cse 248 skew 1kahng, ucsd 2011 cse248 spring 2011 skew

CSE 248 Skew 1 Kahng, UCSD 2011

CSE248Spring 2011

Skew


Agenda Special trees (H, Y, X)

Zero-skew tree problem and DME (BK92, CHHBK92) BST-DME (bounded-skew trees)

Lower bounds (Charikar et al.) and provably good ZST construction (Zelikovsky-Mandoiu)

Planar zero-skew trees (ZD94, KT96)

Prescribed or useful distance – and LP (OPP96)


λ-Geometry Routing

Introduced by [Burman et al. 1991]• λ uniformly distributed routing directions• Approximates Euclidean routing as λ approaches infinity

λ = 2Manhattan “M”

λ = 4Octilinear “X”

λ = 3Hexagonal “Y”


Y Trees

Distance between adjacent sinks = 1

N-level clock tree: path length =

total wire length =

Covers all slots


Y Clock Tree on Square Mesh


21% less than H-tree total wire length =

9% less than H tree, 3% less than X tree

No self-overlapping between parallel wire segments


Y Clock Tree on Hexagonal Mesh


total wire length =

Covers 3/4 of slots


Some Clock Tree Types


Zero-Skew Tree (ZST) Problem Zero Skew Clock Routing Problem (S,G): Given a set S of sink

locations and a connection topology G, construct a ZST T(S) with topology G and having minimum cost.

Skew = maximum value of |td(s0,si) – td(s0,sj)| over all sink pairs si, sj in S.

Td = signal delay (from source s0)

Connection topology G = rooted binary tree with nodes of S as leaves Edge ea in G is the edge from a to its parent |ea| is the (assigned) length of edge ea

Cost = total edge length


Zero-Skew Example (555 sinks, 40 Obstacles)


Manhattan Arcs, TRRs and Merging Segments Manhattan Arc: line segment with

slope +1 or –1

Tilted Rectangular Region (TRR): collection of points within a fixed distance of a Manhattan arc Core = Manhattan arc Radius = distance

Merging segment = locus of feasible locations for a node v in the topology, consistent with minimum wirelength If v is a sink, then ms(v) = {v} If v is an internal node, then ms(v) is

the set of all points within distance |ea| of ms(a), and within distance |eb| of ms(b)


Bottom-Up: Tree of Merging Segments

Goal: Construct a tree of merging segments corresponding to topology G Merging segment of a node depends on merging segment of

its children bottom-up construction Let a, b be children of v. We want placements of v that allow

TSa and TSb to be merged with minimum added wire while preserving zero skew

Merging cost = |ea| + |eb|

Lemma: The intersection of two TRRs is also a TRR and can be found in constant time.


Tree of Merging Segments Constant time per each new merging segment linear

time (in size of S) to construct entire tree


Top-Down: Embedding Nodes of Topology Goal: Find exact locations

(“embeddings”) pl(v) of internal nodes v in the ZST topology.

If v is the root node, then any point on ms(v) can be chosen as pl(v)

If v is an internal node other than the root, and p is the parent of v, then v can be embedded at any point in ms(v) that is at distance |ev| or less from pl(p) Detail: create square TRR trrp with

radius ev and core equal to pl(p); placement of v can be any point in ms(v) trrp


Find_Exact_Placements Each instruction executed at most once for each node

in G, and TRR intersection is O(1) time Find_Exact_Placements is O(n) DME is O(n)


Facts About DME Construction

DME produces a ZST under linear delay model with minimum cost over all ZSTs with topology G and sink set S

For any sink set S and topology G, DME will construct a ZST with minimum feasible source-sink delay, equal to one-half the diameter of S (Theorem 2, BK92)

Corollary: We know the merging segments top-down!


Vertex Loci in a Bounded-Skew Tree

skew0 2 4 6

2

4

6

0246

2

4

6

skew

v

s4

va b

s1 s2 s3

Topologys0 b

a

Given a skew bound, where can internal nodes of the given topology (e.g., a, b, v) be placed?


