considered, wire shaping is not important [Alpert considered, wire shaping is not important [Alpert TCAD TCAD’ ’01] 01]
In post-
routing stage, wire shaping could improve timing, reduce timing, reduce vias vias and reduce coupling and so and reduce coupling and so forth forth
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A Fully Polynomial Time Approximation Scheme Approximation Scheme
Provably good
Within (1+ ɛ
ɛ)
)
ɛ ɛ> 0
> 0
Runs in time polynomial in n polynomial in n (segments), m (layers) (segments), m (layers) and 1/ and 1/ ɛ
ɛ
Ultimate solution for an NP an NP-
hard problem in theory in theory
Highly practical
1 1X X 2 2X X 4 4X X
It depends on M and uses a DP of O(m and uses a DP of O(mn n3
3/
/ ɛ
2) time
) time
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Bound independent oracle query Our DP needs one run for all W
2/
Ratio between upper and Ratio between upper and lower bounds of the cost of lower bounds of the cost of
An iterative DP with An iterative DP with incremental W incremental W
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W* : the cost of optimal solution W* : the cost of optimal solution
Check it Make guess on W* Return the solution Good (close to W* ) Not Good Key 2: Smart guess Key 1: Efficient checking
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Benefit of guess Benefit of guess
Only maintain the solutions with cost solutions with cost no greater than the no greater than the guessed cost guessed cost
Accelerate DP
Oracle (x): the checker, able to decide whether x> > W* or not W* or not – – Without knowing W* Without knowing W* – – Answer efficiently Answer efficiently
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Oracle (x) Guess x within the bounds Setup upper and lower bounds of cost W* Update the bounds
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Scale and round Scale and round each wire cost each wire cost
⎥ ⎦ ⎥ ⎢ ⎣ ⎢ = n x w w / ε Only interested in Only interested in whether there is a whether there is a solution with cost solution with cost up to x satisfying up to x satisfying timing constraint timing constraint
Dynamic Dynamic Programming Programming Perform DP to Perform DP to scaled problem scaled problem with cost bound with cost bound n/ n/ ɛ
ɛ. Time
. Time polynomial in n/ polynomial in n/ ɛ
ɛ
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xɛ
ɛ/n
2xɛ
ɛ/n
3xɛ
ɛ/n
4xɛ
ɛ/n
Wire cost
≤xɛ
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Yes, there is a solution satisfying timing constraint No, no such solution
With cost rounding back, the solution has cost at most n/ɛ • xɛ/n + xɛ= (1+ɛ)x > W* With cost rounding back, the solution has cost at least n/ɛ • xɛ/n = x ≤ W*
DP result w/ all w are integers ≤ n/ɛ
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To model effect to upstream, a upstream, a candidate solution candidate solution is associated with is associated with
v: a node
Q: required arrival time time
W: cumulative wire cost wire cost
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Candidate solutions are propagated toward the source
are generated
– Subtree processing – Solution update at buffer
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Three paths
– – p pa
a: a
: a -
> u – – P Pb
b: b
: b -
> u – – P Pc
c: c
: c -
> u
Qu
u(l)= min{
(l)= min{ Q Qa
a-
d(pa
a,l),
,l),Q Qb
b-
d(p pb
b,l),Q
,l),Qc
c-
d(pc
c,l)}
,l)}
Wu
u(l)=
(l)= W Wa
a+
+ W Wb
b+
+ W Wc
c+ w(T,l)
+ w(T,l)
Wires are in the same layer l layer l
( (Q Qu
u,W
,Wu
u)
) ( (Q Qa
a,
,W Wa
a)
) ( (Q Qb
b,
,W Wb
b)
) ( (Q Qc
c,
,W Wc
c)
)
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Naï ïve ve merging takes merging takes O(W O(Wk
k) time with k
) time with k branches branches
( (Q Qa
a,1 ,1,
,W Wa
a,1 ,1)
) ( (Q Qa
a,2 ,2,
,W Wa
a,2 ,2)
) ( (Q Qa
a,3 ,3,
,W Wa
a,3 ,3)
) ( (Q Qa
a,4 ,4,
,W Wa
a,4 ,4)
) ( (Q Qu
u,
,W Wa
a)
) ( (Q Qb
b,1 ,1,
,W Wb
b,1 ,1)
) ( (Q Qb
b,2 ,2,
,W Wb
b,2 ,2)
) ( (Q Qb
b,3 ,3,
,W Wb
b,3 ,3)
) ( (Q Qb
b,4 ,4,
,W Wb
b,4 ,4)
) ( (Q Qc,1
c,1,
,W Wc
c,1 ,1)
) (Q (Qc,2
c,2,
,W Wc
c,2 ,2)
) (Q (Qc,3
c,3,
,W Wc
c,3 ,3)
) (Q (Qc,4
c,4,
,W Wc
c,4 ,4)
)
k k
k) solutions
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1 i1 i1), Q(a
2 i2 i2),..., Q(a
k ik ik)
1+ i
2+
k= w,
max max 1
1 ≤ r r ≤ k k min { Q(a
min { Q(a1
1 i1 i1),Q(a
),Q(a2
2 i2 i2),..., Q(a
),..., Q(ar
r ir+ 1 ir+ 1), ...,Q(a
), ...,Q(ak
k ik ik)}
)}
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Solution(w= 8, Q= 9) is shown. Solution(w= 8, Q= 9) is shown. To compute Solution (w= 9, Q) To compute Solution (w= 9, Q)
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Candidate Solution (w= 9, Q= 8) Candidate Solution (w= 9, Q= 8)
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Candidate Solution (w= 9, Q= 4) Candidate Solution (w= 9, Q= 4)
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Candidate Solution (w= 9, Q= 5) Candidate Solution (w= 9, Q= 5)
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Candidate Solution (w= 9, Q= 7) Candidate Solution (w= 9, Q= 7)
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Lemma: given a subtree subtree with m layers, k with m layers, k branches and W non branches and W non-
dominated solutions at each downstream buffer, one can merge them in downstream buffer, one can merge them in O( O(mkW mkW) time. ) time.
