Binomial Heaps. Bryan M. Franklin

Transcription

1 Binomial Heaps Bryan M. Franklin 1

2 Tradeoffs Worst Case Operation Binary Heap Binomial Heap Make-Heap Θ(1) Θ(1) Insert Θ(lg n) O(lg n) Minimum Θ(1) O(lg n) Extract-Min Θ(lg n) Θ(lg n) Union Θ(n) Θ(lg n) Decrease-Key Θ(lg n) Θ(lg n) Delete Θ(lg n) Θ(lg n) 2

3 Binomial Trees Definition: B0: A binomial tree with a single node. Bk: Two Bk-1 binomial trees connected such that one tree has the other as its leftmost child. 3

4 Binomial Trees B0 B1 B2 B3 B4 4

5 Binomial Tree Properties Properties of binomial tree Bk: 2 k nodes Height of tree is k ( ) k Exactly nodes at depth i i Root has degree k Children of the root from left to right are binomial trees Bk-1, Bk-2, Bk-3,..., B0 5

6 Binomial Tree Properties k=4 2 4 =16 nodes B4 2 k Nodes 6

7 Binomial Tree Properties k=4 height is B4 Height of k 7

8 Binomial Tree Properties ( ) 4 0 ( ) 4 1 ( ) 4 2 ( ) 4 3 ( ) 4 4 =1 =4 =6 =4 =1 B4 Exactly ( ) k i nodes at depth i 8

9 Binomial Tree Properties 4 k=4 Root s degree is B0 Root s children: B1 B3, B2, B1, B0 0 B3 B2 Root has degree k 9

10 Representation Left-child, Right-sibling (ch. 10) Parent Right-sibling Left-child 10

11 Representation Example B4 11

12 Binomial Heap A binomial heap is a set, H, of binomial trees, such that: 1) Each tree in H satisfies the min-heap property. 2) The root of every tree in H has a unique degree. 12

13 Binomial Heaps head[h1] head[h2]

14 Heap Operations Make-Heap( ) Minimum(H) Union(H1, H2) Insert(H, x) Extract-Min(H) Decrease-Key(H, x, k) Delete(H, x) 14

15 Make-Binomial-Heap Make-Binomial-Heap() 1 H new Heap 2 head[h] NIL 3 return H 15

16 Binomial-Heap- Minimum Binomial-Heap-Minimum(H) 1 y NIL 2 x head[h] 3 min 4 while x NIL 5 do if key[x] < min 6 then min key[x] 7 y x 8 x sibling[x] 9 return y 16

17 Binomial-Heap- Minimum x y head[h1] NIL min= 17

18 Binomial-Heap- Minimum y x head[h1] NIL min=12 18

19 Binomial-Heap- Minimum y x head[h1] NIL min=7 19

20 Binomial-Heap- Minimum y x head[h1] NIL min=7 20

21 Binomial-Link Binomial-Link(y, z) 1 p[y] z 2 sibling[y] child[z] 3 child[z] y 4 degree[z] degree[z]+1 21

22 Binomial-Link y z 22

23 Binomial-Link p[y] y z 23

24 Binomial-Link sibling[y] y z 24

25 Binomial-Link child[z] y z 25

26 Binomial-Heap-Union Binomial-Heap-Union(H 1,H 2 ) 1 H Make-Binomial-Heap() 2 head[h] Binomial-Heap-Merge(H 1,H 2 ) 3 free H 1 and H 2, but not the lists they point to 4 if head[h] =NIL 5 then return H 6 prev-x NIL 7 x head[h] 8 next-x sibling[x] 9 while next-x NIL 10 do if degree[x] degree[next-x] or (sibling[next-x] NIL and degree[sibling[next-x]] = degree[x]) 11 then prev-x x 12 x next-x 13 else if key[x] key[next-x] 14 then sibling[x] sibling[next-x] 15 Binomial-Link(next-x,x) 16 else if prev-x = NIL 17 then head[h] next-x 18 else sibling[prev-x] next-x 19 Binomial-Link(x, next-x) 20 x next-x 21 next-x sibling[x] 22 return H 26

27 Binomial-Heap-Union head[h1] head[h2]

28 Binomial-Heap-Union x next-x head[h]

29 Binomial-Heap-Union x next-x head[h]

30 Binomial-Heap-Union prev-x x next-x head[h]

31 Binomial-Heap-Union prev-x x next-x head[h]

32 Binomial-Heap-Union prev-x head[h] 12 x 3 next-x

33 Binomial-Heap-Insert Binomial-Heap-Insert(H, x) 1 H Make-Binomial-Heap() 2 p[x] NIL 3 child[x] NIL 4 sibling[x] NIL 5 degree[x] 0 6 head[h ] x 7 H Binomial-Heap-Union(H, H ) 33

34 Extract-Min Binomial-Heap-Extract-Min(H) 1 find root in H with minimum key value, remove as x 2 H Make-Binomial-Heap() 3 fill heap H by linking x s children in reverse order 4 H Binomial-Heap-Union(H, H ) 5 return x 34

35 Extract-Min head[h]

36 Extract-Min x head[h]

37 Extract-Min x 1 head[h]

38 Extract-Min x 1 head[h] head[h ]

39 Extract-Min x 1 head[h]

40 Decrease-Key Binomial-Heap-Decrease-Key(H, x, k) 1 if k>key[x] 2 then error new key is larger than current key 3 key[x] k 4 y x 5 z p[y] 6 while z NIL and key[y] < key[z] 7 do exchange key[y] key[z] 8 exchange satellite data as well 9 y z 10 z p[y] 40

41 Decrease-Key head[h] x

42 Decrease-Key head[h] x

43 Decrease-Key head[h] z y

44 Decrease-Key head[h] z y

45 Decrease-Key head[h] z y

46 Decrease-Key head[h] z y

47 Decrease-Key head[h] z 6 y

48 Decrease-Key head[h] z 6 y

49 Delete Binomial-Heap-Delete(H, x) 1 Binomial-Heap-Decrease-Key(H, x, ) 2 Binomial-Heap-Extract-Min() 49

50 The End 50