Fibonacci Heaps Lecture slides adapted from: Chapter 20 of - - PowerPoint PPT Presentation
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 COS 423 Theory of Algorithms Kevin
COS 423 Theory of Algorithms • Kevin Wayne • Spring 2007 Adapted by Cheng Li and Virgil Pavlu
Lecture slides adapted from:
Ingenious data structure and analysis. Original motivation: improve Dijkstra's shortest path algorithm
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.
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V insert, V extract-min, E decrease-key
Set of heap-ordered trees. Maintain pointer to minimum element. Set of marked nodes.
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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
Set of heap-ordered trees. Maintain pointer to minimum element. Set of marked nodes.
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7 23 30 17 35 26 46 24 Heap H 39 41 18 52 3 44
min find-min takes O(1) time
Set of heap-ordered trees. Maintain pointer to minimum element. Set of marked nodes.
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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
= number of nodes in heap. degree(x) = number of children of node x. = upper bound on the maximum degree of any node.
= number of trees in heap H. = number of marked nodes in heap H.
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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
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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:
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Create a new singleton tree. Add to root list; update min pointer (if necessary).
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7 23 30 17 35 26 46 24 39 41 18 52 3 44 21
min
Heap H
Create a new singleton tree. Add to root list; update min pointer (if necessary).
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39 41 7 23 18 52 3 30 17 35 26 46 24 44 21
min
Heap H
H’ = the heap after insert
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39 41 7 18 52 3 30 17 35 26 46 24 44 21 23
min
Heap H
potential of heap H
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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
Delete min; meld its children into root list; update min. Consolidate trees so that no two roots have same degree.
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39 41 18 52 3 44 17 23 30 7 35 26 46 24
min
Delete min; meld its children into root list; update min. Consolidate trees so that no two roots have same degree.
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39 41 17 23 18 52 30 7 35 26 46 24 44
min
Delete min; meld its children into root list; update min. Consolidate trees so that no two roots have same degree.
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39 41 17 23 18 52 30 7 35 26 46 24 44
min
current
Delete min; meld its children into root list; update min. Consolidate trees so that no two roots have same degree.
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39 41 17 23 18 52 30 7 35 26 46 24 44 1 2 3 current
min degree
Delete min; meld its children into root list; update min. Consolidate trees so that no two roots have same degree.
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39 41 17 23 18 52 30 7 35 26 46 24 44 1 2 3
min
current
degree
Delete min; meld its children into root list; update min. Consolidate trees so that no two roots have same degree.
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39 41 17 23 18 52 30 7 35 26 46 24 44 1 2 3
min
current
degree
Delete min; meld its children into root list; update min. Consolidate trees so that no two roots have same degree.
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39 41 17 23 18 52 30 7 35 26 46 24 44 1 2 3
min
current
degree
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 1 2 3
min
current
degree
Delete min; meld its children into root list; update min. Consolidate trees so that no two roots have same degree.
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39 41 7 30 18 52 17 35 26 46 24 44 1 2 3 23 current
min degree
Delete min; meld its children into root list; update min. Consolidate trees so that no two roots have same degree.
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39 41 7 30 18 52 23 17 35 26 46 24 44 1 2 3
min
current
degree
Delete min; meld its children into root list; update min. Consolidate trees so that no two roots have same degree.
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39 41 7 30 18 52 23 17 35 26 46 24 44 1 2 3
min
current
degree
Delete min; meld its children into root list; update min. Consolidate trees so that no two roots have same degree.
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39 41 7 30 18 52 23 17 35 26 46 24 44 1 2 3
min
current
degree
Delete min; meld its children into root list; update min. Consolidate trees so that no two roots have same degree.
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39 41 7 30 18 52 23 17 35 26 46 24 44 1 2 3
min
current
degree
Delete min; meld its children into root list; update min. Consolidate trees so that no two roots have same degree.
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39 41 7 30 18 52 23 17 35 26 46 24 44 1 2 3
min
current
degree
Delete min; meld its children into root list; update min. Consolidate trees so that no two roots have same degree.
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7 30 52 23 17 35 26 46 24 1 2 3
min degree
39 41 18 44 current
Delete min; meld its children into root list; update min. Consolidate trees so that no two roots have same degree.
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7 30 52 23 17 35 26 46 24
min
39 41 18 44
to meld min's children into root list. (at most
to update min.(the size of the root list is at most
to consolidate trees.(one of the roots is linked to
(at most roots with distinct
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potential function
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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,
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min marked node:
Decrease key of x. Change heap min pointer (if necessary).
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min x
decrease-key of x from 46 to 29
Decrease key of x. Change heap min pointer (if necessary).
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min x
decrease-key of x from 46 to 29
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;
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min
decrease-key of x from 29 to 15
p x
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;
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min
decrease-key of x from 29 to 15
p x
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;
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min
decrease-key of x from 29 to 15
p
x
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;
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min
decrease-key of x from 29 to 15
p
x mark parent
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;
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min x p
decrease-key of x from 35 to 5
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;
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min x p
decrease-key of x from 35 to 5
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;
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decrease-key of x from 35 to 5
x p min
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;
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decrease-key of x from 35 to 5
x p second child cut min
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;
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decrease-key of x from 35 to 5
x p min
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;
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decrease-key of x from 35 to 5
x p p' second child cut min
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;
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decrease-key of x from 35 to 5
x p p' min don't mark parent if it's a root p''
time for changing the key. time for each of c cuts, plus melding into root list.
(the original trees and c trees produced by
(c-1 nodes were unmarked by the first c-1
Difference in potential .
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potential function
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39 41 7 17 18 52 3 30 23 35 26 46 24 44 21
min min
Heap H' Heap H''
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39 41 7 17 18 52 3 30 23 35 26 46 24 44 21
min
Heap H
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potential function
39 41 7 17 18 52 3 30 23 35 26 46 24 44 21
min
Heap H
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decrease-key of x to -∞. extract-min element in heap.
amortized for decrease-key. amortized for extract-min.
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potential function
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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