MST Project

8/8/2019 MST Project

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Minimum SpanningMinimum Spanning

TreesTrees

CS 146CS 146

Prof. SinProf. Sin--Min LeeMin Lee

Regina WangRegina Wang


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Spanning Tree properties:Spanning Tree properties:

On a connected graph G=(V, E), aspanning tree:

is a connected subgraph

acyclic is a tree (|E| = |V| - 1)

contains all vertices of G

(spanning)


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Spanning treesSpanning trees

Suppose you have a connected undirected graph

Connected: every node is reachable from every other node

Undirected: edges do not have an associated direction

...then a spanning tree of the graph is a connected subgraph in

which there are no cycles Total of 16 different spanning trees for the graph below

A connected,

undirected graph

Four of the spanning trees of the graph


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Finding a spanning treeFinding a spanning tree

BFS

DFS

Greedy algorithm

An undirected graph Result of a BFS

starting from top

Result of a DFS

starting from top


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Minimum weight spanning tree (MST)Minimum weight spanning tree (MST)

On a weighted graph, A MST:

connects all vertices through edges withleast weights.

has w(T) w(T) for every other spanningtree T in G

Adding one edge not in MST will create acycle


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Finding Minimum Spanning TreeFinding Minimum Spanning Tree

Two greedy algorithms

Kruskals and Prims

make best decision at every stage

locally optimal choice globally optimal

solution.

Greedy algorithms do not always yields

optimal solutions.


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Kruskals algorithmKruskals algorithm

EdgeEdge firstfirst

1. Arrange all edges in a list (L)1. Arrange all edges in a list (L)

in nonin non--decreasing orderdecreasing order

2.2. Select edges from L, and includeSelect edges from L, and includethat in set T, avoid cycle.that in set T, avoid cycle.

3.3. Repeat 3 until T becomes a treeRepeat 3 until T becomes a tree

that covers all verticesthat covers all vertices


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KruskalKruskals Algorithms Algorithm

