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An Optimal Jumper An Optimal Jumper Insertion Algorithm for Antenna Insertion Algorithm for Antenna Avoidance/Fixing on General Routing Avoidance/Fixing on General Routing Trees with Obstacles Trees with Obstacles Bor- -Yiing Yiing Su and
An Optimal Jumper Insertion Algorithm for Antenna Avoidance/Fixing on General Routing Trees with Obstacles An Optimal Jumper Insertion Algorithm for Antenna Avoidance/Fixing on General Routing Trees with Obstacles
Bor Bor-
Yiing Su and Su and Yao Yao-
Wen Chang Chang
National Taiwan University
Jiang Jiang Hu Hu
Texas A&M University
Introduction
Problem Definition
An Optimal Algorithm for Jumper Insertion
Complexity Analysis
Experimental Results
Conclusions
The process antenna effect is a phenomenon of plasma plasma-
induced gate oxide degradation gate oxide degradation caused caused by by charge accumulation charge accumulation on conductors.
During metallization, each time an additional layer of interconnect is added. While the metal layer of interconnect is added. While the metal line is being manufactured, the line is being manufactured, the floating floating interconnect interconnect acts as a temporary acts as a temporary capacitor capacitor to to store charges induced from plasma etching. store charges induced from plasma etching.
a b Metal 1 Metal 2 Metal 3 A two-pin net
If the charged conductor is connected only to the the gate oxide gate oxide, Fowler , Fowler-
Nordheim (F (F-
N) tunneling current will discharge through the thin oxide current will discharge through the thin oxide and damage the gate. and damage the gate.
– – No harm to diffusion source (discharge
No harm to diffusion source (discharge through substrate) through substrate)
a b Metal 1 Metal 2 Metal 3 A two-pin net a b Metal 1 Metal 2 Metal 3 Poly Cross section view source terminalF-N tunneling current
Jumper Insertion
– – Break the signal wires with antenna violations and
Break the signal wires with antenna violations and route them to the top route them to the top-
most layer
– – Induce
Induce vias vias
Embedded Protection Diode
– – Add a protection diode on every input port of a cell
Add a protection diode on every input port of a cell
– – May consume unnecessary areas
May consume unnecessary areas
Diode Insertion
– – Insert
Insert “ “under under-
the-
wire” ” diodes to fix the violation diodes to fix the violation
– – Need extra silicon space to place the diodes
Need extra silicon space to place the diodes
Jumper Insertion
– – Break the signal wires with antenna violations and
Break the signal wires with antenna violations and route them to the highest layer route them to the highest layer
– – Induce
Induce vias vias
Embedded Protection Diode
– – Add a protection diode on every input port of a cell
Add a protection diode on every input port of a cell
– – May consume unnecessary areas
May consume unnecessary areas
Diode Insertion
– – Insert
Insert “ “under under-
the-
wire” ” diodes to fix the violation diodes to fix the violation
– – Need extra silicon space to place the diodes
Need extra silicon space to place the diodes
Jumper insertion reduces the charge amount for violated nets during manufacturing for violated nets during manufacturing
– – Side effects
Side effects: : Delay and congestion Delay and congestion
a b
a b
Metal 1 Metal 2 Metal 3 Metal 1 Metal 2 Metal 3 Poly a b
a b
A two-pin net A two-pin net with jumper insertion Jumper
Steiner points are just wire junctions. They cannot help discharge the wire. cannot help discharge the wire.
s is a Steiner point. is a Steiner point. The charges accumulated The charges accumulated
e(u, s), , s), e(s e(s, v , v1
1), and
), and e(s e(s, v , v2
2)
) will all will all cause antenna effect on the gate terminal cause antenna effect on the gate terminal u u. .
u v1
s
v2
gate terminal gate terminal
Since a jumper routes a signal wire to the topmost layer, we must consider the routing topmost layer, we must consider the routing with with obstacles
in the active layers active layers, e.g., pre , e.g., pre-
routed nets, power/ground nets, clock nets.
