1. Digital Search Trees & Binary Tries
Analog of radix sort to searching.
Keys are binary bit strings.
Fixed length – 0110, 0010, 1010, 1011.
Variable length – 01, 00, 101, 1011.
Application – IP routing, packet classification, firewalls.
IPv4 – 32 bit IP address; |key| <= 32.
IPv6 – 128 bit IP address; |key| <= 128.
Will discuss IP application in detail later. IPv4 (v6) the keys in the search structure are up to 32 bits (128) long.
Application of variable length binary keysHuffman codes/tree.

2. Digital Search Tree Assume fixed number of bits. Not empty =>
Root contains one dictionary pair (any pair).
All remaining pairs whose key begins with a 0 are in the left subtree.
All remaining pairs whose key begins with a 1 are in the right subtree.
Left and right subtrees are digital search trees on remaining bits.
So, on level I branching is done on bit I of the key.

3. Example
Start with an empty digital search tree and insert a pair whose key is 0110.
0110
Now, insert a pair whose key is 0010.
0110
0010

4. Example
Now, insert a pair whose key is 1001.
0110
0010
1001
0110
0010

5. Example Now, insert a pair whose key is 1011. 1001 0110 0010 1011 1001

6. Example Now, insert a pair whose key is 0000. 1001 0110 0010 1011 0000

7. Search/Insert/Delete
1001
0110
0010
1011
0000
Delete from nonleaf => replace with key in subtree leaf. Note can’t delete as in binary search tree as that would change levels of nodes and mess up relationship between level and branching bit.
Complexity of each operation is O(#bits in a key).
#key comparisons = O(height).
Expensive when keys are very long.

8. Binary Trie Information Retrieval.
At most one key comparison per operation.
Fixed length keys.
Branch nodes.
Left and right child pointers.
No data field(s).
Element nodes.
No child pointers.
Data field to hold dictionary pair.
Huffman trees have same structure but for variable length keys that satisfy the prefix property (no key is a proper prefix of another)

9. Example At most one key comparison for a search. 1 1100 0001 0011 1000
1001 1
Every branch node has >=2 keys/pairs in its subtree.
At most one key comparison for a search.

10. Variable Key Length Left and right child fields.
Left and right pair fields.
Left pair is pair whose key terminates at root of left subtree or the single pair that might otherwise be in the left subtree.
Right pair is pair whose key terminates at root of right subtree or the single pair that might otherwise be in the right subtree.
Field is null otherwise.

11. Example At most one key comparison for a search. null 1 00 01100 10
null 1 00
01100 10
11111
0000
001
1000
101 1
Alternative is to store only length k keys at level k. Then, 0000, for example, would be stored one level down from shown figure. Longest prefix matching.
00100
001100
At most one key comparison for a search.

12. Fixed Length Insert Insert 0111. Zero compares. 1 1 1100 0111 0001
0011
1100
1000
1001 1
0111 1
Insert 0111.
Zero compares.

13. Fixed Length Insert
0001
0011
1100
1000
1001 1
0111 1
Insert 1101.

14. Fixed Length Insert
1100
0001
0011
1000
1001 1
0111 1
Insert 1101.

15. Fixed Length Insert Insert 1101. One compare. 0001 0011 1000 1001 11
0111 1 1
1100
1101
Insert 1101.
One compare.

16. Fixed Length Delete Delete 0111. 1 0001 0011 1000 1001 0111 1100 11011
0001
0011
1000
1001
0111
1100
1101
Sibling is branch node.
Delete 0111.

17. Fixed Length Delete Delete 0111. One compare. 1 0001 0011 1000 10011
0001
0011
1000
1001
1100
1101
Delete 0111.
One compare.

18. Fixed Length Delete Delete 1100. 1 0001 0011 1000 1001 1100 11011
0001
0011
1000
1001
1100
1101
Sibling is element node.
Delete 1100.

19. Fixed Length Delete 1 1 1
0001
0011 1 1
1000
1001
1101
Delete 1100.

20. Fixed Length Delete
1101 1 1 1
0001
0011 1
1000
1001
Delete 1100.

21. Fixed Length Delete
1101 1 1 1
0001
0011 1
1000
1001
Delete 1100.

22. Fixed Length Delete Delete 1100. One compare. 1 1 1101 1 0001 0011 11 1
1101 1
0001
0011 1
1000
1001
Delete 1100.
One compare.

23. Fixed Length Join(S,m,B)
Convert 3-way join to 2-way join
Insert m into B to get B’.
S empty => B’ is answer; done.
S is element node => insert S element into B’; done;
B’ is element node => insert B’ element into S; done;
If you get to this step, the roots of S and B’ are branch nodes.

24. Fixed Length Join(S,m,B)
S has empty right subtree. a S b c B’ c
J(S,B’)
J(a,b) + =
J(X,Y) ajoin X and Y, all keys in X < all in Y.

25. Fixed Length Join(S,m,B)
S has nonempty right subtree.
Left subtree of B’ must be empty, because all keys in B’ > all keys in S. a S b c B’ a
J(S,B’)
J(b,c) + =
Complexity = O(height).