{{1,2}1,2} 1122

{{3,4}3,4} 1212

{{1,8}1,8} 1313

{{4,5}4,5} 1313{{2,7}2,7} 1414

{{3,6}3,6} 1414

{{7,8}7,8} 1414

{{5,6}5,6} 1414

{5,8}{5,8} 1515

{6,7}{6,7} 1515

{1,4}{1,4} 1616

{2,3}{2,3} 1616

8 5

1 4

2 3

7 6

14

15

15

14

13 13

16

16

1212

1414


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{{1,2}1,2} 1122

{{3,4}3,4} 1212

{{1,8}1,8} 1313

{{4,5}4,5} 1313{{2,7}2,7} 1414

{{3,6}3,6} 1414

{{7,8}7,8} 1414

{{5,6}5,6} 1414

{5,8}{5,8} 1515

{6,7}{6,7} 1515

{1,4}{1,4} 1616

{2,3}{2,3} 1616

8 5

1 4

2 3

7 6

14

15

15

14

13 13

16

16

1212

1414


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{{1,2}1,2} 1122

{{3,4}3,4} 1212

{{1,8}1,8} 1313

{{4,5}4,5} 1313{{2,7}2,7} 1414

{{3,6}3,6} 1414

{{7,8}7,8} 1414

{{5,6}5,6} 1414

{5,8}{5,8} 1515

{6,7}{6,7} 1515

{1,4}{1,4} 1616

{2,3}{2,3} 1616

8 5

1 4

2 3

7 6

14

15

15

14

13 13

16

16

1212

1414


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{{1,2}1,2} 1122

{{3,4}3,4} 1212

{{1,8}1,8} 1313

{{4,5}4,5} 1313{{2,7}2,7} 1414

{{3,6}3,6} 1414

{{7,8}7,8} 1414

{{5,6}5,6} 1414

{5,8}{5,8} 1515

{6,7}{6,7} 1515

{1,4}{1,4} 1616

{2,3}{2,3} 1616

8 5

1 4

2 3

7 6

14

15

15

14

13 13

16

16

1212

1414


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{{1,2}1,2} 1122

{{3,4}3,4} 1212

{{1,8}1,8} 1313

{{4,5}4,5} 1313{{2,7}2,7} 1414

{{3,6}3,6} 1414

{{7,8}7,8} 1414

{{5,6}5,6} 1414

{5,8}{5,8} 1515

{6,7}{6,7} 1515

{1,4}{1,4} 1616

{2,3}{2,3} 1616

8 5

1 4

2 3

7 6

14

15

15

14

13 13

16

16

1212

1414


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{{1,2}1,2} 1122

{{3,4}3,4} 1212

{{1,8}1,8} 1313

{{4,5}4,5} 1313{{2,7}2,7} 1414

{{3,6}3,6} 1414

{{7,8}7,8} 1414

{{5,6}5,6} 1414

{5,8}{5,8} 1515

{6,7}{6,7} 1515

{1,4}{1,4} 1616

{2,3}{2,3} 1616

8 5

1 4

2 3

7 6

14

15

15

14

13 13

16

16

1212

1414


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{{1,2}1,2} 1122

{{3,4}3,4} 1212

{{1,8}1,8} 1313

{{4,5}4,5} 1313{{2,7}2,7} 1414

{{3,6}3,6} 1414

{{7,8}7,8} 1414

{{5,6}5,6} 1414

{5,8}{5,8} 1515

{6,7}{6,7} 1515

{1,4}{1,4} 1616

{2,3}{2,3} 1616

8 5

1 4

2 3

7 6

14

15

15

14

13 13

16

16

1212

1414


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{{1,2}1,2} 1122

{{3,4}3,4} 1212

{{1,8}1,8} 1313

{{4,5}4,5} 1313{{2,7}2,7} 1414

{{3,6}3,6} 1414

{{7,8}7,8} 1414

{{5,6}5,6} 1414

{5,8}{5,8} 1515

{6,7}{6,7} 1515

{1,4}{1,4} 1616

{2,3}{2,3} 1616

8 5

1 4

2 3

7 6

14

15

15

14

13 13

16

16

1212

1414

Skip

Skip {7,8} to avoid cycle


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{{1,2}1,2} 1122

{{3,4}3,4} 1212

{{1,8}1,8} 1313

{{4,5}4,5} 1313{{2,7}2,7} 1414

{{3,6}3,6} 1414

{{7,8}7,8} 1414

{{5,6}5,6} 1414

{5,8}{5,8} 1515

{6,7}{6,7} 1515

{1,4}{1,4} 1616

{2,3}{2,3} 1616

8 5

1 4

2 3

7 6

14

15

15

14

13 13

16

16

1212

1414

Skip

Skip {5,6} to avoid cycle

Skip


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{{1,2}1,2} 1122

{{3,4}3,4} 1212

{{1,8}1,8} 1313

{{4,5}4,5} 1313{{2,7}2,7} 1414

{{3,6}3,6} 1414

{{7,8}7,8} 1414

{{5,6}5,6} 1414

{5,8}{5,8} 1515

{6,7}{6,7} 1515

{1,4}{1,4} 1616

{2,3}{2,3} 1616

8 5

1 4

2 3

7 6

14

15

15

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13 13

16

16

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MST is formed


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Prims algorithmPrims algorithm

Start form any arbitraryStart form any arbitrary vertexvertex

Find the edge that has minimumFind the edge that has minimum

weight formweight form allall known verticesknown vertices

Stop when the tree covers allStop when the tree covers all

verticesvertices


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8 5

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2 3

7 6

14

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13 1316

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1

Start from any arbitrary vertex



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2 3

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13 1316

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1

2

Weight:12

The best

choice is {1,2}


12

Whats next?


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8 5

1 4

2 3

7 6

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1316

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1

2

Weight:12+13

After {1,8} we may

choose {2,7} or

{7,8}

There are more

than one MST


12

Whats next?

8

13


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8 5

1 4

2 3

7 6

14

15

15

14

1316

16

12

14

1

2

Lets choose{2, 7} at this

moment


12 We are free to

choose {5,8} or

{6,7} but not {7,8}

because we need

to avoid cycle

8

13

14

7


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8 5

1 4

2 3

7 6

14

15

14

1316

16

12

14

1

2

Lets choose

{5,8} at this

moment


12

The best choice

is now {4,5}

8

13

14

7

155


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8 5

1 4

2 3

7 6

14

15

14

16

16

12

14

1

2


12The best choice

is now {4,3}

8

13

14

7

155

13

4


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8 5

1 4

2 3

7 6

14

15

14

16

16

14

1

2


12The best choice

is now {3,6} or

{5,6}

8

13

14

7

155

13

4

12

3


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8 5

1 4

2 3

7 6

14

15

14

16

16

1

2


12A MST is formed

Weight = 93

8

13

14

7

155

13

4

12

3

6

14


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Prims algorithmPrims algorithm moremore

complexcomplex

0

5

20

54

3

4414

99

32

8100

11

1431

19


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Prims algorithmPrims algorithm moremore

complexcomplex

0

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3

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99

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0

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0

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0

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0

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0

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0

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0

5

20

54

3

4414

99

32

8100

11

1431

19

MST is formed


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Compare Prim and KruskalCompare Prim and Kruskal

Both have the same output MST

Kruskals begins with forest and merge into a tree

Prims always stays as a tree

If you dont know all the weight on edges usePrims algorithm

If you only need partial solution on the graph usePrims algorithm


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Compare Prim and KruskalCompare Prim and Kruskal

Complexity

Kruskal: O(NlogN)

comparison sort for edgesPrim: O(NlogN)

search the least weight edge for

every vertice


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Why do we need MST?Why do we need MST?

a reasonable way for clustering

points in space into natural

groups

can be used to give approximate

solutions to hard problems


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Minimizing costsMinimizing costs

Suppose you want to provide solution to :

electric power, Water, telephone lines,network setup

To minimize cost, you could connectlocations using MST

However, MST is not necessary theshortest path and it does not apply to cycle


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MST dont solve TSPMST dont solve TSP

Travel salesman problem (TSP) can not

be solved by MST :

salesman needs to go home

(whats the cost going home?) TSP is a cycle

Use MST to approximate

solve TSP by exhaustive approach

try every permutation on cyclic graph


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More practiceMore practice

Kruskal's AlgorithmKruskal's Algorithm appletapplet

Prims AlgorithmPrims Algorithm appletapplet


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The EndThe End

Thank youThank you

Documents

MST Project