COS 423 Theory of Algorithms • Kevin Wayne • Spring 2007

Adapted by Cheng Li and Virgil Pavlu

Fibonacci Heaps

Lecture slides adapted from:

•  Chapter 20 of Introduction to Algorithms by Cormen, Leiserson, Rivest, and Stein.

•  Chapter 9 of The Design and Analysis of Algorithms by Dexter Koze

Fibonacci Heaps

History. [Fredman and Tarjan, 1986]   Ingenious data structure and analysis.

  Original motivation: improve Dijkstra's shortest path algorithm

(module 12) from to

Basic idea.

  Similar to binomial heaps, but less rigid structure.

  Binomial heap: eagerly consolidate trees after each insert.

  Fibonacci heap: lazily defer consolidation until next extract-min.

2

V insert, V extract-min, E decrease-key

Fibonacci Heaps: Structure

Fibonacci heap.   Set of heap-ordered trees.

  Maintain pointer to minimum element.

  Set of marked nodes.

3

7 23

30

17

35

26 46

24

Heap H 39

41 18 52

3

44

roots heap-ordered tree

each parent smaller than its children

Fibonacci Heaps: Structure

Fibonacci heap.   Set of heap-ordered trees.

  Maintain pointer to minimum element.

  Set of marked nodes.

4

7 23

30

17

35

26 46

24

Heap H 39

41 18 52

3

44

min

find-min takes O(1) time

Fibonacci Heaps: Structure

Fibonacci heap.   Set of heap-ordered trees.

  Maintain pointer to minimum element.

  Set of marked nodes.

5

7 23

30

17

35

26 46

24

Heap H 39

41 18 52

3

44

min

use to keep heaps flat (stay tuned)

marked

Fibonacci Heaps: Notation

Notation.   = number of nodes in heap.

  degree(x) = number of children of node x.

  = upper bound on the maximum degree of any node.

In fact, . The proof (omitted) uses Fibonacci

numbers.

  = number of trees in heap H.

  = number of marked nodes in heap H.

6

7 23

30

17

35

26 46

24

39

41 18 52

3

44

degree = 3 min

Heap H

t(H) = 5 m(H) = 3

marked

n = 14

Fibonacci Heaps: Potential Function

7

7 23

30

17

35

26 46

24

Φ(H) = 5 + 2⋅3 = 11

39

41 18 52

3

44

min

Heap H

potential of heap H :

t(H) = 5 m(H) = 3

marked

This lecture is not a complete treatment of Fibonacci heaps; in order to implement (code) and use them, more details are necessary (see book). Our main purpose here is to understand how the potential function works. Next: analyze change in potential and amortized costs for heaps operations: •  Insert (easy, required) •  Extract min (medium, required) •  Decrease Key (difficult, optional) •  Union (easy, required) •  Delete (medium, required)

8

Insert

Fibonacci Heaps: Insert

Insert.   Create a new singleton tree.

  Add to root list; update min pointer (if necessary).

9

7 23

30

17

35

26 46

24

39

41 18 52

3

44

21

insert 21

min

Heap H

Fibonacci Heaps: Insert

Insert.   Create a new singleton tree.

  Add to root list; update min pointer (if necessary).

10

39

41

7 23

18 52

3

30

17

35

26 46

24

44

21

min

Heap H

insert 21

Fibonacci Heaps: Insert Analysis

Actual cost.   H’ = the heap after insert

Change in potential.

Amortized cost.

11

39

41

7

18 52

3

30

17

35

26 46

24

44

21 23

min

Heap H

potential of heap H

12

Extract-Min

Linking Operation

Linking operation. Make larger root be a child of smaller root.

13

39

41 18 52

3

44 77

56 24

15

tree T1 tree T2

39

41 18 52

3

44

77

56 24

15

tree T'

smaller root larger root still heap-ordered

Fibonacci Heaps: Extract-Min

Extract-min.   Delete min; meld its children into root list; update min.

  Consolidate trees so that no two roots have same degree.

14

39

41 18 52

3

44

17 23

30

7

35

26 46

24

min

Fibonacci Heaps: Extract-Min

Extract-Min.   Delete min; meld its children into root list; update min.

  Consolidate trees so that no two roots have same degree.

