1. Skip List: Implementation- + S2 - 34 + S1 - 23 34 +
Start of 41. S0 - 12 23 34 45 +

2. Lecture No.41
Data Structure
Dr. Sohail Aslam

3. Implementation: TowerNode40 50 60
head
tail 20 30 26 57
Tower Node
TowerNode will have array of next pointers.
Actual number of next pointers will be decided by the random procedure.
Define MAXLEVEL as an upper limit on number of levels in a node.

4. Implementation: QuadNode
A quad-node stores:
item
link to the node before
link to the node after
link to the node below
link to the node above
This will require copying the key (item) at different levels
quad-node x
Start lecture 41

5. Skip Lists with Quad Nodes- + S2 - 31 + S1 - 23 31 34 64 + S0 - 12 23 26 31 34 44 56 64 78 +

6. Performance of Skip Lists
In a skip list with n items
The expected space used is proportional to n.
The expected search, insertion and deletion time is proportional to log n.
Skip lists are fast and simple to implement in practice

7. Implementation 5: AVL tree
An AVL tree, ordered by key
insert: a standard insert; (log n)
find: a standard find (without removing, of course); (log n)
remove: a standard remove; (log n)
key
entry
key
entry
key
entry
key
entry
and so on

8. Anything better?
So far we have find, remove and insert where time varies between constant logn.
It would be nice to have all three as constant time operations!

9. Implementation 6: Hashing
An array in which TableNodes are not stored consecutively
Their place of storage is calculated using the key and a hash function
Keys and entries are scattered throughout the array.
key
entry 4 10
hash function
array index
Key
123

10. Hashing insert: calculate place of storage, insert TableNode; (1)
find: calculate place of storage, retrieve entry; (1)
remove: calculate place of storage, set it to null; (1)
key
entry 4 10
123
All are constant time (1) !

11. Hashing
We use an array of some fixed size T to hold the data. T is typically prime.
Each key is mapped into some number in the range 0 to T-1 using a hash function, which ideally should be efficient to compute.

12. Example: fruits
Suppose our hash function gave us the following values:
hashCode("apple") = 5 hashCode("watermelon") = 3 hashCode("grapes") = 8 hashCode("cantaloupe") = 7 hashCode("kiwi") = 0 hashCode("strawberry") = 9 hashCode("mango") = 6 hashCode("banana") = 2
kiwi
banana
watermelon
apple
mango
cantaloupe
grapes
strawberry 1 2 3 4 5 6 7 8 9

13. Example Store data in a table array: kiwi banana watermelon apple
table[5] = "apple" table[3] = "watermelon" table[8] = "grapes" table[7] = "cantaloupe" table[0] = "kiwi" table[9] = "strawberry" table[6] = "mango" table[2] = "banana"
kiwi
banana
watermelon
apple
mango
cantaloupe
grapes
strawberry 1 2 3 4 5 6 7 8 9

14. Example Associative array: kiwi
table["apple"] table["watermelon"] table["grapes"] table["cantaloupe"] table["kiwi"] table["strawberry"] table["mango"] table["banana"]
kiwi
banana
watermelon
apple
mango
cantaloupe
grapes
strawberry 1 2 3 4 5 6 7 8 9

15. Example Hash Functions
If the keys are strings the hash function is some function of the characters in the strings.
One possibility is to simply add the ASCII values of the characters: æ
length - 1 ö å h (
str ) = ç
str [ i ] ÷ %
TableSize ç ÷ è ø i =
Example : h (
ABC ) = ( 65 + 66 + 67 )%
TableSize

16. Finding the hash function
int hashCode( char* s ) { int i, sum; sum = 0; for(i=0; i < strlen(s); i++ ) sum = sum + s[i]; // ascii value return sum % TABLESIZE; }

17. Example Hash Functions
Another possibility is to convert the string into some number in some arbitrary base b (b also might be a prime number): æ
length - 1 ö å h (
str ) = ç
str [ i ] b i ÷ % T ç ÷ è ø i =
Example : h (
ABC ) = ( 65 b + 66 b 1 + 67 b 2 )% T

18. Example Hash Functions
If the keys are integers then key%T is generally a good hash function, unless the data has some undesirable features.
For example, if T = 10 and all keys end in zeros, then key%T = 0 for all keys.
In general, to avoid situations like this, T should be a prime number.
End of lecture 41. Start of lecture 42.