computer notes - Data Structures - 28
- 1. Class No.28 Data Structures http://ecomputernotes.com
- 2. Equivalence Relations A binary relation over a set S is called an equivalence relation if it has following properties Reflexivity: for all element xS, x x Symmetry: for all elements x and y, x y if and only if y x Transitivity: for all elements x, y and z, if x y and y z then x z The relation “is related to” is an equivalence relation over the set of people http://ecomputernotes.com
- 3. Equivalence Relations The relationship is not an equivalence relation. It is reflexive, since x x, and transitive, since x y and y z implies x z, it is not symmetric since x y does not imply y x. http://ecomputernotes.com
- 4. Equivalence Relations Electrical connectivity, where all connections are by metal wires, is an equivalence relation. It is clearly reflexive, since any component is connected to itself. It is symmetric because if component a is connected to component b then b must be electrically connected to a . It is transitive, since if component a is connected to component b and b is connected to c then a is connected to c . http://ecomputernotes.com
- 5. The Disjoint Sets View An equivalence relation over a set S can be viewed as a partitioning of S into disjoint sets. Each set of the partition is called an equivalence class of (all elements that are related). http://ecomputernotes.com
- 6. The Disjoint Sets View Every member of S appears in exactly one equivalence class. To decide if a b , we need only to check whether a and b are in the same equivalence class. This provides a strategy to solve the equivalence problem. http://ecomputernotes.com
- 7. Dynamic Equivalence Problem If the relation is stored as a two-dimensional array of booleans, then, of course, this can be done in constant time. The problem is that the relation is usually not explicitly defined, but, rather, implicitly defined. Haaris Haaris Saad Ahmed Omar Asim Qasim Saad Ahmed Omar Asim Qasim T T T T T T T T T T http://ecomputernotes.com
- 8. Dynamic Equivalence Problem As an example, suppose the equivalence relation is defined over the five-element set { a 1 , a 2 , a 3 , a 4 , a 5 }. There are 25 pairs of elements, each of which is related or not. (30 pairs – 5 self-pairs=25). http://ecomputernotes.com
- 9. Dynamic Equivalence Problem However, given the information a 1 a 2 , a 3 a 4 , a 5 a 1 , a 4 a 2 , Implies that all pairs are related. We would like to be able to infer this information quickly. a 2 a 3 a 1 a 5 a 4 http://ecomputernotes.com
- 10. Dynamic Equivalence Problem The input is initially a collection of n sets, each with one element. This initial representation is that all relations (except reflexive relations) are false. Each set has a different element so that S i S j = ; this makes the sets disjoint . http://ecomputernotes.com
- 11. Dynamic Equivalence Problem There are two permissible operations: find and union . Find returns the name of the set (equivalence class) that contains a given element, i.e., S i find( a ) Union merges two sets to create a new set S k = S i S j . http://ecomputernotes.com
- 12. Dynamic Equivalence Problem If we want to add the relation a b , then we first see if a and b are already related. This is done by performing finds on both a and b to check whether they are in the same set. If they are not, we apply union which merges the two sets a and b belong to. The algorithm to do this is frequently known as Union / Find for this reason. http://ecomputernotes.com
- 13. Dynamic Equivalence Problem Notice that we do not perform any operations comparing the relative values of set elements. We merely require knowledge of their location, i.e., which set an element belongs to. We can thus assume that all elements are numbered sequentially from 1 to n . Thus, initially we have S i = { i } for i = 1 through n . http://ecomputernotes.com
- 14. Dynamic Equivalence Problem Our second observation is that the name of the set returned by find is fairly arbitrary. All that really matters is that find ( x ) = find ( y ) if an only if x an y are in the same set. We will now discuss a solution to the union / find problem that for any sequence of at most m finds and upto n -1 unions will require time proportional to ( m + n ). http://ecomputernotes.com