15

39

41 17 23 18 52

30

7

35

26 46

24

44

min

Fibonacci Heaps: Extract-Min

Extract-Min.   Delete min; meld its children into root list; update min.

  Consolidate trees so that no two roots have same degree.

16

39

41 17 23 18 52

30

7

35

26 46

24

44

min current

Fibonacci Heaps: Extract-Min

Extract-Min.   Delete min; meld its children into root list; update min.

  Consolidate trees so that no two roots have same degree.

17

39

41 17 23 18 52

30

7

35

26 46

24

44

0 1 2 3

current min

degree

Fibonacci Heaps: Extract-Min

Extract-Min.   Delete min; meld its children into root list; update min.

  Consolidate trees so that no two roots have same degree.

18

39

41 17 23 18 52

30

7

35

26 46

24

44

0 1 2 3

min current

degree

Fibonacci Heaps: Extract-Min

Extract-Min.   Delete min; meld its children into root list; update min.

  Consolidate trees so that no two roots have same degree.

19

39

41 17 23 18 52

30

7

35

26 46

24

44

0 1 2 3

min

current

degree

Fibonacci Heaps: Extract-Min

Extract-Min.   Delete min; meld its children into root list; update min.

  Consolidate trees so that no two roots have same degree.

20

39

41 17 23 18 52

30

7

35

26 46

24

44

0 1 2 3

min

current

degree

link 23 into 17

Fibonacci Heaps: Extract-Min

Extract-Min.   Delete min; meld its children into root list; update min.

  Consolidate trees so that no two roots have same degree.

21

39

41 17

23

18 52

30

7

35

26 46

24

44

0 1 2 3

min

current

degree

link 17 into 7

Fibonacci Heaps: Extract-Min

Extract-Min.   Delete min; meld its children into root list; update min.

  Consolidate trees so that no two roots have same degree.

22

39

41 7

30

18 52

17

35

26 46

24

44

0 1 2 3

23

current

min

degree

link 24 into 7

Fibonacci Heaps: Extract-Min

Extract-Min.   Delete min; meld its children into root list; update min.

  Consolidate trees so that no two roots have same degree.

23

39

41 7

30

18 52

23

17

35

26 46

24 44

0 1 2 3

min

current

degree

Fibonacci Heaps: Extract-Min

Extract-Min.   Delete min; meld its children into root list; update min.

  Consolidate trees so that no two roots have same degree.

24

39

41 7

30

18 52

23

17

35

26 46

24 44

0 1 2 3

min

current

degree

Fibonacci Heaps: Extract-Min

Extract-Min.   Delete min; meld its children into root list; update min.

  Consolidate trees so that no two roots have same degree.

25

39

41 7

30

18 52

23

17

35

26 46

24 44

0 1 2 3

min

current

degree

Fibonacci Heaps: Extract-Min

Extract-Min.   Delete min; meld its children into root list; update min.

  Consolidate trees so that no two roots have same degree.

26

39

41 7

30

18 52

23

17

35

26 46

24 44

0 1 2 3

min

current

degree

link 41 into 18

Fibonacci Heaps: Extract-Min

Extract-Min.   Delete min; meld its children into root list; update min.

  Consolidate trees so that no two roots have same degree.

27

39 41

7

30

18 52

23

17

35

26 46

24

44

0 1 2 3

min

current

degree

Fibonacci Heaps: Extract-Min

Extract-Min.   Delete min; meld its children into root list; update min.

  Consolidate trees so that no two roots have same degree.

28

7

30

52

23

17

35

26 46

24

0 1 2 3

min

degree

39 41

18

44

current

Fibonacci Heaps: Extract-Min

Extract-Min.   Delete min; meld its children into root list; update min.

  Consolidate trees so that no two roots have same degree.

29

7

30

52

23

17

35

26 46

24

min

39 41

18

44

stop

Fibonacci Heaps: Extract-Min Analysis

Extract-Min.

Actual cost.

  to meld min's children into root list. (at most

children of min)

  to update min.(the size of the root list is at most

)

  to consolidate trees.(one of the roots is linked to

another in each merging, and thus the total number of iterations

is at most the number of roots in the root list.)

