Revisiting Fidelity: A Case of
Elmore-based Y-routing Trees
Tuhina Samanta*, Prasun Ghosal*, Hafizur Rahaman*
and
Parthasarathi Dasgupta†
*Bengal Engineering & Science University, India
†Indian Institute of Management Calcutta, India
`
SLIP 2008, Newcastle
Outline
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Y-Routing and Y-Routing Trees
Fidelity of Delay Estimators
Some Previous works
Motivation of our work
Construction of Elmore-based Y-Routing tree
Fidelity and Rank Correlation
Experimental Results
Conclusions
SLIP 2008, Newcastle
Routing – a critical step
• For a given set of terminals, connect different subsets
(nets) of these terminals
• Cost components to be minimized
• wire length
• number of layers
• via
• delay
• Some recent Challenges in DSM Physical Design
• Dominance of interconnect length
• Congestion
• Parasitic effects
• Thermal effects
SLIP 2008, Newcastle
Steiner trees for Routing
• Involves the addition of a set of Steiner points to a given set of
demand points (terminals) to generate an interconnection (with
minimum wire-length, etc.)
• Traditional Manhattan routing allows edges in only two directions viz.
horizontal and vertical
• Objectives:
– Minimizing total/maximum sink delay
– Maximizing Required-Arrival-Time at the source for a
given set of Required-Arrival-Times at sink terminals
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Generalized (λ-) Steiner trees
• λ = 2 for Manhattan Steiner trees, λ = 3 for Y-Steiner
trees, λ = 4 for X- Steiner trees
• λ extends from 2 to 4 => number of metal layers
increases from two to four
• More the number of layers => more the number of vias
• Cost to implement increases with increasing λ
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Y-routing
• Routing wires along 0-, 120- and 240-degree
(hexagonal) orientations
a) X-routing grids
b) Y-routing grids
H. Chen et al., The Y-Architecture for On-Chip Interconnect: Analysis
and Methodology, IEEE TCAD/ICAS, 2004
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Y-routing : Pros and Cons
• Pros:
 number of layers not too large
 uniform routing grid
 more throughput / routability
 simple DRC
• Cons:
 more number of layers than in Manhattan routing
 more wire-length than in X-routing
 …..
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Y-routing Trees
An Example Y-routing tree
T. Samanta et al. A Heuristic Method for Constructing Hexagonal
Steiner Minimal Trees for Routing in VLSI, ISCAS - 2006
SLIP 2008, Newcastle
Outline
•
•
•
•
•
•
•
•
Y-Routing and Y-Routing Trees
Fidelity of Delay Estimators
Some Previous works
Motivation of our work
Construction of Elmore-based Y-Routing tree
Fidelity and Rank Correlation
Experimental Results
Conclusions
SLIP 2008, Newcastle
Accuracy of Heuristics
• Heuristics - used to solve hard problems in
reasonable time
• Quality measured by the degree of nearness to
the Optimal Solution
• How to distinguish good heuristics from bad
heuristics?
• What if the Optimal Solution Cost is Unknown?
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Relative Accuracy of Heuristics
• Measuring relative performances of different
heuristics
• Generate all possible solutions
• Compute the costs of the generated solutions
using each heuristic
• Find the deviations in the performances of the
heuristics based on these costs
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Fidelity of a Heuristic
• hi, hj = cost of heuristic solution for instances i,j
• si, sj = cost of optimal solution for instances i,j
• m = total number of instances considered
Fidelity of heuristic (f) =
|(i,j): 0 < i < j < m, (hi – hj)(si – sj) > 0 or (si = sj)|
(m)
2
J L Ganley, Accuracy and fidelity of fast net length estimates.
Integration, the VLSI Journal, 1997.
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Why bother about Signal Delay?
• Global Routing trees typically constructed with
an objective of minimizing circuit delay to
increase speed of the circuit
• Accurate measurement of signal delay very
important
• Exact signal delay measurement is too complex
and time consuming
• Delay estimators should correctly discriminate
the good solutions from the bad ones
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Estimating Signal Delay
• Linear Delay
• Elmore Delay
• Higher order moments – 2-Pole
• Others – Fitted Elmore delay
• SPICE
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Linear Delay
d1
d2
Delay = d1 + d2
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Elmore Delay
• Delay through an on-path resistor = resistance 
downstream capacitance
• Delay through a path (driver to a sink pin) = Sum of
delays through individual edges on the path
• First moment of interconnect under impulse response
• Based on the 50% delay
Source
C1/2
r
Rest of circuit
C1/2
C2
Delay through r = r.(c1/2 + C2)
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Elmore Delay Characteristics
• Fairly accurate delay estimate at nodes far from source
• Expressible as a closed-form expression involving only
resistors and capacitors
• Provable upper bound on actual delay for all inputs
• Additive
Source, S
A
B
Delay (S, B) = Delay(S, A) + Delay(A, B)
P. Penfield, J. Rubinstein, M. A. Horowitz. Signal Delay in RC tree
networks, IEEE TCAD/ICAS, July 1983.
