Zero-Skew TreesZero-Skew Trees

Zero-Skew Tree: rooted tree in which all root-to-leaf

paths have the same length

Used in VLSI clock routing & network multicasting

The Zero-Skew Tree Problem The Zero-Skew Tree Problem

Zero-Skew Tree Problem

Given: set of terminals in rectilinear plane

Find: zero-skew tree with minimum total length

Previous results [CKKRST 99]• NP-hard for general metric spaces• factor 2e ~ 5.44 approximation

Our results:• factor 4 approximation for general metric spaces• factor 3 approximation for rectilinear plane

ZST Lower-BoundZST Lower-Bound

R

rrNOPT0

d)( (CKKRST 99)

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

ZST Lower-BoundZST Lower-Bound

R

rrNOPT0

d)( (CKKRST 99)

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

ZST Lower-BoundZST Lower-Bound

R

rrNOPT0

d)( (CKKRST 99)

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

ZST Lower-BoundZST Lower-Bound

R

rrNOPT0

d)( (CKKRST 99)

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

ZST Lower-BoundZST Lower-Bound

R

rrNOPT0

d)( (CKKRST 99)

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

Constructive Lower-BoundConstructive Lower-Bound

Computing N(r) is NP-hard, but …

Lemma: For any ordering of the terminals, if

},...,{MinDist2

11 ittr

ntt ,...,1

irN )( then

Constructive Lower-BoundConstructive Lower-Bound

R

rrNOPT0

d)(

r

N(r)

2

n

n-1

Stretching Rooted Spanning TreesStretching 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

Stretching Rooted Spanning TreesStretching Rooted Spanning Trees

T

)(rootdelayTLoop length =

)(rootdelayT

Stretching Rooted Spanning TreesStretching Rooted Spanning Trees

T

)(rootdelayTSum of loop lengths =

)(rootdelayT

Zero-Skew Spanning Tree ProblemZero-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 the MST and Min-Star?How good are the 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 AlgorithmThe 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 AlgorithmThe 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’ 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?How good is Rooted-Kruskal?

• 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’ becomes child of t, root of merged tree is t

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

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

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

• 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?How good is Rooted-Kruskal?

• 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’ becomes child of t, root of merged tree is t

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

Lemma: length(T) 2 OPT

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

Number terminals in reverse order of becoming non-roots

N

iitt

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

Factor 4 ApproximationFactor 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

Stretching Using Steiner PointsStretching Using Steiner Points

TStretchedT

)()( TdelayTlength )()( TdelayTlength )(2

1Tdelay

Factor 3 ApproximationFactor 3 Approximation

• Length after stretching = length(T) + ½ delay(T)

• delay(T) length(T)

• length(T) 2 OPT

ZST length 3 OPT

Algorithm: Rooted-Kruskal + Improved Stretching

Practical ConsiderationsPractical Considerations

• For a fixed topology, minimum length ZST can be found in linear time using the Deferred Merge Embedding (DME) algorithm [Eda91, BK92, CHH92]

• Practical algo: Rooted-Kruskal + Stretching + DME

Theorem: Both stretching algorithms lead to the

same ZST topology when applied to the Rooted-

Kruskal tree

Running TimeRunning Time

• Stretching: O(N logN)

• Rooted-Kruskal: O(N logN) using the dynamic closest-pair data structure of [B98]

• DME: O(N) [Eda91, BK92, CHH92]

O(N logN) overall

Extension to Other Metric SpacesExtension to Other Metric Spaces

Everything works as in rectilinear plane, except:2• Running time of Rooted-Kruskal becomes O(N )

• No equivalent of DME known for other spaces

• The space must be metrically convex to apply second stretching algorithm

Bounded-Skew TreesBounded-Skew Trees

b-bounded-skew tree: difference between length of

any two root-to-leaf paths is at most b

Bounded-Skew Tree Problem: given a set of terminals

and bound b>0, find a b-bounded-skew tree with

minimum total length

Previous approximation guarantees [CKKRST 99]:• factor 16.11 for arbitrary metrics• factor 12.53 for rectilinear plane

Our results: factor 14, resp. 9 approximation

BST construction idea + lower boundBST construction idea + lower bound

Two stage BST construction:

• Cover terminals by disjoint b-bounded-skew trees

• Connect roots via a zero-skew tree

Lemma: For any set of terminals, and anyS SW

)1|(|)( WbBSTOPTWZSTOPT

Constructing the tree coverConstructing the tree cover

• T MST on terminals, rooted arbitrarily

• W • While T do:

• Find leaf of T furthest from the root

• Find its highest ancestor u that still has delay b

• Add u to W

• Add T to the tree cover and delete it from Tu

Lemma: )()1|(| MSTlengthWb

BST ApproximationBST Approximation

Algorithm: Output tree cover approximate ZST on W

)factorion approximat ZST(

)()()(

WZSTOPTMSTlengthBSTlength

)(

)1|(|)(

MSTlengthBSTOPT

WbBSTOPTWZSTOPT

)ratioSteiner (

)(

BSTOPTSMTOPTMSTlength

BST ApproximationBST Approximation

BSTOPTBSTlength )()( Theorem:

ionapproximat 9factor 5.1,3 Rectilinear Plane:

ionapproximat 14factor 2,4 Arbitrary metric spaces:

ratioSteiner

factorion approximat ZST

Summary of ResultsSummary of Results

Problem Zero-Skew Bounded-skew

Metric General Rectilinear General Rectilinear

Previous

factor5.44 16.11 12.53

New

factor4 3 14 9

Open ProblemsOpen Problems

• Complexity of ZST problem in rectilinear plane

• Complexity of finding the spanning tree with minimum length+delay?

• Zero-skew Steiner ratio: supremum, over all sets of terminals, of the ratio between minimum ZST length and minimum spanning tree length+delay

• What is the ratio for rectilinear plane?

• What is the ratio for arbitrary spaces? ( 4, 3)

• Planar ZST / BST