Change in potential:

  (at most roots with distinct

degrees remain and no nodes become marked during the

operation)

Amortized cost: 30

potential function

31

Decrease Key

Fibonacci Heaps: Decrease Key

Intuition for deceasing the key of node x.   If heap-order is not violated, just decrease the key of x.

  Otherwise, cut tree rooted at x and meld into root list.

  To keep trees flat: as soon as a node has its second child cut,

cut it off and meld into root list (and unmark it).

32

24

46

17

30

23

7

88

26

21

52

39

18

41

38

72 35

min

marked node: one child already cut

Fibonacci Heaps: Decrease Key

Case 1. [heap order not violated]   Decrease key of x.

  Change heap min pointer (if necessary).

33

24

46

17

30

23

7

88

26

21

52

39

18

41

38

72

29

35

min

x

decrease-key of x from 46 to 29

Fibonacci Heaps: Decrease Key

Case 1. [heap order not violated]   Decrease key of x.

  Change heap min pointer (if necessary).

34

24

29

17

30

23

7

88

26

21

52

39

18

41

38

72 35

min

x

decrease-key of x from 46 to 29

Fibonacci Heaps: Decrease Key

Case 2a. [heap order violated]   Decrease key of x.

  Cut tree rooted at x, meld into root list, and unmark.

  If parent p of x is unmarked (hasn't yet lost a child), mark it;

Otherwise, cut p, meld into root list, and unmark

(and do so recursively for all ancestors that lose a second child).

35

24

29

17

30

23

7

88

26

21

52

39

18

41

38

72

15

35

min

decrease-key of x from 29 to 15

p

x

Fibonacci Heaps: Decrease Key

Case 2a. [heap order violated]   Decrease key of x.

  Cut tree rooted at x, meld into root list, and unmark.

  If parent p of x is unmarked (hasn't yet lost a child), mark it;

Otherwise, cut p, meld into root list, and unmark

(and do so recursively for all ancestors that lose a second child).

36

24

15

17

30

23

7

88

26

21

52

39

18

41

38

72 35

min

decrease-key of x from 29 to 15

p

x

Fibonacci Heaps: Decrease Key

Case 2a. [heap order violated]   Decrease key of x.

  Cut tree rooted at x, meld into root list, and unmark.

  If parent p of x is unmarked (hasn't yet lost a child), mark it;

Otherwise, cut p, meld into root list, and unmark

(and do so recursively for all ancestors that lose a second child).

37

24 17

30

23

7

88

26

21

52

39

18

41

38

35

min

decrease-key of x from 29 to 15

p

15

72

x

Fibonacci Heaps: Decrease Key

Case 2a. [heap order violated]   Decrease key of x.

  Cut tree rooted at x, meld into root list, and unmark.

  If parent p of x is unmarked (hasn't yet lost a child), mark it;

Otherwise, cut p, meld into root list, and unmark

(and do so recursively for all ancestors that lose a second child).

38

24 17

30

23

7

88

26

21

52

39

18

41

38

35

min

decrease-key of x from 29 to 15

p

15

72

x

mark parent

24

Fibonacci Heaps: Decrease Key

Case 2b. [heap order violated]   Decrease key of x.

  Cut tree rooted at x, meld into root list, and unmark.

  If parent p of x is unmarked (hasn't yet lost a child), mark it;

Otherwise, cut p, meld into root list, and unmark

(and do so recursively for all ancestors that lose a second child).

39 35

24

15

17

30

23

7

88

26

21

52

39

18

41

38

72 24

5

min

x

p

decrease-key of x from 35 to 5

Fibonacci Heaps: Decrease Key

Case 2b. [heap order violated]   Decrease key of x.

  Cut tree rooted at x, meld into root list, and unmark.

  If parent p of x is unmarked (hasn't yet lost a child), mark it;

Otherwise, cut p, meld into root list, and unmark

(and do so recursively for all ancestors that lose a second child).

40 5

24

15

17

30

23

7

88

26

21

52

39

18

41

38

72 24

min

x

p

decrease-key of x from 35 to 5

Fibonacci Heaps: Decrease Key

Case 2b. [heap order violated]   Decrease key of x.

  Cut tree rooted at x, meld into root list, and unmark.

