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Unformatted text preview: Chapter 8 Disjoint Set Class CS 340 Chapter 8: Disjoint Set Class 1 Relations A relation R on a set S maps every pair of elements in S to either TRUE or FALSE. For example, the greater than relation, >, is TRUE for the integer pair (5,3) (i.e., 5 > 3) but not for the pair (6,8) (since 6>8 is FALSE). S = (3, 5, 6, 8) F T T T 8 F F T T 6 F F F T 5 F F F F 3 8 6 5 3 > (5,3) 5 R 3 = True, where R is the relation “5 is related to 3” CS 340 Chapter 8: Disjoint Set Class 2 Equivalence Relations An equivalence relation R is a relation satisfying the following three properties: – ( Reflexive ) α R α for every α in set S. – ( Symmetric ) For every pair α and β in set S, if α R β , then β R α . – ( Transitive ) For every triple α , β , γ in set S, if α R β and β R γ , then α R γ . For example… – The greater than relation is not an equivalence relation for integers (since it lacks the symmetric property) S = (3, 5, 6, 8) F T T T 8 F F T T 6 F F F T 5 F F F F 3 8 6 5 3 > F T T T 8 F F T T 6 F F F T 5 F F F F 3 8 6 5 3 > R e f l e x i v e S y m m e t r i c CS 340 Chapter 8: Disjoint Set Class 3 More Equivalence Relation Examples An equivalence relation R is a relation satisfying the following three properties: – ( Reflexive ) α R α for every α in set S. – ( Symmetric ) For every pair α and β in set S, if α R β , then β R α . – ( Transitive ) For every triple α , β , γ in set S, if α R β and β R γ , then α R γ . For example… – Modulo10 equality is an equivalence relation for integers – The does not equal relation is not an equivalence relation for integers (since it lacks both the reflexive and transitive properties) – The electrical connectivity relation is an equivalence relation for network system components (i.e., endstations, servers, switches) – The “electrical connectivity” (or just connectivity) relation is the one we will focus on in this chapter....
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 Spring '09
 Bouvier,D
 Algorithms, Data Structures, Equivalence relation, Disjoint Set ADT, Disjointset data structure, Disjoint Set Union Operations

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