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Fibonacci Heaps
Priority Queues
Operationmake-heap
insertfind-min delete-min
union decrease-key
delete
Binarylog N
log N
N
log Nlog N
Binomial
log Nlog Nlog Nlog Nlog Nlog N
Fibonacci
†
1 1 log N
log N
Relaxed
log N
log N
Linked List
1 N N 11 N
is-empty
Heaps
this time
† amortized
3
Fibonacci Heaps
Fibonacci heap history.
Fredman and Tarjan (1986)
n^
Ingenious data structure and analysis. n^
Original motivation: O(m + n log n) shortest path algorithm.^ –
also led to faster algorithms for MST, weighted bipartite matching n^
Still ahead of its time. Fibonacci heap intuition.^ n^
Similar to binomial heaps, but less structured. n^
Decrease-key and union run in O(1) time. n^
"Lazy" unions.
Fibonacci Heaps: Structure
Fibonacci heap.^ n^
Set of min-heap ordered trees.
7
23
17 30
26 35
24 46
H
39
41
18
(^352)
44
min
marked
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Fibonacci Heaps: Implementation
Implementation.^ n^
Represent trees using left-child, right sibling pointers and circular,doubly linked list.^ –
can quickly splice off subtrees n^
Roots of trees connected with circular doubly linked list.^ –
fast union n^
Pointer to root of tree with min element.^ –
fast find-min
7
23
17 30
26 35
24 46
H
39
41
18
(^352)
44
min
Fibonacci Heaps: Potential Function
Key quantities.^ n^
Degree[x] = degree of node x. n^
Mark[x] = mark of node x (black or gray). n^
t(H)
= # trees.
n^
m(H) = # marked nodes. n^
(H) = t(H) + 2m(H) = potential function.
7
23
17 30
26 35
24 46
H
t(H) = 5, m(H) = 3 Φ
(H) = 11
39
41
18
(^352)
44
min
degree = 3
7
Fibonacci Heaps: Insert
Insert.^ n^
Create a new singleton tree. n^
Add to left of min pointer. n^
Update min pointer.
7
23
17 30
26 35
24 46
H
39
41
18
(^352)
44
min
21
Insert 21
Fibonacci Heaps: Insert
Insert.^ n^
Create a new singleton tree. n^
Add to left of min pointer. n^
Update min pointer.
39
41
7
23
18
(^352)
17 30
26 35
24 46
44
min
H
21
Insert 21
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Fibonacci Heaps: Delete Min
Delete min.^ n^
Delete min and concatenate its children into root list. n^
Consolidate trees so that no two roots have same degree.
39
41
17
23
18
52
(^730)
26 35
24 46
44
current
min
Fibonacci Heaps: Delete Min
Delete min.^ n^
Delete min and concatenate its children into root list. n^
Consolidate trees so that no two roots have same degree.
39
41
17
23
18
52
(^730)
26 35
24 46
44
current
0
1
2
3
min
15
Fibonacci Heaps: Delete Min
Delete min.^ n^
Delete min and concatenate its children into root list. n^
Consolidate trees so that no two roots have same degree.
39
41
17
23
18
52
(^730)
26 35
24 46
44
current
0
1
2
3
min
Fibonacci Heaps: Delete Min
Delete min.^ n^
Delete min and concatenate its children into root list. n^
Consolidate trees so that no two roots have same degree.
39
41
17
23
18
52
(^730)
26 35
24 46
44
current
0
1
2
3
min
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Fibonacci Heaps: Delete Min
Delete min.^ n^
Delete min and concatenate its children into root list. n^
Consolidate trees so that no two roots have same degree.
39
41
17
23
18
52
(^730)
26 35
24 46
44
current
0
1
2
3
Merge 17 and 23 trees.
min
Fibonacci Heaps: Delete Min
Delete min.^ n^
Delete min and concatenate its children into root list. n^
Consolidate trees so that no two roots have same degree.
39
41
17 23
18
52
(^730)
26 35
24 46
44
current
0
1
2
3
Merge 7 and 17 trees.
min
19
Fibonacci Heaps: Delete Min
Delete min.^ n^
Delete min and concatenate its children into root list. n^
Consolidate trees so that no two roots have same degree.
39
41
(^730)
18
52
17
26 35
24 46
44
current
0
1
2
3
23
Merge 7 and 24 trees.
min
Fibonacci Heaps: Delete Min
Delete min.^ n^
Delete min and concatenate its children into root list. n^
Consolidate trees so that no two roots have same degree.
39
41
(^730)
18
52
17 23
26 35
24 46
44
current
0
1
2
3
min
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Fibonacci Heaps: Delete Min
Delete min.^ n^
Delete min and concatenate its children into root list. n^
Consolidate trees so that no two roots have same degree.
39
41
(^730)
18
52
17 23
26 35
24 46
44
current
0
1
2
3
min
Fibonacci Heaps: Delete Min
Delete min.^ n^
Delete min and concatenate its children into root list. n^
Consolidate trees so that no two roots have same degree.
39
41
(^730)
18
52
17 23
26 35
24 46
44
min
Stop.
27
Fibonacci Heaps: Delete Min Analysis
Notation.^ n^
D(n) = max degree of any node in Fibonacci heap with n nodes. n^
t(H)
= # trees in heap H.
n^
(H) = t(H) + 2m(H).
Actual cost.
O(D(n) + t(H))
n^
O(D(n)) work adding min’s children into root list and updating min.^ –
at most D(n) children of min node n^
O(D(n) + t(H)) work consolidating trees.^ –
work is proportional to size of root list since number of rootsdecreases by one after each merging – ≤
D(n) + t(H) - 1 root nodes at beginning of consolidation
Amortized cost. O(D(n))^ n^
t(H’)
D(n) + 1 since no two trees have same degree.
n^
(H)
D(n) + 1 - t(H).
