1. CS200: Algorithm Analysis

2. PRIM’s ALGORITHM
Greedy algorithm. A is the MST being constructed. Thus A is always a Tree.
Starts from an arbitrary root.
At each step find a light edge crossing the cut (Va, V- Va) where Va = vertices that A is incident on. Add this edge to A. (see image, next slide)
How to find the light edge?
Idea: Keep V - A in a priority queue, sorted by weight of min-weight edge connecting to A.

4. Prim’s Algorithm

5. Trace algorithm for example graph.

6. Trace algorithm for example graph.

7. Trace algorithm for example graph.

8. Trace algorithm for example graph.

9. Trace algorithm for example graph.

10. Trace algorithm for example graph.

11. Trace algorithm for example graph.

12. Trace algorithm for example graph.

13. Trace algorithm for example graph.

14. Trace algorithm for example graph.

15. Trace algorithm for example graph.

16. Trace algorithm for example graph.

17. Trace algorithm for example graph.

18. Analysis of Prim’s

19. Analysis of Prim’s

20. Analysis of Prim’s

21. Analysis of Prim’s

22. Analysis of Prim’s

23. Analysis of Prim’s

24. Analysis of Prim’s

25. Analysis of Prim’s

26. Analysis of Prim’s

27. Analysis of Prim’s

28. Analysis of Prim’s

29. 1. Array :(unsorted)-> scan to find min, just index and update to change keys.
2. Binary heap: requires reheapify on ExtractMin and DecreaseKey.

30. KRUSKAL’S ALGORITHM Also a greedy algorithm.
Idea: uses the notion of Disjoint Set Union (used for Lab4)
Disjoint Set: S is a disjoint set = {S1,S2,...,Sn} st Si  Sj = . S is a set of sets, each member set is disjoint.
Each set can be viewed as an equivalence class - where all members are equivalent and are represented by one “unique” member.

31. Operations on a Disjoint Set:
MakeSet(x) S <– S  {{x}}; {x} is not in any set Si of S. x is called the representative of the set.
Union(Si, Sj) S<– S – {Si, Sj}  {Si  Sj}; make new representative from Si and Sj.
FindSet(x) returns Si st x in Si and Si in S.
These operations are used in Kruskal’s algorithm.

32. MSTKruskal(G,W) //uses the Union-Find algorithm
T = empty set //T will be MST when done
for each v in V do
MakeSet(v) //put each vertex in its own equivalence class // make v the class representative
sort edges in W by increasing edge weight w
for each edge (u,v) in W do //in sorted order
if FindSet(u) != FindSet(v) then
T = T union {(u,v)}
Union( FindSet(u),FindSet(v))
return T

33. Trace algorithm on example in html notes.
The above disjoint set operations take O(E* a(V,E)) where a is very slow growing (using best known algorithm for disjoint set union–> SS 21.3).
Overall runtime is O(ElgE) which is almost linear if edges are already sorted.

34. Implementations of a Disjoint Set:
Linked List => lots of traversing

35. Implementations of a Disjoint Set:
Augmented Linked List: weight is used to link smaller list into larger one for Union operation, each node points back to head so it is faster to locate the set representative for Find-Set.

36. Implementations of a Disjoint Set:
Forest of Trees:

37. Implementations of a Disjoint Set:
Forest of Trees: Union by Rank, smaller tree joined into larger tree but this can lead to unbalanced trees.

38. Implementations of a Disjoint Set:
Forest of Trees: Balancing the Trees, the Find-Set only needs to traverse one edge in a flat tree.

39. Implementations of a Disjoint Set:
Forest of Trees: Balancing the Trees using Path Compression

40. Implementations of a Disjoint Set:
The punch-line of this discussion is that, taken together, union by rank and path compression produce a spectacularly efficient implementation of the disjoint-set data structure.
Theorem On a disjoint-set forest with union by rank and path compression, any sequence of m operations, n of which are MAKE-SET operations, has worst-case running time
Θ 􏰞mα(n)􏰟 ,
where α is the inverse Ackermann function. Thus, the amortized worst- case running time of each operation is Θ(α(n)). If one makes the approximation α(n) = O(1), which is valid for literally all conceivable purposes, then the operations on a disjoint-set forest have O(1) amortized running time.
Because the Ackermann function is an extremely rapidly growing function, the inverse Ackermann function α is an extremely slow growing function (though it is true that lim n→∞ α(n) = ∞).

41. Ackermann’s Function

42. Inverse Ackermann’s Function