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Binary and Binomial Heaps
These lecture slides are adaptedfrom CLRS, Chapters 6, 19.
Priority Queues
Supports the following operations.
n^
Insert element x. n^
Return min element. n^
Return and delete minimum element. n^
Decrease key of element x to k.
Applications.
n^
Dijkstra’s shortest path algorithm. n^
Prim’s MST algorithm. n^
Event-driven simulation. n^
Huffman encoding. n^
Heapsort. n^
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Priority Queues in Action
PQinit()for each v
V
key(v)
PQinsert(v)
key(s)
while (!PQisempty())
v = PQdelmin()for each w
Q s.t (v,w)
E
if
π
(w) >
π
(v) + c(v,w)
PQdecrease(w,
π
(v) + c(v,w))
Dijkstra’s Shortest Path Algorithm
Dijkstra/Prim1 make-heap|V| insert|V| delete-min|E| decrease-key
Priority Queues
Operationmake-heap
insert find-min delete-min
union
decrease-key
delete
Binarylog N
log N
N
log Nlog N
Binomial
log Nlog Nlog Nlog Nlog Nlog N
Fibonacci *
log N
log N
Relaxed
log N
log N
Linked List
1 N N 11 N
is-empty
Heaps
O(|E| + |V| log |V|)
O(|E| log |V|)
O(|V|
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Binary Heap: Definition
Binary heap.
n^
Almost complete binary tree.
-^
filled on all levels, except last, where filled from left to right
n^
Min-heap ordered.
-^
every child greater than (or equal to) parent
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Binary Heap: Properties
Properties.
n^
Min element is in root. n^
Heap with N elements has height =
log
2
N
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N = 14Height = 3
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Binary Heaps: Array Implementation
Implementing binary heaps.
n^
Use an array: no need for explicit parent or child pointers.
-^
Parent(i) =
i/
-^
Left(i)
= 2i
-^
Right(i)
= 2i + 1
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Binary Heap: Insertion
Insert element x into heap.
n^
Insert into next available slot. n^
Bubble up until it’s heap ordered.
-^
Peter principle: nodes rise to level of incompetence
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next free slot
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Binary Heap: Delete Min
Delete minimum element from heap.
n^
Exchange root with rightmost leaf. n^
Bubble root down until it’s heap ordered.
-^
power struggle principle: better subordinate is promoted
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Binary Heap: Delete Min
Delete minimum element from heap.
n^
Exchange root with rightmost leaf. n^
Bubble root down until it’s heap ordered.
-^
power struggle principle: better subordinate is promoted
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18
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77
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45
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Binary Heap: Delete Min
Delete minimum element from heap.
n^
Exchange root with rightmost leaf. n^
Bubble root down until it’s heap ordered.
-^
power struggle principle: better subordinate is promoted
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exchange with left child
Binary Heap: Delete Min
Delete minimum element from heap.
n^
Exchange root with rightmost leaf. n^
Bubble root down until it’s heap ordered.
-^
power struggle principle: better subordinate is promoted
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53
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18
81
77
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45
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exchange with right child
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Binary Heap: Delete Min
Delete minimum element from heap.
n^
Exchange root with rightmost leaf. n^
Bubble root down until it’s heap ordered.
-^
power struggle principle: better subordinate is promoted
n^
O(log N) operations.
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stop: heap ordered
Binary Heap: Heapsort
Heapsort.
n^
Insert N items into binary heap. n^
Perform N delete-min operations. n^
O(N log N) sort. n^
No extra storage.
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H
1
H
2
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Binary Heap: Union
Union.
n^
Combine two binary heaps H
1
and H
2
into a single heap.
n^
No easy solution.
-^
(N) operations apparently required
n^
Can support fast union with fancier heaps.
