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Ion Mandoiu (Georgia Tech)

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Zero-Skew Tree: rooted tree in which all root-to-leaf paths have the same length ... Practical algo: Rooted-Kruskal Stretching DME ... – PowerPoint PPT presentation

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Title: Ion Mandoiu (Georgia Tech)


1
Practical Approximation Algorithms forZero- and
Bounded-Skew Trees
  • Ion Mandoiu (Georgia Tech)
  • Alex Zelikovsky (Georgia State)
  • ISMP 2000

2
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
3
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

4
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

5
ZST Lower-Bound
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
ZST Lower-Bound
(CKKRST 99)
N(r)min. of balls of radius r that cover all
sinks
9
ZST Lower-Bound
(CKKRST 99)
N(r)min. of balls of radius r that cover all
sinks
10
Constructive Lower-Bound
Computing N(r) is NP-hard, but
11
Constructive Lower-Bound
12
Stretching Rooted Spanning Trees
  • ZST root spanning tree root

13
Stretching Rooted Spanning Trees
14
Stretching Rooted Spanning Trees
15
Zero-Skew Spanning Tree Problem
16
How good are the MST and Min-Star?
17
The Rooted-Kruskal Algorithm
  • While ? 2 roots remain

18
The Rooted-Kruskal Algorithm
19
How good is Rooted-Kruskal?
Lemma delay(T) ? length(T)
20
How good is Rooted-Kruskal?
Lemma length(T) ? 2 OPT
21
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
22
Stretching Using Steiner Points
23
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
24
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
25
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
26
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

27
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
28
BST construction idea lower bound
Two stage BST construction
  • Cover terminals by disjoint b-bounded-skew trees
  • Connect roots via a zero-skew tree

29
Constructing the tree cover
30
BST Approximation
Algorithm Output tree cover ? approximate ZST on
W
31
BST Approximation
32
Summary of Results
Problem Zero-Skew Zero-Skew Bounded-skew Bounded-skew
Metric General Rectilinear General Rectilinear

Previous factor 5.44 5.44 16.11 12.53
New factor 4 3 14 9
33
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
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