1. Sets and Multimaps
5/9/2018
Presentation for use with the textbook Data Structures and Algorithms in Java, 6th edition, by M. T. Goodrich, R. Tamassia, and M. H. Goldwasser, Wiley, 2014
Sets and Multimaps
Sets and Multimaps

2. Definitions Set an unordered collection of unique elements
Elements of a set are like keys of a map, no auxiliary values.
Multiset (also known as a bag)
a set-like container that allows duplicates.
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 and Multimaps

3. Set ADT
Sets and Multimaps

4. Implementing a Set
Similar techniques from Map
Sets and Multimaps

5. 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 and Multimaps

6. Union as merging two sorted lists
A, D, J, M, N
B, C, D, N, P
Sets and Multimaps

7. Intersection and Subtraction
Similarly…
Sets and Multimaps

8. 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 and Multimaps

9. 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 and Multimaps

10. 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
aIsLess(a,S)
bIsLess(b,S)
bothAreEqual(a,b,S)
Union(A,B)
Intersection(A,B)
subtraction(A,B)
Sets and Multimaps

11. 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
aIsLess(a,S)
bIsLess(b,S)
bothAreEqual(a,b,S)
Union(A,B)
Append a to S
Append b to S
Intersection(A,B)
nothing
Subtraction(A,B)
Sets and Multimaps

12. Mulitset Set-like with duplicates Can use a map Key = element
Value = count
Sets and Multimaps

13. Multimap store multiple entries with the same key
implement a multimap M by means of a map M’
let E(k) be the list of entries of M with key k
The entries of M’ are the pairs (k, E(k))
Ie, each “entry” is a list of entries
Sets and Multimaps

14. Mulitmaps
Sets and Multimaps