Additive Spanners for k-Chordal Graphs

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Additive Spanners for k-Chordal Graphs

• V. D. Chepoi, F.F. Dragan, C. Yan

University Aix-Marseille II, France Kent State
University, Ohio, USA
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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 t-spanner of G)?
G multiplicative 2-
and additive 1-spanner of G
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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 t-spanner of G)?
G multiplicative 2-
and additive 1-spanner of G
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Applications

• in distributed systems and communication
  networks
• synchronizers in parallel systems
• Close relationship were established between the
  quality of spanners for a given undirected graph
  (in terms of the stretch factor t and the number
  of edges E), and the time and communication
  complexities of any synchronizer for the network
  based on this graph
• topology for message routing
• efficient routing schemes can use only the edges
  of the spanner

G
2-spanner for G
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Applications

• in distributed systems and communication
  networks
• synchronizers in parallel systems
• Close relationship were established between the
  quality of spanners for a given undirected graph
  (in terms of the stretch factor t and the number
  of edges E), and the time and communication
  complexities of any synchronizer for the network
  based on this graph
• topology for message routing
• efficient routing schemes can use only the edges
  of the spanner

G
2-spanner for G
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Some Known Results
(multiplicative case)

• general graphs PelegSchaffer89
• given a graph G(V, E) and two integers t, m?1,
  whether G has a t-spanner with m or fewer edges,
  is NP-complete
• chordal graphs PelegSchaffer89
• G is chordal if it has no chordless cycles
  of length gt3
• every n-vertex chordal graph G(V, E) admits a
  2-spanner with O(n1.5) edges
• there exist (infinitely many) n-vertex chordal
  graphs G(V, E) for which every 2-spanner
  requires ?(n1.5) edges
• every n-vertex chordal graph G(V, E) admits a
  3-spanner with O(n logn) edges
• every n-vertex chordal graph G(V, E) admits a
  5-spanner with at most 2n-2 edges

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Some Known Results
(multiplicative case)

• general graphs PelegSchaffer89
• given a graph G(V, E) and two integers t, m?1,
  whether G has a t-spanner with m or fewer edges,
  is NP-complete
• chordal graphs PelegSchaffer89
• G is chordal if it has no chordless cycles
  of length gt3
• every n-vertex chordal graph G(V, E) admits a
  3-spanner with O(n logn) edges
• every n-vertex chordal graph G(V, E) admits a
  5-spanner with at most 2n-2 edges
• tree spanner BDLL2002
• given a chordal graph G(V, E) and an integer
  tgt3, whether G has a t-spanner with n-1 edges
  (tree t-spanner), is NP-complete

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Some Known Results
(multiplicative case)

• general graphs PelegSchaffer89
• given a graph G(V, E) and two integers t, m?1,
  whether G has a t-spanner with m or fewer edges,
  is NP-complete
• chordal graphs PelegSchaffer89
• G is chordal if it has no chordless cycles
  of length gt3
• every n-vertex chordal graph G(V, E) admits a
  3-spanner with O(n logn) edges
• every n-vertex chordal graph G(V, E) admits a
  5-spanner with at most 2n-2 edges ? 2-appr.
  algorithm for any t ? 5
• tree spanner BDLL2002
• given a chordal graph G(V, E) and an integer
  tgt3, whether G has a t-spanner with n-1 edges
  (tree t-spanner), is NP-complete

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This Talk

• From multiplicative to additive
• every chordal graph admits an additive 4-spanner
  with at most 2n-2 edges which can be constructed
  in linear time
• every chordal graph admits an additive 3-spanner
  with O(n logn) edges which can be constructed in
  polynomial time
• Extension to k-chordal graphs
• G is k-chordal if it has no chordless cycle of
  length gtk
• Every k-chordal graph admits an additive
  (k1)-spanner with at most 2n-2 edges which can
  be constructed in O(n?km)
• Better bounds for subclasses of 4-chordal graphs
  
• Every HH-free graph (or chordal bipartite graph)
  admits an additive 4-spanner with at most 2n-2
  edges which can be constructed in linear time
• Note that any additive t-spanner is a
  multiplicative (t1)-spanner

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MethodConstructing Additive 4-Spanner

• Given a chordal graph G(V, E) and an arbitrary
  vertex u

u
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BFS-Ordering and BFS-Tree up-phase

• We start from u and construct a BFS tree. The red
  edges are tree edges.
• First layer.

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u
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BFS-Ordering and BFS-Tree up-phase

• Second layer

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BFS-Ordering and BFS-Tree up-phase

• Third layer

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BFS-Ordering and BFS-Tree up-phase

• Fourth Layer

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Constructing Spanner down-phase

• Start from the last layer. For vertices of each
  connected component in the layer create a star
  for the fathers.

