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Definitions

Common Algorithms

Applications

Minimum Spanning TreesAlgorithms and Applications

Varun Ganesan

18.304 Presentation

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Definitions

Common Algorithms

Applications

Outline

1 Definitions

Graph Terminology

Minimum Spanning Trees

2 Common Algorithms

Kruskals Algorithm

Prims Algorithm

3 Applications

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Definitions

Common Algorithms

Applications

Graph Terminology

Minimum Spanning Trees

Outline

1 Definitions

Graph Terminology

Minimum Spanning Trees

2 Common Algorithms

Kruskals Algorithm

Prims Algorithm

3 Applications

Varun Ganesan MSTs

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Definitions

Common Algorithms

Applications

Graph Terminology

Minimum Spanning Trees

Graphs

In graph theory, agraphis an ordered pairG= (V,E)comprising a set of verticesor nodes together with a set

ofedges.

Edgesare 2-element subsets ofVwhich represent aconnection between two vertices.

Edges can either be directed or undirected.Edges can also have aweightattribute.

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D fi i i

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Definitions

Common Algorithms

Applications

Graph Terminology

Minimum Spanning Trees

Graphs

In graph theory, agraphis an ordered pairG= (V,E)comprising a set of verticesor nodes together with a set

ofedges.

Edgesare 2-element subsets ofVwhich represent aconnection between two vertices.

Edges can either be directed or undirected.Edges can also have aweightattribute.

Varun Ganesan MSTs

D fi iti

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Definitions

Common Algorithms

Applications

Graph Terminology

Minimum Spanning Trees

Graphs

In graph theory, agraphis an ordered pairG= (V,E)comprising a set of verticesor nodes together with a set

ofedges.

Edgesare 2-element subsets ofVwhich represent aconnection between two vertices.

Edges can either be directed or undirected.Edges can also have aweightattribute.

Varun Ganesan MSTs

Definitions

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Definitions

Common Algorithms

Applications

Graph Terminology

Minimum Spanning Trees

Connectivity

A graph isconnectedwhen there is a path between everypair of vertices.

Acycleis a path that starts and ends with the same

vertex.

Atreeis a connected, acyclic graph.

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Definitions

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Definitions

Common Algorithms

Applications

Graph Terminology

Minimum Spanning Trees

Connectivity

A graph isconnectedwhen there is a path between everypair of vertices.

Acycleis a path that starts and ends with the same

vertex.

Atreeis a connected, acyclic graph.

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Definitions

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Definitions

Common Algorithms

Applications

Graph Terminology

Minimum Spanning Trees

Outline

1 Definitions

Graph Terminology

Minimum Spanning Trees

2 Common Algorithms

Kruskals Algorithm

Prims Algorithm

3 Applications

Varun Ganesan MSTs

DefinitionsG h T i l

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Definitions

Common Algorithms

Applications

Graph Terminology

Minimum Spanning Trees

Spanning Trees

Formally, for a graphG= (V,E), the spanning tree isE Esuch that:

uV : (u, v) E (v,u) E vVIn other words: the subset of edges spans all vertices.

|E|= |V| 1In other words: the number of edges is one less than thenumber of vertices, so that there are no cycles.

Varun Ganesan MSTs

DefinitionsG h T i l

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Common Algorithms

Applications

Graph Terminology

Minimum Spanning Trees

Spanning Trees

Formally, for a graphG= (V,E), the spanning tree isE Esuch that:

uV : (u, v) E (v,u) E vVIn other words: the subset of edges spans all vertices.

|E|= |V| 1In other words: the number of edges is one less than thenumber of vertices, so that there are no cycles.

Varun Ganesan MSTs

DefinitionsGraph Terminology

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Common Algorithms

Applications

Graph Terminology

Minimum Spanning Trees

Spanning Trees

Formally, for a graphG= (V,E), the spanning tree isE Esuch that:

uV : (u, v) E (v,u) E vVIn other words: the subset of edges spans all vertices.

Linear Graph

|E|= |V| 1In other words: the number of edges is one less than thenumber of vertices, so that there are no cycles.

