1. Zero-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

2. The 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

3. ZST Lower-Bound (CKKRST 99)
N(r)=min. # of balls of radius r that cover all sinks

4. ZST Lower-Bound (CKKRST 99)
N(r)=min. # of balls of radius r that cover all sinks

5. ZST Lower-Bound (CKKRST 99)
N(r)=min. # of balls of radius r that cover all sinks

6. ZST Lower-Bound (CKKRST 99)
N(r)=min. # of balls of radius r that cover all sinks

7. ZST Lower-Bound (CKKRST 99)
N(r)=min. # of balls of radius r that cover all sinks

8. Constructive Lower-Bound
Computing N(r) is NP-hard, but …
Lemma: For any ordering of the terminals, if
then

9. Constructive Lower-Bound
N(r) 2 n
n-1

10. Stretching Rooted Spanning Trees
ZST root = spanning tree root
where = max path length from to a leaf of
ZST root-to-leaf path length =

11. Stretching Rooted Spanning Trees
Loop length =

12. Stretching Rooted Spanning Trees
Sum of loop lengths =

13. Zero-Skew Spanning Tree Problem
Theorem: Every rooted spanning tree can be stretched to a ZST of total length
where
Zero-Skew Spanning Tree Problem: Find rooted spanning tree minimizing

14. How good are the MST and Min-Star?.
MST: min length, huge delay 1 2
N-1 3
N-2 …
Star: min delay, huge length

15. The Rooted-Kruskal Algorithm
Initially each terminal is a rooted tree; d(t)=0 for all t
While  2 roots remain:
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

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

17. How good is Rooted-Kruskal?
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 ) T
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’) } T
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’)
Lemma: delay(T)  length(T)

18. 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’) }
Pick closest two roots, t & t’, where d(t)  d(t’)
Number terminals in reverse order of becoming non-roots
length(T) =
Lemma: length(T)  2 OPT

19. Factor 4 Approximation  ZST length  4 OPT
Algorithm: Rooted-Kruskal + Stretching
Length after stretching = length(T) + delay(T)
delay(T)  length(T)
length(T)  2 OPT
 ZST length  4 OPT

20. Stretching Using Steiner Points

21. Factor 3 Approximation  ZST length  3 OPT
Algorithm: Rooted-Kruskal + Improved Stretching
Length after stretching = length(T) + ½ delay(T)
delay(T)  length(T)
length(T)  2 OPT
 ZST length  3 OPT

22. Practical 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

23. Running Time  O(N logN) overall 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

24. Extension 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

25. Bounded-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 for arbitrary metrics
factor for rectilinear plane
Our results: factor 14, resp. 9 approximation

26. BST 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 any

27. Constructing 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 T u
Lemma:

28. BST Approximation
Algorithm: Output tree cover  approximate ZST on W

29. BST Approximation
Theorem:
Rectilinear Plane:
Arbitrary metric spaces:

30. Summary of Results 5.44 16.11 12.53 4 3 14 9 Problem Zero-Skew
Bounded-skew
Metric
General
Rectilinear
Previous
factor
5.44
16.11
12.53
New 4 3 14 9

31. Open 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