Advanced Interconnect Optimizations

Buffers Improve Slack

RAT = 300Delay = 350Slack = -50

RAT = 700Delay = 600Slack = 100

RAT = 300Delay = 250Slack = 50

RAT = 700Delay = 400Slack = 300

slackmin = -50

slackmin = 50Decouple capacitive load from critical path

RAT = Required Arrival Time

Slack = RAT - Delay

Timing Driven Buffering Problem Formulation

• Given– A Steiner tree– RAT at each sink– A buffer type– RC parameters– Candidate buffer locations

• Find buffer insertion solution such that the slack at the driver is maximized

Candidate Buffering Solutions

Candidate Solution Characteristics

• Each candidate solution is associated with– vi: a node

– ci: downstream capacitance

– qi: RAT

vi is a sinkci is sink capacitance

v is an internal node

Van Ginneken’s Algorithm

Candidate solutions are propagated toward the source

Dynamic Programming

Solution Propagation: Add Wire

• c2 = c1 + cx

• q2 = q1 – rcx2/2 – rxc1

• r: wire resistance per unit length

• c: wire capacitance per unit length

(v1, c1, q1)(v2, c2, q2)x

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Solution Propagation: Insert Buffer

• c1b = Cb

• q1b = q1 – Rbc1 – tb

• Cb: buffer input capacitance

• Rb: buffer output resistance

• tb: buffer intrinsic delay

(v1, c1, q1)(v1, c1b, q1b)

Solution Propagation: Merge

• cmerge = cl + cr

• qmerge = min(ql , qr)

(v, cl , ql) (v, cr , qr)

Solution Propagation: Add Driver

• q0d = q0 – Rdc0 = slackmin

• Rd: driver resistance

• Pick solution with max slackmin

(v0, c0, q0)(v0, c0d, q0d)

Example of Solution Propagation

(v1, 1, 20)22

v1 v1

(v2, 3, 16)

• r = 1, c = 1• Rb = 1, Cb = 1, tb = 1• Rd = 1

(v2, 1, 12)

v1

(v3, 5, 8)

v1

(v3, 3, 8)

slack = 5

slack = 3

Add wire

Add wire

Insert buffer Add wire

Add driver

Add driver

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Example of Merging

Left candidates

Right candidates

Merged candidates

Solution Pruning

• Two candidate solutions– (v, c1, q1)

– (v, c2, q2)

• Solution 1 is inferior if – c1 > c2 : larger load

– and q1 < q2 : tighter timing

Pruning When Insert Buffer

They have the same load cap Cb, only the one with max q is kept

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Generating Candidates

(1)

(2)

(3)

From Dr. Charles Alpert

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Pruning Candidates

(3)

(a) (b)

Both (a) and (b) “look” the same to the source.Throw out the one with the worst slack

(4)

17

Candidate Example Continued

(4)

(5)

18

Candidate Example ContinuedAfter pruning

(5)

At driver, compute which candidate maximizesslack. Result is optimal.

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Merging Branches

Right Candidates

Left Candidates

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Pruning Merged Branches

Critical

With pruning

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Van Ginneken Example

(20,400)

(20,400)(30,250)(5, 220)

WireC=10,d=150

BufferC=5, d=30

(20,400)

BufferC=5, d=50C=5, d=30

WireC=15,d=200C=15,d=120

(30,250)(5, 220)

(45, 50)(5, 0)(20,100)(5, 70)

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Van Ginneken Example Cont’d

(20,400)(30,250)(5, 220)

(45, 50)(5, 0)(20,100)(5, 70)

(5,0) is inferior to (5,70). (45,50) is inferior to (20,100)

(20,400)(30,250)(5, 220)

(20,100)(5, 70)(30,10)

(15, -10)

Pick solution with largest slack, follow arrows to get solution

Wire C=10

Basic Data Structure

(c1, q1) (c2, q2) (c3, q3)

Sorted list such that

• c1 < c2 < c3

• If there is no inferior candidates q1 < q2 < q3

Worse load cap

Better timing

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Prune Solution List

(c1, q1) (c2, q2) (c3, q3)

Increasing c

q1 < q2 ?

(c4, q4)

q3 < q4 ?

Y

NPrune 2 q1 < q3 ?

q2 < q3 ?

Y

q3 < q4 ?

Y

Prune 3 q1 < q4 ?

N Prune 3

N

N Prune 4N Prune 4

q2 < q4 ?

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Pruning In Merging

(cl1, ql1)

(cl2, ql2)

(cl3, ql3)

(cr1, qr1)

(cr2, qr2)

ql1 < ql2 < qr1 < ql3 < qr2

Merged candidate

s

(cl1+cr1, ql1)

(cl2+cr1, ql2)

(cl3+cr1, qr1)

(cl3+cr2, ql3)

(cl1, ql1)

(cl2, ql2)

(cl3, ql3)

(cr1, qr1)

(cr2, qr2)

(cl1, ql1)

(cl2, ql2)

(cl3, ql3)

(cr1, qr1)

(cr2, qr2)

(cl1, ql1)

(cl2, ql2)

(cl3, ql3)

(cr1, qr1)

(cr2, qr2)

Left candidate

s

Right candidate

s

Van Ginneken Complexity

• Generate candidates from sinks to source

• Quadratic runtime

– Adding a wire does not change #candidates

– Adding a buffer adds only one new candidate

– Merging branches additive, not multiplicative

– Linear time solution list pruning

• Optimal for Elmore delay model

Multiple Buffer Types

(v1, 1, 20)22

v1

v1

(v2, 3, 16)

• r = 1, c = 1

• Rb1 = 1, Cb1 = 1, tb1 = 1

• Rb2 = 0.5, Cb2 = 2, tb2 = 0.5

• Rd = 1

(v2, 1, 12)

v1

(v2, 2, 14)