Dictionary - Data Structures and Algorithm - Lecture Slides, Slides of Data Structures and Algorithms

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Data Structures

& Algorithm Analysis

Dictionaries

• A dictionary is a collection of elements each of which has a unique search key - Uniqueness criteria may be relaxed (multiset) - (I.e. do not force uniqueness)
• Keep track of current members, with periodic insertions and deletions into the set
• Examples
  • Membership in a club, course records
  • Symbol table (contains duplicates)
  • Language dictionary (WordSmith, Webster, WordNet)
• Similar to database

Dictionary ADT

• simple container methods: size()

isEmpty() elements()

• query methods: findElement(k)

findAllElements(k)

• update methods: insertItem(k, e)

removeElement(k) removeAllElements(k)

• special element NO_SUCH_KEY , returned by an unsuccessful search

How to Implement a Dictionary?

• Sequences / Arrays

• ordered
• unordered

• Binary Search Trees

• Skip lists

• Hashtables

More Arrays

• Ordered array

• searching takes O(log n) time (binary search)

• inserting and removing takes O(n) time

• application to look-up tables (frequent

searches, rare insertions and removals)

• Apply binary search

Binary Searches

• narrow down the search range in stages
• “high-low” game
• findElement(22)

An Alternative to Arrays

• Unordered Array:

• insertion: O(1)
• search: O(n)

• Ordered Array

• insertion: O(n)
• search: O(log n)

• Skip Lists:

• insertion: O(log n)
• search: O(log n)
• And avoid the fixed-size drawback of arrays!

Skip Lists

• good implementation for a dictionary

• a series of lists {S0, S1, …, Sk} • each list Si stores a sorted subset of the dictionary D

- ∞ 12 18 25 28 72 74 ∞

• ∞ 18 25 74 ∞
• ∞ 18 ∞
• ∞ ∞ S S S S

Traversing Positions in a Skip List

• Assume a node P in the skip list

• after(p)
• before(p)
• below(p)
• above(p)

• Running time of each operation?

Operations in a Skip List

• Use skip lists to implement dictionaries 

• Need to deal with

• Search
• Insert
• Remove

Searching

• Search for 27

S

S

S

S

More Searching

• Search for 74

S

S

S

S

Running Time Analysis

• log n levels  O(log n) for going down in the

skip list

• at each level, O(1) for moving forward

• why? works like a binary search
• in skip lists, the elements in list S(i+1) play the role

of search dividers for elements in S(i)

• (in binary search: mid-list elements to divide the

search)

• total running time: O(log n) Docsity.com

Insertion in Skip Lists

• First: identify the place to insert new key k

•  node p in S0 with largest key less or equal than

k

• Insert new item(k,e) after p

• with probability 50%, the new item is inserted

in list S

• with probability 25% , the new item is inserted in

list S

• with probability 12.5% , the new item is inserted in list S - with probability 6.25% , the new item is inserted in list S