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connected components
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Constructing Spanner down-phase

• Third Layer

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connected components
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Constructing Spanner down-phase

• Second layer

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connected components
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Final Spanner

• The final spanner is showed in red

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Final Spanner

• The final spanner is showed in red

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1 vs 3
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Analysis Of The Algorithm

• Given a chordal graph G(V, E), we produce a
  spanning graph H(V,E) such that
• H is an additive 4-spanner of G
• H contains at most 2n-2 edges
• H can be constructed in O(nm) time

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Analysis Of The Algorithm

• Given a chordal graph G(V, E), we produce a
  spanning graph H(V,E) such that
• H is an additive 4-spanner of G
• H contains at most 2n-2 edges
• H can be constructed in O(nm) time

y
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Layer i
Layer i-1
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u
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Constructing Additive 3-Spanner

• G is a chordal graph with n vertices and with a
  BFS ordering (started at u)
• Take all the edges of the additive 4-spanner
• in each connected component S induced by layer r,
  we run the algorithm presented in
  PelegSchaffer89, to construct a
  multiplicative 3-spanner for S

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Constructing Additive 3-Spanner

• G is a chordal graph with n vertices and with a
  BFS ordering (started at u)
• Take all the edges of the additive 4-spanner
• in each connected component S induced by layer r,
  we run the algorithm presented in
  PelegSchaffer89, to construct a
  multiplicative 3-spanner for S

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Analysis Of The Algorithm

• Given a chordal graph G(V, E) with n vertices
  and m edges, we produce a spanning graph H(V,E)
  such that
• H is an additive 3-spanner of G
• H contains O(n logn) edges
• H can be constructed in polynomial time

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MethodConstructing Additive (k1)-Spanner
Given a k-chordal graph G(V, E) and an arbitrary
vertex u
u
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BFS-Ordering and BFS-Tree up-phase

• We start from u and construct a BFS tree. The red
  edges are tree edges.

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Constructing Spanner down-phase

• Start from the last layer. For vertices of each
  component,choose
• the smallest one. Then try to connect others to
  it or its ancestor.

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a component on layer 3
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Constructing Spanner down-phase

• Start from the last layer. For vertices of each
  component,choose
• the smallest one. Then try to connect others to
  it or its ancestor.

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a component on layer 3
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edge used to connect 3 and 5
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Constructing Spanner down-phase

• Start from the last layer. For vertices of each
  component,choose
• the smallest one. Then try to connect others to
  it or its ancestor.

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a component on layer 3
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edge used to connect 3 and 5
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Final Spanner

• Final spanner is shown in red.

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Analysis Of The Algorithm

• G is k-chordal if it has no chordless cycles of
  length gtk
• The spanner constructed by the above algorithm
  has the following properties
• It is an additive (k1)-spanner
• It contains at most 2n-2 edges
• It can be constructed in O(knm) time

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Open questions and future directions

• Can these ideas be applied to other graph
  families to obtain good sparse additive spanners?
  
• Can one get a constant approximation for the
  additive 3-spanner problem on chordal graphs?
• so far,
• only a log-approximation for t3
• 2-approximation for tgt3
• What about t2 (additive)?
• so far, (from PelegSchaffer89)
• a log-approximation for multiplicative 3-spanner
• for t1, the lower bound is ?(n1.5) edges (as
  multiplicative 2-spanner)

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• Thank You

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Layering

• Given a graph G(V, E) and an arbitrary vertex
  u?V, the
• sphere of u is defined as

• The ball of radius centered at u is defined
  as

• A layering of G with respect to some vertex u
  is a partition
• of V into the spheres

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BFS Ordering

• G(V, E) is a graph with n vertices
• In Breadth-First-Search (BFS), started at vertex
  u, we number the vertices from n to 1 as follows
• u is numbered by n and is put on an initially
  empty queue
• a vertex v is repeatedly removed from the head of
  the queue and the neighbors of v which are still
  unnumbered are consequently numbered and placed
  onto the queue
• we call v the father of those vertices which are
  placed onto the queue when v is removed from the
  queue. We use f(v) to denote the father of v
• An ordering generated by BFS is called
  BFS-ordering

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Layering and BFS-ordering, an example

• The vertices are numbered in BFS-ordering and the
  BFS tree is shown in red

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Layer 2
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Layer 1
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Constructing Additive 4-Spanner

• Method of constructing spanner H(V, E)
• let be arbitrary vertex of G and
  , we use Breadth-First-Search
  (BFS) rooted at to label all the vertices of
  G.
• start from the layer of , for each vertex
  we add into
• start from the layer and for each connected
  component induced by we find its
  projection on layer and make
  it star and put all the edges in the star into

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Example

• The following is an example. The think red lines
  consists
• the spanners for the chordal graph

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Definitions and Symbols for k-chordal graph

• G is k-chordal if it has no chordless cycles of
  lengthgtk
• Let u be an arbitrary vertex of G. We define a
  graph with the lth sphere as a vertex
  set. Two vertices are adjacent in if and
  only if they can be connected by a path outside
  the ball . We use to
  denote all the connected component of
• Also we define

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Constructing Spanners

• G is k-chordal if it has no chordless cycles of
  lengthgtk
• Method of constructing a spanner H(V, E)
• for each vertex , we add into E
• for each connected component we identify a
  vertex such that is the minimum in
  BFS-ordering among all vertices in
• check if then we add to
  
• check if then we add
  to
• If none of the above is true, we let
  and repeat 3 and 4