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Elmore Delay Computation
A possible scheme of traversing a RC tree:
• Pass 1: Compute the effective capacitance at each
node of the RC tree
• Pass 2: At a node, compute the actual Elmore delay
(from the source) from the sum of (a) delay up to the
predecessor node, and (b) the product of the resistance
between the predecessor node and the current node,
and the effective capacitance at current node obtained in
Pass 1.
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Fidelity of Delay Estimators
Degree to which an optimal or near-optimal solution
according to a delay estimator will also be optimal or
near-optimal according to the actual delay.
For a set of possible solutions obtained using the
estimator, how close are the ranks correlated to those
for the solutions obtained by the actual delay
measurement?
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Computing Fidelity
• Enumerate all possible routing solutions.
• Rank all tree topologies using the estimator.
• Rank all tree topologies by SPICE delay model (actual).
• Find the average difference between the two sets of
ranks for each topology.
SLIP 2008, Newcastle
Outline
•
•
•
•
•
•
•
•
Y-Routing and Y-Routing Trees
Fidelity of Delay Estimators
Some Previous works
Motivation of our work
Construction of Elmore-based Y-Routing tree
Fidelity and Rank Correlation
Experimental Results
Conclusions
SLIP 2008, Newcastle
Some Previous Works
• Construction of near-optimal routing trees based on
Elmore delay (Boese et al, ICCAD. 1993)
• Optimum wire sizing in routing trees (Cong et al,
ACMTODAES, 1996)
• P-tree – Cheng et al, Based on Permutation Trees
• Q-Tree – Kahng et al
SLIP 2008, Newcastle
Outline
•
•
•
•
•
•
•
•
Y-Routing and Y-Routing Trees
Fidelity of Delay Estimators
Some Previous works
Motivation of our work
Construction of Elmore-based Y-Routing tree
Fidelity and Rank Correlation
Experimental Results
Conclusions
SLIP 2008, Newcastle
Existing works: Caveats
• Fidelity definitions and experiments
– Mostly Common-sense based (average, standard
deviation, etc.)
– Relevant data sets rarely considered
– Tested only for Manhattan Architectures
– Presence of tie solutions
SLIP 2008, Newcastle
Objectives of our work
• Verifying the fidelity relations of Linear and Elmore Delay
for Non-Manhattan routing architectures, specifically the
Y-Routing architectures
• Studying the fidelity values for relevant routing
topologies
• Defining some new fidelity metrics based on nonparametric statistical quantities, and studying the
performances of Linear and Elmore Delay based on
these new metrics
SLIP 2008, Newcastle
Outline
•
•
•
•
•
•
•
•
Y-Routing and Y-Routing Trees
Fidelity of Delay Estimators
Some Previous works
Motivation of our work
Construction of Elmore-based Y-Routing tree
Fidelity and Rank Correlation
Experimental Results
Conclusions
SLIP 2008, Newcastle
Proposed Algorithm: Overview
Input: A set of given terminals P.
Output: A Y-routed Elmore Routing Tree over P
Step 1.
Construct the Hanan grid
Step 2.
For each edge e Є E
Find the sum (dtot(e)) of the perpendicular
distances of all the terminals (in P) from e
Step 3.