BST-DME Bottom-Up Phase

s4

va b

s1 s2 s3

Topology

s0

s1

s3

s4

s2

mr(a)mr(b)mr(v)

B = 4

Bottom-Up: build tree of merging regions corresponding to given topology

s0


BST-DME Top-Down Phase

s4

va b

s1 s2 s3

Topology

s0

s1

s3

s4

s2

a bv

B = 4

s0


ZST Approximation

Charikar et al. [CKKRST99]• NP-hard for general metric spaces• factor 2e ~ 5.44 approximation

Zelikovsky-Mandoiu [ZM02] (J. Disc. Math.)• factor 4 approximation for general metric spaces• factor 3 approximation for rectilinear plane


ZST Lower-Bound

R

rrNOPT0

d)( (CKKRST 99)

N(r)=min. # of balls of radius r that cover all sinks


ZST Lower-Bound

R

rrNOPT0

d)( (CKKRST 99)



Constructive Lower-Bound

Computing N(r) is NP-hard, but …

Lemma: For any ordering of the terminals, if

},...,{MinDist21

1 ittr

ntt ,...,1

irN )( then

Where MinDist{A} = minu,v in A, u != v d(u,v)

Explanation: If r is less than half the minimum separation of any two of the first i terminals, then at least i balls are needed to cover the terminal set. In particular, at least i balls are needed to cover just the first i terminals in the ordering.


Constructive Lower Bound

R

rrNOPT0

d)(

r

N(r)

2

n

n-1

{i = 2 to n} ½ MinDist{t1, …, ti}

½ MinDist{t1, …, tn} ½ MinDist{t1, t2}

Explanation: From the previous lemma, N(r) is at least i when r < ½ MinDist(t1,…, ti). So, we are making a lower sum for the given integral. The original version of this slide was missing the ½ factor above.


Stretching Rooted Spanning Trees

• ZST root = spanning tree root

)(rootdelayT

where = max path length from to a leaf of

)(rootdelayT• ZST root-to-leaf path length =

)(vdelayT v vT

T



T

)(rootdelayTLoop length =

)(rootdelayT



T

)(rootdelayTSum of loop lengths =

)(rootdelayT


Zero-Skew Spanning Tree Problem

)()( TdelayTlength

Theorem: Every rooted spanning tree can be stretched to a ZST of total length

where

T

)()()(

TVv T vdelayTdelay

Zero-Skew Spanning Tree Problem: Find rooted spanning tree minimizing )()( TdelayTlength T


How Good are MST and Min-Star?

)(Nlength

.

..

MST: min length, huge delay

)( 2Ndelay

0

1

2

N-1

3

N-2

)log( NNOPT

)(Nlength …

…

Star: min delay, huge length

)1(delay

)1(OPT


The Rooted-Kruskal Algorithm

• While 2 roots remain:• Initially each terminal is a rooted tree; d(t)=0 for all t

• Pick closest two roots, t & t’, where d(t) d(t’)

t’ t

• t’ becomes child of t, root of merged tree is t• d(t) max{ d(t), d(t’) + dist(t ,t’) }

t’ t


The Rooted-Kruskal Algorithm


• Pick closest two roots, t & t’, where d(t) d(t’) • t’ becomes child of t, root of merged tree is t• d(t) max{ d(t), d(t’) + dist(t ,t’) }


How Good is Rooted-Kruskal?



Lemma: delay(T) length(T)

• Initially each terminal is a rooted tree; d(t)=0 for all t

• d(t) max{ d(t), d(t’) + dist(t ,t’) }

At the end of the algorithm, d(t)=delay (t )TT


• Initially each terminal is a rooted tree; d(t)=0 for all t

At the end of the algorithm, d(t)=delay (t )

When edge (t ,t’) is added to T:• length(T) increases by dist(t ,t’)• delay(T) increases by at most dist(t ,t’)


How Good is Rooted-Kruskal?



