an o(bn 2 ) time algorithm for optimal buffer insertion with b buffer types authors: zhuo li and...

An O(bn2) Time Algorithm for Optimal Buffer Insertion with b Buffer Types

Authors: Zhuo Li and Weiping ShiPresenter: Sunil Khatri

Department of Electrical EngineeringTexas A&M University

College Station, Texas 77845, USA

2DATE 2005, MUNICH03/10/2005

Outline• Introduction• O(b2n2) Algorithm• New O(bn2) Algorithm• Experimental Results• Extension• Conclusion

3DATE 2005, MUNICH03/10/2005

• Buffer insertion and sizing is one of the most effective method for reducing interconnect delay.

Introduction

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Saxena, et al.

[TCAD 2004]

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• Modern libraries contain hundreds of different buffers with different characteristics.

• Polarity, input capacitance, driving resistance, intrinsic delay, noise margin, power, area, etc.

• Buffer library size has quadratic effect on running time in traditional algorithms.

• With such large number of buffers and buffer types, fast algorithms for buffer insertion are crucial for timing closure.

Introduction(cont.)

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Problem Formulation• Given: A routing tree, n possible buffer positions, sink

capacitances and required arrival times (RAT), a buffer library, wire resistance and capacitance.

• Delay model: Elmore delay for interconnect and linear delay model for buffers.

s0

s1s2

s3

s4

buffer library

source

sinks

possible buffer positions

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Maximum Slack Problem• Find: Where to insert buffers so that the slack

at the source Q(s0) is maximized.

)},()({min)( 000 iiissdelaysRATsQ

s0

s1

s3

s4

without buffer, Q(S0)= – 50 ps

s2

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Maximum Slack Problem• Find: Where to insert buffers so that the slack

at the source Q(s0) is maximized.

)},()({min)( 000 iiissdelaysRATsQ

s0

s1

s3

s4

with 2 buffers, Q(S0)= 100 ps

s2

8DATE 2005, MUNICH03/10/2005

Previous Research• Maximum Slack

• van Ginneken [ISCAS 90]: O(n2) time and space, where n is the number of buffer positions.

• Lillis, Cheng and Lin [TCAS 96]: O(b2n2) time and space for b buffer types.

• Shi and Li [DAC 03]: O(nlogn) time for 2-pin nets, O(nlog2n) time for multi-pin nets. O(nlogn) space.

• Minimum Buffer Cost (Area, Power, etc.)• Lillis, Cheng and Lin [TCAS 96]: pseudo-

polynomial time algorithm.• Shi, Li and Alpert [ASPDAC 04]: buffer cost

minimization is NP-hard if b is a variable.

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Dynamic Programming• Each candidate solution of a sub-tree is represented by a

(Q, C) pair, where Q is slack and C is downstream capacitance.

• For any two candidates A1 and A2 of the same sub-tree, if Q(A1)Q(A2) and C(A1)C(A2), then A1 is redundant.

• O(b2n2) time dynamic programming algorithm (Lillis-Cheng-Lin)

• For b buffer types, the number of candidates is at most bn+1• For a wire, update (Q, C) value for every candidate in O(bn) time• For a buffer position, add b new candidates in O(b2n) time• For a branch point, merge two sets of candidates in O(bn1+bn2)

time

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Dynamic Programming• Each candidate solution of a sub-tree is represented by a

(Q, C) pair, where Q is slack and C is downstream capacitance.

• For any two candidates A1 and A2 of the same sub-tree, if Q(A1)Q(A2) and C(A1)C(A2), then A1 is redundant.

• O(bn2) time dynamic programming algorithm (This paper)• For b buffer types, the number of candidates is at most bn+1• For a wire, update (Q, C) value for every candidate in O(bn)

time• For a buffer position, add b new candidates in O(bn) time• For a branch point, merge two sets of candidates in O(bn1+bn2)

time

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Data Structure: Linked List• Use linked list to store non-redundant

candidates• Sorted in decreasing Q and decreasing C order• Each entry also contains the list of buffer positions

(Q1,C1) (Q2,C2) (Q3,C3)

Less CapacitanceBetter Slack

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Best Candidates• For each buffer Bi, R(Bi) is buffer driver resistance, C(Bi)

is buffer input capacitance, and t(Bi) is buffer intrinsic delay. Label buffers according to non-decreasing order of resistance R(B1)R(B2) … R(Bb).

• For each buffer type Bi

• Define the best candidate i as the candidate that maximizes slack among all candidates after Bi is inserted.

• The new slack is Q(i)–R(Bi)C(i) –t(Bi).• Define the new candidate i as the candidate formed by i with

buffer type Bi.

• How to find all best candidates quickly is the key addressed in this paper.