– – Active layers: the layers from the current routing
Active layers: the layers from the current routing layer up to the topmost layer layer up to the topmost layer
Need to avoid obstacles when adding jumpers
u1 u2 Obstacle Jumper
Ho, Chang & Chen, ISPD-
04
– – Bottom
Bottom-
up dynamic programming in loglinear loglinear time time
– – Insert jumpers
Insert jumpers right beside right beside nodes of a nodes of a spanning spanning tree tree
Wu, Hu Hu & & Mahapatra Mahapatra, ISPD , ISPD-
05
– – Insert jumpers
Insert jumpers at arbitrary positions at arbitrary positions of edges of a
Steiner Steiner tree in linear time tree in linear time
– – Optimal
Optimal only for some special tree topologies
Su and Chang, DAC-
05
– – Insert jumpers
Insert jumpers at arbitrary positions at arbitrary positions of edges of a
spanning spanning tree in tree in loglinear loglinear time time
– – Optimal for a
Optimal for a spanning spanning tree with arbitrary topologies tree with arbitrary topologies
Previous works are optimal only for restricted Previous works are optimal only for restricted cases & do not consider obstacles!! cases & do not consider obstacles!!
Yes Yes N N N N N N DAC DAC-
05 Yes Yes N N N N Consider Consider
Yes Yes Yes Yes N N Allow Allow arbitrary arbitrary insertion insertion positions positions? ? Yes Yes Yes Yes N N Consider Consider Steiner Steiner trees? trees? Yes Yes N N N N Optimal Optimal for general for general routing tree? routing tree? This This work work ISPD ISPD-
05 ISPD ISPD-
04
Introduction to Antenna Effect to Antenna Effect
Problem Definition
An Optimal Algorithm for Jumper Insertion
Complexity Analysis
Experimental Results
Conclusions
Formulate the problem of jumper insertion on a routing tree for antenna avoidance/fixing as a routing tree for antenna avoidance/fixing as a tree tree-
cutting problem
T = ( (V VG
G∪
∪V VN
N , E
, E): a Steiner tree ): a Steiner tree
– – Set
Set V VG
G of nodes represents all
gate terminals
– – Set
Set V VN
N of nodes represents other nodes
– – Set
Set E E of edges denotes the
wires connecting the connecting the circuit terminals or junctions circuit terminals or junctions
D: set of set of obstacles
in the active layers
Projection of the obstacles in D D defines the defines the forbidden regions forbidden regions for jumper insertion for jumper insertion
F: : set of the forbidden regions. set of the forbidden regions.
– – f(u
f(u) = 1 ) = 1 if node if node u u is in a forbidden region is in a forbidden region (u (u∈ ∈ F); F); f(u f(u) = 0, ) = 0, otherwise
The edge weight edge weight in a Steiner tree models the in a Steiner tree models the strength of antenna effect caused by the strength of antenna effect caused by the corresponding wire. corresponding wire.
The edge weight can be
– – the
the length length of the wire
– – the
the area area of the wire
– – the
the perimeter perimeter of the wire
– – the
the ratio ratio of antenna strength to gate size, or
– – any other reasonable models.
any other reasonable models.
Problem JIROA JIROA ( (J Jumper umper I Insertion nsertion on a
Routing tree with tree with O Obstacles for bstacles for A Antenna ntenna avoidance/fixing avoidance/fixing) )
Input: A routing tree A routing tree T T = ( = (V VG
G∪
∪V
VN
N, E
, E), an upper ), an upper bound bound Lmax Lmax, and a set , and a set D D of rectangular obstacles
Goal: : find the minimum set find the minimum set C C of cutting nodes
Constraint: : c c ≠ ≠ u u for any for any c c ∈ ∈ C C and and u u ∈ ∈ V V, , f(c f(c) = 0, ) = 0, ∀ ∀c c ∈ ∈ C C, such that , such that L L( (u u) ) ≤ ≤ Lmax Lmax, , ∀ ∀u u ∈ ∈ V VG
G .