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DefinitionsKruskals Algorithm

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Common Algorithms

Applications

Kruskal s Algorithm

Prims Algorithm

Outline

1 Definitions

Graph Terminology

Minimum Spanning Trees

2 Common Algorithms

Kruskals Algorithm

Prims Algorithm

3 Applications

Varun Ganesan MSTs

Definitions

C Al i hKruskals Algorithm

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Common Algorithms

Applications

Kruskal s Algorithm

Prims Algorithm

Main Idea

Start with|V|disjoint components.

Consider lesser weight edges first to incrementally connectcomponents.

Make certain to avoid cycles.

Continue until spanning tree is created.

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Definitions

Common AlgorithmsKruskals Algorithm

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Common Algorithms

Applications

g

Prims Algorithm

Main Idea

Start with|V|disjoint components.

Consider lesser weight edges first to incrementally connectcomponents.

Make certain to avoid cycles.

Continue until spanning tree is created.

Varun Ganesan MSTs

Definitions

Common AlgorithmsKruskals Algorithm

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Common Algorithms

ApplicationsPrims Algorithm

Main Idea

Start with|V|disjoint components.

Consider lesser weight edges first to incrementally connectcomponents.

Make certain to avoid cycles.

Continue until spanning tree is created.

Varun Ganesan MSTs

DefinitionsCommon Algorithms

Kruskals Algorithm

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Common Algorithms

ApplicationsPrims Algorithm

Pseudocode

Kruskals Algorithm

1A= 02foreachvG.V:3MAKE-SET(v)

4foreach(u, v)ordered byweight(u, v), increasing:5 ifFIND-SET(u)=FIND-SET(v):6A= A (u, v)

7UNION(u,

v)8 returnA

Varun Ganesan MSTs

DefinitionsCommon Algorithms

Kruskals Algorithm

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Common Algorithms

ApplicationsPrims Algorithm

Example

Varun Ganesan MSTs

DefinitionsCommon Algorithms

Kruskals Algorithm

P i Al i h

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Common Algorithms

ApplicationsPrims Algorithm

Example

Varun Ganesan MSTs

DefinitionsCommon Algorithms

Kruskals Algorithm

P i Al ith

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g

ApplicationsPrims Algorithm

Example

Varun Ganesan MSTs

DefinitionsCommon Algorithms

Kruskals Algorithm

Prims Algorithm

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g

ApplicationsPrim s Algorithm

Example

Varun Ganesan MSTs

DefinitionsCommon Algorithms

Kruskals Algorithm

Prims Algorithm

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ApplicationsPrim s Algorithm

Example

Varun Ganesan MSTs

DefinitionsCommon Algorithms

Kruskals Algorithm

Prims Algorithm

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ApplicationsPrim s Algorithm

Example

Varun Ganesan MSTs

DefinitionsCommon Algorithms

A li i

Kruskals Algorithm

Prims Algorithm

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ApplicationsPrim s Algorithm

Proof Of Correctness

Spanning Tree Validity

By avoiding connecting two already connected vertices,output has no cycles.

IfGis connected, output must be connected.Minimality

Consider a lesser total weight spanning tree with at leastone different edgee= (u, v).Ifeleads to less weight, thenewould have been

considered before some edge that connectsuandv in ouroutput.

Varun Ganesan MSTs

DefinitionsCommon Algorithms

A li ti

Kruskals Algorithm

Prims Algorithm

http://find/

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Applicationss go t

Proof Of Correctness

Spanning Tree Validity

By avoiding connecting two already connected vertices,output has no cycles.

IfGis connected, output must be connected.Minimality

Consider a lesser total weight spanning tree with at leastone different edgee= (u, v).Ifeleads to less weight, thenewould have been

considered before some edge that connectsuandv in ouroutput.

Varun Ganesan MSTs

DefinitionsCommon Algorithms

Applications

Kruskals Algorithm

Prims Algorithm

http://find/

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Applicationsg

Proof Of Correctness

Spanning Tree Validity

By avoiding connecting two already connected vertices,output has no cycles.

