Minimum Spanning Trees

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1 Minimum Spanning Trees Algorithms and Presentation

2 Outline 1 Graph Terminology Minimum Spanning Trees 2 3

3 Outline Graph Terminology Minimum Spanning Trees 1 Graph Terminology Minimum Spanning Trees 2 3

4 Graphs Graph Terminology Minimum Spanning Trees In graph theory, a graph is an ordered pair G = (V, E) comprising a set of vertices or nodes together with a set of edges. Edges are 2-element subsets of V which represent a connection between two vertices. Edges can either be directed or undirected. Edges can also have a weight attribute.

5 Graphs Graph Terminology Minimum Spanning Trees In graph theory, a graph is an ordered pair G = (V, E) comprising a set of vertices or nodes together with a set of edges. Edges are 2-element subsets of V which represent a connection between two vertices. Edges can either be directed or undirected. Edges can also have a weight attribute.

6 Graphs Graph Terminology Minimum Spanning Trees In graph theory, a graph is an ordered pair G = (V, E) comprising a set of vertices or nodes together with a set of edges. Edges are 2-element subsets of V which represent a connection between two vertices. Edges can either be directed or undirected. Edges can also have a weight attribute.

7 Graphs Graph Terminology Minimum Spanning Trees In graph theory, a graph is an ordered pair G = (V, E) comprising a set of vertices or nodes together with a set of edges. Edges are 2-element subsets of V which represent a connection between two vertices. Edges can either be directed or undirected. Edges can also have a weight attribute.

8 Connectivity Graph Terminology Minimum Spanning Trees A graph is connected when there is a path between every pair of vertices. A cycle is a path that starts and ends with the same vertex. A tree is a connected, acyclic graph.

9 Connectivity Graph Terminology Minimum Spanning Trees A graph is connected when there is a path between every pair of vertices. A cycle is a path that starts and ends with the same vertex. A tree is a connected, acyclic graph.

10 Connectivity Graph Terminology Minimum Spanning Trees A graph is connected when there is a path between every pair of vertices. A cycle is a path that starts and ends with the same vertex. A tree is a connected, acyclic graph.

11 Outline Graph Terminology Minimum Spanning Trees 1 Graph Terminology Minimum Spanning Trees 2 3

12 Spanning Trees Graph Terminology Minimum Spanning Trees Formally, for a graph G = (V, E), the spanning tree is E E such that: u V : (u, v) E (v, u) E v V In other words: the subset of edges spans all vertices. E = V 1 In other words: the number of edges is one less than the number of vertices, so that there are no cycles.

13 Spanning Trees Graph Terminology Minimum Spanning Trees Formally, for a graph G = (V, E), the spanning tree is E E such that: u V : (u, v) E (v, u) E v V In other words: the subset of edges spans all vertices. E = V 1 In other words: the number of edges is one less than the number of vertices, so that there are no cycles.

14 Spanning Trees Graph Terminology Minimum Spanning Trees Formally, for a graph G = (V, E), the spanning tree is E E such that: u V : (u, v) E (v, u) E v V In other words: the subset of edges spans all vertices. Linear Graph E = V 1 In other words: the number of edges is one less than the number of vertices, so that there are no cycles.

15 Graph Terminology Minimum Spanning Trees What Makes A Spanning Tree The Minimum? MST Criterion: When the sum of the edge weights in a spanning tree is the minimum over all spanning trees of a graph Figure: Suppose (b, f ) is removed and (c, f ) is added...

16 Outline 1 Graph Terminology Minimum Spanning Trees 2 3

17 Main Idea Start with V disjoint components. Consider lesser weight edges first to incrementally connect components. Make certain to avoid cycles. Continue until spanning tree is created.

18 Main Idea Start with V disjoint components. Consider lesser weight edges first to incrementally connect components. Make certain to avoid cycles. Continue until spanning tree is created.

19 Main Idea Start with V disjoint components. Consider lesser weight edges first to incrementally connect components. Make certain to avoid cycles. Continue until spanning tree is created.

20 Main Idea Start with V disjoint components. Consider lesser weight edges first to incrementally connect components. Make certain to avoid cycles. Continue until spanning tree is created.

21 Pseudocode 1 A = 0 2 foreach v G.V : 3 MAKE-SET(v) 4 foreach (u, v) ordered by weight(u, v), increasing: 5 if FIND-SET (u) FIND-SET (v): 6 A = A (u, v) 7 UNION(u, v) 8 return A

22 Example

23 Example

24 Example

25 Example

26 Example

27 Example

28 Proof Of Correctness Spanning Tree Validity By avoiding connecting two already connected vertices, output has no cycles. If G is connected, output must be connected. Minimality Consider a lesser total weight spanning tree with at least one different edge e = (u, v). If e leads to less weight, then e would have been considered before some edge that connects u and v in our output.

29 Proof Of Correctness Spanning Tree Validity By avoiding connecting two already connected vertices, output has no cycles. If G is connected, output must be connected. Minimality Consider a lesser total weight spanning tree with at least one different edge e = (u, v). If e leads to less weight, then e would have been considered before some edge that connects u and v in our output.

30 Proof Of Correctness Spanning Tree Validity By avoiding connecting two already connected vertices, output has no cycles. If G is connected, output must be connected. Minimality Consider a lesser total weight spanning tree with at least one different edge e = (u, v). If e leads to less weight, then e would have been considered before some edge that connects u and v in our output.

31 Proof Of Correctness Spanning Tree Validity By avoiding connecting two already connected vertices, output has no cycles. If G is connected, output must be connected. Minimality Consider a lesser total weight spanning tree with at least one different edge e = (u, v). If e leads to less weight, then e would have been considered before some edge that connects u and v in our output.

32 Proof Of Correctness Spanning Tree Validity By avoiding connecting two already connected vertices, output has no cycles. If G is connected, output must be connected. Minimality Consider a lesser total weight spanning tree with at least one different edge e = (u, v). If e leads to less weight, then e would have been considered before some edge that connects u and v in our output.

33 Proof Of Correctness Spanning Tree Validity By avoiding connecting two already connected vertices, output has no cycles. If G is connected, output must be connected. Minimality Consider a lesser total weight spanning tree with at least one different edge e = (u, v). If e leads to less weight, then e would have been considered before some edge that connects u and v in our output.

34 Outline 1 Graph Terminology Minimum Spanning Trees 2 3

35 Main Idea Start with one (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.

36 Main Idea Start with one (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.

37 Main Idea Start with one (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.

38 Main Idea Start with one (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.

39 Main Idea Start with one (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.

40 Example

41 Proof of Correctness Both the spanning tree and minimality argument are nearly identical for Prim s as they are for Kruskal s.

42 Some Taxonomy Clustering Analysis Traveling Salesman Problem Approximation

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