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

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

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

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

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

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

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

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

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

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

Stretching Rooted Spanning Trees Loop length =

Stretching Rooted Spanning Trees Sum of loop lengths =

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

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

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

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’) }

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)

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

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

Stretching Using Steiner Points

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

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

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

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

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

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

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:

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

BST Approximation Theorem: Rectilinear Plane: Arbitrary metric spaces:

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

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