Priority Queues
Operationmake-heap
insert find-min delete-min
union
decrease-key
delete
Binarylog N
log N
N
log Nlog N
Binomial
log Nlog Nlog Nlog Nlog Nlog N
Fibonacci *
log N
log N
Relaxed
log N
log N
Linked List
1 N N 11 N
is-empty
Heaps
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Binomial Heap: Implementation
Implementation.
n^
Represent trees using left-child, right sibling pointers.
-^
three links per node (parent, left, right)
n^
Roots of trees connected with singly linked list.
-^
degrees of trees strictly decreasing from left to right
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(^337)
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3
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heap
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Leftist Power-of-2 Heap
Binomial Heap
Parent
Left
Right
Binomial Heap: Properties
Properties of N-node binomial heap.
n^
Min key contained in root of B
, B 0
,... , B 1
.k
n^
Contains binomial tree B
i^
iff b
= 1 where bi^
⋅n
b
b 2
b 1
0
is binary
representation of N. n^
At most
log
2
N
^
n^
Height
log
2
N
B
4
B
0
B
1
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30 32
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(^337)
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N = 19# trees = 3height = 4binary = 10011
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Binomial Heap: Union
Create heap H that is union of heaps H’ and H’’.
n^
"Mergeable heaps." n^
Easy if H’ and H’’ are each order k binomial trees.
-^
connect roots of H’ and H’’
-^
choose smaller key to be root of H
H’’
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H’
Binomial Heap: Union
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(^3 )
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(^7 )
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Binomial Heap: Union
45 55
30 32
23 24
22
48 50
31
17
44
8
29
10
6
(^3 )
18
41
33
28
15
(^7 )
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Binomial Heap: Union
45 55
30 32
23 24
22
48 50
31
17
44
8
29
10
6
(^337)
41
33
28
15
(^725)
12 18
18 12
31
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30 32
23 24
22
48 50
31
17
44
8
29
10
6
(^337)
41
33
28
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(^725)
12 18
25
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7
3
12 18
18 12
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30 32
23 24
22
48 50
31
17
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29
10
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(^337)
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12 7 25
18
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3
28 41
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Binomial Heap: Delete Min
Delete node with minimum key in binomial heap H.
n^
Find root x with min key in root list of H, and delete n^
H’
broken binomial trees
n^
H
Union(H’, H)
Running time. O(log N)
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30 32
23 24
22
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31
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(^637)
18
44
8
29
10
H
H’
3
(^637)
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x 55
30 32
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H
Binomial Heap: Decrease Key
Decrease key of node x in binomial heap H.
n^
Suppose x is in binomial tree B
.k
n^
Bubble node x up the tree if x is too small.
Running time. O(log N)
n^
Proportional to depth of node x
log
2
N
^
depth = 3
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Binomial Heap: Delete
Delete node x in binomial heap H.
n^
Decrease key of x to -
n^
Delete min.
Running time. O(log N)
Binomial Heap: Insert
Insert a new node x into binomial heap H.
n^
H’
MakeHeap(x)
n^
H
Union(H’, H)
Running time. O(log N)
3
(^637)
18
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H
x H’
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Binomial Heap: Sequence of Inserts
Insert a new node x into binomial heap H.
n^
If N =
, then only 1 steps.
n^
If N =
, then only 2 steps.
n^
If N =
, then only 3 steps.
n^
If N =
, then only 4 steps.
Inserting 1 item can take
(log N) time.
n^
If N =
, then log
2
N steps.
But, inserting sequence of N items takes O(N) time!
n^
(N/2)(1) + (N/4)(2) + (N/8)(3) +...
2N
n^
Amortized analysis. n^
Basis for getting most operationsdown to constant time.
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(^637)
x
1
1
−
=^
N
N
N n^
n
N
n
Priority Queues
Operationmake-heap
insert find-min delete-min
union
decrease-key
delete
Binarylog N
log N
N
log Nlog N
Binomial
log Nlog Nlog Nlog Nlog Nlog N
Fibonacci *
log N
log N
Relaxed
log N
log N
Linked List
1 N N 11 N
is-empty
Heaps
just did this
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