Initializing A Max Heap

1
Initializing A Max Heap

• input array -, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10,
  11

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Initializing A Max Heap
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3
2
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6
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5
11
8
7
8
9
7
10

• Start at rightmost array position that has a
  child.

Index is n/2.
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Initializing A Max Heap
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3
2
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6
7
11
5
8
7
8
9
7
10

• Move to next lower array position.

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Initializing A Max Heap
1
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11
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8
7
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9
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10
5
Initializing A Max Heap
1
3
2
9
6
7
11
5
8
7
8
4
7
10
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Initializing A Max Heap
1
3
2
9
6
7
11
5
8
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8
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10
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Initializing A Max Heap
1
7
2
9
6
3
11
5
8
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8
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7
10
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Initializing A Max Heap
1
7
2
9
6
3
11
5
8
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10
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Initializing A Max Heap
1
7
11
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6
3
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Find a home for 2.
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Initializing A Max Heap
1
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11
9
6
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5
Find a home for 2.
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Initializing A Max Heap
1
7
11
9
6
3
10
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5
2
Done, move to next lower array position.
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Initializing A Max Heap
1
7
11
9
6
3
10
2
8
7
8
4
7
5
Find home for 1.
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Initializing A Max Heap
11
7
9
6
3
10
2
8
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8
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7
5
Find home for 1.
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Initializing A Max Heap
11
7
10
9
6
3
2
8
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8
4
7
5
Find home for 1.
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Initializing A Max Heap
11
7
10
9
6
3
5
2
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7
8
4
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Find home for 1.
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Initializing A Max Heap
11
7
10
9
6
3
5
2
8
7
8
4
7
1
Done.
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Time Complexity
11
7
9
6
3
5
8
8
7
4
7
2
1
10
Height of heap h. Number of subtrees with root
at level j is lt 2 j-1. Time for each subtree is
O(h-j1).
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Complexity
Time for level j subtrees is lt 2j-1(h-j1)
t(j). Total time is t(1) t(2) t(h-1)
O(n).
19
Leftist Trees

• Linked binary tree.
• Can do everything a heap can do and in the same
  asymptotic complexity.
• Can meld two leftist tree priority queues in
  O(log n) time.

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Extended Binary Trees

• Start with any binary tree and add an external
  node wherever there is an empty subtree.
• Result is an extended binary tree.

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A Binary Tree
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An Extended Binary Tree
number of external nodes is n1
23
The Function s()

• For any node x in an extended binary tree, let
  s(x) be the length of a shortest path from x to
  an external node in the subtree rooted at x.

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s() Values Example
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s() Values Example
2
2
1
2
1
1
0
1
0
0
1
1
0
0
0
0
0
0
0
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Properties Of s()

• If x is an external node, then s(x) 0.
• Otherwise,
• s(x) min s(leftChild(x)),
• s(rightChild(x)) 1

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Height Biased Leftist Trees

• A binary tree is a (height biased) leftist tree
  iff for every internal node x, s(leftChild(x)) gt
  s(rightChild(x))

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A Leftist Tree
2
2
1
2
1
1
0
1
0
0
1
1
0
0
0
0
0
0
0
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Leftist Trees--Property 1

• In a leftist tree, the rightmost path is a
  shortest root to external node path and the
  length of this path is s(root).

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A Leftist Tree
2
2
1
2
1
1
0
1
0
0
1
1
0
0
0
0
0
0
0
Length of rightmost path is 2.
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Leftist TreesProperty 2

• The number of internal nodes is at least
• 2s(root) - 1
• Because levels 1 through s(root) have no external
  nodes.
• So, s(root) lt log(n1)

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A Leftist Tree
2
2
1
2
1
1
0
1
0
0
1
1
0
0
0
0
0
0
0
Levels 1 and 2 have no external nodes.
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Leftist TreesProperty 3

• Length of rightmost path is O(log n), where n is
  the number of nodes in a leftist tree.
• Follows from Properties 1 and 2.

34
Leftist Trees As Priority Queues
Min leftist tree leftist tree that is a min
tree. Used as a min priority queue. Max leftist
tree leftist tree that is a max tree. Used as a
max priority queue.
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A Min Leftist Tree
2
4
3
6
8
5
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8
6
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Some Min Leftist Tree Operations
put() remove() meld() initialize() put() and
remove() use meld().
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Put Operation

• put(7)

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Put Operation

• put(7)

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Create a single node min leftist tree.
7
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Put Operation

• put(7)

2
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6
Create a single node min leftist tree. Meld the
two min leftist trees.
7
40
Remove Min
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Remove Min
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6
Remove the root.
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Remove Min
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6
Remove the root. Meld the two subtrees.
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Meld Two Min Leftist Trees
Traverse only the rightmost paths so as to get
logarithmic performance.
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Meld Two Min Leftist Trees
4
3
6
8
5
6
9
8
6
Meld right subtree of tree with smaller root and
all of other tree.
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Meld Two Min Leftist Trees
4
3
6
8
5
6
9
8
6
Meld right subtree of tree with smaller root and
all of other tree.
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Meld Two Min Leftist Trees
6
4
6
8
8
6
Meld right subtree of tree with smaller root and
all of other tree.
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Meld Two Min Leftist Trees
6
8
Meld right subtree of tree with smaller root and
all of other tree. Right subtree of 6 is empty.
So, result of melding right subtree of tree with
smaller root and other tree is the other tree.
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Meld Two Min Leftist Trees
Make melded subtree right subtree of smaller root.
Swap left and right subtree if s(left) lt s(right).
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Meld Two Min Leftist Trees
Make melded subtree right subtree of smaller root.
Swap left and right subtree if s(left) lt s(right).
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Meld Two Min Leftist Trees
Make melded subtree right subtree of smaller root.
Swap left and right subtree if s(left) lt s(right).
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Meld Two Min Leftist Trees
3
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Initializing In O(n) Time

• create n single node min leftist trees and place
  them in a FIFO queue
• repeatedly remove two min leftist trees from the
  FIFO queue, meld them, and put the resulting min
  leftist tree into the FIFO queue
• the process terminates when only 1 min leftist
  tree remains in the FIFO queue
• analysis is the same as for heap initialization