Collective Additive Tree Spanners of Homogeneously Orderable Graphs

1
Collective Additive Tree Spanners of
Homogeneously Orderable Graphs

• F.F. Dragan, C. Yan and Y. Xiang
• Kent State University, USA

2
Well-known Tree t -Spanner Problem

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

(a multiplicative tree t-spanner of G)
or
(an additive tree r-spanner of G)?
multiplicative tree 4-, additive tree 3-spanner
of G
G
T
3
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.)
• easy to construct for some special families of
  graphs.

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Well-known Sparse t -Spanner Problem
Given unweighted undirected graph G(V,E) and
integers t,m,r. Does G admit a spanning graph H
(V,E) with E ? m s.t.
(a multiplicative t-spanner of G)
or
(an additive r-spanner of G)?
H
G
multiplicative 2- and additive 1-spanner of G
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Some known results for sparse spanner problems

• general graphs
• t, m?1 is NP-complete PS89
• multiplicative (2k-1)-spanner with n11/k edges
  TZ01, BS03
• n-vertex chordal graphs (multiplicative case)
  PS89
• (G is chordal if it has no chordless cycles
  of length 3)
• multiplicative 3-spanner with O(n logn) edges
• multiplicative 5-spanner with 2n-2 edges
• n-vertex c-chordal graphs (additive case)
  CDY03, DYL04
• (G is c-chordal if it has no chordless cycles
  of length c)
• additive (c1)-spanner with 2n-2 edges
• additive (2 ?c/2? )-spanner with n log n edges
• ? For chordal graphs additive 4-spanner with
  2n-2 edges, additive 2-spanner with n log n edges

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Collective Additive Tree r -Spanners Problem (a
middle way)

• Given unweighted undirected graph G(V,E) and
  integers ?, r.
• Does G admit a system of ? collective additive
  tree r-spanners T1, T2, T?
  such that

(a system of ? collective additive tree
r-spanners of G )?
surplus
,
collective multiplicative tree t-spanners can be
defined similarly
2 collective additive tree 2-spanners
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Collective Additive Tree r -Spanners Problem

• Given unweighted undirected graph G(V,E) and
  integers ?, r.
• Does G admit a system of ? collective additive
  tree r-spanners T1, T2, T?
  such that

(a system of ? collective additive tree
r-spanners of G )?
,
2 collective additive tree 2-spanners
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Collective Additive Tree r -Spanners Problem

• Given unweighted undirected graph G(V,E) and
  integers ?, r.
• Does G admit a system of ? collective additive
  tree r-spanners T1, T2, T?
  such that

(a system of ? collective additive tree
r-spanners of G )?
,
2 collective additive tree 2-spanners
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Collective Additive Tree r -Spanners Problem

• Given unweighted undirected graph G(V,E) and
  integers ?, r.
• Does G admit a system of ? collective additive
  tree r-spanners T1, T2, T?
  such that

(a system of ? collective additive tree
r-spanners of G )?
,
,
2 collective additive tree 0-spanners or
multiplicative tree 1-spanners
2 collective additive tree 2-spanners
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Applications of Collective Tree
Spannersrepresenting complicated graph-distances
with few tree-distances

• message routing in networks
• Efficient routing schemes are known for
  trees
• but not 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.
• - (? log2n/ log log n)-bit labels,
• - O(? ) initiation, O(1) decision
• solution for sparse t-spanner problem
• If a graph admits a system of ? collective
  additive tree r-spanners, then the graph admits a
  sparse additive r-spanner with at most ?(n-1)
  edges, where n is the number of nodes.

2 collective tree 2-spanners for G
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Previous results on the collective tree spanners
problem(Dragan, Yan, Lomonosov
SWAT04)(Corneil, Dragan, Köhler, Yan WG05)

• chordal graphs, chordal bipartite graphs
• log n collective additive tree 2-spanners in
  polynomial time
• ?(n1/2) or ?(n) trees necessary to get 1
• no constant number of trees guaranties 2 (3)
• circular-arc graphs
• 2 collective additive tree 2-spanners in
  polynomial time
• c-chordal graphs
• log n collective additive tree 2 ?c/2? -spanners
  in polynomial time
• interval graphs
• log n collective additive tree 1-spanners in
  polynomial time
• no constant number of trees guaranties 1

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Previous results on the collective tree spanners
problem(Dragan, Yan, Corneil WG04)