Fibonacci Heaps: Delete Min Analysis
Is amortized cost of O(D(n)) good?^ n^
Yes, if only Insert, Delete-min, and Union operations supported.^ –
in this case, Fibonacci heap contains only binomial trees sincewe only merge trees of equal root degree – this implies D(n)
log
N 2
n^
Yes, if we support Decrease-key in clever way.^ –
we’ll show that D(n)
log
Nφ
, where
φ
is golden ratio
-^ φ
φ
-^ φ
-^ limiting ratio between successive Fibonacci numbers!
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Decrease key of element x to k.^ n^
Case 0: min-heap property not violated.^ –
decrease key of x to k – change heap min pointer if necessary
24 46
17 30
23
7
26 88
21 52
18 39
38 41
Decrease 46 to 45.
72
Fibonacci Heaps: Decrease Key
45
35
min
Decrease key of element x to k.^ n^
Case 1: parent of x is unmarked.^ –
decrease key of x to k – cut off link between x and its parent – mark parent – add tree rooted at x to root list, updating heap min pointer
24 45
17 30
23
7
26 88
21 52
18 39
38 41
Decrease 45 to 15.
72
Fibonacci Heaps: Decrease Key
15
35
min
31
Decrease key of element x to k.^ n^
Case 1: parent of x is unmarked.^ –
decrease key of x to k – cut off link between x and its parent – mark parent – add tree rooted at x to root list, updating heap min pointer
24 15
17 30
23
7
26 88
21 52
18 39
38 41
Decrease 45 to 15.
24 72
Fibonacci Heaps: Decrease Key
35
min
Decrease key of element x to k.^ n^
Case 1: parent of x is unmarked.^ –
decrease key of x to k – cut off link between x and its parent – mark parent – add tree rooted at x to root list, updating heap min pointer
24
17 30
23
7
26 88
21 52
18 39
38 41
Decrease 45 to 15.
24
Fibonacci Heaps: Decrease Key
35
min
15 72
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Notation.^ n^
t(H)
= # trees in heap H.
n^
m(H) = # marked nodes in heap H. n^
(H)
= t(H) + 2m(H).
Actual cost. O(c)^ n^
O(1) time for decrease key. n^
O(1) time for each of c cascading cuts, plus reinserting in root list. Amortized cost. O(1)^ n^
t(H’)
= t(H) + c
n^
m(H’)
m(H) - c + 2
-^
each cascading cut unmarks a node
-^
last cascading cut could potentially mark a node
n^
c + 2(-c + 2) = 4 - c.
Fibonacci Heaps: Decrease Key Analysis
Delete node x.^ n^
Decrease key of x to -
n^
Delete min element in heap. Amortized cost. O(D(n))^ n^
O(1) for decrease-key. n^
O(D(n)) for delete-min. n^
D(n) = max degree of any node in Fibonacci heap.
Fibonacci Heaps: Delete
39
Fibonacci Heaps: Bounding Max Degree
Definition. D(N) = max degree in Fibonacci heap with N nodes.Key lemma. D(N)
log
N, whereφ
φ
Corollary. Delete and Delete-min take O(log N) amortized time.Lemma. Let x be a node with degree k, and let y
,... , y 1
denote thek
children of x in the order in which they were linked to x. Then:Proof.^ n^
When y
is linked to x, yi
,... , y 1
i-
already linked to x,
degree(x) = i - 1 ⇒
degree(y
) = i - 1 since we only link nodes of equal degreei
n^
Since then, y
has lost at most one childi
-^ otherwise it would have been cut from x n^
Thus, degree(y
) = i - 1 or i - 2i
≥^
if
if
degree
i
i
i
y^ i
Fibonacci Heaps: Bounding Max Degree
Key lemma. In a Fibonacci heap with N nodes, the maximum degree ofany node is at most log
N, whereφ
φ
Proof of key lemma.^ n^
For any node x, we show that size(x)
≥ φ
degree(x)
-^ size(x) = # node in subtree rooted at x –^ taking base
φ^
logs, degree(x)
log
(size(x))φ
log
N.φ
n^
Let s
be min size of tree rooted at any degree k node.k^
-^ trivial to see that s
= 1, s 0
-^ s
monotonically increases with kk^
n^
Let x* be a degree k node of size s
,k^
and let y
,... , y 1
be children in orderk
that they were linked to x*.
Assume k
=^ − =
− = =^220
2 2
] deg[ 2
size
k i^
i k i
i k i
y
k i
i
k
s s s
y
x size
s
i
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Fibonacci Facts
Definition. The Fibonacci sequence is:^ n^
•^
Slightly nonstandard definition. Fact F1.
F^ k
≥ φ
k, where
φ
Fact F2.Consequence.
s^ k
F
k^
≥^
kφ
n^
This implies that size(x)
≥ φ
degree(x)
for all nodes x.
if
F
F
if
if
F
(^2) - k
(^1) - k
k
k k k
≥^
−^2 =^0
For
k i^
i
k^
F
F
k
=^ − =
− = =^220
2 2
] deg[ 2
size
k i^
i k i
i k i
y
k i
i
k
s s s
y
x size
s
i
Golden Ratio
Definition. The Fibonacci sequence is: 1, 2, 3, 5, 8, 13, 21,.. .Definition. The golden ratio
φ
n^
Divide a rectangle into a square and smaller rectangle such that thesmaller rectangle has the same ratio as original one.
Parthenon, Athens Greece
43
Fibonacci Facts
Fibonacci Numbers and Nature
Pinecone
Cauliflower
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