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After merging, one non-
dominated solution per layer per cost, totally layer per cost, totally O( O(mW mW) solutions ) solutions
For each cost, find largest Q for all layers largest Q for all layers after buffer and after buffer and propagate it propagate it
( (Q Qu
u,W
,Wu
u)
) ( (Q Qa
a,
,W Wa
a)
) ( (Q Qb
b,
,W Wb
b)
) ( (Q Qc
c,
,W Wc
c)
)
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2/
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U (L): upper (lower) bound on W*
Naive binary search style approach
Runtime depends on the initial bounds U and L
Oracle (x) x= (U+ L)/2 and W= n/ ɛ
ɛ
Set U and L on W*
U= (1+ (1+ɛ ɛ)x )x L= x
W* < (1+ W* < (1+ ɛ
ɛ)x
)x W* W* ≥
≥ x
x
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Rounding factor xɛ
ɛ/n for cost
Larger ɛ
ɛ: faster with
: faster with rough estimation rough estimation
Smaller ɛ
ɛ: slower with
: slower with accurate estimation accurate estimation
Adapt ɛ
ɛ and relate it with U and L
and relate it with U and L
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Wire cost U/L
xɛ/n xɛ/n
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Begin with large ɛ
’ and progressively reduce it and progressively reduce it according to U/L as x approaches W* according to U/L as x approaches W*
Set ɛ
ɛ’
’ as a geometric sequence of …, 8, 4, 2, 1, 1/2, …, ɛ
ɛ
ɛ) time. Total
Total runtime is O( runtime is O(… … + n/8 + n/4 + n/2 + + n/8 + n/4 + n/2 + … … + n/ + n/ ɛ
ɛ) =
) = O(n/ O(n/ ɛ
ɛ). Independent of # of iterations
). Independent of # of iterations
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' * , * , * , * , '
1 , 1
i i l i u i l i u i
W W x W W ε ε + ⋅ = − = ) ( ) ( ) 1 (
) 3 / 4 ( 2 / 1 1 * , * , 2 1 * , * , 2 1 ' 2
i t
t i i u i l t i i u i l t i i
W W mn O W W mn O mn O
−
⋅ ≤ ≤ ≤ ≤ ≤ ≤
⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = = ε
) ( ) 59 . ( ) (
2 ) 3 / 4 ( 2 / 1 2 ) 3 / 4 ( 2 / 1 * , * , 2
mn O mn O W W mn O
t j t j i u i l
j j
= = ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛
< ≤ ⋅ ⋅ < ≤
i t
t u t l i u i l i u i l i u i l i l i u i l i u
W W W W W W W W W W W W
−
⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = ⇒ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = ⇒ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ =
+ + ) 3 / 4 ( * , * , * , * , 3 / 4 * , * , * , * , 4 / 3 * , * , * 1 , * 1 ,
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At least one feasible solution, feasible solution,
solution w/ cost solution w/ cost 2n/ 2n/ ɛ
Lɛ
/n = 2L
U
Runs in O(mn2
2/
/ ɛ
) time time
Pick min cost solution satisfying Pick min cost solution satisfying timing at driver timing at driver W= 2n/ W= 2n/ ɛ
ɛ
Scale and round each cost by L Scale and round each cost by Lɛ
ɛ/n
/n Run DP
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2/
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Oracle (x) Adapting ɛ = [U/L-1] 1/2 Set U and L of W* Set x= [UL/(1+ ɛ)] 1/2 Update U or L U/L< 2 Compute final solution
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– – 1000 industrial nets 1000 industrial nets
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Approximation Ratio ɛ Wire Cost Ratio
0.1 0.2 0.3 0.4 0.5 . 5 . 1 . 2 . 3 . 4 . 5 Old FPTAS New FPTAS
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Approximation Ratio ɛ Speedup
1 2 3 4 5 6 7 0.05 0.1 0.2 0.3 0.4 0.5 Old FPTAS New FPTAS
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FPTAS always achieves the theoretical guarantee
Larger ɛ
ɛ leads to more speedup
leads to more speedup
3.9x faster with 2.2% additional wire area compared to DP
Up to 6.5x faster than DP
On average about 2x faster than previous FPTAS
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Propose a (1+ ɛ
ɛ) approximation for timing
) approximation for timing constrained layer assignment for any constrained layer assignment for any ɛ
ɛ > 0 running
> 0 running in O(mn in O(mn2
2/
/ ɛ
ɛ) time
) time
– – Linear time DP running in O( Linear time DP running in O(mnW mnW) time ) time – – Bound independent oracle query Bound independent oracle query
– – Up to 6.5x faster than DP and 2x faster than Up to 6.5x faster than DP and 2x faster than previous FPTAS previous FPTAS – – Few percent additional wire area compared to Few percent additional wire area compared to DP as guaranteed theoretically DP as guaranteed theoretically
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