.
– – L
L( (u u): ): sum of effective edge sum of effective edge weights on node u weights on node u
– – Lmax
Lmax : : antenna upper bound antenna upper bound
Routing tree T Lmax L(u) Cutting nodes u
Introduction to Antenna Effect to Antenna Effect
Problem Definition
An Optimal Algorithm for Jumper Insertion
Complexity Analysis
Experimental Results
Conclusion
A subleaf subleaf is a node for which all its children are is a node for which all its children are leaf nodes, and all the children have been leaf nodes, and all the children have been processed. processed.
Let u, v u, v be two adjacent nodes with be two adjacent nodes with f(v f(v) = 1 ) = 1. . Then Then r(u r(u, v) , v) denotes the cutting node denotes the cutting node c c on edge
e = (u, v) e = (u, v) with with f(c f(c) = 0 ) = 0 and and l(u l(u, c) , c) (weight between (weight between nodes u and c) nodes u and c) being the being the maximum maximum among among every node on edge every node on edge e. e.
u v Obstacle c = r(u,v)
f f( (v v) = 1 ) = 1
Algorithm BUJIO BUJIO ( (B Bottom
Up p J Jumper umper I Insertion nsertion with with O Obstacles) bstacles) for jumper insertion applies a for jumper insertion applies a bottom bottom-
up approach to insert cutting nodes on a to insert cutting nodes on a routing tree. routing tree.
– – Step 1: Sort the obstacles in
Step 1: Sort the obstacles in D D by the by the x x-
axes and then the then the y y-
axes
– – Step 2: Compute the initial
Step 2: Compute the initial weight weight for every node for every node
– – Step 3: Makes every
Step 3: Makes every leaf node leaf node satisfy the antenna satisfy the antenna rule rule
– – Step 4: Make every
Step 4: Make every subleaf subleaf node node satisfy the satisfy the antenna rule, and then cut the corresponding leaf antenna rule, and then cut the corresponding leaf nodes to turn a nodes to turn a subleaf subleaf node into a new leaf node node into a new leaf node
Algorithm BUJIO (Bottom Up Jumper Insertion with Obstacles) for jumper insertion uses a with Obstacles) for jumper insertion uses a bottom bottom-
up approach to insert cutting nodes on a routing tree. routing tree.
– – Step 1: Sort the obstacles in
Step 1: Sort the obstacles in D D by the by the x x-
axes and then the then the y y-
axes
– – Step 2: Compute the weight of every node
Step 2: Compute the weight of every node
– – Step 3: Let every
Step 3: Let every leaf node leaf node satisfy the antenna rule satisfy the antenna rule
– – Step 4: Make every
Step 4: Make every subleaf subleaf node node satisfy the satisfy the antenna rule, and then cut the corresponding leaf antenna rule, and then cut the corresponding leaf nodes to turn a nodes to turn a subleaf subleaf node into a new leaf node node into a new leaf node
