Title: Collective%20Tree%20Spanners%20and%20Routing%20in%20AT-free%20Related%20Graphs
1Collective 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
5Applications 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
6Some 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
7Some 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
8Previous 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
9New 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
10Permutation, 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
11Trapezoid Graphs
- Observ There are trapezoid graphs which do not
admit any additive tree 2-spanners.
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12Collective 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
13Collective Tree Spanners For AT-free Graphs
- 2 collective additive tree 2-spanners of G
cactus-tree
caterpillar-tree
14Collective 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
152 Collective Tree 3-Spanners for DSP Graphs
- 2 collective additive tree 3-spanners of G
165 Collective Tree 2-Spanners for DSP Graphs
- 5 collective additive tree 2-spanners.
17Graphs 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
18Collective Trees for Graphs with Bounded
Asteroidal Number
(collective additive tree 4-spanners)
19Routing 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.
20Future 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 22K4, 4