• AT-free graphs
• include interval, permutation, trapezoid,
  co-comparability
• 2 collective additive tree 2-spanners in linear
  time
• an additive tree 3-spanner in linear time
  (before)
• graphs with a dominating shortest path
• an additive tree 4-spanner in polynomial time
  (before)
• 2 collective additive tree 3-spanners in
  polynomial time
• 5 collective additive tree 2-spanners in
  polynomial time
• graphs with asteroidal number an(G)k
• k(k-1)/2 collective additive tree 4-spanners in
  polynomial time
• k(k-1) collective additive tree 3-spanners in
  polynomial time

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Previous results on the collective tree spanners
problem(Gupta, Kumar,Rastogi SICOMP05)

• the only paper (before) on collective
  multiplicative tree spanners in weighted planar
  graphs
• any weighted planar graph admits a system of
  O(log n) collective multiplicative tree
  3-spanners
• they are called there the tree-covers.
• it follows from (Corneil, Dragan, Köhler, Yan
  WG05) that
• no constant number of trees guaranties c (for
  any constant c)

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Some results on collective additive tree spanners
of weighted graphs with bounded parameters
(Dragan, Yan ISAAC04)
to get 0
No constant number of trees guaranties r for any
constant r even for outer-planar graphs
to get 1

• w is the length of a longest edge in G

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Some results on collective additive tree spanners
of weighted c-chordal graphs (Dragan, Yan
ISAAC04)
No constant number of trees guaranties r for any
constant r even for weakly chordal graphs
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(This paper)Homogeneously orderable Graphs

• A graph G is homogeneously orderable if G has an
  h-extremal ordering Brandstädt et.al.95.
• Equivalently A graph G is homogeneously
  orderable if and only if the graph L(D(G)) of G
  is chordal and each maximal two-set of G is
  join-split.
• L(D(G)) is the intersection graph of D(G).
• Two-set is a set of vertices at pair-wise
  distance 2.

join-split
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Hierarchy of Homogeneously Orderable Graphs (HOGs)
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Our results on Collective additive tree spanners
of n-vertex homogeneously orderable graphs
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To get 1 one needs trees
trivial
n-1 BFS-trees
trees
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Our results on Collective additive tree spanners
of n-vertex homogeneously orderable graphs
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Layering and Clustering

• The projection of each cluster is a two-set.
• The connected components of projections are
  two-sets and have a common neighbor down.

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Additive Tree 3-spanner
Linear Time
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Our results on Collective additive tree spanners
of n-vertex homogeneously orderable graphs
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H and H2
HOG
Chordal
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15
16
13
14
12
25
11
23
1
21
19
7
8
2
3
24
22
5
9
4
10
18
20
6
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H2 (chordal graph) and its balanced decomposition
tree
1, 2, 3, 4, 5, 6, 7, 9, 11, 12
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15
16
13
14
12
25
11
8, 10
23
13, 14, 15, 16, 17
18, 19, 20, 21, 22, 23, 24
1
21
19
8
7
2
3
24
22
4
5
9
10
18
20
25
6
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Constructing Local Spanning Trees for H

• For each layer of the decomposition tree,
  construct local spanning trees of H (shortest
  path trees in the subgraph).
• Here, we use the second layer for illustration.

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1, 2, 3, 4, 5, 6, 7, 9, 11, 12
15
16
13
14
12
25
11
23
8, 10
13, 14, 15, 16, 17
18, 19, 20, 21, 22, 23, 24
1
21
19
7
8
2
3
24
22
5
9
4
10
18
20
25
6
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Local Additive Tree 2-spanner
Theorem
must hold
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Our results on Collective additive tree spanners
of n-vertex homogeneously orderable graphs
One tree cannot give 2
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No constant number d of trees can guarantee
additive stretch factor 2
root
gadget
clique
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No constant number d of trees can guarantee
additive stretch factor 2
Tree of gadgets



The depth is a function of d
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Open questions and future plans

• Given a graph G(V, E) and two integers ? and r,
  what is the complexity of finding a system of ?
  collective additive (multiplicative) tree
  r-spanner for G? (Clearly, for most ? and r, it
  is an NP-complete problem.)
• Find better trade-offs between ? and r for planar
  graphs, genus g graphs and graphs w/o an h-minor.
  
• We may consider some other graph classes. Whats
  the optimal ? for each r?
• More applications of collective tree spanner

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• Thank You