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Efficient Set Operations Using Generic Merge Algorithm

Implement set operations like union, intersection, and subtraction using the generic merge algorithm. Running time is O(nA + nB) where nA and nB are the sizes of sets A and B.

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Efficient Set Operations Using Generic Merge Algorithm

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  1. Sets Sets

  2. 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

  3. Generic Merging (§10.2) AlgorithmgenericMerge(A, B) S empty sequence whileA.isEmpty() B.isEmpty() a A.first().element(); b B.first().element() ifa<b aIsLess(a, S); A.remove(A.first()) else if b<a bIsLess(b, S); B.remove(B.first()) else { b = a } bothEqual(a, b, S) A.remove(A.first()); B.remove(B.first()) whileA.isEmpty() aIsLess(a, S); A.remove(A.first()) whileB.isEmpty() bIsLess(b, S); B.remove(B.first()) return S • Generalized merge of two sorted lists A and B • Template method genericMerge • Auxiliary methods • aIsLess • bIsLess • bothEqual • Runs in O(nA+nB) time provided the auxiliary methods run in O(1) time Sets

  4. 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. Using Generic Merge for Set Operations Sets

  5. We represent a set by the sorted sequence of its elements By specializing the auxliliary methods he generic merge algorithm can be used to perform basic set operations: union intersection subtraction The running time of an operation on sets A and B should be at most O(nA+nB) Set union: aIsLess(a, S) S.insertFirst(a) bIsLess(b, S) S.insertLast(b) bothAreEqual(a, b, S) S. insertLast(a) Set intersection: aIsLess(a, S) { do nothing } bIsLess(b, S) { do nothing } bothAreEqual(a, b, S) S. insertLast(a) Set Operations Sets

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