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

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

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O(log n)= tw x log n collective additive tree 0-spanners. New results on the collective tree spanners problem on AT-free related graphs ... – PowerPoint PPT presentation

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


1
Collective Tree Spanners and Routing in AT-free
Related Graphs
  • F.F. Dragan, C. Yan, D. Corneil
  • Kent State University
  • University of Toronto

2
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
3
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
4
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
5
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
6
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

7
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

8
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

9
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

10
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

11
Trapezoid Graphs
  • Observ There are trapezoid graphs which do not
    admit any additive tree 2-spanners.

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12
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
13
Collective Tree Spanners For AT-free Graphs
  • 2 collective additive tree 2-spanners of G

cactus-tree
caterpillar-tree
14
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
15
2 Collective Tree 3-Spanners for DSP Graphs
  • 2 collective additive tree 3-spanners of G

16
5 Collective Tree 2-Spanners for DSP Graphs
  • 5 collective additive tree 2-spanners.

17
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
18
Collective Trees for Graphs with Bounded
Asteroidal Number
(collective additive tree 4-spanners)
19
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.

20
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

21
  • Thank You

22
K4, 4
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