IfGis connected, output must be connected.Minimality

Consider a lesser total weight spanning tree with at leastone different edgee= (u, v).Ifeleads to less weight, thenewould have been

considered before some edge that connectsuandv in ouroutput.

Varun Ganesan MSTs

DefinitionsCommon Algorithms

Applications

Kruskals Algorithm

Prims Algorithm

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Applications

Proof Of Correctness

Spanning Tree Validity

By avoiding connecting two already connected vertices,output has no cycles.

IfGis connected, output must be connected.Minimality

Consider a lesser total weight spanning tree with at leastone different edgee= (u, v).Ifeleads to less weight, thenewould have been

considered before some edge that connectsuandv in ouroutput.

Varun Ganesan MSTs

DefinitionsCommon Algorithms

Applications

Kruskals Algorithm

Prims Algorithm

http://find/

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Applications

Proof Of Correctness

Spanning Tree Validity

By avoiding connecting two already connected vertices,output has no cycles.

IfGis connected, output must be connected.Minimality

Consider a lesser total weight spanning tree with at leastone different edgee= (u, v).Ifeleads to less weight, thenewould have been

considered before some edge that connectsuandv in ouroutput.

Varun Ganesan MSTs

DefinitionsCommon Algorithms

Applications

Kruskals Algorithm

Prims Algorithm

http://find/

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Applications

Proof Of Correctness

Spanning Tree Validity

By avoiding connecting two already connected vertices,output has no cycles.

IfGis connected, output must be connected.Minimality

Consider a lesser total weight spanning tree with at leastone different edgee= (u, v).Ifeleads to less weight, thenewould have been

considered before some edge that connectsuandv in ouroutput.

Varun Ganesan MSTs

DefinitionsCommon Algorithms

Applications

Kruskals Algorithm

Prims Algorithm

http://find/

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Applications

Outline

1 Definitions

Graph Terminology

Minimum Spanning Trees

2 Common Algorithms

Kruskals Algorithm

Prims Algorithm

3 Applications

Varun Ganesan MSTs

DefinitionsCommon Algorithms

Applications

Kruskals Algorithm

Prims Algorithm

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pp

Main Idea

Start withone(any) vertex.

Branch outwards to grow your connected component.

Consider only edges that leave the connected component.

Add smallest considered edge to your connected

component.

Continue until a spanning tree is created.

Varun Ganesan MSTs

DefinitionsCommon Algorithms

Applications

Kruskals Algorithm

Prims Algorithm

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pp

Main Idea

Start withone(any) vertex.

Branch outwards to grow your connected component.

Consider only edges that leave the connected component.

Add smallest considered edge to your connected

component.

Continue until a spanning tree is created.

Varun Ganesan MSTs

DefinitionsCommon Algorithms

Applications

Kruskals Algorithm

Prims Algorithm

http://find/

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Main Idea

Start withone(any) vertex.

Branch outwards to grow your connected component.

Consider only edges that leave the connected component.

Add smallest considered edge to your connected

component.

Continue until a spanning tree is created.

Varun Ganesan MSTs

DefinitionsCommon Algorithms

Applications

Kruskals Algorithm

Prims Algorithm

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Main Idea

Start withone(any) vertex.

Branch outwards to grow your connected component.

Consider only edges that leave the connected component.

Add smallest considered edge to your connected

component.

Continue until a spanning tree is created.

Varun Ganesan MSTs

DefinitionsCommon Algorithms

Applications

Kruskals Algorithm

Prims Algorithm

http://find/

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Main Idea

Start withone(any) vertex.

Branch outwards to grow your connected component.

Consider only edges that leave the connected component.

Add smallest considered edge to your connected

component.

Continue until a spanning tree is created.

Varun Ganesan MSTs

DefinitionsCommon Algorithms

Applications

Kruskals Algorithm

Prims Algorithm

http://find/

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Example

Varun Ganesan MSTs

DefinitionsCommon Algorithms

Applications

Kruskals Algorithm

Prims Algorithm

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Proof of Correctness

Both the spanning tree and minimality argument are nearlyidentical for Prims as they are for Kruskals.

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