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Interconnect Topology Optimization

- Problem: given a source and a set of sinks, build the best interconnect topology to minimize different design objectives:
- Wire length: traditional (lower capacitance load, and overall congestion)
- Performance: Nanoscale circuits
- New interest: speed and other non-traditional routing architecture
- In most cases, topology means tree
- Because tree is the most compact structure to connect everything without redundancy
- Delay analysis is easy

Terminology

- For multi-terminal net, we can easily construct a tree (spanning tree) to connect the terminals together.
- However, the wire length may be unnecessarily large.
- Better use Steiner Tree:
- A tree connecting all terminals as well as other added nodes (Steiner nodes).
- Rectilinear Steiner Tree:
- Steiner tree such that edges can only run horizontally and vertically.
- Manhattan planes
- Note: X (or Y)-architecture (non-Manhanttan)

Steiner Node

Prim’s Algorithm for Minimum Spanning Tree

- Grow a connected subtree from the source, one node at a time.
- At each step, choose the closest un-connected node and add it to the subtree.

Y

X

s

Conventional Routing Algorithms Are Not Good Enough

Minimum spanning tree may have very long source-sink path.

Shortest path tree may have very large routing cost.

Want to minimize path lengths and routing cost at the same time.

Interconnect Topology Optimization Under Linear Delay ModelPerformance-Driven Interconnect Topology Design

- BPRIM & BRBC (bounded-radius bounded-cost) algorithm
- [Cong et al, ICCD’91, TCAD’92]
- RSA algorithm (for Min. Rectilinear Steiner Arborescences)
- [Rao-Sadayappan-Hwang-Shor, Algorithmica’92]
- Prim-Dijkstra tradeoff algorithm
- [Alpert et al, TCAD 1995]
- SERT algorithm (Steiner Elmore Routing Tree)
- [Boese-Kahng-McCoy-Robins, TCAD-95]
- MVERT algorithm (Minimum Violation Elmore Routing Tree)
- [Hou-Hu-Sapatnekar, TCAD-99]
- BINO alg. (Buffer Insertion and Non-Hanan Optimization)
- [Hu-Sapatnekar, ISPD-99]
- ……

Given Net N with source s and connected by tree T.

Radius of net N: distance from the source to the furthest sink.

Radius of a routing tree r(T): length of the longest path from the root to a leaf.

Cost of an edge: distance between two.

Cost of a routing tree cost(T): sum of the edge costs in T.

minpathG(u,v): shortest path from u to v in G

distG(u,v): cost of minpathG(u,v).

Definitionsr(T)

radius of the net

source

s

routing tree

First Idea: Bounded Radius Minimum Spanning Tree [Cong-Kahng-Robin-Sarrafzadeh-Wong, ICCD-91]

- Basic Idea: Restrict the tree radius while minimizing the routing cost
- Bounded radius minimum spanning tree problem (BRMST):

Given a net N with radius R,

find a minimum cost tree with radius r(T)(1+ )R

source

=

radius = 4.03

cost = 4.03

source

= 1

radius = 1.77

cost = 4.26

source

= 0

radius = 1

cost = 4.95

trade-off between radius and the cost of routing trees

- Parameter controls the trade-off between radius and cost
- = minimum spanning tree; = 0 shortest path tree

x

x

y

y

s

s

x’

distT(s,x)+cost(x,y)

(1+ )R

distT(s,x’)+cost(x’,y)

R

BPRIM Algorithm forBounded-Radius Minimum Spanning Trees- Given net N with source s and radius R, and parameter .
- Grow a connected subtree T from the source, one node at a time
- At each step, choose the closest pair x T and y N-T
- If distT(s, x) + cost(x,y) (1+)R, add (x,y)
- Else backtrack along minpathT(s,x) to find x’ such that

distT(s, x’) + cost(x’, y) R, then add (x’, y)

- Slack R is introduced at each backtrace so we do not have

to backtrace too often.

A Pathological Example for BPRIM Algorithm

- BPRIM can be arbitrarily bad.

x

x

y

y

all leaves connect

directly to the

source

source s

source s

Optimal Solution

Solution by BPRIM

s

s

10

s

9

8

L

4

3

7

Li’

Li

2

6

1

5

Graph Q

if S=distL(Li’,Li) distSPT(S, Li)

then add minpathSPT(s, Li) to Q and reset S =0, Li’ = Li

depth-first tour L

for MST

An Improved Algorithm -- BRBC[Cong-Kahng-Robin-Sarrafzadeh-Wong, T-CAD’92]- Construct MST and SPT, Q = MST;
- Construct list L -- a depth-first tour of MST;
- Traverse L while keeping a running total S of traversed edge costs, when reaching Li

If S distSPT(s, Li) then add minpathSPT(s, Li) to Q and reset S = 0,

Else continue traverse L;

- Construct the shortest path tree T of Q.

BRBC Trees Have Bounded Radius

Theorem 1: r(T)(1+ ) R

Proof: For any vertex x, let y be the last vertex before x in L that we add minpathSPT(s, L).