Iteratively apply Steps 3 & 4 until Iteratively apply Steps 3 & 4 until all gate terminals satisfy the antenna rule all gate terminals satisfy the antenna rule
u1 u3 u2 Obstacle d1 d2 d3
Step 1: Sort the obstacles in D D by the by the x x-
axes and and then the then the y y-
axes
– – Determine
Determine f(u f(u) ) for some node u in for some node u in O( O(lg lg D) D) time time
Step 2: Compute the initial weight weight of every node
– – w(u
w(u) ): accumulated edge weights on node : accumulated edge weights on node u u
u w(u) = l(u,d1) + l(u,d2) + l(u,d3)
Step 3: Make every leaf node leaf node which is a gate which is a gate terminal satisfy the antenna rule terminal satisfy the antenna rule
Tree node Cutting node
Lmax - w(u)
p(u) u l (u, p(u)) + w(u) > Lmax f(c) = 0 Lmax: upper bound on antenna C : cutting set p(u) : parent of u l(e), l(u,v) : weight of the edge e = (u,v) Cut c l (u, p(u)) + w(u) > Lmax f(c) = 1 p(u) u Cut c Cut c1 = r(u,c)
Step 4: Make every subleaf subleaf node node satisfy antenna rule satisfy antenna rule
totallen(up)
): total weights of the edges
: total weights of the edges between the node between the node up and its children and its children
Classify the subleaf subleaf nodes according to nodes according to totallen totallen
– – Case 1: u
Case 1: up
p and all its children are not gate terminals
and all its children are not gate terminals
– – Case 2:
Case 2: totallen( totallen(up) ) + + w(u w(up
p)
) ≤ ≤ L Lmax
max
– – Case 3:
Case 3: totallen( totallen(up) ) + + w(u w(up
p)
) > > L Lmax
max
totallen( totallen(up) ) = ∑i=1 l(ui , up) + w(ui)
k
up : a subleaf node ui : subleaf’s children, ∀1 ≤ i ≤ k
up u1 u2
u1 u2
Case 1: up and All Its Children Are in VN Case 1: up and All Its Children Are in VN
up
p and all its children are
and all its children are not gate terminals not gate terminals, so , so no antenna violation occurs. no antenna violation occurs.
– – w(u
w(up
p)
) ← ← w(u w(up
p) +
) + totallen( totallen(up) )
– – Cut off
Cut off u up
p’
’s children to turn s children to turn u up
p into a
into a leaf node leaf node
up
w(u w(up
p)
)← ← w(u w(up
p) +
) + totallen( totallen(up) ) Steiner point Steiner point
u1 u2
If up
p has a parent
has a parent
– – If
If totallen( totallen(up) ) + + w(u w(up
p) +
) + l(u l(up
p,
, p(u p(up
p))
)) ≤ ≤ L Lmax
max
If u up
p∈
∈ V VN,
N, assign
assign w(u w(up
p)
) ← ← w(u w(up
p) +
) + totallen( totallen(up) )
Cut u up
p’
’s children from the tree, make s children from the tree, make u up
p a leaf node
a leaf node
– – Else insert the cutting node to make
