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

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

Set ADT Sets and Multimaps

Implementing a Set Similar techniques from Map Sets and Multimaps

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

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

Intersection and Subtraction Similarly… Sets and Multimaps

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

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

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

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

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

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

Mulitmaps Sets and Multimaps