Zero-Skew Trees

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 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 4approximation for general metric spaces • factor 3approximation for rectilinear plane

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

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

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

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

11. Loop length = Stretching Rooted Spanning Trees

12. Sum of loop lengths = Stretching Rooted Spanning Trees

13. 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 Zero-Skew Spanning Tree Problem

14. MST: min length, huge delay Star: min delay, huge length N-1 … N-2 . . . 3 2 1 … 0 How good are the MST and Min-Star?

15. 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’ t’ t t The Rooted-Kruskal Algorithm • While  2 roots remain:

16. 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’ becomes child of t, root of merged tree is t • d(t)  max{ d(t), d(t’) + dist(t ,t’) } The Rooted-Kruskal Algorithm

17. Initially each terminal is a rooted tree; d(t)=0 for all t • Initially each terminal is a rooted tree; d(t)=0 for all t • 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’) • Pick closest two roots, t & t’, whered(t)  d(t’) • t’ becomes child of t, root of merged tree is t • d(t)  max{ d(t), d(t’) + dist(t ,t’) } • d(t)  max{ d(t), d(t’) + dist(t ,t’) } At the end of the algorithm, d(t)=delay (t ) At the end of the algorithm, d(t)=delay (t ) T 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: delay(T)  length(T)

18. 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’) • 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’) } Number terminals in reverse order of becoming non-roots length(T) = How good is Rooted-Kruskal? Lemma:length(T)  2 OPT

19. Factor 4 Approximation 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 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 • 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. 2 • Running time of Rooted-Kruskal becomes O(N ) Extension to Other Metric Spaces Everything works as in rectilinear plane, except: • 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 16.11 for arbitrary metrics • factor 12.53 for rectilinear plane Our results: factor 14, resp. 9 approximation

26. Lemma: For any set of terminals, and any BST construction idea + lower bound Two stage BST construction: • Cover terminals by disjoint b-bounded-skew trees • Connect roots via a zero-skew tree

27. 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: Constructing the tree cover

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

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

30. Summary of Results

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