1. Sequences
11/12/2018 5:23 AM
Sets
Sets

2. Definitions
A set is an unordered collection of elements, without duplicates that typically supports efficient membership tests.
Elements of a set are like keys of a map, but without any auxiliary values.
A multiset (also known as a bag) is a set-like container that allows duplicates.
A multimap is similar to a traditional map, in that it associates values with keys; however, in a multimap the same key can be mapped to multiple values.
For example, the index of a book maps a given term to one or more locations at which the term occurs.
Sets

3. Set ADT
Sets

4. Boolean Set Operations
Sets

5. Set Update Operations
Sets

6. Nodes storing set elements in order
Storing a Set in a List
We can implement a set with a list
Elements are stored sorted according to some canonical ordering
The space used is O(n)
Nodes storing set elements in order
List 
Set elements
Sets

7. Generic Merging Generalized merge of two sorted lists A and B
Template method genericMerge
Auxiliary methods
aIsLess
bIsLess
bothAreEqual
Runs in O(nA + nB) time provided the auxiliary methods run in O(1) time
Algorithm genericMerge(A, B)
S  empty sequence
while A.isEmpty()  B.isEmpty()
a  A.first().element(); b  B.first().element()
if a < b
aIsLess(a, S); A.remove(A.first())
else if b < a
bIsLess(b, S); B.remove(B.first())
else { b = a }
bothAreEqual(a, b, S)
A.remove(A.first()); B.remove(B.first())
while A.isEmpty()
while B.isEmpty()
return S
Sets

8. Using Generic Merge for Set Operations
Any of the set operations can be implemented using a generic merge
For example:
For intersection: only copy elements that are duplicated in both list
For union: copy every element from both lists except for the duplicates
All methods run in linear time
Sets

9. Assignment #10
R-10.9 Draw the 11-entry hash table that results from using the hash function, h(i) = (3i+5) mod 11, to hash the keys 12, 44, 13, 88, 23, 94, 11, 39, 20, 16, and 5, assuming collisions are handled by chaining.
Sets

10. R Draw the 11-entry hash table that results from using the hash function, h(i) = (3i+5) mod 11, to hash the keys 12, 44, 13, 88, 23, 94, 11, 39, 20, 16, and 5,, assuming collisions are handled by linear probing?
Sets

11. Mimic the style of the figures in the book
R Draw an example skip list S that results from performing the following series of operations on the skip list shown in Figure 10.13: del S[38], S[48] = x , S[24] = y , del S[55]. Record your coin flips, as well.
Mimic the style of the figures in the book
Sets