1. Bit Vector Linked List (Sorted and Unsorted) Hash Table Search Trees
Set Implementations
Bit Vector
Linked List (Sorted and Unsorted)
Hash Table
Search Trees
9/21/2018
CS 303 – Implement Set
Lecture 9

2. Bit Vector
A =
U = ai, 0 <= i <= 15
A = {a0,a3,a4,a7,a9,a12,a13,a15}
1 Bit per possible element
If |U| <= wordlength, then very efficient. If not...NOT
Easy: UNION, INTERSECTION, DIFFERENCE, MEMBER, NULL, EQUAL
Hard: MIN, MAX
Impossible: nothing – this is a “direct model”
9/21/2018
CS 303 – Implement Set
Lecture 9

3. Linked List A a0 a3
a15
U = ai
Unsorted: ai’s kept in any order (making insert easy? NO! why?)
Easy: NULL
Hard: Everything else! (motivation for linear order, <)
Sorted (by <): ai’s kept sorted
Hard: MEMBER
Easy: Everything else (most operations are 1 pass through data)
9/21/2018
CS 303 – Implement Set
Lecture 9

4. Dictionary ADT MakeNull(A): A = NULL Member(A): (x in A)?
Insert(A): A = A UNION {x}
Delete(x,A): A = A – {x}
Note: no requirement for Ordering, UNION, INTERSECTION, DIFFERENCE, ...
Applications: Phone listings, symbol tables, words (duh!)
Note that applications may SORT as a matter of implementation convenience
9/21/2018
CS 303 – Implement Set
Lecture 9

5. Dictionary Implementations
Bit Vector – almost never useful
Arrays or linked list (unsorted)
MEMBER – scan entire list!
INSERT – if not MEMBER, add
DELETE – if MEMBER, remove (for Array, move everything)
All are O(n), because of MEMBER
Arrays or linked list (sorted)
speeds search (O(n) becomes O(log n) for array)
slows INSERT (O(1) becomes O(1))
but perhaps the application doesn’t do a lot of INSERTs
9/21/2018
CS 303 – Implement Set
Lecture 9

6. Binary Search Trees
An implementation technique aimed at Sets with INSERT, DELETE, MEMBER, MIN
MEMBER, MIN – Binary Search
INSERT – Search until x or L is found; add if necessary
DELETE – Find x. If a leaf, remove it. If not a leaf
find largest(smallest) y in left(right) subtree
move y to where x is
Special cases at root! [exercise!] a b c
b < a < c
9/21/2018
CS 303 – Implement Set
Lecture 9

7. Analysis of Binary Search Tree
Best Case: BST is balanced. All operations take O(log n) time
Worst Case: BST is linear. All operations take O(n) time
Average Case: BST is “random”. The BST is “almost balanced” and operations take
O(c log n) time. Worse than fully balanced, but only by a constant factor.
Next time – how to avoid the worst case...read about AVL, Splay, and B-trees!
But...I’ll talk about 2-3 Trees
9/21/2018
CS 303 – Implement Set
Lecture 9