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Analysis of Algorithms. Directed Minimum Spanning Trees. Uri Zwick February 2014. Directed minimum spanning trees. 11. 16. 22. 17. 5. 8. 1. 13. 3. 18. 30. 12. 25. 9. 2. 15. Directed minimum spanning trees. 11. 16. 22. 17. 5. 8. 1. 13. 3. 18. 30. 12. 25. 9. 2.

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analysis of algorithms

Analysis of Algorithms

Directed Minimum Spanning Trees

Uri Zwick

February 2014

slide2

Directed minimum spanning trees

11

16

22

17

5

8

1

13

3

18

30

12

25

9

2

15

slide3

Directed minimum spanning trees

11

16

22

17

5

8

1

13

3

18

30

12

25

9

2

15

slide4

Directed minimum spanning trees

11

16

22

17

5

8

1

13

3

18

30

12

25

9

2

15

slide5

Directed spanning trees

Lemma:The following conditions are equivalent:

(i) T is a directed spanning tree of G rooted at r.

(ii) The indegree of r in T is 0, the indegree of every other vertex of G in T is 1, and T is acyclic,

i.e., contains no directed cycles.

(iii) The indegree of r in T is 0, the indegree of every other vertex of G in T is 1, and there are

directed paths in T from r to all other vertices.

slide7

Cheapest entering edges

Select the cheapest edge entering each vertex, other than the specified root r.

If, magically, we get a directed spanning tree rooted at r, then it is of course an optimal tree

slide8

Cheapest entering edges

Select the cheapest edge entering each vertex, other than the specified root r.

We may, of course, get cycles…

slide9

Cheapest entering edges

Lemma: If C is a cycle composed of cheapest entering edges, then there is a MDST T that contains all edges of C, except for one.

slide10

Cheapest entering edges

Lemma: If C is a cycle composed of cheapest entering edges, then there is a MDST T that contains all edges of C, except for one.