單元7: Heap
1.
2.
3.
4.
(Chapter 9)
定義 for priority queues
Leftist trees
Binomial heap
Min-Max heap
1
1.定義
A double-ended priority queue is a data structure
that supports the following operation:
1.
2.
3.
4.
5.
Return an element with minimum priority.
Return an element with maximum priority.
Insert an element with arbitrary priority.
Delete an element with minimum priority.
Delete an element with maximum priority.
2
2.Leftist Trees
An extended binary tree
Leftist trees are defined using the concept of an extended
binary tree. An extended binary tree is a binary tree in
which all empty binary subtrees have been replaced by a
square node.
The square nodes in an
extended binary tree are
called external nodes. The
original nodes of the
binary tree are called
internal nodes.
A
B
D
C
E
3
2.Leftist Trees
An extended binary tree
Define shortest(x) to be the length of a shortest
path from x to an external node.
2
A
2
1
B
D
E
1
0
0
0 0
1
C
0
0
shortest(x) = 1+ min {shortest(leftChild(x)), shortest(rightChild(x))}
4
2.Leftist Trees
leftist tree 定義:
A leftist tree is a binary tree such that if it is not
empty, then shortest(LeftChild(x)) ≥
hortest(RightChild(x)) for every internal node x.
2
2
1
D
A
B
2
C
E
1
1
1
A
B
C
2
1
D
E
1
5
2.Leftist Trees
Lemma : Let x be the root of a leftist tree that has
n (internal) nodes
(a) n ≥ 2shortest(x) – 1
(b) The rightmost root to external node path is
the shortest root to external node path. Its length
is shortest(x) ≤ log2(n+1).
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2.Leftist Trees
A min leftist tree (max leftist tree)
-is a leftist tree in which the key value in each
node is no larger (smaller) than the key
values in its children (if any). In other words,
a min (max) leftist tree is a leftist tree that is
also a min (max) tree.
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2.Leftist Trees
A min leftist tree ( operation) :
Insert; delete; combine (meld)
meld
1. 找兩者之root 相比, 較小者為root, 且其左child 留下。
2. 其右child 與另一較大之tree 以相同步驟combine, 結果作為
right child。
3. 調整為符合leftist tree
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2.Leftist Trees
meld
2
1
1
1
13
11
2
2
50 1
7
80
1
2 12
1
1
20
5
12
9
8
1 10
1
18
1
15
50 1
80
1
9
2.Leftist Trees
meld
2
1
1
1
2
2
12
8
1
15
9
1
11
1 10
13
5
7
1 50
1
80
1
2 12
20
1
18
10
2.Leftist Trees
meld
2
2
12
1
15
1
5
8
1 10
2
1
1 50
1
80
1
2 12
20
1
9
1
18
7
11
13
1
The complexity O(log n)
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2.Leftist Trees
A min leftist tree ( operation) :
insert(7)
32
2
7
2
4
1
2
6
1
6
3
1
1
8
1
8
21
5
7
1
9
12
2.Leftist Trees
A min leftist tree ( operation) :
DELETE(2)
2
2
2
4
1
2
6
1
6
3
1
1
8
1
8
21
5
7
1
9
13
2.Leftist Trees
A min leftist tree ( operation) :
DELETE(2)
2
3
2
1
4
5
1
6
9
8
7
8
1
1
1
1
2
6
1
8
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2.Leftist Trees
A min leftist tree ( operation) :
DELETE(2)
2
3
2
1
5
4
1
2
6
7
1
1
8
6
1
9
1
8
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2.Leftist Trees
Weight-Biased Leftist tree
Define w(x) to be the number of internal nodes in
the subtree with root x
4
A
2
A
2
1
B
D
1
1
0
0
0 0
1
C
E
0
3
B
D
E
1
0
0
0
0 0
1
C
0
0
w(x) = 1+ w(leftChild(x))+w(rightChild(x))
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3. Binomial heap
• A binomial heap is a data structure that supports the same
functions as those supported by leftist trees.
• Unlike leftist trees, where an individual operation can be
performed in O(log n) time, it is possible that certain individual
operations performed on a binomial heap may take O(n) time.
• By amortizing part of the cost of expensive operations over
the inexpensive ones, then the amortized complexity of an
individual operation is either O(1) or O(log n) depending on
the type of operations.