By the choice of y, we have

distL(y,x)distSPT(s,x) R

Therefore,

distQ(s,x)distQ(s,y) +distQ(y,x)

R+distL(y,x)

R+R =(1+ )R

Graph Q

s

x

y

s

tour L

distSPT(s,y)

R

y

x

distL(y,x) distSPT(S, x)

R

BRBC Trees Have Bounded Cost

- Theorem 2: cost(T) (2+2/ ) cost(MST).
- Proof: Let v1 v2 … vk be the vertices that we add minpathSPT(s,vi)

Note that T is a subgraph of Q

Graph Q

s

s

tour L

vi-1

vi

distL(vi-1,vi) distSPT(S, vi)

- Idea borrowed from [Awarbuch - Baratz - Peleg, PODC-90]

Experimental Results of BRBC Algorithm

Radius, as fraction of MST radius

Cost, as fraction of MST cost

Prim’s and Dijkstra’s Algorithms

- d(i,j): length of the edge (i, j)
- p(j): length of the path from source to j
- Prim: d(i,j) Dijkstra: d(i,j) + p(j)

p(j)

d(i,j)

The Prim-Dijkstra Trade-off: AHHK Tree

- Prim: add edge minimizing d(i,j)
- Dijkstra: add edge minimizing p(i) + d(i,j)
- Trade-off: c(p(i)) + d(i,j) for 0 <= c <= 1
- When c=0, trade-off = Prim
- When c=1, trade-off = Dijkstra

Conventional Rectilinear Steiner Tree

- Extensive studies, even outside the VLSI design community
- Minimize total wire length
- 1-Steiner and iterated 1-Steiner [Kahng-Robins, 1992]
- One steiner point is added at each step
- Good performance but slow: O(n4logn)
- Recent result
- Efficient Steiner Tree Construction Based on Spanning Graphs [p. 152] , H. Zhou ISPD 2003
- O(nlogn)
- Highly Scalable Algorithms for Rectilinear and Octilinear Steiner Trees [p. 827] by A.B. Kahng, I.I. Mándoiu, A.Z. ZelikovskyASPDAC 2003
- O(nlog2n)

Hanan’s Result on Rectilinear Steiner Tree

- [Hanan, SIAM J. Appl. Math. 1966]
- For rectilinear Steiner tree construction, there exists a routing tree with minimum total wire length on the grid formed by horizontal and vertical lines passing through source and sinks.

Hanan nodes

source

Hanan Grid

Rectilinear Steiner Arborescence Algorithm[Rao-Sadayappan-Hwang-Shor, Algorithmica’92]

- Given n nodes lying in the first quadrant
- Purpose is to maintain shortest paths from source to sink and minimize total wire length
- RSA algorithm
- Start with a forest of n single-node A-trees.
- Iteratively substituting min(p,q) for pair of nodes p, q

where min(p,q) = (min{xp, xq}, min{yp, yq}).

- The pair p, q are chosen to maximize ||min(p,q)|| over all current nodes.

p

q

min(p,q)

Performance of RSA Algorithm

- Time Complexity O(n log n) when implemented using a plane-sweep technique.
- Wirelength of the tree by RSA algorithm 2 x Optimal solution (i.e., 2 x wirelength of minimumRectilinear Steiner Arborescence

Interconnect Designs Under Distributed RC Delay Model [Cong-Leung-Zhou, DAC’93]

- Routing Tree T: connects the source with a set of sinks
- plk(T): pathlength from sink k to source in T

Z0

Z0

driver

Z0

Z0

Fd

Z0

Z0

Z0

R0

C0

Interconnect Topology Design Formulation Under Distributed RC Delay Model

- Objective: Minimize

Comparison of Three Types of Trees

MST

SPT

QMST

MST cost 9(optimal) 11 10

SPT cost 37 29 (optimal) 31

QMST cost 45 36 34 (optimal)

Impact of Resistance Ratio

- Definition: Rd/R0

Driver resistance versus unit wire resistance

- Determined by the Technology:

Reduce device dimension

R0

Rd

Rd/R0

- Impact on Interconnect Optimization:
- Why Minimum-cost shortest path tree is useful

Steiner Elmore Routing Tree (SERT) Heuristic[Boese-Kahng-McCoy-Robins, TCAD’95]

- Use Elmore Delay Model directly in construction of routing tree T.
- Add nodes to T one-by-one like Prim’s MST algorithm.
- Two versions:
- SERT Algorithm:
- At each step, choose v T and uT s.t. the maximum Elmore-delay to any sink has minimum increase.
- SERT-C Algorithm:
- SERT with identified critical sink
- First connect the critical sink to the source by a shortest path,
- then continues as in SERT, except that we minimize the Elmore delay of the critical sink rather than the max. delay.

Steps of SERT Algorithm

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source

source

source

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Examples of SERT-C Construction

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c) Node 5 critical

a) Node 2 or 4 critical

b) Node 3 or 7 critical

(also 1-Steiner tree)

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d) Node 6 critical

f) Node 9 critical

e) Node 8 critical

(also SERT)

Maze Routing

- http://foghorn.cadlab.lafayette.edu/MazeRouter.html

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