L feb abhi shelat

Transcription

1 L feb abhi shelat

2 General-MST-Strategy(G =(V, E)) 1 A 2 repeat V 1 times: 3 Pick a cut (S, V S) that respects A 4 Let e be min-weight edge over cut (S, V S) 5 A A {e}

3 prim General-MST-Strategy(G =(V, E)) 1 A 2 repeat V 1 times: 3 Pick a cut (S, V S) that respects A 4 Let e be min-weight edge over cut (S, V S) 5 A A {e} A is a subtree edge e is lightest edge that grows the subtree

4 prim b 8 d 8 g a 9 e 5 9 i c 1 f 6 h

5 prim e d f c b a i g h 8 7 e d f c b a i g h 8 7 e d f c b a i g h 8 7 e d f c b a i g h 8 7 e d f c b a i g h 8 7 e d f c b a i g h 8 7 e d f c b a i g h 8 7 e d f c b a i g h 8 7 e d f c b a i g h 8 7

6 implementation idea: maintain the set A set key of all nodes not adjacent to A to be

7 implementation idea: use a priority queue to keep track of light edges makequeue: insert: extractmin: decreasekey:

8 implementation prim(g =(V, E)) 1 Q Q is a Priority Queue 2 Initialize each v V with key k v, π v nil 3 Pick a starting node r and set k r 0 4 Insert all nodes into Q with key k v. 5 while Q 6 do u extract-min(q) 7 for each v Adj (u) 8 do if v Q and w(u, v) <k v 9 then π v u 10 decrease-key(q, v, w(u, v)) Sets k v w(u, v) A = {(v, π v ) : v V {r} Q}

9 prim b 8 8 d 0 8 g a 9 e i prim(g =(V, E)) c 1 Q Q is a Priority Queue 2 Initialize each v V with key k v, π v nil 3 Pick a starting node r and set k r 0 4 Insert all nodes into Q with key k v. 5 while Q 6 do u extract-min(q) 7 for each v Adj (u) 8 do if v Q and w(u, v) <k v 9 then π v u 10 decrease-key(q, v, w(u, v)) Sets k v w(u, v) 1 f 6 6 h 5

10 prim b 8 8 d 0 8 g a 9 e i prim(g =(V, E)) c 1 Q Q is a Priority Queue 2 Initialize each v V with key k v, π v nil 3 Pick a starting node r and set k r 0 4 Insert all nodes into Q with key k v. 5 while Q 6 do u extract-min(q) 7 for each v Adj (u) 8 do if v Q and w(u, v) <k v 9 then π v u 10 decrease-key(q, v, w(u, v)) Sets k v w(u, v) 1 f 6 6 h 5

11 implementation prim(g =(V, E)) 1 Q Q is a Priority Queue 2 Initialize each v V with key k v, π v nil 3 Pick a starting node r and set k r 0 4 Insert all nodes into Q with key k v. 5 while Q 6 do u extract-min(q) 7 for each v Adj (u) 8 do if v Q and w(u, v) <k v 9 then π v u 10 decrease-key(q, v, w(u, v)) Sets k v w(u, v)

12 implementation prim(g =(V, E)) 1 Q Q is a Priority Queue 2 Initialize each v V with key k v, π v nil 3 Pick a starting node r and set k r 0 4 Insert all nodes into Q with key k v. 5 while Q 6 do u extract-min(q) 7 for each v Adj (u) 8 do if v Q and w(u, v) <k v 9 then π v u 10 decrease-key(q, v, w(u, v)) Sets k v w(u, v) O(V log V + E log V) =O(E log V)

13 faster implementation prim(g =(V, E)) 1 Q Q is a Priority Queue 2 Initialize each v V with key k v, π v nil 3 Pick a starting node r and set k r 0 4 Insert all nodes into Q with key k v. 5 while Q 6 do u extract-min(q) 7 for each v Adj (u) 8 do if v Q and w(u, v) <k v 9 then π v u 10 decrease-key(q, v, w(u, v)) Sets k v w(u, v) O(E + V log V)

14 fibonacci heaps insert: findmin: extractmin: decreasekey:

15 amortized analysis

16 binary counter

17 binary counter Increment(A) 1 i 0 2 while i < len(a) and A i =1 3 do A i 0 4 i i +1 5 if i < len(a) 6 then A i 1

18 binary counter better analysis Increment(A) 1 i 0 2 while i < len(a) and A i =1 3 do A i 0 4 i i +1 5 if i < len(a) 6 then A i 1

19

20 potential method data structure A

21 potential method data structure A ĉ i = c i + Φ(A i ) Φ(A i 1 )

22 image: edupics.com

23 amortized cost of n ops: n i=1 ĉ i =

24 binary counter let let! b i " t i be the... be the... Φ(A i )= ĉ i =

25 heaps

26 binomial heaps B 4 B 4 B 5

27 fibonacci heap min picture courtesy of jeff erickson

28 each node has 4 pointers 2 fields: degree marked D(n)

29 fibonacci heap min

30 fibheap potential min Φ(H) =

31 key idea: be lazy put off the work until last minute

32 create new heap

33 insert node min

34 findmin min

35 extractmin min two steps: split consolidate

36 extractmin example min[h]

37 min[h] Q Q@ A~

38 A// ),x4 -- I

39 A 30 e,' AN

40 A/ We) X 0-- xw30@ R

41 x *-I w A// / : 38 2 :32

42 A I

43 A-- l 10ltf0l

44 ) lb 38 41

45 min[h]

46 analysis min cost 1. extract 2. add children 3. consolidate potential before: potential after: amortized cost:

47 decrease key operation what changes in the heap: A 30 e,' AN

48 decrease key decreasekey(v) promote node v to the root list update min of heap if necessary if 2 children of a node have been promoted, promote parent as well (recursively)

49 min[h] W~tgW decreasekey(46, 15)

50 min[hm decreasekey(35,5)

51 -- - 4

52 min[h]

53 min[h] V

54 analysis decreasekey(v) promote node v to the root list update min of heap if necessary if 2 children of a node have been promoted, promote parent as well (recursively) actual cost: potential before: potential after:

55 finally prove: D(n) = log φ n size(x) =

56 lemma: for node x, consider children (degrees)

57 lemma: F k+2 = 1 + k i=0 F i

58 lemma: size(x) F deg(x)+2 s i :

59 thm: D(n) log φ n