An Algorithm for Solving the Minimum Vertex-Ranking Spanning Tree Problem on Series-Parallel Graphs

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An Algorithm for Solving the Minimum
Vertex-Ranking Spanning Tree Problem on
Series-Parallel Graphs

• ICECE 2006
• Md. Abul Kashem, Chowdhury Sharif Hasan, and
  Anupam Bhattacharjee

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Graphs, Cycles and Trees
root
parent
nodes
child
A Graph
A cycle
A tree
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Spanning Trees

• A connected graph with no cycles is a spanning
  tree
• In a connected cyclic graph, if we delete edges
  to remove cycles.
• If there remains no cycle, we call it a spanning
  tree of the graph.

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SP graphs Series Connection
s
t
s
t
G1
G2
s
t
G
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SP Graphs Parallel Connection
s
t
s
t
G1
G2
s
t
G
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Vertex-Ranking

• A labeling (ranking) of the vertices of G with
  positive integers such that every path in G with
  end vertices of the same label i contains an
  internal vertex with label j gt i.

Valid Ranking
Invalid Ranking
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Minimum Vertex-Ranking

• A Vertex-Ranking is minimum if least number of
  ranks are needed to rank the graph.

A minimum vertex-ranking
A non-optimal Vertex-Ranking
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Minimum Vertex-Ranking Spanning Tree

• The problem is to find a spanning tree of a graph
  whose vertex-ranking needs least number of ranks.
• Input A graph
• Output A tree with minimum vertex-ranking

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Binary Decomposition Tree
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Solution types

• Two types of partial solutions
• A one-tree type solution a spanning tree is kept
• A two-tree type solution a spanning forest
  having exactly two components (trees) with
  terminal vertices in different trees is kept.

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Steps of the algorithm

• An SP graph is given
• Step1 Binary decomposition tree
• Step2 Equivalence class computation for each
  leaf node

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For s-node Solution Computation
s
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For p-node Solution Computation
DESIRED
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Points to note

• Total running time of the algorithm is
• O(n5 log4 n).
• Some open problems still
• Develop a polylog-time parallel algorithm for
  solving the minimum vertex- ranking spanning tree
  problem on series-parallel graphs.
• Develop a polynomial-time sequential algorithm
  for solving the minimum edge-ranking spanning
  tree problem on series-parallel graphs

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Thank you