ω(e) = √(length of e × dtot(e))
• ω(e) allows resolving tie cases during shortest path construction
• ω(e) enables construction of trees of shorter lengths and thus of reduced Elmore delays
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Proposed Algorithm
Step 4. Choose source vertex n0 and arbitrarily any one sink vertex n
Find the shortest (in total ω) path between n0 and n
This forms the partial Steiner Tree Spart
Step 5. while not all terminals of P have been selected
Step 6.
for all the terminals n not yet selected
Step 7.
find the terminal which when connected to Spart
with the shortest path yields minimum Elmore delay
Step 8.
augment Spart with this shortest path
Step 9. return Spart
SLIP 2008, Newcastle
Proposed Algorithm - Example
n0
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Proposed Algorithm - Example
n0
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Proposed Algorithm - Example
n0
SLIP 2008, Newcastle
Proposed Algorithm - Example
n0
SLIP 2008, Newcastle
Proposed Algorithm - Example
n0
SLIP 2008, Newcastle
Proposed Algorithm - Example
n0
SLIP 2008, Newcastle
Outline
•
•
•
•
•
•
•
•
Y-Routing and Y-Routing Trees
Fidelity of Delay Estimators
Some Previous works
Motivation of our work
Construction of Elmore-based Y-Routing tree
Fidelity and Rank Correlation
Experimental Results
Conclusions
SLIP 2008, Newcastle
Rank Correlation for Fidelity
Does fidelity computation have anything to do
with rank correlation?
Should fidelity base on discrimination of ALL
candidate solutions or only the good solutions?
How would the tie cases be handled?
Should fidelity be dimensionless?
SLIP 2008, Newcastle
Spearman’s Rank Correlation
• Two variables x and y
• n pairs of observations on x and y are ordered in
magnitude
• Define rank of ordered observation xj as rank(xj) = j
• Spearman’s Rank Correlation is R
n
2
R = 1 - 6i=1 di /(n(n2 – 1)),
where di = |rank(xi) – rank(yi)|
SLIP 2008, Newcastle
Kendall’s Tau
• Two variables x and y
• n pairs of observations on x and y are ordered in
magnitude
• Q = total number of interchanges required for the y
values to order them as x values
where 0 < Q < n(n-1)/2
• Kendall’s Tau = 
 = 1 – 4Q / n(n-1)
 = (# concordant pairs - # discordant pairs) / total # of pairs
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Kendall’s Tau: Properties
• If the agreement between the two rankings is perfect (i.e., the two
rankings are the same) the coefficient has value 1.
• If the disagreement between the two rankings is perfect (i.e., one
ranking is the reverse of the other) the coefficient has value -1.
• For all other arrangements the value lies between -1 and 1, and
increasing values imply increasing agreement between the rankings.
• If the rankings are completely independent, the coefficient has value
0 on average.
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Kendall’s Tau: Example
Delay
Topo1
Topo2
Topo3
Topo4
Topo5
Topo6
Topo7
Topo8
Elmore
rank
1
2
3
4
5
6
7
8
SPICE
rank
3
4
1
2
5
7
8
6
Total no. of pairs = 28
No. of concordant pairs = 22
No. of discordant pairs = 6
 = 0.57
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Normalized Delay
δ = absolute delay value for an estimator
µ = the average of all absolute values for the estimator
σ = the standard deviation
Normalized value for delay is given by
δnorm = (δ – μ)/σ
Fidelity of Elmore (Linear) = ratio of the
• normalized delay of Elmore (Linear) to Spice
SLIP 2008, Newcastle
Outline
•
•
•
•
•
•
•
•
Y-Routing and Y-Routing Trees
Fidelity of Delay Estimators
Some Previous works
Motivation of our work
Construction of Elmore-based Y-Routing tree
Fidelity and Rank Correlation
Experimental Results
Conclusions
SLIP 2008, Newcastle
Technology Node Parameters
B A McCoy K D Boese, A B Kahng and G Robins. Fidelity and nearoptimality of elmore-based routing constructions. In Proc. of the IEEE
International Conference on Computer Design, pp. 81–84, October 1993
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Spearman’s Correlation Coefficient
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Kendall’s Tau
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Normalized Delay Ratios
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Outline
•
•
•
•
•
•
•
•
Y-Routing and Y-Routing Trees
Fidelity of Delay Estimators
Some Previous works
Motivation of our work
Construction of Elmore-based Y-Routing tree
Fidelity and Rank Correlation
Experimental Results
Conclusions
SLIP 2008, Newcastle
What did we achieve?
• An efficient algorithm for constructing
– Y-routing tree of minimum Elmore delay
• Efficient schemes for
– Computing fidelities of Elmore and Linear delay estimates
against Spice delay values
• Confirmation that Fidelity of Elmore is better than fidelity
of Linear for unbuffered routing trees
SLIP 2008, Newcastle
Possible extensions
• Extending the algorithm for X- and M-routing trees and
comparing fidelity results for these architectures
• Experiments with better delay metrics such as 2-Pole
• Experiments with larger nets with variations of several
parameters
SLIP 2008, Newcastle