Else insert the cutting node to make totallen( totallen(up) ) + + w(u w(up
p)+
)+ l(u l(up
p, c)
, c) = = L Lmax
max
If f(c f(c) = ) = 1 1, use , use c c1
1=
= r(u r(up
p,c
,c) ) to replace to replace c c
Reduce the tree to make c c or
c1
1 a leaf node
a leaf node
Tree node Cutting node
p(up) up
w(u w(up
p)
) ← ← w(u w(up
p) +
) + totallen( totallen(up) )
p(up) u1 u2 up
totallen( totallen(up) ) +
+ w(u w(up
p) +
) + l(u l(up
p, c)
, c) = = L Lmax
max
c c1 = r(up,c)
Case 3: totallen( totallen(up) ) + + w(u w(up
p)
) > > L Lmax
max
– – Step 1:
Step 1: S = S =∪
∪i=1
i=1{l(e(u
{l(e(ui
i,u
,up
p)) +
)) + w(u w(ui
i)}
)}
– – Step 2: Find
Step 2: Find S = S = S Sl
l∪
∪S Sh
h,
, S
Sl
l∩
∩S Sh
h=
=φ φ, such that , such that
For any a a∈
∈S Sl
l and b
and b ∈ ∈S Sh
h, a
, a ≤
≤ b b
∑
s s ≤
≤ L Lmax
max-
w(up
p)
)
For any b
b ∈ ∈S Sh
h,
, ∑ s + b > s + b > L
Lmax
max-
w(up
p)
)
– – Step 3: Add cutting nodes
Step 3: Add cutting nodes c c1
1,
, … …, , c cSh
Sh on every
s ∈
∈S Sh
h
– – Step 4: Use
Step 4: Use Case 2 Case 2 to cut u to cut up
p into a leaf node
into a leaf node
Tree node for Sl Tree node for Sh up
s s ∈ ∈
S S
l l
C1, C2, … CSh k
s s ∈ ∈
S S
l l
Cutting node
u u1
1, u
, u2
2, ..,
, .., u uk
k
u u1
1
u u2
2
u u3
3
u u4
4
u u5
5
u u6
6
u u7
7
u u8
8
s s1
1
s s2
2
1
2
3
u u1
1
u u2
2
u u3
3
u u4
4
u u5
5
u u6
6
u u7
7
u u8
8
s s1
1
s s2
2
1
3
w(u w(u5
5) = 5
) = 5
u u1
1
u u2
2
u u3
3
u u4
4
u u5
5
u u6
6
u u7
7
u u8
8
s s1
1
s s2
2
1
3
6 6 15 15 7 7 7 7 5 5 6 6 7 7 13 13 6 6
Lmax Lmax = 10 = 10
u u1
1
u u2
2
u u3
3
u u4
4
u u5
5
u u6
6
u u7
7
u u8
8
s s1
1
s s2
2
1
3
6 6 15 15 7 7 7 7 5 5 6 6 7 7 13 13 6 6 3 3 10 10 5 5 10 10 2 2 13 13 c c3
3
c c c c2
2
w(u w(u5
5)=5
)=5 c c1
1
5 5 2 2
Lmax Lmax = 10 = 10
u u1
1
u u2
2
u u5
5
u u6
6
u u8
8
s s1
1
s s2
2
1
4
6 6 7 7 5 5 6 6 7 7 6 6 3 3 13 13 c c3
3
c c2
2
c c4
4
1 1 5 5 c c1
1
2 2
Lmax Lmax = 10 = 10
u u1
1
u u2
2
u u5
5
u u6
6
s s1
1
1
4
6 6 7 7 5 5 7 7 13 13 c c2
2
c c4
4
5 5 c c5
5
c c6
6
Lmax Lmax = 10 = 10
u u1
1
u u2
2
1
6 6 7 7 13 13 c c2
2
c c6
6
c c7
7 c
c8
8
Lmax Lmax = 10 = 10
u u1
1
u u2
2
u u3
3
u u4
4
u u5
5
u u6
6
u u7
7
u u8
8
s s1
1
s s2
2
1
3
6 6 15 15 7 7 7 7 5 5 6 6 7 7 13 13 6 6 c c1
1
c c2
2
c c3
3
c c4
4
c c5
5
c c6
6
c c8
8
c c7
7
Totally we insert Totally we insert 8 jumpers 8 jumpers
Steiner Steiner tree! tree!
Introduction to Antenna Effect to Antenna Effect
Problem Definition
An Optimal Algorithm for Jumper Insertion
Complexity Analysis
Experimental Results
Conclusions
The tree-
cutting problem JIROA exhibits the properties of optimal substructures and properties of optimal substructures and greedy choices. greedy choices.