Lemma: length(T) 2 OPT


Number terminals in reverse order of becoming non-roots

N

iitt

21 },...,{MinDistlength(T) =

Explanation: Kruskal is adding connections which, in reverse order, give the MinDist’s of the latest I+1 terminals in the tree. For example, the last connection is the MinDist of the latest two terminals. The first connection is the MinDist of all n terminals. The summation is exactly twice the lower bound from Slide 30.


Factor 4 Approximation

• Length after stretching = length(T) + delay(T)• delay(T) length(T)• length(T) 2 OPT

ZST length 4 OPT

Algorithm: Rooted-Kruskal + Stretching


Planar Zero-Skew Tree (ZST) Problem Planar Zero Skew Clock Routing Problem (S,G): Given a set S of

sink locations and a connection topology G, construct a ZST T(S) with topology G and having minimum cost, which is embeddable in the plane with no crossing edges.

Zhu-Dai 1992: Iteratively: (1) Connect free sink to its minimal balance point (shortest wire, maintaining exact zero pathlength skew); (2) Choose the “free” (unused) sink whose minimum balance distance is maximized.


Planar Zero-Skew Tree (ZST) Problem Planar Zero Skew Clock Routing Problem (S,G): Given

a set S of sink locations and a connection topology G, construct a ZST T(S) with topology G and having minimum cost, which is embeddable in the plane with no crossing edges.

Enabling observations: Theorem 2 of BK92 allows us to build the tree of merging

segments top-down Binary tree topology = recursive bipartitioning of the sink set S

Planar-DME: Can achieve planarity by choosing the sink partitions and edge embeddings carefully Topology is constructed on the fly (it is not prescribed a priori)


Planar Zero-Skew Tree (ZST) Problem Planar Zero Skew Clock Routing Problem (S,G): Given

a set S of sink locations and a connection topology G, construct a ZST T(S) with topology G and having minimum cost, which is embeddable in the plane with no crossing edges.

Enabling observations: Theorem 2 of BK92 allows us to build the tree of merging

segments top-down “Single-Pass DME” Binary tree topology = recursive bipartitioning of the sink set S

Planar-DME: Can achieve planarity by choosing the sink partitions and edge embeddings carefully Topology is constructed on the fly (it is not prescribed a priori)


Top-Down, Single-Pass Partitioning+Embedding

Entire routing region Pa is divided into convex polygons until only one sink lies within each convex polygon

Boundaries of polygons are given by thick dashed lines and tree edges

Tree of merging segments is given by thin dotted lines


Embedding Rules

In each recursive call, given: a subset of sinks S’,

contained in convex polygon P(S’)

a point p inside P(S’) which is the embedding of the parent of the root of the subtree over S’

Must connect p to a point v on ms(v) = center(S’)

“Embedding Rules” ensure that we find an embedding point for v in O(1) time


Partitioning Rules

Goal: find a splitting line Divides P(S’) into two convex polygons (thus partitioning the sink set

between the two subtrees of v) Allows the routing from p to v to be on the boundary between the two

polygons

“Partitioning Rules” ensure that we find the splitting line in Theta(|S’|) time


Prescribed Delays by Linear Programming Oh, Pyo, Pedram 1996

Topology and sink locations are given Use LP to determine edge lengths ( implicitly determining

Steiner point locations)Notes. (1) Having path lengths greater than or equal to lower bounds (= endpoint to endpoint distances) is a sufficient condition for there to exist Steiner point locations that satisfy the pathlength bounds. (2) Solving for the Steiner point locations in the straightforward way (i.e., as (x,y) coordinates) does not work, but solving for the locations implicitly does work.

cse 248 skew 1kahng, ucsd 2011 cse248 spring 2011 skew

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