Minimum Spanning Tree

Only available on StudyMode
  • Topic: Graph theory, Minimum spanning tree, Greedy algorithm
  • Pages : 29 (2477 words )
  • Download(s) : 60
  • Published : September 26, 2012
Open Document
Text Preview
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 real-time face verification – reducing data storage in sequencing amino acids in a protein – model locality of particle interactions in turbulent fluid flows – autoconfig protocol for Ethernet bridging to avoid cycles in a network Cluster analysis.

6

http://ginger.indstate.edu/ge/gfx

8

Two Greedy Algorithms

Weighted Graph Interface

Kruskal's algorithm. Consider edges in ascending order of cost. Add the next edge to T unless doing so would create a cycle.

public class WeightedGraph

(graph data type)

WeightedGraph(int V)

Prim's algorithm. Start with any vertex s and greedily grow a tree T from s. At each step, add the cheapest edge to T that has exactly one endpoint in T.

void insert(Edge e)
Iterable adj(int v)
int V()
String toString()

create an empty graph with V vertices
insert edge e
return an iterator over edges incident to v
return the number of vertices
return a string representation

Theorem. Both greedy algorithms compute an MST.
for (int v = 0; v < G.V(); v++)
{
for (Edge e : G.adj(v))
{
int w = e.other(v);
// edge v-w
}
}

Greed is good. Greed is right. Greed works. Greed
clarifies, cuts through, and captures the essence of the
evolutionary spirit." - Gordon Gecko

iterate through all edges (once in each direction)

9

11

Edge data type

public class Edge implements Comparable
{
public final int v, int w;
public final double weight;
public Edge(int v, int w, double weight)
{
this.v = v;
this.w = w;
this.weight = weight;
}

introduction
weighted graph API
cycles and cuts
Kruskal’s algorithm
Prim’s algorithm
advanced algorithms
clustering

public int either()
{ return v; }
public int other(int vertex)
{
if (vertex == v) return w;
else return v;
}

10

public int compareTo(Edge f)
{
Edge e = this;
if
(e.weight < f.weight) return -1;
else if (e.weight > f.weight) return +1;
else if (e.weight > f.weight) return 0;
}
}

12

Weighted graph: Java...
tracking img