Collective%20Tree%20Spanners%20and%20Routing%20in%20AT-free%20Related%20Graphs

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Collective Tree Spanners and Routing in AT-free
Related Graphs

• F.F. Dragan, C. Yan, D. Corneil
• Kent State University
• University of Toronto

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Well-known Tree t -Spanner Problem

• Given unweighted undirected graph G(V,E) and
  integers t, s.
• Does G admit a spanning tree T (V,E) such that

(a multiplicative tree t-spanner of G) or
(an additive tree s-spanner of G)?
G
multiplicative tree 4- and additive tree

3- spanner of G
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Well-known Sparse t -Spanner Problem
Given unweighted undirected graph G(V,E) and
integers t, m. Does G admit a spanning graph H
(V,E) with E ? m such that
(a multiplicative t-spanner of G) or
(an additive s-spanner of G)?
G multiplicative 2-
and additive 1-spanner of G
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New Collective Additive Tree r -Spanners
Problem

• Given unweighted undirected graph G(V,E) and
  integers m, r.
• Does G admit a system of m spanning trees
  T1,T2,, Tm such that

(a system of m collective additive tree
r-spanners of G)?
2 collective additive tree 2-spanners
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Applications of Collective Tree Spanners

• message routing in networks
• Efficient routing scheme is known for trees
  but is hard for general graphs. For any two
  nodes, we can route the message between them in
  one of the trees which approximates the distance
  between them.
• solution for sparse s-spanner problem
• If a graph admits a system of m collective
  additive tree s-spanners, then the graph will
  have an additive graph s-spanner with at most
  m(n-1) edges, where n is the number of nodes.

2 collective additive tree 2-spanners of G
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Some known results for the tree spanner problem
(mostly multiplicative case)

• general graphs CC95
• t ? 4 is NP-complete. (t3 is still open, t ? 2
  is P)
• approximation algorithm for general graphs
  EP04
• O(logn) approximation algorithm
• chordal graphs BDLL02
• t ? 4 is NP-complete. (t3 is still open.)
• planar graphs FK01
• t? 4 is NP-complete. (t3 is polynomial time
  solvable.)
• AT-free graphs and their subclasses
• 1 additive tree 3-spanner Pr99, PKLMW03
• a permutation graph admits a multiplicative tree
  3-spanner MVP96
• an interval graph admits an additive tree
  2-spanner

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Some known results for the sparse spanner problem

• general graphs PS89
• t, m?1 is NP-complete
• n-vertex chordal graphs (multiplicative case)
  PS89
• (G is chordal if it has no chordless cycles
  of length gt3)
• multiplicative 3-spanner with O(n logn) edges
• multiplicative 5-spanner with 2n-2 edges
• n-vertex k-chordal graphs (additive case)
  CDY03,DYL04
• (G is k-chordal if it has no chordless cycles
  of length gtk)
• additive 2 ?k/2? -spanner with O(n logn) edges
• additive (k1)-spanner with 2n-2 edges

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Previous results on the collective tree spanners
problem DYL2004

• n-vertex chordal graphs
• log n collective additive tree 2-spanners
• n-vertex chordal bipartite graphs
• log n collective additive tree 2-spanners
• n-vertex k-chordal graphs
• log n collective additive 2 ?k/2? -spanners
• n-vertex planar graphs
• vn log n collective additive tree 0-spanners
• n-vertex graphs of bounded tree width tw
• O(log n) tw x log n collective additive tree
  0-spanners

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New results on the collective tree spanners
problem on AT-free related graphs

• n-vertex AT-free graphs
• 2 collective additive tree 2-spanners
• n-vertex permutation graphs
• 1 additive tree 2-spanner
• n-vertex DSP-graphs
• 2 collective additive tree 3-spanners
• 5 collective additive tree 2-spanners
• n-vertex graphs of bounded asteroidal number an
• an(an-1)/2 collective additive tree 4-spanners
• an(an-1) collective additive tree 3-spanners

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Permutation, Trapezoid and Co-comparability Graphs

• Thm Every permutation graph admits an additive
  tree 2-spanner, constructable in linear time.
• ? admits a multiplicative tree 3-spanner
  MVP96
• Observ There are bipartite permutation graphs on
  2n vertices for which any system of collective
  additive tree 1-spanners will need to have at
  least ?(n) spanning trees.
• ? the result of the previous Thm cannot be
  improved
• Observ There are trapezoid graphs which do not
  admit any additive tree 2-spanners.
• ? disprove of the conjecture from PKLMW03
  that any co-comparability graph admits an
  additive tree 2-spanner

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Trapezoid Graphs

• Observ There are trapezoid graphs which do not
  admit any additive tree 2-spanners.

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Collective Tree Spanners For AT-free Graphs

• Any AT-free graph G admits an additive tree
  3-spanner PKLMW03
• Thm Any AT-free graph G admits a system of 2
  collective additive tree 2-spanners which can be
  constructed in linear time.
• To get 2, one needs at least 2 spanning trees
• To get 1, one needs at least ?(n) spanning trees

an AT-free graph with its backbone
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Collective Tree Spanners For AT-free Graphs

• 2 collective additive tree 2-spanners of G

cactus-tree
caterpillar-tree
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Collective Tree Spanners for DSP Graphs

• Any DSP-graph admits an additive tree 4-spanner
  PKLMW03
• Thm Any DSP-graph admits a system of 2
  collective additive tree 3-spanners and a system
  of 5 collective additive tree 2-spanners.

an DSP graph G with its dominating path
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2 Collective Tree 3-Spanners for DSP Graphs

• 2 collective additive tree 3-spanners of G

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5 Collective Tree 2-Spanners for DSP Graphs

• 5 collective additive tree 2-spanners.

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Graphs With Bounded Asteroidal Number

• Any graph with asteroidal number an(G) admits an
  additive tree (3an(G) -1)-spanner
  PKLMW03
• Thm Any graph with asteroidal number an(G)
  admits a system of an(G)(an(G)-1)/2 collective
  additive tree 4-spanners and a system of
  an(G)(an(G)-1) collective additive tree
  3-spanners.

A graph with its dominating target
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Collective Trees for Graphs with Bounded
Asteroidal Number
(collective additive tree 4-spanners)
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Routing Schemes for AT-free Graphs

• FG01,TZ01 For family of n-node trees there is
  a routing labeling scheme with labels of size
  O(log²n/loglogn)- bits per node and
  constant time routing decision.
• For AT-free graphs, O(log²n/loglogn)-bits per
  node, constant time routing decision and
  deviation at most 2.
• Thm Every AT-free graph of diameter Ddiam(G)
  and of maximum vertex degree ? admits a
  (3log2D6log2?O(1))-bit routing labeling scheme
  of deviation at most 2. Moreover, the scheme is
  computable in linear time, and the routing
  decision is made in constant time per vertex.

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Future Plans

• Find best possible trade-off between number of
  trees and additive stretch factor for planar
  graphs (currently vn log n collective additive
  tree 0-spanners).
• Consider the collective additive tree spanners
  problem for other structured graph families.
• Complexity of the collective additive tree
  spanners problem for different m and r on general
  graphs and special graph classes.
• More applications of

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• Thank You

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K4, 4