  If parent p of x is unmarked (hasn't yet lost a child), mark it;

Otherwise, cut p, meld into root list, and unmark

(and do so recursively for all ancestors that lose a second child).

41

24 17

30

23

7

26

21

52

39

18

41

38

24

5

88

15

72

decrease-key of x from 35 to 5

x

p

min

Fibonacci Heaps: Decrease Key

Case 2b. [heap order violated]   Decrease key of x.

  Cut tree rooted at x, meld into root list, and unmark.

  If parent p of x is unmarked (hasn't yet lost a child), mark it;

Otherwise, cut p, meld into root list, and unmark

(and do so recursively for all ancestors that lose a second child).

42

24 17

30

23

7

26

21

52

39

18

41

38

24

5

88

15

72

decrease-key of x from 35 to 5

x

p

second child cut

min

Fibonacci Heaps: Decrease Key

Case 2b. [heap order violated]   Decrease key of x.

  Cut tree rooted at x, meld into root list, and unmark.

  If parent p of x is unmarked (hasn't yet lost a child), mark it;

Otherwise, cut p, meld into root list, and unmark

(and do so recursively for all ancestors that lose a second child).

43

24

26

17

30

23

7

21

52

39

18

41

38

88 24

5 15

72

decrease-key of x from 35 to 5

x p min

Fibonacci Heaps: Decrease Key

Case 2b. [heap order violated]   Decrease key of x.

  Cut tree rooted at x, meld into root list, and unmark.

  If parent p of x is unmarked (hasn't yet lost a child), mark it;

Otherwise, cut p, meld into root list, and unmark

(and do so recursively for all ancestors that lose a second child).

44

24

26

17

30

23

7

21

52

39

18

41

38

88 24

5 15

72

decrease-key of x from 35 to 5

x p

p'

second child cut

min

Fibonacci Heaps: Decrease Key

Case 2b. [heap order violated]   Decrease key of x.

  Cut tree rooted at x, meld into root list, and unmark.

  If parent p of x is unmarked (hasn't yet lost a child), mark it;

Otherwise, cut p, meld into root list, and unmark

(and do so recursively for all ancestors that lose a second child).

45

26

17

30

23

7

21

52

39

18

41

38

88

5 15 24

72

decrease-key of x from 35 to 5

x p p' min

don't mark parent if it's a root

p''

Fibonacci Heaps: Decrease Key Analysis

Decrease-key.

Actual cost. , where c is the number of cascading cuts

  time for changing the key.

  time for each of c cuts, plus melding into root list.

Change in potential.

  (the original trees and c trees produced by

cascading cuts)

  (c-1 nodes were unmarked by the first c-1

cascading cuts and the last cut may have marked a node)

  Difference in potential .

Amortized cost.

46

potential function

47

Union

Fibonacci Heaps: Union

Union. Combine two Fibonacci heaps.

Representation. Root lists are circular, doubly linked lists.

48

39

41

7 17

18 52

3

30

23

35

26 46

24

44

21

min min

Heap H' Heap H''

Fibonacci Heaps: Union

Union. Combine two Fibonacci heaps.

Representation. Root lists are circular, doubly linked lists.

49

39

41

7 17

18 52

3

30

23

35

26 46

24

44

21

min

Heap H

Fibonacci Heaps: Union

Actual cost:

Change in potential: 0 ( and remain the same)

Amortized cost:

50

potential function

39

41

7 17

18 52

3

30

23

35

26 46

24

44

21

min

Heap H

51

Delete

Fibonacci Heaps: Delete

Delete node x.   decrease-key of x to -∞.

  extract-min element in heap.

Amortized cost.

  amortized for decrease-key.

  amortized for extract-min.

52

potential function

Priority Queues Performance Cost Summary

53

make-heap

Operation

insert

find-min

extract-min

union

decrease-key

delete

1

Binary Heap

log n

1

log n

n

log n

log n

1

Binomial Heap

log n

log n

log n

log n

log n

log n

1

Fibonacci Heap †

1

1

log n

1

1

log n

1

Relaxed Heap

1

1

log n

1

1

log n

1

Linked List

1

n

n

1

n

n

is-empty 1 1 1 1 1

† amortized n = number of elements in priority queue