Minimum Spanning Tree
Minimum Spanning Tree
MST. Given connected graph G with positive edge weights, find a min weight set of edges that connects all of the vertices.
Minimum Spanning Tree
24
4
23
6
9
18
•
•
•
•
•
•
•
5
introduction
Weighted graph API cycles and cuts
Kruskal’s algorithm
Prim’s algorithm advanced algorithms clustering 11
16
8
10
14
7
21
G
References: Algorithms in Java (Part 5), Chapter 20
Intro to Algs and Data Structures, Section 5.4
Copyright © 2007 by Robert Sedgewick and Kevin Wayne.
3
Minimum Spanning Tree
MST. Given connected graph G with positive edge weights, find a min weight set of edges that connects all of the vertices.
24
4
introduction weighted graph API cycles and cuts
Kruskal’s algorithm
Prim’s algorithm advanced algorithms clustering 23
6
9
18
5
11
16
8
10
14
7
21
cost(T) = 50
Brute force: Try all possible spanning trees problem 1: not so easy to implement problem 2: far too many of them
!
!
Ex: [Cayley, 1889]: VV-2 spanning trees on the complete graph on V vertices.
2
4
MST Origin
Medical Image Processing
Otakar Boruvka (1926).
Electrical Power Company of Western Moravia in Brno.
Most economical construction of electrical power network.
Concrete engineering problem is now a cornerstone problem in combinatorial optimization.
MST describes arrangement of nuclei in the epithelium for cancer research
!
!
!
Otakar Boruvka
http://www.bccrc.ca/ci/ta01_archlevel.html
5
7
Applications
MST is fundamental problem with diverse applications.
!
!
!
!
Network design.
– telephone, electrical, hydraulic, TV cable, computer, road
Approximation algorithms for NP-hard problems.
– traveling salesperson problem, Steiner tree
Indirect applications.
– max bottleneck paths
– LDPC codes for error correction
– image registration with Renyi entropy
– learning salient features for
MST. Given connected graph G with positive edge weights, find a min weight set of edges that connects all of the vertices.
Minimum Spanning Tree
24
4
23
6
9
18
•
•
•
•
•
•
•
5
introduction
Weighted graph API cycles and cuts
Kruskal’s algorithm
Prim’s algorithm advanced algorithms clustering 11
16
8
10
14
7
21
G
References: Algorithms in Java (Part 5), Chapter 20
Intro to Algs and Data Structures, Section 5.4
Copyright © 2007 by Robert Sedgewick and Kevin Wayne.
3
Minimum Spanning Tree
MST. Given connected graph G with positive edge weights, find a min weight set of edges that connects all of the vertices.
24
4
introduction weighted graph API cycles and cuts
Kruskal’s algorithm
Prim’s algorithm advanced algorithms clustering 23
6
9
18
5
11
16
8
10
14
7
21
cost(T) = 50
Brute force: Try all possible spanning trees problem 1: not so easy to implement problem 2: far too many of them
!
!
Ex: [Cayley, 1889]: VV-2 spanning trees on the complete graph on V vertices.
2
4
MST Origin
Medical Image Processing
Otakar Boruvka (1926).
Electrical Power Company of Western Moravia in Brno.
Most economical construction of electrical power network.
Concrete engineering problem is now a cornerstone problem in combinatorial optimization.
MST describes arrangement of nuclei in the epithelium for cancer research
!
!
!
Otakar Boruvka
http://www.bccrc.ca/ci/ta01_archlevel.html
5
7
Applications
MST is fundamental problem with diverse applications.
!
!
!
!
Network design.
– telephone, electrical, hydraulic, TV cable, computer, road
Approximation algorithms for NP-hard problems.
– traveling salesperson problem, Steiner tree
Indirect applications.
– max bottleneck paths
– LDPC codes for error correction
– image registration with Renyi entropy
– learning salient features for
References: ! Ex: [Cayley, 1889]: VV-2 spanning trees on the complete graph on V vertices. Otakar Boruvka (1926).