On the Power of BFS to Determine a Graph

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On the Power of BFS to Determine a Graphs
Diameter
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The Diameter Problem (find a longest shortest
path in a graph)

• G (V,E) is a connected, finite, and undirected
  graph
• The length of a path from a vertex v to a vertex
  u is the number of edges in the path
• The distance d(u,v) is the length of a shortest
  (u,v)-path
• The eccentricity ecc(v) of a vertex v is the
  maximum distance from v to a vertex in G

• The radius r(G) is the minimum eccentricity of a
  vertex in G and the diameter d(G) is the maximum
  eccentricity
• The diameter problem find d(G) and x,y such
  that d(x,y)d(G) (in other words, find
  a vertex of maximum eccentricity)

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Applications

• The diameter problem is a basic problem in
  algorithmic graph theory and computational
  geometry.
• It naturally arises in communication and
  transportation networks (the linkage structure
  of communication networks is
  usually modeled by the graph).
• if the number of links in a path is roughly
  proportional to the time delay or signal
  degradation encountered by messages sent along
  the path, the diameter is then involved in the
  complexity analysis for the performance of the
  network
• the diameter of a communication network gives a
  lower bound on the time needed to transmit a
  message from an arbitrary source node to all
  other nodes.

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Known General Results

• Determining the diameter of a graph is a basic
  but seemingly quite time consuming operation.
• No efficient algorithms for the diameter problem
  in general graphs, avoiding the computation of
  whole distance matrix, has been designed (Can the
  diameter be computed easier than the whole
  distance matrix?).
• A ratio 2/3 approximation to the diameter in
  time
• Diameter with an additive error 2

• naïve approach O(nm) ( for
  dense graphs)
• via matrix multiplications O(M(n) log n)
  Seidel92

Coppersmith/Winograd 87

• not practical, large hidden constants

Aingworth/Chekuri/Indyk/Motwani 96
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Our Approach

• Examine the naïve algorithm of
• choosing a vertex
• performing some version of BFS from this vertex
  and then
• showing a nontrivial bound on the eccentricity
  of the last vertex visited in this search.
• This approach has already received considerable
  attention
• (classical result Handler73) for trees this
  method produces a vertex of maximum eccentricity
• Dragan et al 97 if LexBFS is used for chordal
  graphs, then whereas
  for interval graphs and Ptolemaic graphs
• Corneil et al99 if LexBFS is used on AT-free
  graphs, then
• Dragan99 if LexBFS is used, then
  for HH-free graphs,
  for HHD-free graphs and
  for HHD-free and AT-free graphs
• Corneil et al01 considers multi sweep LexBFSs
  

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Motivation for this paper

• Those results motivate a number of interesting
  questions
• Is it an inherent property of LexBFS to end in a
  vertex of high eccentricity for the various
  restricted graph families mentioned above? What
  happens if we use other variants of BFS?
• Why do AT-free and chordal graphs, two families
  with very disparate structure, exhibit such
  similar behavior with respect to the efficacy of
  LexBFS to find vertices of high eccentricity?
• Although LexBFS fails'' to find vertices of
  high eccentricity for graphs in general, all
  known examples that exhibit such failure have
  large induced cycles. If we bound the size of
  the largest induced cycle, can we get a bound on
  the eccentricity of the vertex that appears last
  in an LexBFS?
• If the previous question is answered in the
  affirmative, is the full power of LexBFS
  needed? What happens if we just use BFS?
• This paper addresses these questions.

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Variants of BFS used
Can be implemented to run in linear time
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Our Results on Restricted Families of Graphs
No induced cycles of length gt3
No asteroidal triples
No asteroidal triples and
The intersection graph of intervals of a line
No induced cycles of length gt4
c
b
a
asteroidal triple a,b,c
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k-Chordal Graphs

• To capture the notion of small induced cycles,
  we define a graph to be k-chordal if it has no
  induced cycles of length greater than k.
• Chordal graphs are exactly 3-chordal
• Hole-free graphs are exactly 4-chordal
• AT-free graphs are 5-chordal
• We have
• Fig. 14 shows a 4k-chordal graph with
  ecc(v)2k1 and diam(G)4k.
• We conjecture that
• It is true for k3 (chordal graphs), k4
  (hole-free graphs) and k5
• For k5 we have

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The Method Chordal Graphs and BFS
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The Method (cont.)
If
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The Method (cont.)
If