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Example• Three buffer types

• R(B1)=1, C(B1), t(B1)• R(B2)=3, C(B2), t(B2)• R(B3)=5, C(B3), t(B3)

Candidates (Q, C):(21, 5)(19, 4)(15, 3)(7, 2)(6, 1)

Best candidate for B1 is 1, and the new candidate is 1

Insert B1:(16t(B1), C(B1))(15t(B1), C(B1))(12t(B1), C(B1))(5t(B1), C(B1))(5t(B1), C(B1))

1 1


22


3 3

Best candidate for B3 is 3, and the new candidate is 3 Best candidate for B2 is 2, and the new candidate is 2

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(Q, C) Plane

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Q

• Non-redundant (Q, C) list is a monotonically decreasing sequence

• As resistance is added, Q values changeA5

(6, 1)

A4(7, 2)

A3(15, 3)

A2(19, 4)

A1(21, 5)

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R(B1) = 1, Q=Q–R(B1)*C

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Q

A1(21-5,

5)

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R(B2) = 3, Q=Q–R(B2)*C

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Q

A1(21-15, 5)

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R(B3) = 5, Q=Q–R(B3)*C

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Q

A1(21-25, 5)

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As R Increases, Q Decreases

-50510152025

Q

1 2 3 4 5

R=5R

=3R=1R

=0

C

R=5R=3R=1R=0

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Best Q Values Move to Left

-50510152025

Q

1 2 3 4 5

R=5R

=3R=1R

=0

C

R=5R=3R=1R=0

Best Q foreach R

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Q

Best Candidates are in Decreasing Order of C

• Lemma 1: C(1) C(2) … C(b)

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• Not enough for an O(bn) algorithm to find all best candidates.

• Need global search

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Q

Pruned

Convex Pruning

A5

A4

A3

A2

• Convex pruning prune candidates like A4

A1

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Before Convex Pruning

-50510152025

Q

1 2 3 4 5

R=5R

=3R=1R

=0

C

R=5R=3R=1R=0

Non-Convex

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After Convex Pruning

-50510152025

Q

1 2 3 4 5

R=5R

=3R=1R

=0

C

R=5R=3R=1R=0

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1

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Convex Hull• After convex pruning, remaining list is a convex hull• Lemma 3: Best candidates must be on the convex hull

• A candidate is on the convex hull if and only if there exists an resistance R such that when R is added, this candidate gives maximum Q

• Lemma 4: On convex hull, if Ai gives maximum Q among neighboring candidates, Ai gives maximum Q among all candidates

• The slope (Qi Qj)/(CiCj) between candidates Ai and Aj (i>j) is the extra resistance value that makes Aj to have better slack than Ai

• On convex hull, slopes are in sorted order• Local Optimal Global Optimal

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Q

Local Optimal Global Optimal

A5

A2

A1 • For any R(Bi), if A2

gives better slack than A1 and A3, then A2 is the best candidate for Bi.

A3

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Q

C

Find Convex Hull: Graham’s Scan • Since the points are sorted, Graham’s scan can

perform convex pruning in linear time

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Q

C



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Q

C



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Q

C

32DATE 2005, MUNICH03/10/2005 O(bn)

O(blogb)

O(bn)

O(bn)

New Subroutine for Adding Buffer• At each buffer position, given the (Q, C) list N in decreasing C order and the

buffer library, where R(B1) R(B2) … R(Bb).

• Generate new (Q, C) list A1, A2, …, with Convex Pruning

• Generate new candidates 1 , 2 … with the following loop• Initialize j = 1, then for i = 1 to b do

If Aj gives better slack than Aj+1 thenGenerate new candidates i for buffer Bi

Q(i ) = Q(Aj)–R(Bi)C(Aj) –t(Bi)C(i) = C(Bi)elsej = j + 1

• Sort i s in non-increasing C order.

• Insert i s into original list N

33DATE 2005, MUNICH03/10/2005

O(bn2) Algorithm• Dynamic programming

• For b buffer types, the number of candidates is at most bn+1

• For a wire, update (Q, C) value for every candidate in O(bn) time

• For a buffer position, add b new candidates in O(bn) time

• For a branch point, merge two sets of candidates in O(bn1+bn2) time

• Total complexity is O(bn2).

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35DATE 2005, MUNICH03/10/2005

Speedup over O(b2n2) Algorithm

024681012

RunningTime

Speedup

8 16 32 64

net1ne

t2net3

Buffer Library Size

net1: 337 sinksnet2: 1944 sinks net3: 2676 sinks

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Speedup vs. Buffer Positions

024681012

RunningTime

Speedup

1x 10x 20xnet1net2

net3

Normalized Buffer Positions

Buffer Library Size: 64

37DATE 2005, MUNICH03/10/2005

Outline• Introduction• O(b2n2) Algorithm• New O(bn2) Algorithm• Experimental Results• Extension and Conclusion

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Extension to Min Buffer Cost• Buffer cost is associated with area and power• Find a solution satisfying the slack requirement

and at the same time, has minimum buffer cost• Each candidate solution is represented by a (Q,

C, W) triple, where Q is slack, C is capacitance, and W is buffer cost

• Worst-case NP-hard• Our algorithm can reduce the operation of

adding a buffer from O(bN) to O(N), where N is the number of non-redundant candidates

39DATE 2005, MUNICH03/10/2005

Conclusion• New O(bn2) algorithm for optimal buffer

insertion with b buffer types• Best candidates must be in decreasing order of C• Best candidates must be on the convex hull• Local optimal global optimal

• Applicable to cost minimization and inverting buffer types

40DATE 2005, MUNICH03/10/2005

Thank You!

an o(bn 2 ) time algorithm for optimal buffer insertion with b buffer types authors: zhuo li and...

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