Algorithm BUJIO optimally solves the JIROA problem in O((V + D) problem in O((V + D) lg lg D) time using O(V) D) time using O(V) space space
– – V: # of tree nodes
V: # of tree nodes
– – D: # of obstacles
D: # of obstacles
Introduction to Antenna Effect to Antenna Effect
Problem Definition
An Optimal Algorithm for Jumper Insertion
Complexity Analysis
Experimental Results
Conclusions
Implemented BUJIO in C++ on a 2.4 GHz Intel
Pentium PC with 256 MB memory in Windows XP
Generated gate terminals on 1 cm x 1 cm planes with a number of nodes and obstacles
Compared with ISPD-04, ISPD-05, DAC-05 works,
extending their works to handle obstacles using the same obstacle processing in this paper
Our BUJIO and the ISPD-05 algorithms work on
Steiner trees while the ISPD-04 and DAC-05 ones
Experiment #1: #Jumpers for Random Cases Experiment #1: #Jumpers for Random Cases
+0.2% +0.2% +0.2% +0.3% +0.5% +0.5% +0.4% +0.5% +0.3% %More 435 518 601 708 827 966 1149 1348 1537 #J ISPD-05 562 649 779 919 1050 1210 1378 1607 1806 #J DAC-05 +29.5% +25.5% +30.0% +30.2% +27.6% +26.0% +20.5% +19.8% +17.8% %More +48.6% 645 434 300 +48.4% 767 517 290 +55.0% 928 599 280 +57.1% 1109 706 270 +55.4% 1279 823 260 +56.2% 1501 961 250 +52.3% 1742 1144 240 +50.6% 2020 1341 230 +53.9% 2359 1533 220 %More #J #J ISPD-04 BUJIO Lmax (um)
Random cases (10,000 nodes, 500 obstacles) Quality rating: BUJIO > ISPD-05 > DAC-05 > ISPD-04 Runtimes are about the same for all methods (0.15 sec)
+0.3% +0.3% +52% +52% +25% +25%
Experiment #2: #Jumpers for Synthetic Cases Experiment #2: #Jumpers for Synthetic Cases
+26.0% +26.8% +27.1% +27.0% +27.3% +27.4% +27.6% +27.8% +27.6% %More 4541 4637 4738 4844 4951 5061 5174 5311 5455 #J ISPD-05 3728 3809 3895 3983 4076 4187 4305 4421 4557 #J DAC-05 +3.4% +4.2% +4.5% +4.4% +4.8% +5.4% +6.1% +6.4% +6.6% %Mor e +33.0% 4796 3605 300 +34.4% 4915 3657 290 +34.9% 5028 3728 280 +35.2% 5157 3814 270 +36.3% 5299 3889 260 +37.3% 5455 3972 250 +38.2% 5604 4056 240 +38.6% 5762 4157 230 +39.1% 5943 4272 220 %More #J #J ISPD-04 BUJIO Lmax (um)
Synthetic cases: make gate terminals adjacent to each other
Quality rating: BUJIO > DAC-05 > ISPD-05 > ISPD-04
+27% +27% +36% +36% +5% +5%
1000 nodes, 500 obstacles, 426 jumpers, Lmax = 500 um
X
Tree node Tree node Steiner point Steiner point Obstacle Obstacle Jumper Jumper
Introduction to Antenna Effect to Antenna Effect
Problem Definition
An Optimal Algorithm for Jumper Insertion
Complexity Analysis
Experimental Results
Conclusions
We have presented the first the first optimal
algorithm to solve the JIROA problem in solve the JIROA problem in O((V+D) O((V+D) lg lg D) D) time time and and O(V) O(V) space space, where V is the number of , where V is the number of vertices and D is the number of obstacles. vertices and D is the number of obstacles.
Experimental results have shown that our algorithm leads to more robust and much better algorithm leads to more robust and much better solutions than the previous works. solutions than the previous works.
Our work can be applied to any routing trees, and thus readily be incorporated into a router and thus readily be incorporated into a router for antenna avoidance or a post for antenna avoidance or a post-
layout optimizer for antenna fixing. for antenna fixing.
Metal 1 Metal 1 Layer Layer Metal 2 Metal 2 Layer Layer
Gate terminal Gate terminal Nets Nets Nets Nets Jumper Jumper
Antenna violation occurs! Antenna violation occurs! Jumper Inserted Jumper Inserted Collision! Cannot insert a jumper here! Collision! Cannot insert a jumper here! No collision occurs. No collision occurs. We can insert a jumper here. We can insert a jumper here.