Minimum Spanning Trees. COMPSCI 355 Fall 2016

Transcription

1 Minimum Spanning Trees COMPSCI all 06

2 Spanning Tree

3 Spanning Tree

4 Spanning Tree Algorithm A Add any edge that can be added without creating a cycle. Repeat until the edges form a spanning tree. Algorithm B Remove any edge that can be removed without disconnecting the graph. Repeat until the edges form a spanning tree.

5 epth-irst Search 0 4 6

6 epth-irst Search 0 4 6

7 epth-irst Search 0 4 6

8 epth-irst Search 0 4 6

9 epth-irst Search 0 4 6

10 epth-irst Search 0 4 6

11 epth-irst Search 0 4 6

12 epth-irst Search 0 4 6

13 Minimum Spanning Tree

14 or each edge in sorted order: if safe, add it.

15 or each edge in sorted order: if safe, add it.

16 or each edge in sorted order: if safe, add it.

17 or each edge in sorted order: if safe, add it.

18 or each edge in sorted order: if safe, add it.

19 6 4 0 or each edge in sorted order: if safe, add it.

20 6 4 0 or each edge in sorted order: if safe, add it.

21 4 0 or each edge in sorted order: if safe, add it.

22 4 0 or each edge in sorted order: if safe, add it.

23 4 0 or each edge in sorted order: if safe, add it.

24 4 0 or each edge in sorted order: if safe, add it.

25 4 0 or each edge in sorted order: if safe, add it.

26 4 0 or each edge in sorted order: if safe, add it.

27 4 0 or each edge in sorted order: if safe, add it.

28 4 or each edge in sorted order: if safe, add it.

29 Proof of Correctness 7 Given two connected components, consider all edges that join them. Only one of these edges can be part of a minimum spanning tree. Of these, an edge of lowest weight is part of a minimum spanning tree.

30 Complexity O(m log m) to sort the edges. O(mn) checking for cycles. But a more careful analysis shows that the algorithm runs in time O(m log n). To see how this works, we reinterpret the action of the algorithm...

31 We start with a forest of n trees.

32 or each edge in sorted order: if it joins two trees, add it.

33 or each edge in sorted order: if it joins two trees, add it.

34 or each edge in sorted order: if it joins two trees, add it.

35 or each edge in sorted order: if it joins two trees, add it.

36 6 4 0 or each edge in sorted order: if it joins two trees, add it.

37 6 4 0 or each edge in sorted order: if it joins two trees, add it.

38 4 0 or each edge in sorted order: if it joins two trees, add it.

39 4 0 or each edge in sorted order: if it joins two trees, add it.

40 4 0 or each edge in sorted order: if it joins two trees, add it.

41 or each edge in sorted order: if it joins two trees, add it.

42 Assuming no parallel edges or self-loops. dges can be sorted in O(m log m) time. Better: use a heap-based priority queue. O(m) time to build. O(log m) time for each extraction. Total: O(n log m), which in fact is O(n log n). What about the time to merge trees?

43 isjoint Set Also called Union-ind Operations: find: determine which set contains a given item. union: join two sets. The find operation returns a representative of the set to which an item belongs. By comparing the result of two find operations, we can check if two items are in the same set.

44 Using Linked Lists Head of each list used as representative. ach node in the graph stores a reference to its corresponding node in one of the disjoint sets.

45 Using Linked Lists find takes O() time. union takes O() time to merge the lists, but then the pointesr-to-head must be updated.

46 Using Linked Lists Merge smaller list onto the tail of the larger list. A node is moved to a larger set at most log(n) times, so total time merging sets is O(n log n).

47 Complexity Build queue and extract edges: O(m log n) Searching for set representatives: O(n) Merging sets: O(n log n) Total: O((m+n) log n) = O(m log n)

48 Complexity Trees can be used instead of linked lists. Root used as representative. Merging more complicated.

49 Complexity Kruskal s algorithm can be shown to run in O(m α(n)) time, where α(n) is the inverse of the single-valued Ackermann function. If n is the number of atoms in the visible universe, then α(n) = 4. ack(4, 4) 0 7

50 Theoretically Optimal It was shown in that Ω(α(n)) time needed on average for isjoint Set operations, no matter what data structures are used.