Transcript Weeping Willow in the Fog

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
Graphs, Cycles and Trees
root
parent
nodes
child
Anupam Bhattacharjee, CSE, BUET
2
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.
Anupam Bhattacharjee, CSE, BUET
3
SP graphs: Series Connection
s
G1
s
Anupam Bhattacharjee, CSE, BUET
t
s
t
G2
G
t
4
SP Graphs: Parallel Connection
s
t
G1
s
s
G2
t
t
G
Anupam Bhattacharjee, CSE, BUET
5
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 > i.
Anupam Bhattacharjee, CSE, BUET
6
Minimum Vertex-Ranking
• A Vertex-Ranking is
minimum if least
number of ranks are
needed to rank the
graph.
Anupam Bhattacharjee, CSE, BUET
7
Minimum Vertex-Ranking Spanning Tree
• The problem is to find
a spanning tree of a
graph whose vertexranking needs least
number of ranks.
• Input: A graph
• Output: A tree with
minimum vertexranking
Anupam Bhattacharjee, CSE, BUET
8
Binary Decomposition Tree
Anupam Bhattacharjee, CSE, BUET
9
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.
Anupam Bhattacharjee, CSE, BUET
10
p
Steps of
the algorithm
• An SP graph is given:
• Step#1: Binary
decomposition tree
• Step#2: Equivalence
class computation for
each leaf node
Anupam Bhattacharjee, CSE, BUET
s
ac
ab
bc
11
For s-node Solution Computation
s
ab
Anupam Bhattacharjee, CSE, BUET
bc
12
For p-node Solution Computation
p
ac
s
Anupam Bhattacharjee, CSE, BUET
13
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
Anupam Bhattacharjee, CSE, BUET
14
Anupam Bhattacharjee, CSE, BUET
15