Efficient Minimum Spanning Tree Algorithms: A Greedy Approach for Network Optimization

1. MINIMUMSPANNING TREE A NEW APPROACH

2. Types of network

3. THE INTERPRETATION

4. BACKUP PATHS b c a d f e

5. THE PROBLEM Occurring of loops

6. EXISTING SOLUTIONS Prim’s algorithm Kruskal algorithm Both of them take time O(n lg n)

7. A SPANNING TREE • All the routers are included • Cycles are eliminated • Each router has knowledge of • some Spanning Tree

8. THE TASK Generating a Spanning Tree having least cost Different MST algorithms are used

9. THE CHALLENGE DESIGNING A MST ALGORITHM HAVING LEAST COMPLEXITY

10. Prerequisites • The algorithm must not • exceed the order O(n lg n) • , desirable. • It must be practically • implementable

11. KEY IDEA • GREEDY APPROACH • At every stage make a • locally optimal choice • Leads to globally optimal • solution

12. IMPLEMENTATION • Each router maintains 2 • data structures • A Priority Queue for all the • routers connected to it • A Disjoint Set for maintaining • set of connected routers

13. ON A GRAPH • Each vertex maintains a Min- • Priority Queue, based on a • key field • A Disjoint-Set data structure • is maintained

14. The algorithm - MBA The steps S<- NULL FOR EACH VERTEX, vV[G] MAINTAIN A MINIMUM PRIORITY QUEUE VG MAKE SET(v)

15. 5. FOR EACH VERTEX V V[G] U <- EXTRACT-MIN (QV) IF(u,v) S S<- S U {(u,v)} FOR EACH EDGE (u,v) E(S) IF FIND-SET(u) ≠ FIND-SET(v) UNION(u,v)

16. 12. FOR EACH EDGE, (u,v) E(G) IF FIND-SET(u)= FIND-SET(v) w(u,v)=∞ FOR EACH VERTEX, v V[G] u<- EXTRACT-MIN( QV) IF (u,v) S S<- S U {(u,v)}

17. IF FIND-SET (u)≠ FIND-SET (v) UNION (u,v) ELSE S<- MAX{(x,y)}, (xy) SET(u) RETURN S

18. WORKING 9 7 c b d 4 9 2 14 e a 11 4 i 7 6 10 8 g f h 1 2

19. 9 7 c b d 4 9 2 14 e a 11 4 i 7 6 10 8 g f h 1 2

20. 9 7 c b d 4 9 2 14 e a 11 4 i 7 6 10 8 g f h 1 2

21. 9 7 c b d 4 9 2 14 e a 11 4 i 7 6 10 8 g f h 1 2

22. 9 7 c b d 4 9 2 14 e a 11 4 i 7 6 10 8 g f h 1 2

23. 9 7 c b d 4 9 2 14 e a 11 4 i 7 6 10 8 g f h 1 2

24. 9 7 c b d 4 9 2 14 e a 11 4 i 7 6 10 8 g f h 1 2

25. 9 7 c b d 4 9 2 14 e a 11 4 i 7 6 10 8 g f h 1 2

26. 9 7 c b d 4 9 2 14 e a 11 4 i 7 6 10 8 g f h 1 2

27. 9 7 c b d 4 9 2 14 e a 11 4 i 7 6 10 8 g f h 1 2

28. 9 7 c b d 4 9 2 14 e a 11 4 i 7 6 10 8 g f h 1 2

29. 9 7 ∞ c b d 4 9 2 ∞ ∞ ∞ 14 e a 11 4 i 7 6 10 8 g f h 1 2 ∞ ∞

30. 9 7 c b d ∞ 4 9 2 ∞ ∞ ∞ 14 e a 11 4 i 7 6 10 8 g f h 1 2 ∞ ∞

31. 9 7 c b d ∞ 4 9 2 ∞ ∞ ∞ 14 e a 11 4 i 7 6 10 8 g f h 1 2 ∞ ∞

32. 9 7 c b d ∞ 4 9 2 ∞ ∞ 14 e a 11 4 i 7 6 10 8 g f h 1 2

33. 9 7 c b d ∞ 4 9 2 ∞ ∞ ∞ 14 e a 11 4 i 7 6 10 8 g f h 1 2 ∞ ∞

34. 9 7 c b d ∞ 4 9 2 ∞ ∞ ∞ 14 e a 11 4 i 7 6 8 10 g f h 1 2 ∞ ∞

35. 9 7 c b d ∞ 4 9 2 ∞ ∞ ∞ 14 e a 11 4 i 7 6 8 10 g f h 1 2 ∞ ∞

36. 9 7 c b d ∞ 4 9 2 ∞ ∞ ∞ 14 e a 11 i 4 7 6 8 10 g f h 1 2 ∞ ∞

37. 9 7 c b d ∞ 4 9 2 ∞ ∞ ∞ 14 e a 11 4 i 7 6 8 10 g f h 1 2 ∞ ∞

38. 9 7 c b d ∞ 4 9 2 ∞ ∞ ∞ 14 e a 11 4 i 7 6 8 10 g f h 1 2 ∞ ∞

39. ANALYSIS • Running time depends upon • Implementation of Disjoint Set Data Structure • Implementation of Priority • Queue

40. ANALYSIS • Disjoint-Set uses : • Path Compression • Union by Rank • Priority Queue : • Min-Priority Queue

41. ANALYSIS • Time taken at each step : • Line 1 : O(1) • Lines 2 & 5 : O(V) • Make Set operation : O(V) • Build Heap : O(log2V) • For ‘V’ vertices : O(V log2V)

42. Extract – min : O(log2V) • For all vertices : O(V log2V) • for loops in lines 9-11 & 12-14 : • O(E)

43. Order till line 14 : O ( V log2V )

44. Same process is repeated once more Overall complexity : O ( V log2V )

45. NEWER DIMENSIONS WHEN USED WITH PARALLEL PROCESSORS O(E) CAN BE ACHIEVED

46. BACKUP PATHS CAN BE FOUND FASTER CYCLES CAN BE ELIMINATED MORE EFFICIENTLY