Searching

1
Searching Sorting

• Binary Trees and Treesort
• Lecture 3

2
Start with tournament ladder
a is the winner
a
a
d
b
a
d
e
b
c
a
e
h
g
d
f
How can we also find the second in
order? Solution Remove a and re-compute
positions in ladder vacated by a as shown in
the next slide.
3
Update the tournament ladder
b is the winner 2nd time
b
b
d
b
g
d
e
b
c
a
e
h
g
d
f
How can we also find the third in
order? Solution Repeat the process that was
used to find 2nd in order.
4
Use tournament ladder for k best

• Count (n -1) comparisons to find the ladder.
• At most log n comparisons to update ladder.
• Conclusion (n - 1) (k - 1) log n to find k
  best
• Sorting All can be ordered in O(n log n)
  time. Sorting
• algorithm is TOURNAMENT SORT.
• In text Tree Sort (
  since Tournament
• ladder is a tree )

5
Some basic definitions about trees

• A graph that is both acyclic connected is
  called a tree.
• A binary tree is a tree structure defined on a
  finite set of nodes that either
• contains no nodes or
• has three disjoint set of nodes a root node,
  binary tree called left subtree and binary tree
  called right subtree
• Binary trees are drawn upside down with root at
  the top.
• Depth of tree maximum depth of any vertex from
  the root.
• A binary tree has at most 2d vertices at depth d.

6
Some examples of binary trees
Complete binary tree with 7 vertices and depth
2
Incomplete binary tree with 11 vertices and depth
4
Left-to-rtight binary tree with 5 vertices and
depth 2
7
Tournament Sort (Tree-Sort)

• Algorithm
• k log(n). Construct a complete binary tree of
  depth k. Assign each element of the array to a
  leaf.
• Working up from the leaves, assign to each node
  the minimum of the values at the root nodes of
  the left and right sub-trees.
• Remove the value at root.
• Put some larger number M at the leaf node of
  the removed number.
• Update the node values traversing up the tree
  along the path of the removed value.

General idea based on binary trees.
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Trace of Tournament Sort

• Assign each element to leaf
• Elevate the minimum
• of two values at any
• level. Identify the smallest
• number at the root.
• Remove 2 from all
• nodes on the path to root.
• Elevate the next
• number in a similar fashion

9
Trace of Tournament Sort cont...

• In the next iteration
• 3 comes to root and
• is removed.
• In the next iteration
• 4 comes to root and
• is removed.

Sorted numbers
2 3
Highest path
10
Trace of Tournament Sort cont...
Highest path

• In the next iteration
• 5 comes to root and
• is removed.
• In the next iteration
• 7 comes to root and
• is removed.

2 3 4 5
Sorted numbers
11
Trace of Tournament Sort cont...

• In the next iteration
• 8 comes to root and
• is removed.
• In the next iteration
• 9 comes to root and
• is removed.

Highest path
9
12
Trace of Tournament Sort cont...

• In the next iteration
• 12 comes to root and
• is removed.

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Complexity

• Time complexity
• - Average case O(n log n)
• - Worst case O(n log n)

14
k th largest elements

• Using the algorithm described for tournament sort
  we can find the k largest elements from a large
  list in
• O( n k log n).
• Eg. Lets say we want to find the largest 20
  numbers from a list of 1000 numbers. The order of
  this search would be
• - O(n k log n) ? 1000 20log(1000) 1,200 as
  compared to
• - O(n log n) ?1000log(1000) 10,000 for
  complete sorting
• - O(log n) ? 20,000 for sequentially
  finding 20 largest.