binomial heaps jyh-shing roger jang ( 張智星 ) csie dept, national taiwan university
TRANSCRIPT
Binomial Heaps
Jyh-Shing Roger Jang (張智星 )CSIE Dept, National Taiwan University
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Memory Allocation in Classes
The binary heap is good for simple operations such as insertion, deletion, and min extraction, but it is bad for union operation.
Union operation requires building up a new heap from scratch using the elements from the old heaps, which is expensive
Binomial heaps support union operations more efficiently.
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Binomial Trees
A binomial heap is a collection of binomial trees A binomial tree Bk is an ordered tree defined recursively
B0 consists of a single node Bk consists of two binomial tree Bk-1, with root of one is the
leftmost child of the root of the other
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Properties of Binomial Trees
For a binomial tree Bk There are 2k nodes Height = k Degree of root = k There are exactly nodes at depth i for i = 0, 1, …, k The root has a k subtrees, with each subtree being Bi with i =
k-1, k-2, …, 0. Deleting root yields binomial trees Bk-1, Bk-2, …, B0 .
Proof byinduction on k
i
k
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Binomial Trees: Why It’s Named So
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Proof of the “Binomial” Property
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Properties of Binomial Trees
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Binomial Heap
Binomial heap by Vuillemin, 1978 Composed of a sequence of binomial trees
Each tree is min-heap ordered 0 or 1 binomial tree of order k
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Binomial Heap: Implementation
Common implementation of binomial heaps Represent trees via left-child, right-sibling pointers
Three links per node: parent, left, right Roots of trees connected with singly linked list
Degrees of trees strictly decreasing from left to right
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Binomial Heap: Properties
Properties of N-node binomial heap Min key contained in roots of Bk-1, Bk-2, …, B0
Contains Bi if bi=1 for binary representation bk-1bk-2…b2b0 . At most binomial trees Height ≤
1log2 N
N2log
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Binomial Heap: Proof of Properties
Properties of N-node binomial heap Assume its binary representation is bk-1bk-2…b1b0, with bk-1 ≠ 0
At most binomial trees Height = k-1 ≤
1log1log
1log1log
122
2222
22
22
1
0211
NkN
NkN
N
Nkk
kkk
1log2 N
N2log
111…1100…0
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Binomial Heap: Union of Same-order Binomial Trees
Create Binomial Tree H that is union of H’ and H’’ Easy if H’ and H’’ are of the same order (degree)
Connect roots of H’ and H’’, with smaller key to be root of H Choose smaller Min key contained in roots of Bk-1, Bk-2, …, B0
Keep min-heap order
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Binomial Heap: Union
Union of two binomial heaps Reference:
http://www.cs.princeton.edu/~wayne/cs423/lectures/heaps.ppt
Just like adding two binary numbers
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Binomial Heap: Union
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Binomial Heap: Union
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Binomial Heap: Union
Create heap H that is union of heaps H' and H''. Analogous to binary addition.
Running time. O(log N) Proportional to number of trees in root lists
001 1
100 1+
011 1
11
1
1
0
1
19 + 7 = 26
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3
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H
Binomial Heap: Delete Min
Delete node with minimum key in binomial heap H. Find root x with min key in root list of H, and delete H' broken binomial trees H Union(H', H)
Running time. O(log N)
Comparison: O(1) for finding minif we always keep a pointer to the min
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Binomial Heap: Delete Min
Delete node with minimum key in binomial heap H. Find root x with min key in root list of H, and delete H' broken binomial trees H Union(H', H)
Running time. O(log N)
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H
H'
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Binomial Heap: Decrease Key
Decrease key of node x in binomial heap H. Suppose x is in binomial tree Bk.
Bubble node x up the tree if x is too small.
Running time. O(log N) Proportional to depth of node x log2 N .
depth = 3
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Binomial Heap: Delete
Delete node x in binomial heap H. Decrease key of x to -. Bubble up to the root. Delete min.
Running time. O(log N)
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Binomial Heap: Insert
Insert a new node x into binomial heap H. H' MakeHeap(x) H Union(H', H)
Running time. O(log N)
3
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x
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Binomial Heap: Sequence of Inserts
Insert a new node x into binomial heap H. If N = .......0, then only 1 steps. If N = ......01, then only 2 steps. If N = .....011, then only 3 steps. If N = ....0111, then only 4 steps.
Inserting 1 item can take (log N) time. If N = 11...111, then log2 N steps.
But, inserting sequence of N items takes O(N) time! (N/2)(1) + (N/4)(2) + (N/8)(3) + . . . 2N Amortized analysis. Basis for getting most operations
down to constant time.
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NN
N
nn
Nn
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Priority Queues
make-heap
Operation
insert
find-min
delete-min
union
decrease-key
delete
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Binary
log N
1
log N
M*log(M+N)
log N
log N
1
Binomial
log N
1
log N
log N
log N
log N
1
Fibonacci *
1
1
log N
1
1
log N
1
Relaxed
1
1
log N
1
1
log N
1
Linked List
1
N
N
1
1
N
is-empty 1 1 1 11
Heaps
just did this
With extra book keeping
M is the size of the smaller heap