Minimum spanning tree on Networks

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Minimum spanning tree on Networks
?????

• basic concept and something else

Seung-Woo Son Complex System and Statistical
Physics Lab. KAIST
2003.9.27.
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Index

• Basic concept of graphs
• Various problems in graph theory
• Minimum spanning tree
• Algorithm of minimum spanning tree
• Applications of minimum spanning tree
• Minimum spanning tree on networks
• Reference

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Basic concept of graphs
What is the graphs?
Mathematics
vertices, nodes
v1
e1
v2
v5
e3
e4
edges, links
e2
v3
v4
In real world
??? ??? ???? ?? ?? ??, ??? ???? ?? ????. ???
????? ?? ???.
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Examples of graphs
ancient Greek gods graph
aspirin, C9H8O4
chemicals
vitamin A, C20H30O
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Examples of graphs
highway system
time and work study
Too much arrange them kill you !!
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Representation of graphs
a
With equation
Graphically
b
e
c
d
With table
a b c d e
a - 1 0 0 0
b 1 - 1 1 1
c 0 1 - 1 0
d 0 1 1 - 1
e 0 1 0 1 -
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Approach of the computer scientist
Programmer
Graph is one of the data structures. It can be
represented with 2 dimensional array or
linked-list.
a
b
e
2D matrix - 5x5 array
c
d
-1 1 0 0 0
1 -1 1 1 1
0 1 -1 1 0
0 1 1 -1 1
0 1 0 1 -1
linked-list
G55
Adjacency List
weight
Adjacency Matrix
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Various problems in graph theory

• Traveling salesperson problem
• Konigsberg bridge problem Euler cycle
• Hamilton cycles problem
• Shortest path algorithm
• Isomorphisms of graphs
• Planar graphs problem
• Spanning tree minimum spanning tree
• Search algorithm
• Maximal flow problem
• Chromatic number problem

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Minimum spanning tree
weighted graph !!
A graph G (V,E) with weighted edges. The subset
of E of G of minimum weight which forms a tree on
V MST .
- ??? ???? ?? ???? ???? ??. (V-1 ?? edge ??)
?????, ???, ????, ????? ??? ??? ???? ???? ???? ??
?? ??.
- ??? ???? ??? ??? ??????(MST) ? ????.
- ??? ??? ?? ?? ??? ??? ??? ???.
?? ? ??? ??? ??? ????? ??? ???.
- ? ??? ???? ??????(shortest path)? ?? ???? ?? ??.
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Algorithm of minimum spanning tree
Prim algorithm
- greedy algorithm
- O(EV2) complexity. It can be improved to O( E
log V )
- find only connected vertices
- at each iteration, adds to current treea
minimum-weight edge that does not complete a cycle
Kruskal algorithm
- use priority queue
- O( E log E )
- find MST for all vertices including
non-connected part
- first sort all edges by weight and add edge
that does not complete a cycle
Sollin algorithm
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Applications of MST(1)
IP ????? ???? ?? ???? ?? ???? ??? ???? ??? ??? ?
? ??? ????? ???? ????? ???? ?? ?? ???? ??? ?? ??
???? ????? ?????? ??? ????? ????? ??? ??? ? ???
?????(Spanning Tree)? ??? ???? ??? ??.
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Applications of MST(2)
Windstar Aerospace Company
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Applications of MST(3)
MST using the correlation of price return and
volatility
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Minimum spanning tree on networks
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Reference

• Nora Hartsfield, Gerhard Ringel, Pearls in Graph
  Theory a comprehensive introduction, Academic
  press (1990)
• Richard Johnsonbaugh, Discrete mathematics 5th
  ed, Prentice Hall (2001)
• ???, C? ??? ???? 2, ???? ?? (2002)
• ?? ?? ?, ?? ?, Algorithms with C C? ??? ????,
  ????? (2001)
• ????? http//kedb.gsnu.ac.kr/kedb/lectures/lecture
  06/lecture_note.htm
• ??? (Graph) http//dcss.kunsan.ac.kr/kmjung/Lectu
  res/DS/chap6.htm
• http//cat1.snut.ac.kr/ykkang/lecture_right_graph
  1.html

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Basic Concepts of Network
Degree 3
Links
A shortest path with path length3
(Equivalent with 3 clicks in WWW)
Nodes
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Clustering Coefficient

• Clustering coefficient for a node represent how
  many links are there between neighbors
• Clustering coefficient for a network is the
  average of all nodess clustering coefficient

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Clustering Coefficient
A clique or a community C1
C0
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Clustering Coefficient
Triangle ? the building block.
Alternative definition of clustering coefficient
3 x of triangle
C
of connected triples
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