Ion Mandoiu Georgia Tech

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Practical Approximation Algorithms forZero- and
Bounded-Skew Trees

• Ion Mandoiu (Georgia Tech)
• Alex Zelikovsky (Georgia State)
• ISMP 2000

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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
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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

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Overview

• Constructive lower-bound on optimum ZST length
• Converting spanning tree to zero-skew trees
• Finding spanning trees with small conversion
  cost
• Improved conversion using Steiner points
• Approximation algorithms for bounded-skew trees
• Conclusions and open problems

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ZST Lower-Bound
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ZST Lower-Bound
(CKKRST 99)
N(r)min. of balls of radius r that cover all
sinks
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ZST Lower-Bound
(CKKRST 99)
N(r)min. of balls of radius r that cover all
sinks
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ZST Lower-Bound
(CKKRST 99)
N(r)min. of balls of radius r that cover all
sinks
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ZST Lower-Bound
(CKKRST 99)
N(r)min. of balls of radius r that cover all
sinks
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Constructive Lower-Bound
Computing N(r) is NP-hard, but
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Constructive Lower-Bound
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Stretching Rooted Spanning Trees

• ZST root spanning tree root

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Stretching Rooted Spanning Trees
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Stretching Rooted Spanning Trees
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Zero-Skew Spanning Tree Problem
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How good are the MST and Min-Star?
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The Rooted-Kruskal Algorithm

• While ? 2 roots remain

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The Rooted-Kruskal Algorithm
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How good is Rooted-Kruskal?
Lemma delay(T) ? length(T)
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How good is Rooted-Kruskal?
Lemma length(T) ? 2 OPT
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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
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Stretching Using Steiner Points
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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
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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
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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
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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

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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 bgt0, 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
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BST construction idea lower bound
Two stage BST construction

• Cover terminals by disjoint b-bounded-skew trees
• Connect roots via a zero-skew tree

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Constructing the tree cover
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BST Approximation
Algorithm Output tree cover ? approximate ZST on
W
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BST Approximation
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Summary of Results
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Open Problems

• Complexity of ZST problem in rectilinear plane
• Complexity of finding the spanning tree with
  minimum lengthdelay?
• Zero-skew Steiner ratio supremum, over all
  sets of terminals, of the ratio between minimum
  ZST length and minimum spanning tree lengthdelay
• What is the ratio for rectilinear plane?
• What is the ratio for arbitrary spaces? ( ?4,
  ?3)
• Planar ZST / BST