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3. Binomial heap
定義:
The binomial tree Bk, of degree k is a tree such
that
if k = 0, then the tree has exactly one node,
and if k > 0, then the tree consists of a root
whose degree is k and whose subtrees are B0,
B1, …, Bk-1.
B0
B1
B2
B3
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3. Binomial heap
operation:Insert, Delete-min, Meld
Insert 6, 5, 3, 4, 7, 9,
3
5
4
6
Delete-min
7
9
1
7
12
15
20
30
16
9
19
3. Binomial heap
Analysis Of Binomial Heaps
Insert
Leftist trees Binomial heaps
Actual Amortized
O(log n)
O(1) O(1)
Delete min (or max) O(log n)
O(n)
O(log n)
Meld
O(1)
O(1)
O(log n)
20
4. Min-Max heap
A double-ended priority queue is a data structure that
supports the following operations:
– inserting an element with an arbitrary key
– deleting an element with the largest key
– deleting an element with the smallest
A Min-Max Heap supports all of the above operations.
21
4. Min-Max heap
Definition:
A min-max heap is a complete binary tree such that if it
is not empty, each element has a data member called key.
Alternating levels of this tree are min levels and max
levels, respectively. The root is on a min level. Let x be
any node in a min-max heap.
If x is on a min (max) level then the element in x has
the minimum (maximum) key from among all elements in
the subtree with root x. A node on a min (max) level is
called a min (max) node.
22
4. Min-Max heap
The min-max heap stores in a one-dimension array h
min
7
40
70
30
45
Insert 5
9
50
7
30
10
20
8
12
max
15 min
max
23
4. Min-Max heap
Insert x
i
Check parent i 為max 或min
(1) min
7
8x
If x<i then x 和i 互換
對x執行verifymin 以找正確位置
(2) max
If x>i then x 和i 互換
對x執行verifymax 以找正確位置
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4. Min-Max heap
Insert 5
min
7
70
verifymin
30
45
40
9
50
7
30
15 min
10
20
8
12
max
5
max
25
4. Min-Max heap
Insert 80
min
5
30
45
9
50
max
40
70
7
30
7
20
8
12
verifymax
10
80
15 min
max
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4. Min-Max heap
Delete of min element
(1)
(2)
(3)
(4)
(5)
將root (最小值)拿掉
準備delete array最後元素x,再重新insert 最後元素x
尋找第2小元素k
If x < k then root 設為x
else if k 為root 之child
k
x
7
8k
max
7
8x
max
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4. Min-Max heap
Delete of min element
(6) else if k 為root 之grandchild
k
k
x
p
7
7
max
8k
min
p
7
7
p>x
max
8x
x
7
min
check
7
p<x
max
8p
min
check
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4. Min-Max heap
Delete of min element 7
7
x
min
99
70
30
45
9
50
7
30
k
20
8
10
90
max
15 min
max
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4. Min-Max heap
Deaps
• A deap is a double-ended heap that supports
the double-ended priority operations of insert, deletemin, and delete-max.
• Similar to min-max heap but deap is faster on these
operations by a constant factor, and
the algorithms are simpler.
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4. Min-Max heap
Def:
A deap is a complete binary tree that is either
empty or satisfies the following properties:
(1) The root contains no element
(2) The left subtree is a min heap.
(3) The right subtree is a max heap.
(4) If the right subtree is not empty, then let i be any node in the left
subtree. Let j be the corresponding node in the right subtree. If such
a j does not exist, then let j be the node in the right subtree that
corresponds to the parent of i. The key in node i is less than or equal
to that of j.
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4. Min-Max heap
Deaps
小
大
45
5
10
15
8
19
97
Min heap
25
30
8
40
20
Max heap
5 < 45 ; 10 <25 ; 8 <40; 15 < 20; 19 < 25; 9 < 40; 30 < 40
32
4. Min-Max heap
Deaps-Insert 4
45
5
10
15
8
19
97
Min heap
25
30
8
比較
20
40
4
Max heap
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4. Min-Max heap
Deaps-Insert 60
45
5
10
15
8
19
97
Min heap
25
30
8
20
40
4
60
比較
Max heap
34
4. Min-Max heap
Deaps-delete 5
45
5
10
15
8
18
97
Min heap
25
10
8
19
20
Last element
Max heap
比較
35