Skip Lists with Quad Nodes - Data Stuctures - Lecture Slides, Slides of Data Structures and Algorithms

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Skip List: Implementation

−∞ +∞

S 0

S 1

S 2

S 3

−∞ 12 23 34 45 +∞

−∞ 34 +∞ −∞ 23 34 +∞

Implementation: TowerNode

 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.

40 50 60

head (^) tail

20 26 30 57

Tower Node

Skip Lists with Quad Nodes

−∞ 12 23 26 31 34 44 56 64 78 +∞

−∞ +∞

−∞^31 +∞

−∞ 23 31 34 64 +∞

S 0

S 1

S 2

S 3

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

Anything better?

 So far we have find, remove and insert where time varies between constant log n.

 It would be nice to have all three as constant time operations!

 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.

Implementation 6: Hashing

key entry

Key (^) functionhash^ arrayindex

4

10

123

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.

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

0 1 2 3 4 5 6 7 8 9

Example

 Associative array:

table["apple"] table["watermelon"] table["grapes"] table["cantaloupe"] table["kiwi"] table["strawberry"] table["mango"] table["banana"]

kiwi

banana watermelon

apple mango cantaloupe grapes strawberry

0 1 2 3 4 5 6 7 8 9

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:

Example h ABC TableSize

h str str i TableSize

length

i : ( ) ( 65 66 67 )%

( ) [ ] %

1

0 = + +

 

  



−

=

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):

Example h ABC b b b T

h str str i b T

length

i

i

: ( ) ( 65 66 67 )%

( ) [ ] %

0 1 2

1

0 = + +

 

  



= ∑ ×

−

=

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.